Sep 2023

Application Catalog

Application
Catalog
How To Use the Application Catalog“Case studies performed using JMAG electromagnetic ﬁeld simulation software”
JMAG is an electromagnetic ﬁeld analysis software that is applicable to many analysis targets such as motors.
It handles magnetic ﬁeld analyses, and can also couple thermal and structural analyses thanks
to enhancements in its multiphysics functions.
The Application Catalog on the JMAG website contains various case studies to introduce the applications
and analysis functions that can be performed with JMAG to our customers.
We will introduce a few of them in this pamphlet.
We have also prepared Application Notes and model data on the JMAG users web site for those
who are just beginning to work with analysis or are trying to work on a new analysis ﬁeld.
Please try it if there are any analyses that you are considering or are interested in.
We will continue to introduce new cases and improve the content of the Application Notes
and model data mentioned above in order to support our customersʼ analysis work.

https://www.jmag-international.com/

2

JSOL Corporation

Contents
02

How To Use the Application Catalog
SR Motor
IPM Motor

14
15
15
17

Case 017
Case 018
Case 019
Case 022

Inductance Analysis of an IPM Motor [DP]Thermal Analysis of an IPM Motor [HT, LS, TR]
Analysis of the Centrifugal Force in an IPM motor [DS]Analysis of the Eddy Current in the Magnet of an IPM Motor [TR]17

Case 023

Eccentricity Analysis of an IPM Motor [DP]24

Case 037

Vector Control Analysis of an IPM Motor Using Control Simulator and the JMAG-RT[DP, RT]

35

Case 055

Integrated Magnetization Analysis of an IPM Motor [DP, ST]

37

Case 058

Eﬃciency Analysis of an IPM Motor [DP, LS]

38

Case 059

Iron Loss Analysis of an IPM Motor Accounting for PWM ‒Direct Link‒ [DP, LS]

43

Case 069

Iron Loss Analysis of an IPM Motor [DP, LS]

53

Case 087

Iron Loss Analysis of an IPM Motor Including the Eﬀect of Shrink Fitting [DP, DS, LS]

55

Case 090

Analysis of the Eﬀect of PWM on the Iron Loss of an IPM Motor [DP, LS]

55

Case 091

Iron Loss Analysis of an IPM Motor Including the Eﬀects of Press Fitting Stress[DP, DS, LS]

09

Case 006

Analysis of the SR Motor Torque Ripple [DP]79

Case 138

Vibration Analysis of an SR Motor [DP, DS]

89

Case 162

Drive Simulation of an SR Motor using a Control Simulator and JMAG-RT [DP, RT]

96

Case 178

Analysis of SR Motor I-Psi Characteristics [DP]97

Case 179

Analysis of SR Motor Static Characteristics [DP]98

Case 180

Analysis of SR Motor Dynamic Characteristics [DP]99

Case 181

Analysis of SR Motor Drive Characteristics [DP, LS]

113

Case 201

Iron Loss Analysis of an SR Motor [DP, LS]

Axial Gap Type Motor
10

Case 008

127

Case 219

Axial Gap Type Motor Cogging Torque Analysis [TR]157

Case 260

Creating Axial Gap-Type Motor Eﬃciency Maps [LS, TR]

164

Case 271

Axial Gap Type Motor AC Copper Loss Analysis [TR]170

Case 277

Vibration Analysis of Axial Gap-Type Motor [DS, TR]

Analysis of an Axial Gap Motor [TR]73

Case 122

Inductance Analysis of an IPM Motor-d/q-axis Inductance Obtained by Actual Measurement- [DP]77

Case 131

Stray Capacitance Analysis of a Motor [EL]81

Case 142

Press Fit Analysis of a Divided Core [DS, DP, LS]

13Case 015

Cogging Torque Analysis of an SPM Motor with a Step Skewed Magnet [TR]87

Case 156

Segregation Analysis of Torque Components for an IPM Motor [DP]16

Case 020

Sound Pressure Analysis of an SPM Motor [DP, DS]

87

Case 157

Analysis of Eddy Currents in an IPM Motor Using the Gap Flux Boundary [DP, FQ]

16Case 021

Iron Loss Analysis of an SPM Motor Including the Eﬀect of Shrink Fitting [DS, DP, LS]

90

Case 165

Creating IPM Motor Eﬃciency Maps [DP, LS]

18

Case 024

Cogging Torque Analysis of an SPM Motor with a Skewed Stator [TR]94

Case 173

Basic Characteristic Analysis of an IPM Motor [DP, LS]

20

Case 029

Iron Loss Analysis of an SPM Motor with Overhanging Magnet [LS, TR]

102

Case 188

Thermal Demagnetization Analysis of IPM Motors Accounting for Coercive ForceDistribution of Magnets [DP]21

Case 030

Magnetization Field Evaluation of an SPM motor [ST]108

Case 194

Analysis of High-Speed Rotation Motors Accounting for Eddy Current [DP, LS]

21

Case 031

Iron Loss Analysis of an SPM Motor Including the Eﬀect of Press-ﬁtting Stress [DS, DP, LS]

23Case 034

Demagnetization Analysis of an SPM Motor [DP]25

Case 040

Cogging Torque Analysis of an SPM Motor [DP]30

Case 046

Sensitivity Analysis of the Magnetization Pattern of an SPM Motor [DP, ST]

112
116

Case 200
Case 205

Iron Loss Analysis of IPM Motor using Hysteresis Model [DP, LS]
Analyzing the Torque Characteristics of
IPM Motors Using a Thermal Equivalent Circuit [DP]SPM Motor

121

Case 211

IPM Motor Wire Joule Loss Analysis [DP]49

Case 080

Cogging Torque Analysis of an SPM Motor with Skewed Magnetization [TR]123

Case 214

Monitoring the Radial Force Acting on the Teeth of IPM Motors Using Circuit Control
Simulation [DP, RT]

61

Case 103

Eﬃciency Analysis of a Permanent Magnet Synchronous Motor [DP, LS]

64

Case 108

Centrifugal Force Rupture Analysis of a Ring Magnet [DS]68

Case 114

Vibration Analysis of an Outer Rotor Motor [DP, DS]

69

Case 115

Eccentricity Analysis of an SPM Motor [DP]72

Case 120

Thermal Demagnetization Analysis of an SPM Motor [DP]88

Case 159

Sensitivity Analysis of Dimensional Tolerance in an SPM Motor [DP]

124
129
130

Case 215
Case 222
Case 223

Simulation of an IPM Motor with a Delta Connection
Using a Control Simulator and JMAG-RT [DP, RT]
Irreversible Thermal Demagnetization Analysis of Incompletely Magnetized Magnets[DP, ST]
Multi-Objective Optimization of an IPM Motor Accounting
for Stress at High-Speed Rotation [DP, DS]

131

Case 225

Induced Voltage Analysis on Memory Motors Using Variable Magnets [ST, DP]

90

Case 163

Torque-Current Curve Analysis of an SPM Motor [DP]141

Case 237

AC Loss Analysis of an IPM Motor [TR]132

Case 226

Vibration Characteristics Analysis of SPM Motors [DP, DS]

142

Case 239

Fault Analysis in an IPM Motor [DP, LS, RT]

135

Case 230

6-Phase SPM Motor Inverter Fault Simulation [DP, RT]

143

Case 240

NT Characteristics Analysis of an IPM Motor Using PAM Drive [DP]143

Case 241

Single-Phase Synchronous Motor N-T-I Characteristics Analysis [DP]153

Case 255

Evaluating the N-T Curve of an IPM motor Using 3D Correction [TR]158

Case 261

154

Case 257

IPM Motor Topology Optimization [DP, DS]

SPM Motor Magnet Coercive Force Optimization For Obtaining Sinusoidal Magnetization
[DP]

156

Case 259

Automatically Searching for Current Phase that Obtains
Maximum Torque and Evaluating Torque Characteristics [DP]163

Case 268

SPM Motor Teeth Geometry Topology Optimization [DP]167

Case 274

SPM Motor Thermal Demagnetization Analysis Using Bonded NdFeB Magnets [DP]174

Case 285

Integrated Magnetization Analysis of an SPM Motor by Magnetizing Multiple Times [DP]159

Case 262

IPM Motor Thermal Demagnetization Analysis Accounting
for Dy-Diﬀusion Magnet Incomplete Magnetization [DP, ST]

160

Case 263

IPM Motor Stress Analysis Taking Into Account Friction
between Magnets and the Rotor Core [DS]160

Case 264

Creating 6-Phase IPM Motor Eﬃciency Maps [DP, LS]

162

Case 267

IPM Motor Topology Optimization Using the Density Method [DP, DS]

166

Case 273

Creating IPM Motor Eﬃciency Maps Accounting for AC Loss [DP, LS]

168

Case 275

IPM Motor Iron Loss Analysis Accounting for the Eﬀect of Residual Strain [DP]169

Case 276

Six-Phase IPM Motor Harmonic Iron Loss Analysis [DP, LS]

172

Case 279

Thermal Analysis Accounting for Cooling of the IPM Motor [ExpressFR]

175

Case 286

IPM Motor Iron Loss Analysis Accounting for PWM: Evaluation with Diﬀerent Modulation Methods [DP, LS]

178

Case 290

Design Exploration of IPM Motors, Including Evaluating Part Temperature and Stress[ExpressFR]

Basic Geometry
75

Case 128

Structural Analysis of a Cantilever [DS]76

Case 130

Thermal Conductivity Analysis of Basic Geometry [HT]80

Case 140

Analysis of Electromagnetic Repulsion Produced in Small Contact Bridges [TR]

RFID
47

Case 077

Inductance Analysis of an RFID Tag [FQ]

https://www.jmag-international.com

3High-Frequency Preheater
149
150

Case 249
Case 250

Dielectric Heating Analysis of a High-Frequency Preheater with Parallel Plate Electrodes[EL, HT]

Sensor20

Case 028

Magnetic Field Analysis of a Speed Sensor [TR]60

Case 102

Magnetic Field Analysis of a Magnetic Sensor [ST]145

Case 244

Magnetic Field Strength Analysis at a Sensor Position [DP]Dielectric Heating Analysis of a High-Frequency Preheater with Roller Electrodes [EL, HT]

Condenser
Electroplating
45

Case 073

Analysis of the Capacitance of a Parallel Plate Capacitor [EL]110

Case 197

Electric ﬁeld distribution Analysis of a Electroplating [EL]

Magnetic Head
Case 027

Head Field Analysis of a Recording Write Head That Accounts for Eddy Currents [TR]Electromagnetic Forming

19
51

Case 084

Electromagnetic Forming Analysis of a Tube [DP]Magnetic Shield
Electromagnetic Brake
34

Case 053

Magnetic Shielding Analysis of a Shield Room [FQ]

155

Case 258

Magnetic Shield Geometry Optimization [ST]19

Case 026

Piezoelectric Actuator
26

Case 042

Displacement Analysis of a Piezoelectric Actuator [DS]Electromagnetic Relay
65

Case 109

Magnetic Gear
118

Case 207

Transfer Torque and Efficiency Analysis of Magnetic Gear [DP]Magnet45

Case 072

Analysis of Attractive Force between Steel Plates and a Magnet [ST]48

Case 079

Analysis of an Eﬀect of Magnetic Field Orientation on Magnetization [ST]74

Case 126

Magnetization Analysis Accounting for Eddy Currents [ST, TR]

140

Case 236

Magnetic Circuit Optimization for Plastic Magnets in an Axial Magnetic Field [ST]147

Case 246

Analysis of Electromagnetic Force Acting Between a Coil and a Magnet [ST]148

Case 247

Magnetic Circuit Optimization for Plastic Magnets in a Radial Magnetic Field [ST]174

Case 284

Analysis of Minute Electromagnetic Forces Acting on a Magnet [TR]Switching Gear / Breaker

Operating Time Analysis of an Electromagnetic Relay Accounting for Eddy Currents [TR]Transformer / Reactor
8

Case 004

Sound Pressure Analysis of a Reactor [DS, TR]

22

Case 032

Analysis of a Transformer [FQ]

33

Case 052

Inductance Analysis of a Sheet Coil Transformer [FQ]

39

Case 060

Superimposed Direct Current Characteristic Analysis of a Reactor [TR]39

Case 061

Current Distribution Analysis of a Choke Coil [DP, TS, TRANSFORMER̲TEMPLATE]

46

Case 075

Iron Loss Analysis of a Reactor [FQ, LS]

47

Case 078

Loss Analysis of a Sheet Coil Transformer [FQ, LS]

49

Case 081

AL-value Analysis of a Choke Coil [ST, TRANSFORMER̲TEMPLATE]

57

Case 097

Sound Pressure Analysis of a Transformer [DS, FQ]

58

Case 099

Superimposed Direct Current Characteristics Analysis of a High Current Reactor [TR]60

Case 101

AL-Value Current Characteristics Analysis of a Choke Coil [ST, TRANSFORMER̲TEMPLATE]

62

Case 105

Leakage Inductance Analysis of a Transformer [FQ, TS, TRANSFORMER̲TEMPLATE]

64

Case 107

Current Voltage Characteristics Analysis of Flyback Converters [DP]83

Case 149

Analysis of Magnetic Force Acting on the Arc of a Vacuum Circuit Breaker [TR]66

Case 110

Loss Analysis of a Choke Coil [FQ, LS, TS, TRANSFORMER̲TEMPLATE]

84

Case 150

Electromagnetic Repulsion Analysis of a Vacuum Circuit Breaker [TR]70

Case 117

Iron Loss Analysis of a Transformer [FQ, LS]

74

Case 123

Thermal Analysis of a Choke Coil [HT, TRANSFORMER̲TEMPLATE]

77

Case 132

Loss Analysis of a Three-phase Transformer [FQ, LS]

78

Case 133

Thermal Analysis of a Three-phase Transformer [HT]81

Case 143

Inductance Analysis of an Air Core Coil [ST]82

Case 146

Analysis of Stray Loss in a Transformer [FQ, HT, LS]

83

Case 148

Loss Analysis of a Power Transformer (Flyback Converter) [TS, DP, ST, LS]

84

Case 151

Insulation Evaluation Analysis of a Power Transformer [EL]85

Case 152

Electromagnetic Force Analysis of Short-circuited Power Transformer Coils [FQ]88

Case 158

Superimposed Direct Current Characteristic Analysis of a Reactor That Accounts
for Minor Hysteresis Loops [FQ, ST]

Pull-in/pull-out Analysis of a PM Stepper Motor Using a Control Simulator
and the JMAG-RT [RT, TR]

100

Case 184

Reactor Sound Pressure Analysis taking Magnetostriction into Account [DS, TR]

Analysis of a Hybrid Stepper Motor [TR]101

Case 185

Analysis of Stray Loss in a Power Transformer [FQ, LS]

Stiﬀness Torque Analysis of a PM Stepper Motor [TR]114

Case 202

Core Stray Loss Analysis of Power Transformer [FQ, LS]

Analysis of Detent Torque of a PM Stepper Motor [TR]115

Case 203

No-Load Test Analysis of Power Transformer [FQ, LS]

Analysis of a PM Stepper Motor Accounting for Magnetization [ST, TR]

115

Case 204

Copper Loss Analysis of Switching Transformers using Litz Wire [TS]117

Case 206

Temperature Analysis of an Oil-Immersed Transformer Using a Coupled
Magnetic Field and Thermal Fluid Analysis [FQ, LS]

128

Case 221Conﬁrmation of the Inﬂuence from the Reduction of High-Frequency Resistanceof a Magnetically Plated Wire Choke Coil [TS, FQ]

Synchronous Reluctance Motor
50

Case 082

Analysis of a Synchronous Reluctance Motor [DP]136

Case 231

Vector Control Simulation of a Synchronous Reluctance Motor [DP, RT]

161

Case 265

Creating Synchronous Reluctance Motor Eﬃciency Maps [DP]176

Case 288

Synchronous Reluctance Motor Topology Optimization [DP]Stepper Motor
11
14
54
56
76

Case 011
Case 016
Case 089
Case 094
Case 129

Speaker / Voice Coil Motor
10

Case 009

Sound Pressure Analysis of a Loudspeaker [DS, TR]

41

Case 065

Static Thrust Analysis of a Voice Coil Motor [TR]145

Case 243

Voice Coil Motor Control Simulation Using a Control Simulator and JMAG-RT [DP, RT]

4

Braking Torque Analysis of an Electromagnetic Brake [TR]

JSOL Corporation

137

Case 232

Reactor Joule Loss Analysis [FQ]

139

Case 235

Reactor Core Stray Loss Analysis [LS, TR]

141

Case 238

Hysteresis Loss Analysis in a Reactor with DC Bias [DP, LS]

153

Case 254

Vibration Analysis of a Large Oil-Filled Transformer [DS, FQ]

Induction Heating

Resistance Heating
28

Case 044

Resistance Heating Analysis of a Steel Sheet [HT ,TR]

12

Case 013

High-Frequency Induction Heating Analysis of a Shaft [FQ, HT]

75

Case 127

Resistance Heating Analysis of Steel [FQ, HT]

29

Case 045

High-Frequency Induction Heating Analysis of an IH Cooking Heater [FQ, HT]

31

Case 047

High-Frequency Induction Heating Analysis of a Crankshaft [FQ, HT]

31

Case 048

High-Frequency Induction Heating Analysis of a Printer Roller [FQ, HT]

32

Case 049

High-Frequency Induction Heating Analysis of a Steel Sheet [FQ, HT]

32

Case 050

High-Frequency Induction Heating Analysis of a Steel Wire
(Translational Induction Hardening) [FQ, HT]Generator
18

Case 025

Analysis of a Claw Pole Alternator [TR]73

Case 121

Output Analysis of a Salient-Pole Synchronous Generator [DP]122

Case 212

AC Loss Analysis of a Claw Pole Alternator [TR]33

Case 051

High-Frequency Induction Heating Analysis of a Gear [FQ, HT]

Simulation of a Claw-Pole Type Alternator Using a Control Simulator and JMAG-RT [RT, TR]

50

Case 083

Magnetic Shielding Analysis of an Induction Furnace [FQ]

Alternator Vibration Analysis [DS, TR]

51

Case 085

High-Frequency Induction Heating Analysis of a Constant Velocity Joint [FQ, HT]

59

Case 100

Surface Heating Analysis of a Steel Plate [FQ, HT]

93

Case 172

High-Frequency Induction Heating Analysis of a Test Piece(Rotational Induction Hardening) [FQ, HT]

125
151

Case 216
Case 251

Heater11

120

Case 010

Case 209

99

Case 183

Agitation Force Analysis of an Induction Furnace [FQ]

102

Case 186

Gear Induction Hardening Analysis Using Two Frequencies [FQ, HT]

Physical Property Testing

119

Case 208

High Frequency Induction Hardening Analysis Accounting for Deformation (Abaqus Link)[FQ, ABAQUS2]

Magnetic Measurement Analysis Accounting for Eddy Currents [TR]126

Case 218

High-Frequency Induction Hardening Analysis Accounting for Deformation [DS, FQ, HT]

127

Case 220

Analysis of the High-Frequency Induction Hardening and Cooling of a Gear [FQ, HT]

138

Case 233

Induction Heating Coil Optimization [FQ, HT]

173

Case 280

Topology Optimization of Induction Heating Coil [FQ, HT]

Thermal Analysis of a Radiant Heater [HT, TR]

Brush Motor / Universal Motor
8

Case 003

Analysis of a Permanent Magnet Brush Motor [DP]44

Case 071

Basic Characteristic Analysis of a Motor with 2 Brushes, 6 Poles, and 19 Slots [DP]56

Case 095

Analysis of a Universal Motor [DP]63

Case 106Iron Loss Analysis of a Brush Motor [DP, LS]

7Case 001

Torque Characteristic Analysis of a Three Phase Induction Motor [DP]Starting Performance Analysis of a Universal Motor [DP]24

Case 038

Starting Performance Analysis of a Single Phase Induction Motor [DP]25

Case 039

Torque Analysis of a Three Phase Induction Motor Accounting for the Skew [DP, TR]

36

Case 056

Torque Characteristics Analysis of a Self Starting Type Permanent Magnet Motor [DP]40

Case 063

Analysis of Torque Characteristics of a Cage Induction Motor [FQ]

42

Case 068

Speed Versus Torque Characteristic Analysis of a Three-Phase Induction Motor [DP]46

Case 074

Speed Versus Torque Analysis of a Single-Phase Induction Motor [DP]71

Case 119

Torque Characteristic Analysis of a Three Phase Wound Rotor Induction Motor [DP]86

Case 154

Calculation of Equivalent Circuit Parameters in a Three-Phase Induction Motor [DP]89

Case 161

Line Start Analysis of a Three-phase Induction Motor [DP]91

Case 166

Line Start Simulation of an Induction Machine Using a Control Simulator and
the JMAG-RT [FQ, LS, RT]

66

Case 111

Wound-Field Synchronous Motor
122

Case 213

Circuit/Control Simulation of a Wound-Field Synchronous Motor [DP, RT]

175

Case 287

Analysis of Current Conditions for the Maximum Eﬃciency of a WFSM [DP, LS]

177

Case 289

Eﬃciency Map Creation for Wound-Field Synchronous Motors [DP, LS]

Linear Solenoid / Linear Actuator

Induction Motor

40

Case 062

Attraction Force Analysis of a Solenoid Valve [DP]42

Case 066

Operating Time Analysis of an Injector [DP]92

Case 167

Iron Loss Analysis of a Three Phase Induction Motor [DP, LS]

57

Case 098

Response Analysis of a Solenoid Valve Using a Control Simulator and the JMAG-RT [DP, RT]

95

Case 176

Drive Characteristic Analysis of a Three-Phase Induction Motor [DP, LS]

69

Case 116

Operating Time Analysis of an Injector by Evaluating the Reduction in Eddy Currents [TR]96

Case 177

Torque Characteristic Analysis of a Three Phase Induction Motor [DP, LS]

103

Case 189

Response Characteristics Analysis of Injectors Accounting for Collision [DP]105

Case 191

106

Case 192

Response Characteristics Analysis of Solenoid Valves Accounting for Residual
Magnetization [DP]Torque Analysis of Three-Phase Induction Motors Accounting for the Skew Using
the Multi-slice Method [DP]107

Case 193

Calculation of Equivalent Circuit Parameters in a Single-Phase Induction Motor [DP]144

Case 242

Solenoid Valve Attractive Force Analysis Taking into Account Movable Part Motion [ST]109

Case 195

146

Case 245

Injector Control and Eddy Current Effects [DP]Torque characteristics analysis of three-phase induction motors using 3D correction
function [DP, TR]

152

Case 252

Geometry Optimization of a Solenoid Valve [DP]125

Case 217

Creating Efficiency Maps for 3-Phase Induction Motors [FQ, LS, EFFICIENCYMAP]

133

Case 227

Circuit Control Simulation for Three-Phase Induction Motor [FQ, RT]

133

Case 228

Induction Motor Thermal Analysis [HT, LS, TR]

134

Case 229

Analysis of the Effect of PWM on the Iron Loss of an Induction Motor [DP, LS]
Vibration Characteristics Analysis of an Induction Motor [DP, DS]

Linear Motor
7

Case 002

Cogging Analysis of a PM Linear Motor [TR]139

Case 234

26

Case 041

Positioning Control Analysis of a Permanent Magnet Linear Motor Using theControl Simulator and the JMAG-RT [DP, RT]

164

Case 269

Creating Three-Phase Induction Motor Efficiency Maps Accounting for Harmonics [DP, LS]

171

Case 278

Creating Six-Phase Induction Motor Efficiency Maps [FQ, LS]

179

Case 291

Control Simulation of Switching Number of Poles in a 6-Phase Induction Machine [DP, Pi]

34

Case 054

Analysis of the Cogging Force of a Moving Coil Linear Motor [TR]41

Case 064

Thrust Force Analysis of a Coreless Linear Motor [TR]61

Case 104

Thrust Force Analysis of a Linear Induction Motor [DP]67

Case 112

Starting Thrust Force Analysis of a Linear Induction Motor [FQ]

78

Case 134

Speed Control Analysis of a Permanent Magnet Linear Motor Using the Control
Simulator and the JMAG-RT [DP, RT]

165

Case 272

Thrust Analysis of Linear Actuator with a Curved Section [TR]Bus Bar
12

Case 014

Inductance Analysis of Bus Bar [FQ, Pi]

70

Case 118

Thermal Analysis of a Busbar [FQ, HT]

110

Case 198

Busbar Thermal Stress Analysis [FQ, DS, HT]

https://www.jmag-international.com

5Coreless Motor
27

Case 043

Torque Analysis of a Coreless Motor [TR]Spindle Motor
9

Case 007Analysis of a Spindle Motor [TR]

Cable
44

Case 070

Analysis of Impedance-Frequency Characteristics of a Cable [FQ]

Bearing
77

Case 131

Stray Capacitance Analysis of a Motor [EL]

Magnet Coupling
121

Case 210

Torque Decoupling of In-out Magnet Coupling [DP]Wireless Power Transfer
52

Case 086

Power Transmission Analysis of a Wireless Power Transfer System [FQ]

67

Case 113

Power Transmission Analysis of a Wireless Power Transfer System with Opposing Cores
[FQ]

80

Case 139

Power Transmission Analysis Using Magnetic Resonance Phenomena [FQ]104

Case 190

Arrangement Optimization of Wireless Power Supply System [FQ]

109

Case 196

Analysis of coil stray capacitance [EL]111

Case 199

Efficiency Analysis of Wireless Power Transfer via Electromagnetic Induction Method
using Resonance Circuits [FQ]

131

Case 224

Wireless Power Transfer Device Stray Loss Analysis [FQ]

The brackets [ ] contain the module name. Please see the JMAG Website or pamphlet for more details.

6

JSOL CorporationJMAG Application Catalog

Torque Characteristic Analysis of a Three Phase
Induction Motor

Case

1

Stator core

Bar
Shaft

module: DP

Overview

Coil
Rotor core

An induction motor is a motor in which the rotating magnetic field of the stator coils causes induced current to flow in an auxiliary conductor,
exerting force on the rotor in the rotational direction and causing it to spin. Induction motors are widely used in everything from industrial
machines to home appliances because they have a simple construction and are small, light, affordable, and maintenance-free.
In an induction motor, the current induced by the auxiliary conductor exerts a large influence on its characteristics. It also causes strong
magnetic saturation in the vicinity of the gap, in particular. With Finite Element Analysis (FEA), it is possible to investigate the characteristics
that accurately evaluate the features listed above, so preliminary design evaluations are effective.
This Application Note introduces a case example of how to obtain the current density distribution of an auxiliary conductor and its rotation
speed versus torque characteristics.

Current Density Distribution

Speed-Torque Curve

The current density distribution at a rotation speed of 1050 r/min is shown in fig. 1. The
torque in an induction motor is generated between the rotational flux stemming from thecurrent flowing in the stator coil and the current induced in the auxiliary conductor, so the
induced current has a large influence on the torque characteristics.

The Speed-Torque curve is shown in fig. 2. The maximum torque is found to be near 1000
r/min. Based on the fact that the torque in the high speed region gets higher, it can also
be seen that the auxiliary conductor's electric resistance in this induction motor is low. In
the range labeled b in fig. 2, the speed change is only about 5%, with a 100% change inthe torque, so the motor is rotating in a stable fashion.

Torque characteristics (Nm)

2.5

At 0.11 (s)

2.0
1.5
1.0
0.5
0.0
a

b

-0.5
0

(Unit: A/m^2)

150

300

450

600

750

900 1050 1200 1350 1500

Rotation speed (r/min)

At 0.114 (s)Fig.1. Current density distribution in the auxiliary
conductor (rotation speed: 1050 r/min)Fig. 2. Speed versus Torque

Case

Mover yoke

Cogging Analysis of a PM Linear Motor2

module: TR

Direction of translation

Coil

Overview
Linear motors have been widely used in carrier devices and the drive units of machine tools due to their capability for high acceleration anddeceleration, as well as their accurate positioning. As an issue for improving performance, people are trying to obtain a large thrust force inorder to enhance responsiveness, but on the other hand it is also necessary to fulfill the demand for the trade-off of wanting to reduce thrustforce variations and the attraction force.
In order to obtain a large thrust force, the material's nonlinear magnetic properties and the magnet's demagnetization characteristics need tobe accounted for, and in order to evaluate thrust force variations, they need to be analyzed after modeling a detailed geometry. This is why
they need to be studied with a magnetic field analysis simulation based on the finite element method (FEM).
This note presents how to obtain cogging, a cause of thrust variation, and evaluate the thrust force and attraction force during drive.

Magnet(4 poles)
Stator yoke

Thrust and Attraction Forces

Fig. 1 shows the cogging force waveform. To examine the cause of the cogging force, theflux density distribution at the position (timing) circled in blue in fig. 1 is shown in fig. 2.
Fig. 2 shows that the flux density at the end of the mover is not evenly distributed. This
imbalance of the magnetic circuit is the key factor that causes large cogging ripples.

The position versus thrust force is shown in fig. 3, and the position versus attraction force
is shown in fig. 4.
The graphs show that the periods of thrust and attraction forces have the same intervalas the period of cogging force. The cogging ripple influences the thrust variations, so the
cogging ripple needs to be made smaller in order to reduce the thrust variations. The
attraction force is also the same as the radial force in a rotating machine, meaning that it
is influenced by the magnetic flux flow in the gap, so it can be reduced by lessening themagnetic flux that flows perpendicular to the gap by using changes in the slot geometry.

800

4500

150

700

4000

100

600

3500

50

500

0
-50

400
300

-100

200

-150

100

-200
0

10

20

30

40

Position (mm)Fig. 1. Cogging Waveform

50

60(Unit: T)70
mover position: 50 mm

Fig. 2. Magnetic flux density distribution

Attraction force (N)200

Thrust (N)Cogging (N)Cogging

3000
2500
2000
1500
1000
500

0
0

10

20

30

40

Position (mm)Fig. 3. Position versus Thrust Force

50

60

70

0
0

10

20

30

40

50

60

70

Position (mm)Fig. 4. Position versus Attraction Forcehttps://www.jmag-international.com/

7

Case

Coil

Rotor core

MagnetAnalysis of a Permanent Magnet Brush Motor

3

module: DP

Overview
A brush motor generates torque through the electromagnetic attraction and repulsion between its rotor and stator. They do not have many
parts and do not require drive circuits, so they are widely used as a power source for compact equipment. A brush motor is composed of amagnetic circuit part, which actually generates torque via electromagnetic phenomena, and the brush/commutator part, which corresponds
to the drive circuit. In order to aim at improving the performance of a brush motor, it is necessary to raise the usage efficiency of the magnetic
circuit in each part and expertly utilize the nonlinear material characteristics. Proper placement of the brush/commutator that correspond to
the drive circuit is also vital.
In order to evaluate the usage efficiency of the magnetic circuit, torque variations, current waveforms, etc. at the design stage, it is best to
first do a detailed calculation of the magnetic flux density in each part, and then perform an electromagnetic field simulation using the finite
element method (FEM), which can evaluate torque with high accuracy.
This document introduces how the characteristics of the brush-type PM motor can be obtained, including torque versus current (T-I), torqueversus speed (T-N), and magnetic flux density distribution.

Stator core

Magnetic Flux Distribution

Fig. 1 shows the Speed-Torque curve, and fig. 2 shows the Torque-Current curve. From
both characteristic graphs it is possible to confirm that it has obtained the characteristics
of a DC motor, such as the fact that the rotation speed and current are proportional to thetorque. The controllable speed range in this motor can also be seen on the Speed-Torquecurve.

The magnetic flux density distribution at a rotation speed of 12000 rpm is shown in fig. 3.
The magnetic flux density is particularly high in the regions circled in red. The magnetic
flux from the rotor and the stator concentrate there. This magnetic flux produces torque.
When magnetic flux leakage occurs due to magnetic saturation, the torque may be
reduced.

14000

70

12000

60

10000

50
Current (A)Rotation speed (rpm)Speed-Torque Curve/Torque-Current Curve

8000
6000

40
30
20

4000

10

2000

0

0
0.0

0.2

0.4

0.6

0.0

0.8

0.2

0.4

0.6

0.8

Torque (Nm)Torque (Nm)Fig. 1 Speed versus TorqueFig. 2 Torque versus Current(Unit: T)

Fig. 3. Magnetic Flux Distributions at 12000 rpm

Case

Sound Pressure Analysis of a Reactor

4

Core

module: DS, TR

Overview

Coil

Reactors are used in a variety of electric power systems. For instance, they fill the role of making the current pulsation between an inverter anda motor more smooth. On the other hand, the sound that originates from a resonance phenomenon between an electromagnetic force andan eigenfrequency can become a problem. The reactor in this analysis has a gap in the magnetic circuit to prevent magnetic saturation. Dueto the magnetic fields that occur with high frequency currents, electromagnetic force generates near the gap, and this electromagnetic
excitation force in turn causes noise. Vibration and sound grow larger when the electromagnetic excitation force and the transformer's
eigenmode resonate. In order to evaluate this phenomenon with good accuracy, it is necessary to find the electromagnetic force distribution
and eigenmode in the high frequencies that become particular problems by using the finite element method (FEM).
This document introduces an example of an evaluation of a reactor's sound pressure when a part of a spacer has been removed.

SpacerElectromagnetic Force Distribution

Acceleration

Fig. 1 shows the distribution of the electromagnetic forces. The spacer is made ofnonmagnetic material, which means that there are no changes in the magnetic circuit
depending on whether or not it has been removed, so the electromagnetic force
distribution is the same regardless of whether or not the spacer is present.
From the electromagnetic force distribution shown in fig. 1, it is apparent that theelectromagnetic force is concentrated in the gap. This is because the air permeability inthe core is different from the gap.

Fig. 3 shows the distribution of the acceleration. The electromagnetic force distribution
does not change depending on the presence of the spacer, so differences in theacceleration distribution depend on differences in the eigenmode. Vibrations becomelarger in the viscinity of the missing spacer due to the spacer being stripped.

Eigenmode
Fig. 2 shows the eigenmode in the vicinity of the fundamental frequency (10 kHz) thatcarries electromagnetic force. Eigenmode in the vicinity of 10 kHz differs due to thespacers being stripped.

Sound Pressure Distribution
The distribution of the sound pressure is shown in fig. 4. The sound pressure distributionchanges depending on the presence of the spacer.
While the core vibrates vertically when the spacer is present, when it is absent, the gap
where it is missing vibrates longitudinally, so the distribution becomes dispersed. The
vibration is also bigger when the spacer is absent, so the sound pressure gets bigger.

The part where the spacer is missing(Unit: m/s^2)(Unit: N)

all spacers attached
all spacers attachedFig. 1 Electromagnetic force distribution

8

JSOL Corporation

one spacer removed

one spacer removed

Fig. 2 The eigenmode of approximately 10 kHz

Fig. 3 Acceleration distribution

all spacers attached

one spacer removed

Fig. 4 Sound pressure distribution(Unit: dB)

JMAG Application Catalog

Coil

Case

Analysis of the SR Motor Torque Ripple

6

module: DP

Overview
With the skyrocketing prices of rare earth magnets, expectations have been rising for SR (switched reluctance) motors because they have a
motor format that does not use permanent magnets. SR motors have a simple structure that can achieve solid performance at a low price.
However, torque generation depends only upon the saliency between the stator and rotor, so torque variations are extremely large and cause
vibration and noise, meaning that the use applications are limited. On the other hand, because of the skyrocketing prices of rare earth metals,
the improvement in current control technology, the possibility of optimized designs thanks to magnetic field analysis, and the rising ability to
reduce challenges, SR motors are being reexamined. SR motors operate using the nonlinear region of a magnetic steel sheet, so the
inductance displays nonlinear behavior that distorts the excitation current waveform a great deal, making it impossible to carry out advanced
projections that are accurate with calculation methods that follow linear formulas. Consequently, it becomes necessary to use the finite
element method (FEM), which can handle nonlinear magnetic properties in material and minute geometry as well as transient currents.
This Application Note explains how to carry out a torque analysis that changes the switch conversion timing and evaluate both the torqueripples and average torque in an SR motor.

Rotor Core

Stator Core

Torque Waveform

Torque Ripple Comparison

Fig. 1 shows the torque waveform when voltage is applied to U-phase at 50 deg andturned off at 80 deg.
It is apparent that torque ripple peaks coincide with declines in torque that occur when
the phase torque switches.

Fig. 2 shows a graph comparing a load analysis at voltage switching angles of 55, 56, and57 degrees in the U-phase with the torque waveform of a switching angle of 50 deg. A
comparison of each of the average torques and the torque ripples at a voltage switchingangle of 50 deg is listed in Table 1. Fig. 2 shows that the switch timing can significantly
reduce the torque ripple. Comparing a 56 degree voltage switching angle with one of 50
degrees, the torque ripple has been contained by approximately 20%, but at the sametime there has been a decline in the average torque, so further studies are necessary inorder to improve the torque ripple without allowing the average torque to decline, suchas considering increases in the overlap of the torque in each phase by changing either thenumber of phases or the width of the salient poles.

50deg

12.0

56deg

57deg

10.0

8.0

Torque (Nm)Torque (Nm)

55deg

12.0

10.0

Ripple6.0
4.0

Tave(θs)(%)
Tave(50)

Tr(θs)
Tr(50)

θs(deg)Tave(θs)(Nm)

Tr(θs)(Nm)

50

4.51

8.44

55

1.70

2.17

37.7

25.7

56

1.39

1.27

30.8

15.1

57

1.13

1.13

25.2

13.4

(%)

8.0
6.0
4.0

2.0
2.0
0.0
0

30

60

90

120

150

0.0

180

0

Rotation Angle (deg)Fig. 1. Torque Waveform

30

60

90
120
Rotation angle (deg)

150

180

Fig. 2. A comparison of the torque using differencesin the voltage switching angle in the U-phase(load analysis)Table 1. The average torque Tave (θs) and the average
torque ripple Tr (θs) according to changes
in the voltage switching angle θs

Case

Shaft

Analysis of a Spindle Motor

7

Stator Core

Rotor Core

module: TR

Overview
Spindle motors are often used as drive motors where limited space is an issue, as is the case with storage media like hard disks. They employ
an outer rotor structure in order to obtain a large torque, but to do so they have to use a great deal of permanent magnets while remaining
thin and compact. In order to reduce the number of parts used in their composition, the rotor core has functions that both bear the magnet's
flux path and transfer the generated torque, which supports the magnet, to the shaft. For this reason the rotor core is composed of materials
that are easy to produce, meaning that there is a possibility that its efficiency as a magnetic circuit will decrease. As motors get smaller, theyrequire a design that accounts for flux leakage because it begins to affect the disc in the rotor.
For this reason, spindle motors need electromagnetic field simulations that use the finite element method (FEM), which can account for
detailed 3D geometry and magnetic saturation in materials, in order to carry out an accurate evaluation.
In this example, how the Speed-Torque curve, the Torque-Current curve and the magnetic flux density distribution of a spindle motor can beobtained.

Coil

MagnetMagnetic Flux Density Distribution

Fig. 1 shows the Speed-Torque curve, and Fig. 2 shows the Torque-Current curve.
The results show that the torque decreases as the speed increases, and increases as thecurrent increases. From the Speed-Torque curve, it becomes apparent that the torque isalmost directly inversely proportional to the rotation speed. From these results, it can beassumed that there is almost no influence from the inductance in the winding.

Fig. 3 shows the magnetic flux density distribution at a rotation speed of 7,200 r/min.
It can be seen that, from the spindle motor's structure, it has magnetic flux density
distribution in the rotor core's rotation axis direction. The magnetic flux density is high inthe rotor core because the rotor core is thin in the magnet's magnetic flux direction. Highmagnetic flux density causes magnetic saturation, which can lead to a decrease in torque.

12000

6

10000

5

8000

4

Current, A

Rotation speed, r/min

Speed-Torque Curve/Torque-Current Curve

6000
4000

3
2
1

2000

0

0
0

0.005

0.01

0.015

0.02

0.025

Torque, Nm

0.03

0.035

0.04

0

0.01

0.02

0.03

0.04

Torque, Nm(Unit:T)

Fig. 1. Speed versus TorqueFig. 2. Torque versus Current

Fig. 3. Magnetic Flux Density Distribution at 7,200 r/min (0.00139 sec)

https://www.jmag-international.com/

9

MagnetCase

Magnet Yoke

Analysis of an Axial Gap Motor

8

module: TR

Overview
Unlike typical cylindrical motors such as radial gap motors, axial gap motors have a structure in which the stator and the rotor, which isarranged on a disk, face each other and produce rotation. For that reason, because it is possible to arrange thinner parts than with a radial gap
motor, they can respond to demands for miniaturization of equipment.
With axial gap motors, evaluations using the magnetic circuit method and empirical data are difficult because the magnetic flux that passes
through the rotor and stator, which face each other, becomes a 3D magnetic circuit, meaning that a 3D electromagnetic field simulation using
the finite element method (FEM) is necessary because it can carry out an accurate analysis.
In this example, how to use JMAG's 3D magnetic field analysis to carry out a load analysis of an axial gap motor, and then obtain the
Torque-Speed curve and the Torque-Current curve.

Coil

Yoke

Magnetic Flux Density Distribution

Torque-Speed Curve/Torque-Current Curve

The magnetic flux density distribution at a rotation speed of 1,000 r/min is shown in Fig.
1. The axial gap motor has magnetic flux density distribution in the rotor coil's rotation
axis direction because of its structure. It is also apparent that the magnetic flux density isgreater in the vicinity between the magnetic poles. High magnetic flux density causes
magnetic saturation, which can lead to a decrease in torque.

Fig. 2 and Fig. 3 show the analysis results for the Torque-Speed and Torque-Current
curves, respectively.
The axial gap motor torque decreases as the rotation speed increases, and increases as
the current increases.

4000

2.0

1.6
1.4
Current, A

Rotation Speed, r/min

1.8
3000

2000

1.2
1.0
0.8
0.6

1000

0.4
0.2
0

(Unit:T)

0

10

20

30

40

50

0.0
0

10

Torque, mN・mFig. 1. Magnetic Flux Density Distribution

20
30
Torque, mN・m

40

50

Fig. 3. Torque-Current Curve

Fig. 2. Torque-Speed Curve

Case

Sound Pressure Analysis of a Loudspeaker

9Cap

module: DS, TR

Vibrator
Edge

Overview
A loudspeaker produces sound when the voice coil makes the vibrator vibrate. The general requirement of the loudspeaker is to produce
uniform sound over a wide range of the frequencies.
Lorentz force is generated in the coil when the magnetic field of a permanent magnet acts on the current flowing through the voice coil, andproduces sound by making the vibrator vibrate. In order to evaluate the sound with good accuracy, it is necessary to handle the resonancephenomenon between the Lorentz force and the speaker's eigenmode properly. The eigenmode and Lorentz force distribution change
depending on the place where the core and coil are wound, so high accuracy calculations need to be carried out using the finite element
method (FEM).
This Application Note presents how the frequency characteristics of sound pressure can be obtained using the constant Lorenz force on thevoice coil, regardless of the frequency.

Sound Pressure Level
The frequency characteristic of the sound pressure level at the front center of theloudspeaker is shown in fig. 1. The sound pressure distributions at frequencies of 200 Hz
and 2000 Hz are shown in fig. 2.
It becomes apparent from the frequency characteristics of the sound pressure level that
this speaker is able to obtain a fairly uniform output at over 100 Hz, and that it is notsuited to usage in the low frequency region of below 100 Hz.

60

Sound pressure level (dB)

50
40
30
20
10
0
-10
10

100

1000

10000
(Unit:dB)

Frequency (Hz)Fig. 1 Frequency Characteristics of the Sound
Pressure Level

10

JSOL Corporation

Fig. 2 Sound pressure level distribution(left: 200 Hz, right: 2,000 Hz)

Yoke

BobbinDamper
Voice coil
MagnetJMAG Application Catalog

Heated Body

Case

10

Thermal Analysis of a Radiant Heater
module: HT, TR

ShieldOverview
Quartz heaters, which are used in semiconductor manufacturing, are a kind of heating device that uses heat radiation phenomena. Thermalradiation is a mechanism of heat transfer, which is defined as the transport of heat through the transmission of electromagnetic waves
between objects that have different temperatures, making transfer possible even through a vacuum. This objective is to transfer heat to aheated body uniformly by placing it near the heater, which has been heated by running current through the coil.
It is necessary to properly handle the effects of the 3D geometry of the heated body or the heat generated from the heater in order to see
whether it is raising the temperature uniformly, so a thermal analysis is carried out.
This Application Note explains how to carry out a thermal analysis using the thermal radiation phenomenon between a heater and a heatedbody to obtain the differences in temperature distribution in the heated body with and without a shield.

HeaterTemperature Distribution
The temperature distributions of the heated body depending on the presence of a shield(the range is the same) are shown in figures 1 and 2, and the temperature difference withthe center of the heated body surface (distance: 150 mm) is shown in fig. 3.
From these results it becomes apparent that, while there is a difference in the absolutevalue of the temperature depending on the presence of a shield, when there is no shield
there are irregularities in the temperature because heat transfer is being carried out
directly to the heated body through thermal radiation from the heater. On the other hand,when there is a shield heat transfer is carried out uniformly to the heated body throughthe shield's even heat distribution.

without shieldwith shield

Temperature diﬀerence (deg C)

12.0
10.0

0

150

300

0

150

300

8.0
6.0
4.0
2.0
0.0
-2.0
0

50

100

150

200

250

300

Distance (mm)(Unit:deg C)(Unit: deg C)

Fig. 1 Temperature distribution in the heated body(without a shield)

Case

11

Fig. 2 Temperature distribution in the heated body(with a shield)

Fig. 3 Temperature difference with the centerof the heated object's surface(distance:150mm)

Pull-in/pull-out Analysis of a PM Stepper Motor
Using a Control Simulator and the JMAG-RT

Yoke
Magnetmodule: RT, TR

Overview
Stepper motors are commonly used for positioning in printers and digital cameras. With a PM stepper motor, there are excitation types suchas one phase excitation, two phase excitation, and one-two phase excitation for the excitation system, and the accuracy for stepper motor
positioning changes depending on which system is used. Pull-in and pull-out torques are important indicators that show the transientcharacteristics of a stepper motor, so it is vital to understand and study them in advance.
To measure them, begin to gradually reduce the load on the stepper motor from a stationary state, measure the pull-in torque when it begins
to rotate, begin to gradually increase the load in sync with the pulses from a rotating state, and measure the pull-out torque when it loses
synchronism. It is necessary to carry out transient analysis while changing the load in order to solve this phenomenon in magnetic field
analysis. While it is possible to calculate it using an equation of motion with JMAG's 3D transient response analysis, such calculations take toomuch time. With JMAG, it is possible to create a motor model that is detailed and conforms to a real machine, and that accounts for spatial
harmonics and magnetic saturation characteristics that are included in a stepper motor. By importing this motor model, a "JMAG-RT model,"to the control/circuit simulator, it is possible to derive the stepper motor's pull-in and pull-out torques quickly and accurately because it
accounts for the motor's magnetic saturation characteristics and spatial harmonics.
This note presents how JMAG-RT can be used to calculate holding torque and coil inductance that varies with current. The result is the
JMAG-RT motor model used as a reference for a circuit / control simulator that runs a transient analysis to obtain pull-in and pull-out torquesof the stepper motor. By also using a single JMAG-RT motor model and changing the circuit on the circuit/control simulator, it is possible toobtain the characteristics of two types of drives: a bifilar winding with a unipolar drive, and a monofilament winding with a bipolar drive. Other
parameters are the same for both analyses.

CoilPull-In and Pull-Out Torques characteristics
Pull-in torquePull-out torque

300

300

250

250

200

Torque (Nm)Torque (Nm)The pull-in and pull-out torque characteristics are shown in fig. 1 and fig 2. Fig. 1 shows
the result with bifilar winding and unipolar drive. Fig. 2 shows the result withmonofilament winding and bipolar drive along with the result of fig. 1. The I-shape in eachfigure represents approximate error range of pull-in and pull-out torques at each point.
Error range is shown in both figures.
The self starting region and slew region can be determined from fig. 1. These results
indicate the correlation between the torque and pulse rate required for the motor to run
on without losing synchronized stepping.
As shown in fig. 2, the result of bifilar winding with unipolar drive and the result of themonofilament winding with bipolar drive have similar pull-in and pull-out torques whenother parameters are the same.

150
100

Pull-in torque (biﬁlar/unipolar)
Pull-out torque (biﬁlar/unipolar)
Pull-in torque (monoﬁlament/bipolar)
Pull-out torque (monoﬁlament/bipolar)

200
150
100

50
50
0
0

50

100

150

200

250

300

350

Pulse rate (pps)

400

0
0

50

100

150

200

250

300

350

400

Pulse rate (pps)Fig. 1 Pull-in and Pull-out Torque Characteristics(bifilar winding /unipolar drive)

Fig. 2 Pull-in and Pull-out Torque Characteristics(monofilament winding /bipolar drive)

https://www.jmag-international.com/

11

Case

13

Work piece (shaft)

High-Frequency Induction Heating Analysis of a Shaft

Coilmodule: FQ, HT

Overview
Shafts are used in parts like axles, which transfer power from the engine to rotate the tires, so the need to have sufficient strength to handle
the torsion. They also need to have increased surface toughness to raise their degree of abrasion resistance in the areas that join with otherparts, and they must maintain their interior toughness in order to obtain strength and fatigue resistance against torsion. By using
high-frequency induction heating, which is a type of surface hardening method, the part is heated rapidly on only its surface by a highfrequency power source. This process also has many other benefits, such as providing a clean working environment because it uses electricalequipment, being very efficient, and providing uniform results for each product. This is why it is being aggressively implemented in the field.
Eddy currents generated by the high-frequency varying magnetic field occur in the surface of the shaft. The material properties also change
due to the rising temperature. Examining detailed phenomena requires handling the phenomena that occur in the interior of the shaft in anumerical analysis based on the finite element method.
This Application Note explains how to create a numerical analysis model and analyze the elevated temperature process in order to use thecoil geometry and current conditions (power supply frequency, current value) to verify whether or not the target temperature distribution is
obtained.

Eddy Current Loss Density Distribution

Temperature Distribution

Fig. 1 shows the eddy current loss density distribution produced in the shaft. The
magnetic field generated by the coil produces eddy currents in the surface of the shaft. In
areas where induction heating makes the temperature exceed the Curie point, themagnetic properties change and both the relative permeability and the electric
conductivity grow smaller, which makes the eddy current loss density shrink as well.

Fig. 2 shows the temperature distribution in the shaft at each time.
From fig. 2, it is apparent that rotating the shaft heats the surface of the shaft facing thecoil uniformly in the circumferential direction.

0(sec)
immediately after heating

immediately after heating

1(sec)2(sec)

3(sec)

4(sec)(Units: deg C)(Unit: W/m^3)

Fig. 1 Eddy current loss density distribution

Fig. 2 Temperature distribution

Case

14

Inductance Analysis of Bus Bar

P-pole Conductor

module: FQ, Pi

N-pole Conductor

Overview

WW-phase
V-phaseV-phaseU-phaseU-phaseVoltage surges can damage the components in electrical equipment such as an inverter.
Busbar inductance can be a cause of surges. Therefore, it is important to reduce it to protect the electrical equipment.
Using FEM allows for the calculation of inductance based on the magnetic field and current distribution obtained from the magnetic field
analysis.
This note presents a case study on the current distribution and the frequency versus inductance characteristic of the busbar.

Current Distribution in the Conductor

InductanceFig.1 shows the current density distribution in the conductor when applying the current
from +U phase to -W phase.
This current distribution is caused by the skin effect in which the current concentrates
near the surface. This current distribution affects resistance and inductance.

Fig. 2 shows self inductance of the N and P pole conductors, Fig. 3 shows the mutual
inductance between the N and P pole conductors, and Fig. 4 shows the Resistance of the
N and P pole conductors.

0.00E+00

4.00E-08

2.00E-08
1.50E-08-3.00E-09-4.00E-09-5.00E-09-6.00E-09-7.00E-09

5.00E-09

12

JSOL Corporation

5.00E-04
Resistance ohm

Inductance H

Inductance H

2.50E-08

1.00E-08

Fig.1 Current Density Distribution of the Conductor
Surface (Upper: N-pole, lower: P-pole)

6.00E-04-2.00E-09

3.00E-08(Unit: A/m)

7.00E-04-1.00E-09

3.50E-08

4.00E-04
3.00E-04
2.00E-04
1.00E-04-8.00E-09

0.00E+00
N-pole conductor

Fig.2 Self-inductanceP-pole conductor-9.00E-09

Fig.3 Mutual inductance

0.00E+00
N-pole conductor

Fig.4 ResistanceP-pole conductorJMAG Application Catalog

Case

15

Cogging Torque Analysis of an SPM Motorwith a Step Skewed Magnet

Stator core
Shaft

module: TR

Overview
Reductions in vibration and noise are being sought after because they are a cause of torque variations in motors, and demands for reduction
are particularly strong with motors that are used in machine tools and power steering. Cogging torque, which is a torque variation that is
produced when there is no current, is generated because the electromagnetic force, which is produced in the gap, changes in relation to therotor's rotation, making it necessary to apply skew to the stator and rotor and improvise with the magnet and stator's geometry in order tolimit said variations in electromagnetic force as a countermeasure for red ucing the torque variations. When applying skew, force in the thrust
direction is produced in exchange for a reduction in the cogging torque, meaning that there is the disadvantage of producing eddy currents
from the magnetic flux that links in the lamination direction.
Consequently, in order to accurately evaluate a motor that has skew applied, one needs a magnetic field analysis simulation that uses thefinite element method (FEM), which can account for a detailed 3D geometry, instead of studies that use the magnetic circuit method or a 2D
magnetic field analysis.
This Application Note presents the use of magnetic field analysis to evaluate the magnetic flux density distribution and cogging torque ineach part of an SPM motor with a step skewed magnet.

MagnetMagnetic Flux Density Distribution

Cogging Torque Waveform

Fig. 1 shows the magnetic flux density distribution at the rotation angle of 60 degrees, andfig. 2 shows the frequency component of the magnetic flux density waveform in the gap.
The upper limit of the frequency component is 1440 Hz, and the primary component of
the frequency is 120 Hz because there are 4 poles and the rotation speed is 30 rps.
As shown in fig. 1, the magnetic circuit is changed by the application of the step skew. As
shown in fig. 2, the ratio of the frequency component at 360 Hz, which is the fundamental
frequency of the cogging torque, is 13 % when the magnet is not step skewed and 4 %when the magnet is step skewed, so the cogging torque may be reduced by applying thestep skew.

Fig. 3 shows the cogging torque waveforms of Model A and Model B. When the magnet isstep skewed, the peak value of the cogging torque is reduced by about 50 percent.

Model A

Model B

(Unit: T)

Model A

0.16

Model A

0.010

0.14

Model B

0.008

0.004

0.1
0.08
0.06

0.002
0.000
-0.0020.04-0.0040.02-0.006

0

10

20

30

40

50

60

-0.0080120 240 360 480 600 720 840 960 1080 1200 1320 1440
Frepuency (Hz)Fig. 1. Magnetic flux density distribution

Model B

0.006

0.12
Torque(Nm)Magnetic Flux Density (T)0.18

Fig. 2. Frequency Componentof the Magnetic Flux Density Waveform in the Gap-0.010
Rotation angle(deg)Fig. 3 Cogging torque waveform

https://www.jmag-international.com/

13

Case

Analysis of a Hybrid Stepper Motor

16

Coil
Stator core
Shaft

module: TR

Rotorcore

Overview

MagnetHybrid stepper motors are used as actuators for equipment where position detection accuracy is required, such as the joints of robots or rotarytables for machine tools. The rotor has a construction that sandwiches a magnet that is magnetized in the axial direction between two rotor
cores that have serrated teeth to create salient poles, and the tips of the stator core's teeth are shaped like gears as well. Because the rotation
resolution is determined by the number of rotor teeth and the number of phases for the drive coil, the design uses large number of teeth, suchas 50 or 100, so the angle resolution can be increased .The most important characteristics for a stepper motor are the controllability, thedetent torque, which is a non-excitation holding torque, and the stiffness torque, which is an excitation holding torque, and not the motor's
output.
The two-plated rotor core of a stepper motor has an N pole on one side and an S pole on the other, so a multipole magnet is achieved bydeviating the saliency of the gear condition by 1/2 pitch. Consequently, the magnetic circuit is 3D. There are also times when the division pitchgeometry of the teeth is complicated, so it is necessary to carry out a 3D electromagnetic field analysis using the finite element method (FEM)
to proceed with an accurate preliminary study.
This document introduces how the detent torque and stiffness torque can be calculated for a hybrid stepper motor.

Detent TorqueStiﬀness TorqueThe detent torque is shown in fig. 1, and the magnetic flux distribution in the XY-plane at
the rotation position of 0.45 degrees is shown in fig. 2. A closer view to show the flux
distribution in the gap is shown in fig. 3. These results reveal that the flux in both the rotor
and stator is saturated even with no excitation. The flux leakage around the tips of theteeth, due to saturation, can be expected to have a large effect on the detent torque.

Fig. 4 shows the stiffness torque with one-phase excited and fig. 5 shows the stiffness
torque with two-phase excited. The flux distribution in the gap of the XY-plane with
one-phase excited is shown in fig. 6, and the flux distribution in the gap of the XY-planewith two-phase excited is shown in fig. 7. As with the detent torque, the magnetic flux
around the teeth is saturated. The flux leakage caused by the saturation can be expected
to also have a large effect on the stiffness torque.

0.020

0.5

0.8

0.015

0.4

0.6

0.3

0.4

0.2

0.000
-0.005-0.0100.1

Torque (Nm)

0.005

Torque (Nm)Torque (Nm)

0.010

0.0
-0.1
-0.2
-0.3

-0.015

-0.5
0.00

0.45

0.90

1.35

Rotation angle (deg)

-0.8
0.00

1.80

0.90

1.80

2.70

Fig. 2 Magnetic flux distribution at the rotation of 0.45 deg

3.60

4.50

5.40

7.20

0.00

0.90

1.80

2.70

3.60

4.50

Fig. 4 Stiffness torque with one-phase excitation

6.30

7.20

Fig. 5 Stiffness torque with two-phase excitation(Unit: T)(Unit: T)

Fig. 6 Magnetic flux distribution of the gap
with one-phase excitation at a rotation of 1.8 deg

Fig. 7 Magnetic flux distribution of the gap
with two-phase excitation at a rotation of 0.9 deg

Stator core

Case

17

5.40

Rotation angle (deg)(Unit: T)

Fig. 3 Flux d istribution in the gap at the rotation of 0.45 deg

6.30

Rotation angle (deg)(Unit: T)

Fig. 1. Detent torque

0.0

-0.4
-0.6

-0.4

-0.0200.2-0.2

Rotor core

Inductance Analysis of an IPM Motormodule: DP

CoilOverview
An IPM motor can use both magnet torque and reluctance torque, so by appropriately choosing the current phase it is possible to improve
efficiency over a broad spectrum of drive range. It is a motor type that is often used in equipment with a wide operational range, from aircompressors in air conditioners to motors that power vehicles. In many cases, strong rare earth magnets are used to increase output density,
so it is necessary to have an IPM motor design that accounts for magnetic circuit saturation.
For this reason a study that considers the influence of saturation needs to be carried out in order to evaluate the IPM motor's design, so thestudy ends up carefully investigating variations in inductance characteristics due to current phase, geometry, or the material's nonlinear
magnetic properties. An analysis based on the magnetic circuit method or a theoretical equation that assumes linear properties cannot
predict these functions with good accuracy and therefore cannot derive them. Consequently, in order to perform an advanced projection of
an IPM motor's design, an electromagnetic field simulation that uses the finite element method (FEM) is necessary.
This Application Note explains how to obtain the current phase angle characteristics of a dq axis inductance that accounts for magnetic
saturation and flux leakage in an IPM motor.

MagnetInductance
3.0

0.040
0.035

Ld

0.030

Lq

Total TorqueReluctance Torque
2.0

0.025
0.020
0.015

JSOL Corporation

1.5
1.0

0.010
0.5

0.005
0.000
0

10

20

30

40

50

60

Current phase angle (deg)Fig. 1. Ld/Lq versus current phase angle

14

Magnet Torque

2.5

Torque (Nm)Inductance (H)Fig. 1, and a current phase angle graph for the torque that was output from theinductance value is shown in fig. 2.
Fig. 1 shows that th e d-axis inductance is smaller because the magnetic flux must passthrough the low permeability magnet. It is less affected by magnetic saturation due tovery large magnetic resistance. On the other hand, the q-axis inductance is larger than thed-axis inductance due to larger permeability, and it is also significantly affected bymagnetic saturation because the magnetic resistance is much smaller.
Fig. 2 shows that the maximum torque for this IPM motor can be obtained when thecurrent phase angle is around 20 deg.

70

80

90

0.0
0

10

20

30

40

50

60

70

Current phase angle (deg)Fig. 2. Torque versus current phase angle

8090

JMAG Application Catalog

Cover

Case

Rotor core

Thermal Analysis of an IPM Motor

18

Magnetmodule: HT, LS, TR

Overview
Exactly how to resolve the problem of rising temperatures is a critical issue when trying to achieve an improvement in a motor's efficiency
and output. In order to solve this problem it is important to investigate a magnetic design that reduces the losses themselves because they
are a source of heat, but it is also important to study a thermal design that improves heat dissipation and does not let the temperature rise.
Copper loss in the coils and iron loss in the core are the dominant heat sources, so this analysis mainly evaluates the effects of this heat.
Changes in the magnet's properties due to temperature are large and its heat resistance is low, so it is necessary to design while paying
careful attention to rising temperatures during operation. During operation, rated evaluations with a continuously operated constant load are
run until a thermal balanced state has been reached. In addition to these rated evaluations, however, thermal transient evaluations that add athermal cycle with an intermittently operated electrical overload are performed, as well.
In order to carry out an accurate thermal design, it is necessary to first correctly understand the heat generation amount and location, so it
would be advantageous to calculate the losses in a magnetic field analysis simulation using the finite element method, and from there tocarry out a thermal analysis using the resulting loss distribution.
In this example, how to evaluate a motor's temperature distribution by creating a thermal analysis model that can investigate the loss analysis
and temperature distribution in order to obtain the motor's total loss distribution, and then analyzing the elevated temperature process.

Coil
Shaft

Stator core

Loss

Temperature Distribution at Steady StateThe values of the losses in each part when a 2 A current is applied are shown in Table 1.
The loss in the coil is relatively large, and it is expected to be the main source of heat.

The distribution of temperature at the steady state when a 2 A current is applied is shownin Fig. 1.
The result shows that the temperature is transferred from the coil to the surroundingparts.

Temperature Variation
The points where the temperature is measured in each part and the temperature changes
when a 2 A current is applied are shown in Fig. 2 and Fig. 3.
Fig. 3 shows that the temperature in each part reaches a steady state.

Iron loss in therotor core

Iron loss in thestator core

Eddy current
loss in themagnetCopper loss inthe coil

0.23 W

4.5 W

0.015W

21W

Consecutive Operating Time and Average Torque with Various Currents(Unit: deg C)

Table 1. Loss in Each Part

Fig. 1. Temperature Distribution at a Steady State

Rotor core

Coil

Magnet

The current was set at three times the rated value to increase the average torque to 1.0
Nm or above. The average torque and temperature variations in the coil are shown in
Table 2 and Fig. 4 (see Fig. 2 for the points of measurement).
Although increasing the current produces a higher torque, at a motor drive time of 20
min the coil's temperature exceeds the upper temperature limit of 180 deg C.
By carrying out a magnetic field-thermal coupled analysis in this way, it is possible toevaluate the motor's properties and constraints with more detail.

Current 2 A

Stator core

Current 6 A

260

45

240
220

35

30

Current, A

2

6

Average torque, Nm

0.36

1.1

Temperature, deg C

Temperature, deg C

40
Measuring points

25

200 The coil's upper
temperature limit
180
160
140
120
100
80
60
40

20

Fig. 2. Points of Measuring Temperature for Each Part

20
0

20

40

60
Time, min

80

100

120

Fig. 3. Temperature variation

10

20

30
40
Time, min

50

60

70

Fig. 4. Temperature Variation of the Coilwith Various Currents

Table 2. Current and Average Torque

Case

19

0

Rotor Core

Analysis of the Centrifugal Force in an IPM motor
module: DS

Overview
While motors have started being combined with motor drives and used in a wide range of velocities, further changes toward high output andhigh efficiency are being demanded of them. While higher speed revolution has been given as a means of attaining higher output, thecentrifugal force becomes larger with higher speed revolution and this requires evaluation from the point of mechanical strength.
IPM motors have a structure that embeds the magnet in the rotor. Centrifugal force kicks in during motor drive, so the magnet becomespressed against the rotor core because it gets peeled off or displaced, making it so that a strong local stress begins to operate. It is necessary tocorrectly handle the phenomena of a magnet peeling off or becoming displaced in order to accurately obtain the local stress distribution. It is
important to account for the adhesion and contact conditions between the magnet and rotor core in an analysis in order to do this.
This Application Note presents examples of cases that obtain changes and stress distribution from the centrifugal force in the rotor.

MagnetMises StressFig. 1 shows the mises stress distribution in the rotor.
Fig. 1 shows that the magnet drastically deforms the rotor core with centrifugal force inthe radial direction. Also, the Mises stress increases at the bridge part of the rotor core.
Mises stress allows the evaluation of the risks of damaged rotor cores and mechanicalstrength.

(Units: MPa)

Fig. 1 Mises stress density distribution

https://www.jmag-international.com/

15

Case

20

Case

Sound Pressure Analysis of an SPM Motor

Bearing
Rotor coremodule: DP, DS

Overview

MagnetStator core

As electric motors are becoming more common, motors which create less noise are in high demand. Sound can be divided into categories ofelectromagnetic noise, mechanical noise, and draft noise, where electromagnetic noise is the most common for medium and small sized
motors. Sound can be divided into categories of electromagnetic noise, mechanical noise, and draft noise, where electromagnetic noise is themost common for medium and small sized motors.
The electromagnetic force in a motor vibrates as an electromagnetic excitation force which creates noise. The vibration and noise are
generated when the electromagnetic excitation force resonates with the motorʼs eigenmodes. In order to evaluate this phenomenon moreaccurately, it is necessary to understand the distribution of electromagnetic force that moves the stator core which is the basis for theradiated sound. The distribution of electromagnetic force or the eigenmodes in a model that depends on the geometry of a stator core isrequired for running an analysis such as for the finite element analysis.
In this example, an evaluation of a reactor's sound pressure, when acquiring electromagnetic force generated by a stator core for a SPM
motor and linking the eigenmodes of a motor.

Flanged bracket
Shaft

Electromagnetic Force

Eigenmode

Fig. 1 shows the distribution of the electromagnetic force distribution. A strong output ofelectromagnetic force can be seen at the tips of the teeth. From these results, it can be
expected that the stator core is generating a strong force in a radial direction.
The results of a resolution in a frequency component for an electromagnetic force in aradial direction on the tips of the teeth is shown in Fig. 2. From these results, thefrequency component in 400 Hz is the most dominant. Also, can see that the frequency
components above 5,000 Hz are very small.

The eigenmode near 3,600 Hz is shown in Fig. 3. It is apparent that a mode in deformation
exists in a stator of ellipse near 3,600 Hz, and a large vibration is going to be generated
from the resonance phenomenon between an eigenmode and an electromagnetic force.

The frequency component measurement point
for the electromagnetic force

Acceleration
Fig. 4 shows the distribution of the acceleration at 3,600 Hz. This is the frequency in which
the mode that deforms elliptically (displayed in the eigenmode analysis) appears. At 3,600
Hz, it can be seen that electromagnetic resonance is occurring between theelectromagnetic force and the eigenmode shown in Fig. 3.

(Unit: N)

Fig. 1. Electromagnetic Force Distribution

Electromagnetic force (amplitude), N

Sound Pressure Distribution
The sound pressure distributions at 3,600 Hz are shown in fig. 5. Like with the distributionof acceleration, at 3,600 Hz an electromagnetic resonance occurs between theelectromagnetic force and the eigenmode shown in fig. 3.

25
20
15
10
5
0

400 1200 2000 2800 3600 4400 5200 6000 6800 7600
Frequency, Hz

Fig. 2. Frequency Components of the Electromagnetic Force

Full modelFig. 3. The Eigenmode of Approximately 3,600 Hz(Red Line: Original Shape)

Case

21

Stator core

(Unit: m/s^2)

Stator core

Full model(Unit: dB)

Fig. 4. Acceleration Distribution of 3,600 Hz

Fig. 5. Sound Pressure Level Distribution in 3,600 Hz

Stator core

Iron Loss Analysis of an SPM Motor Including
the Eﬀect of Shrink Fitting

Frame

Magnetmodule: DS, DP, LS

CoilOverview
A magnetic steel sheet is used for the iron core in a motor. A frame is shrunk into a stator core in order to sustain the laminated structure andto improve the strong joint between the frames. It is know that a compressive stress is generated during the shrinking process which increases
the iron loss process. Therefore, it is important to account the affects of stress during iron loss evaluation. Therefore, it is important to accountthe affects of stress during iron loss evaluation.
An iron loss is generated by the magnetization field in displacement with a steel sheet. The size of the iron loss is dependent on the iron loss
properties of a steel sheet. Iron loss characteristics of a steel sheet deteriorates when there is stress in shrinkage. The stress generated by theshrinkage is distributed in areas in which the section in the back yoke becomes large. In order to evaluate the iron loss with good accuracy, it isnecessary to obtain the stress distribution for the magnetic flux, time variation, and steel sheet with accuracy.
This note presents an analysis to obtain the iron loss density of an SPM motor both including and not including the stress caused by shrink
fitting.

Shaft

Bolt

Compressive Stress Distribution
The compressive stress distribution is indicated in Fig. 1. The compressive stress caused by
the frame shrinking in the radial direction increases around the slots as indicated in Fig. 1.
This is because the stress decreases towards the tips of the teeth while the stress isconcentrated around the slots.

Low

Iron Loss Density Distribution
The iron loss density distribution is indicated in Fig. 2. The iron losses increase up to 7
times depending on whether or not the stress is applied. An analysis that accounts for thethermal stress caused by shrink fitting needs to be performed to evaluate the iron losses
more accurately because the iron losses increase when the thermal stress is applied.

16

JSOL Corporation

High
Iron loss: 0.37W
without stress

Iron loss: 0.64 W
with stress(Unit: MPa)

Fig. 1. Principal stress (compressive stress)

Fig. 2. Iron loss density distribution(Unit: W/m^3)

JMAG Application Catalog

Analysis of the Eddy Current in the Magnet
of an IPM Motor

Case

22

Rotor coreCoilmodule: TR

Overview
More and more permanent magnet motors are starting to use rare earth magnets, which have a high energy product, in order to achieve
higher output density. Neodymium rare earth magnets contain a great deal of iron so they have a high electric conductivity, but when avarying magnetic field is applied they produce Joule loss from eddy currents. Due to the spread of IPM structure adoption and field
weakening controls in recent years to speed up rotation, the frequencies and fluctuation ranges of varying fields applied to magnets haveincreased, and there has been a corresponding increase in Joule loss. By dividing the magnet, like one would a laminated core, to control
eddy currents, one can obtain a method of lowering the apparent electric conductivity and lowering the eddy currents. Armature reactions inthe stator occur before the eddy currents produced in the magnet, so the eddy currents are determined by the slot geometry of the statorcore, the geometry of the rotor, the nonlinear magnetic properties of the core material, and the current waveform that flows through the coil.
In order to examine these kinds of magnet eddy currents ahead of time, one has to account for things like these geometries and material
properties precisely, so a magnetic field simulation using the finite element method (FEM), which can account for them, would be the mosteffective.
In this example, the use of a magnetic field analysis in a state of operation to obtain variations in magnet eddy current losses according to thenumber of divisions in the magnet.

Stator core
MagnetEddy Current Density Distribution in the Magnet

Change of the Loss in Divided Magnet

Fig. 1 shows the eddy current density distribution in the magnet. The eddy currents areconcentrated on the surface of the magnet. Using a divided magnet breaks up the eddy
current paths.

Fig. 2 shows the magnet's eddy current loss density distribution and Fig. 3 shows theeddy current loss.
By increasing the number of divisions in the magnet, the magnet increases the eddy
current flow path and increases the electrical resistance. Because of this, the eddy current
density in each divided magnet is reduced as well, making it so that the total amount ofeddy current loss decreases.

0.07
0.06

Loss (W)

0.05
0.04
0.03
0.02
0.01
0
No division

28 divisions

48 divisions

88 divisions(Unit: A/ｍ^2)

Fig. 1 Eddy current density distribution in the magnet

No division

28 divisions

48 divisions

88 divisions

No division(Unit: W/m^3)

Fig. 2 Eddy current loss density distribution in the magnet

2 divisions

4 divisions

8 divisions

Fig. 3 Eddy current loss in the magnet

Case

Stator core
MagnetEccentricity Analysis of an IPM Motor

23

module: DP

Overview
Rotor eccentricity is one cause of vibration and noise in motors. It is well known that motor torque is produced by electromagnetic attraction
and repulsion between the stator and rotor, but not much attention is paid to the fact that electromagnetic attraction acts in the radial
direction between the rotor and stator. This is because it seems that this electromagnetic force is canceled out and therefore not produced
because the rotor and stator are normally arranged concentrically. However, if there are dimensional errors in the parts that support the shaft
or stator and concentricity is not maintained, in other words if there is eccentricity, the electromagnetic force in the radial direction is notcanceled out. In this case, friction increases due to the constant action of the radial load on the shaft bearings, causing vibration and noise.
A certain amount of error from processing has to be expected. Processing error itself is not so large that the parts cannot be put together, buteven assembly error can cause a minute eccentricity of around 1/10 mm. Analysis that can handle this level of precision is needed to evaluate
this kind of minute geometry difference, and electromagnetic field analysis using the Finite Element Method (FEM) is useful because it has thesensitivity for detailed geometry differences.
This Application Note presents how to obtain variations in electromagnetic force according to changes in the amount of rotor eccentricity.

Shaft
Rotor core
CoilElectromagnetic Force on the Rotor

Electromagnetic Force Acting on the Teeth

For each amount of eccentricity, the eccentricity direction component of theelectromagnetic force acting on the rotor is indicated in fig. 1, and the perpendicular
direction component of the eccentricity is indicated in fig. 2.
The graphs show the eccentricity amount becomes larger while the eccentricity direction
component of the electromagnetic force becomes smaller. The balance for theorthogonal direction component of the eccentricity for an electromagnetic force isdeformed and the amplitude becomes larger.

The electromagnetic force of the radial direction component that acts on the teeth where
the gap length is most narrow is indicated in fig. 3, and the results for the frequency
components of the electromagnetic force displayed in fig 3 is indicated in fig. 4. The radial
direction is the positive center direction for a circle.
Fig. 3 shows the larger the eccentricity amount the faster the electromagnetic forcechanges and is the cause of deformation in the stator. The area in red square as shown infig. 4 indicate that an increase in harmonic components based on eccentricity causes
vibration and noise.

Eccentricity amount 0.0mm
Eccentricity amount 0.4mm
Eccentricity amount 0.2mm

Eccentricity amount 0.0mm
Eccentricity amount 0.4mm
Eccentricity amount 0.2mm
40

500
400
300
200
100
0

20
10
0
-10
-20
-30
-40
-50

-100
0

30

60

90

120

150

180

Rotation angle (deg)Fig .1. Electromagnetic force acting on the rotor(Eccentricity direction component)

Eccentricity 0.0mm

350

30

0

30

60

90

120

150

180

Rotation angle (deg)Fig .2. Electromagnetic force acting on the rotor(Orthogonal direction component of the eccentricity)

Electromagnetic force (N)600Electromagnetic force (N)Electromagnetic force (N)Electromagnetic force (N)700

Eccentricity amount 0.0mm
Eccentricity amount 0.4mm
Eccentricity amount 0.2mm

300
250
200
150
100
50
0
-50
0

30

60

90

120

150

180

Rotation angle (deg)Fig. 3. Electromagnetic force acting on the teeth(radial direction component)

Eccentricity 0.2mm

Eccentricity 0.4mm

100
90
80
70
60
50
40
30
20
10
0
120

240

360

480

600

720

840

960

1080 1200

Frequency (Hz)Fig. 4. Frequency component
for the electromagnetic force acting on the teeth

https://www.jmag-international.com/

17

Case

24

Cogging Torque Analysis of an SPM Motorwith a Skewed Stator

Stator core
Shaft

module: TR

Overview
Reductions in vibration and noise are being sought after because they are a cause of torque variations in motors, and demands for reduction
are particularly strong with motors that are used in machine tools and power steering. Cogging torque, which is a torque variation that is
produced when there is no current, is generated because the electromagnetic force, which is produced in the gap, changes in relation to therotor's rotation, making it necessary to apply skew to the stator and rotor and improvise with the magnet and stator's geometry in order tolimit said variations in electromagnetic force as a countermeasure for reducing the torque variations. When applying skew, force in the thrust
direction is produced in exchange for a reduction in the cogging torque, meaning that there is the disadvantage of producing eddy currents
from the magnetic flux that links in the lamination direction.
Consequently, in order to accurately evaluate a motor that has skew applied, one needs a magnetic field analysis simulation that uses thefinite element method (FEM), which can account for a detailed 3D geometry, instead of studies that use the magnetic circuit method or a 2D
magnetic field analysis.
This note presents the use of magnetic field analysis to evaluate the cogging torque of an SPM motor with a skewed stator.

MagnetMagnetic Flux Density Distribution

Cogging TorqueFig. 1 shows the magnetic flux density distribution of the stator core, and fig. 2 shows themagnetic flux density waveform in the gap. The magnetic flux density waveform is theaverage of the three points that are indicated on the line in the axial direction shown infig. 2. As shown in Fig 1, magnetic circuit changes with the application of the skew.
Applying skew also deviates the phases of the magnetic flux density waveforms at thethree points. Therefore, the variation in the average magnetic flux density waveform issmaller as shown in fig. 2, so the cogging torque may be reduced.

Fig. 3 shows the cogging torque waveform of the SPM motor with the skewed stator andwith the non-skewed stator. The peak value of cogging torque of the skewed stator is
reduced by about 60 percent compared with that of the non-skewed stator.

Measuring point 1

Without skew

With skew

Measuring point 2
0.008

without skew

(Unit: T)with skew

Without skew

0.76

Torque (Nm)Magnetic Flux Density (T)Measuring point 3
With skew

0.72

0.004
0.000
-0.0040.68-0.008
0.64
0

10

0.60
0

10

20

30

40

50

20

30

40

50

60

Rotation angle (deg)60

Rotation Angle (deg)Fig. 1 Magnetic Flux Density Distribution

Fig. 2 Magnetic flux density waveform in the gap

Fig. 3 Cogging Torque Waveform

Shaft

Case

25

Rotor core

Analysis of a Claw Pole Alternator
module: TR

Overview
Demand for high fuel efficiency in vehicles has been growing every year, and auxiliary machines like power steering and coolant pumps havebeen switching to electrical operation to support those needs. This is why the amount of electrical power being used in typical gasoline
vehicles is increasing with each passing year, and manufacturers are looking for high-output alternators that can supply this level of electricity.
They need to increase the output density, however, because they cannot increase the size of the actuator to correspond with the added
generation capacity. They also need to achieve higher efficiency. A claw pole alternator generates electricity in the coil on the stator side withthe rotor side acting as an electromagnet. The excitation coil on the rotor side is a single phase, and the claw pole is arranged so that it wraps
around this coil. The claws that extend from the inside of the coil and the ones that extend from the outside of the coil have poles withdifferent polar characteristics, and they have the same polar structure as a magnet that is arranged with magnetization that alternates
between North and South. Because the alternator needs to be designed with a 3D geometry to account for the claw poles and the analysis
needs to consider eddy currents generated in the surface of the claw poles, which are made from a metal plate, an electromagnetic field
analysis using the finite element method would be the most useful, as it can simulate detailed geometries and account for eddy currents.
This Application Note presents the use of an electromagnetic field analysis to evaluate the output capacity of a claw pole alternator operating
at 1500 r/min while accounting for eddy currents in the rotor core.

Eddy Current Density Distribution/ Magnetic Flux Density
Distribution
Fig. 1 shows the eddy current density distribution and fig. 2 shows the magnetic flux
density distribution.
Eddy current is produced in the surface of the claw pole from the rotor core. These eddy
currents produce eddy current losses and reduce the alternator's efficiency. The eddy
currents also effect the magnetic flux density distribution, so they must be taken into
account when studying a magnetic circuit at high output.

OutputFig. 3 shows the current versus time characteristic taking into account the effect of ashunt resistance.
The transient phenomena that occur until a steady state are caused by influence from thestator coil's inductance and the rotor core's eddy currents, so ripples in a steady state arecaused by the commutator circuit.
The alternator output is approximately 41 Ampere, which can be found from the averagevalue of the current waveform at a steady state.

90.0
80.0
70.0

Current (A)

60.0
50.0
40.0
30.0
20.0
Claw pole cross-section

10.0
0.0
0.00
(Unit: A/m^2)

Fig. 1 Eddy current density distribution at 0.06 sec

18

JSOL Corporation

0.01

0.02

(Unit: T)

Fig. 2 Magnetic flux density distribution at 0.06 sec

Rotor coil
Stator core

0.03
Time (sec)Fig. 3 Current versus time

0.04

0.05

0.06

JMAG Application Catalog

Case

26

Stator Core

Braking Torque Analysis of an Electromagnetic Brake
module: TR

Overview
An electromagnetic brake is an auxiliary brake device for large-scale vehicles like trucks and buses. It is fit onto the propeller shaft and applies abraking force. There are both hydraulic and electromagnetic types. With an electromagnetic brake, a magnetic field is produced in the statorcoil, making eddy currents occur because of time variations in the magnetic flux density linking to the rotor. This, in turn, produces a braking
torque. The range in which eddy currents occur in the rotor and the braking torque can vary a great deal according to the current flowing to
the stator coil and the rotor's rotation speed.
In order to estimate the electromagnetic brake's performance accurately at the design stage, it is best to carry out an electromagnetic field
analysis simulation using the finite element method (FEM) because it can approximate the material's nonlinear magnetic properties and canapproximate the skin effect caused by current distribution, as well.
This Application Note shows how to obtain the braking torque of an electromagnetic brake during drive.

Rotor
Coil

Eddy Current Density Distribution

Dependency of Braking Torque on Rotation Speed

Fig. 1 shows the eddy current density distribution of the rotor at different rotation
speeds. Eddy currents flow on the inner surface of the rotor to interrupt the magnetic
flux produced by the stator coil. When the rotation speed increases, the magnetic flux
flowing in the rotor changes substantially, thereby increasing the eddy currents
generated on the rotor.

Fig. 2 shows the level of dependence of the braking torque on rotation speed.
When the rotation speed increases, eddy currents are concentrated on the surface dueto the skin effect, making the electric resistance grow larger and shrinking the amountthat the eddy currents increase. As a result, increases in the braking torque also become
smaller because it is caused by eddy currents.

140

Braking torque (Nm)

120
100
80
60
40
20
0
Rotation speed:1000 rpm

Rotation speed:
3000 rpm

Rotation Speed:
5000 rpm

0(Unit: A/m^2)

27

2000

3000

4000

5000

6000

Rotation speed (rpm)Fig. 2 Level of dependence of the braking torqueon rotation speed

Fig. 1 Eddy current density distribution
on the rotor at different rotation speeds

Case

1000

Head Field Analysis of a Recording Write Head
That Accounts for Eddy Currents

Upper coilyoke

module: TR

Overview
Magnetic heads are devices that are used to record data on storage media, and are often found in hard disks. A magnetic head has both arecording head that writes data by magnetizing a round magnetic disk and a playing head that reads the data from the magnetic disk's
magnetization pattern. For the recording head, the vital thing is an evaluation of the recording head field's responsiveness toward input
electrical signals. This evaluation comes from a detailed evaluation of the magnetic flux density distribution around the head. To study thesecharacteristics, the analysis needs to include the effect of eddy currents generate on the yoke.
In order to account for eddy current distribution that is produced in the fine part at the tip of the recording write head, a magnetic field
analysis using the finite element method (FEM) is most effective.
This Application Note shows how to obtain the responsiveness of recording head field that is generated in the magnetic head.

Eddy Current Density Distribution / Magnetic Flux Density
Distribution
Fig. 1 shows the eddy current distribution around the head, and fig. 2 shows the magnetic
flux density distribution. As shown in the eddy current density distribution, eddy currents
are generated on the surface of the yoke. This eddy current interrupts the flow of themagnetic flux, resulting in reduced responsiveness of the head field to themagnetomotive force.

Response Characteristics of the Recording Head Field
Fig 3 shows a graph of the response characteristics of the head field. The head field starts
up more slowly than the magnetomotive force waveform. This is caused not by the coilinductance but also by the eddy currents on the yoke. When analyzing the response of
the head field, the analysis must include the effects of eddy currents.

Point to measure
the head ﬁeld

Head ﬁeld, A/m(Unit: A/m^2)

Fig. 1 Eddy current density distribution(Unit: T)

Fig. 2 Magnetic flux density distribution

0.4

2.0E+050.2

0.0E+00

0.0-0.2-2.0E+05-4.0E+05

-0.4
0

2

4

Electromotive force, AT

Head ﬁeld (with eddy currents)
Electromotive force
Head ﬁeld (with no eddy currents)
4.0E+05

Head part

Head partyoke
Lower coil

Reference:
K. Fujiwara, F. Ikeda, A. Kameari, Y. Kanai, K.
Nakamura, N. Takahashi, K. Tani, and T.
Yamada, ”Thin Film Write Head Field Analysis Usinga Benchmark Problem,” IEEE Trans.
Magn., vol.36, no.4, pp.1784-1787, July 2000.

6
8 10 12 14 16
Time, nsec

Fig. 3 Response characteristics of the head field

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Magnetic Field Analysis of a Speed Sensor

Yokemodule: TR

CoilOverview

MagnetAntilock brake systems (ABS) have become a standard feature in vehicles, so speed sensors are attached to each wheel in order to measure
their respective speeds. There are several methods of detecting rotation speed, but magnetic sensors are weather resistant and have a small
number of parts because there only needs to be a gear on the rotation side, so they are widely used. The challenges from a design standpoint
are the angle and relative distance between the gear's teeth and sensor, and how to ensure sensitivity and responsiveness when considering
the magnetic influence of the surrounding air. In order to proceed with an advance study like this that considers a precise geometry andmaterial properties, an electromagnetic field analysis using the finite element method (FEM) is effective.
This Application Note presents the use of magnetic field analysis to evaluate the variation of the voltage signal of a magnetic speed sensor
for a range of air gap distances.

Rotating ring

Magnetic Flux Density Distribution

Voltage Signal at Diﬀerent Gap Widths

Fig. 1 shows the magnetic flux density distribution at a rotation angle of 5.0 deg when the
gap width is 0.5 mm.
The figures show how the magnetic flux from the magnet flows through the yoke andreaches the rotating ring. When the rotating ring rotates, the permeance in the magnetic
circuit changes due to the moving teeth. This changes the amount of magnetic flux thatpasses through the yoke, thus generating the electromotive force in the coil.

Fig. 2 shows the voltage signal waveform at each gap width, and fig. 3 shows the peakvalue of the voltage signal as a function of the gap width.
When the gap width is large, the voltage is small because the permeance is smaller
compared with a small gap width, and then the magnetic flux amount that flows throughthe yoke and the coil becomes smaller.

0.5(mm)1.0(mm)2.0(mm)

2.5(mm)6.0

1.5(mm)5.0

6.0
Voltage (V)Voltage (V)

4.0
2.0
0.0

4.0
3.0
2.0

-2.0
1.0
-4.0
-6.0
0.00E+00(Unit: T)

0.0
2.50E-04

5.00E-04

7.50E-04

1.00E-03

0.0

1.25E-03

0.5

Fig. 1. Magnetic flux density distribution

Case

29

1.0

1.5

2.0

2.5

3.0

Gap width (mm)Time (sec)Fig. 2. Voltage Signal Waveform

Fig. 3. Voltage Signal at Different Gap Widths

Iron Loss Analysis of an SPM Motor with Overhanging
MagnetShaft
MagnetStator coremodule: LS, TR

Overview
There are times when permanent magnet motors are designed with a magnet made with overhang, in other words made longer than thestator's stack length, in order to strengthen the magnetic field that it creates. A space is necessary in the stator core to supply the coil ends,
and there is a wasted space in the rotor if the rotor and stator have the same stack length, so a magnet is placed in this space with theobjective of increasing the magnetic flux without making the magnet thicker. However, the magnetic field produced by the overhanging partof the magnet enters the stator at an angle, so magnetic flux is produced in the lamination direction, which creates a possibility of increasing
eddy current loss by a wide margin. When the overhang is too big, the magnet's magnetic field goes to waste because it does not reach thestator. For this reason it is necessary to set up the overhang amount properly while looking at the trade-off between an increase in torque andan increase in losses. A magnetic field analysis using the finite element method (FEM), which can obtain the relationship between a three
dimensional magnetic field and eddy currents, is an effective method for an advance study.
This Application Note presents the use of a no-load iron loss analysis of an SPM motor with and without an overhanging magnet.

Magnetic Flux Density Distribution, Eddy Current Density Distribution
Fig. 1 shows the flux density distribution of the stator core with and without overhanging
magnet, fig. 2 shows the flux density distribution in the lamination direction, and fig. 3
shows the eddy current density distribution. The flux density is higher around the surfaceof the stator core because the overhanging magnet increased the flux in the in-plane
direction and the lamination direction. The eddy currents generated around the surfaceof the tooth tips also penetrates into the interior of the stator core because of the increasein magnetic flux in the lamination direction.

3.5

2.5

Loss (W)0.0
Without overhang
without overhang

with overhang

Eddy current loss causedby the ﬂux in thelamination direction

0.8

Loss (W)

0.7

Hysteresis loss causedby the ﬂux in thelamination direction

0.6
0.5

Eddy current loss causedby the ﬂux in thein-plane direction

0.4
0.3
0.2

Hysteresis loss causedby the ﬂux in thein-plane direction

0.1
with overhang(Unit: T)

0.0
Loss increases caused by the overhang

Fig. 1 Magnetic flux density distribution at 0.015 s

20

JSOL Corporation

Fig. 2 Magnetic flux density distributionof the Z-component at 0.015 s

With overhang(Unit: A/m^2)

Fig. 3 Eddy current density distribution at 0.015 s

0.9

without overhang

Hysteresis loss causedby the ﬂux in thein-plane direction

0.5

1.0

(Unit: T)

Eddy current loss causedby the ﬂux in thein-plane direction

1.5
1.0

Fig. 4 show the values of the eddy current loss and hysteresis loss from the magnetic fluxin the lamination direction and in-plane direction, which was obtained from a magnetic
field analysis and an iron loss calculation. Fig. 5 shows the amount of increase in thelosses from overhang.
Losses generated by magnetic flux in the lamination direction and in-plane direction bothincreased when overhang was applied.

with overhang

Hysteresis loss causedby the ﬂux in thelamination direction

2.0

Iron Loss Comparison with and without Overhanging Magnet

without overhang

Eddy current loss causedby the ﬂux in thelamination direction

3.0

Fig. 5 Loss increases caused by the overhang

Fig. 4 Iron loss comparison

JMAG Application Catalog

Case

30

Stator core

Magnetization Field Evaluation of an SPM motor

Magnetmodule: ST

Shaft

Overview
A surface magnet type motor (SPM) has direct contact with a stator which generates torque through the interaction of magnetic fields
generated by magnets and excitation coils. The magnetization state of a magnet is the main factor which determines the characteristics of a
motor. It is important that motor designers adjust the magnetization pattern when designing a motor to restrain the higher components of an
induced voltage to confirm the variation in characteristics. A general method of confirming the magnetization direction of a magnet ismeasuring the induced voltage waveform or the surface magnetic flux density of the magnet.
The characteristic such as magnetization device, magnetization of electric current, or magnetizing ration needs to be taken into account whenevaluating the magnetic patterns. To evaluate the characteristics in advance, understanding the entire flow of magnetization is necessary,
where a magnetic field analysis simulation using the finite element method which can obtain details on electromagnetic behavior from results
is very useful.
In this example, carry out SPM motor's magnetization to obtain the magnetization field and magnet's surface magnetic flux density.

Coil

Magnetization Field

Surface Magnetic Flux Density of the Magnet

The magnetization field distribution during a magnetization field evaluation is shown infig. 1. This figure shows the magnetization field distribution that is obtained by flowing
current through the stator coil. The magnetization field between poles is smaller than at
the center of the pole. The effective magnetization field distribution is obtained from thismagnetization field and the orientation of the magnet.

Fig. 3 shows the surface magnetic flux density waveform of the magnet when theuniformly magnetized magnet is used and when the magnetization process is taken into
account. The orientation direction of a magnet is in parallel direction. The surface
magnetic flux density differs substantially between when the uniformly magnetizedmagnet is used and when magnetization process is taken into account near 45 deg whereorientation components of the magnetization field decrease.

Magnetization Ratio
Fig. 2 shows the magnetization distribution of the magnet after the magnetizationinfluence from being used in the motor. It can be seen that near the magnetic poles on
the outside of the magnet is fully magnetized, but between the poles on the inside of themagnet the magnet is not fully magnetized.

90 deg45 degAngle

Magnetic Flux Density, T

0 deg(Unit: A/m)
(Unit: none)

Direction of magnetization ﬁeld
created in the stator coreFig.1. Magnetization Field Distribution

Case

31

Fig. 2. Magnetization Distribution Ratio

0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5

0

Constant magnetization
Account for magnetization process: Orientation of the magnets

15

90

30

45
60
Angle, deg

75

Fig. 3. Surface Magnetic Flux Density Waveform of
the Magnet (at 0.25 mm above the magnet surface)

Iron Loss Analysis of an SPM Motor Including
the Eﬀect of Press-ﬁtting Stress

Frame
Magnetmodule: DS, DP, LS

CoilOverview
The laminated structure of a core in a SPM motor can be sustained using press-fitting or shrink fitting. The press-fitting stress needs to beaccounted for in the iron loss evaluation because the stress caused by press-fitting is known to increase the iron losses when a magnetic steel
sheet is used for the core of the motor.
An iron loss is generated by the magnetization field in displacement with a steel sheet. The size of the iron loss is dependent on the iron loss
properties of a steel sheet. The iron loss characteristics of a steel sheet deteriorates by stress from press fit coupling. The stress generated by
the press fit coupling is distributed in areas in which the section in the back yoke becomes large. In order to evaluate the iron loss with good
accuracy, it is necessary to obtain the stress distribution for the magnetic flux, time variation, and steel sheet with accuracy.
This note presents the use of the press fit condition to model a core and frame and obtains iron loss density for when the generated stress isused and not used.

Rotor core
Stator core

Compressive Stress Distribution

Iron Loss Density Distribution

The compressive force distribution is indicated in fig. 1, and the tensor display is indicatedin fig. 2. The compression stress is large in the yoke when compared to the frame andteeth. The compression stress is concentrated in the yoke facing the slots because thestress decreases towards the tip of the teeth while the stress in the slots does notdecrease. The compression stress is concentrated around the slots, and the tensile
strength is concentrated around the frame as indicated in fig. 2.

The iron loss density distribution is indicated in fig. 3. The iron loss density distribution
varies depending on whether or not the stress is applied. The iron losses are increased by
15 % due to the stress. An analysis that accounts for the compressive stress caused by thepress-fitting needs to be performed to evaluate the iron losses more accurately because
the iron losses increase when the compressive stress is applied.

Low

Contact with the slot
High

Iron loss: 7.6 W
without stress(Units: MPa)

Iron loss: 9.2 W
with stress(Unit: W/m^3)(inward arrow: compressive stress; outward arrow: tensile stress)

Fig. 1 Compressive stress distribution

Fig. 2. Tensor display

Fig. 3. Iron loss density distribution

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Primary coil

Analysis of a TransformerCore

module: FQ

Overview
A transformer is an electrical device that uses electromagnetic induction to convert the voltage level of alternating current power. In an idealtransformer the secondary voltage is constant regardless of the load, but in reality it tends to vary with the size of the load and the power
factor. The size of a transformer's voltage variations is a vital output characteristic when considering constant voltage reception. It is also
important to maintain a balanced state because an imbalance in the voltage and current in each phase can bring about a rise in thetransformer's temperature or a fault in the device using the transformer.
A transformer's output characteristics depend on the leakage flux from the iron core. Leakage flux passes through the air instead of the ironcore, so it is hard to predict accurately during the design phase. It is possible to handle magnetic flux passing through the air in a magnetic
field analysis, meaning that it is also possible to evaluate a transformer's output characteristics, including the effects of leakage flux.
In this example, the use of a magnetic field analysis to evaluate changes in the secondary voltage caused by load variations in a low frequency
transformer.

Secondary coil

Secondary Voltage for the Load

Induced Voltage in the Secondary Coil
Secondary coil's induced voltage amplitude and its phase are indicated in Table 1. Fig. 2
shows the vector plot for the induced voltage in the secondary coil, when the load
resistance is large and small. The amplitude is shown by the vector length, and the phaseis shown by the slope of the vector. When the load resistance is large the amplitude isalmost the same and a phase difference of 120 deg is maintained, so the transformer is ina balanced state. On the other hand, when the load resistance is small there is a gap in
amplitude between each phase, so the transformer is in an unbalanced state because thephase difference does not reach 120 deg.

Magnetic Flux Density - RMS Distribution
Fig. 3 shows the magnetic flux density - RMS distribution of the transformer when theload resistance large and small. When the load resistance is large the magnetic flux
density is distributed almost symmetrically, but when it is small the current in each phase
is unbalanced, which makes the magnetic flux density asymmetric.

16
Ideal secondary voltage

14

JSOL Corporation

Large load resistanceEw(100.0)

10

Small load resistance86120 deg

4
Load resistance

Large

2

Small

Ev(99.6)

115 deg
Ev(93.9)

Eu、Ev、Ew：
Induced voltage
in the secondary coil
The numbers in parentheses
are percentages of the
maximum values

0
0

500

1000

1500

2000

2500

Secondary current, A

U-phaseLoad
Amplitude Phase
resistance
V
deg
ohm
0.005
21.61
132

V-phaseThe amplitudes have been converted to percentages of the maximum
values in each phase to make the balanced state easier to see.

Fig. 2. Induced Voltage in the Secondary CoilFig. 1. Secondary Voltage for the Load

W-phase-Amplitude
V

Phase
deg

Amplitude
V

Phase
deg

22.12

247

23.56

10

0.01

23.12

140

23.59

257

24.29

19

0.02

23.79

145

24.09

263

24.42

25

0.03

23.99

146

24.21

266

24.42

26

0.04

24.09

147

24.26

267

24.41

27

0.05

24.14

148

24.28

267

24.40

28

0.06

24.17

148

24.29

268

24.39

28

Table 1. Induced Voltage in the Secondary Coil22

Eu(91.7)
Eu(99.1)

Ew(100.0)

12
Secondary voltage, V

Fig. 1 shows changes in the secondary voltage for the load. The ideal secondary voltageof the transformer used for this analysis is 14.1 V. When the load resistance decreases, thesecondary current increases. Accordingly, the secondary voltage falls because the voltage
drop increases due to the resistance of the secondary coil and the leakage reactance.

U-phase V-phase W-phase

U-phase V-phase W-phaselarge load

small load

(Unit: T)

Fig. 3. Magnetic Flux Density - RMS Distribution

JMAG Application Catalog

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34

Stator core

Demagnetization Analysis of an SPM Motor

Magnetmodule: DP

Overview
Rare earth magnets has characteristics of large energy product, but decrease when using in an area exceeding a knee point causing
irreversible demagnetization. For motors, the possibility of thermal demagnetization through thermal stress may occur when the magnet
temperature rises due to iron loss or copper loss during rotation. Large amounts of electric currents are run through an excitation coil wheredemagnetization may occur when a reverse magnetic field is applied on a magnet. Demagnetization of magnets in a motor is one of thecauses of decrease in motor performance, where whether or not performance has decreased demagnetization needs to be predicted.
The magnetic field analysis simulation can handle magnetic fields or temperature generated in an internal magnet which can evaluate
demagnetization on the edges of a magnet accurately.
This note presents the use of a magnetic field analysis to evaluate the demagnetizing ratio distribution of an of an SPM motor by changing
the current flow.

Shaft
Coil

Magnetic Flux Density Distribution

Induced Voltage Waveform

Fig. 1 shows the magnetic flux density for current 20 A, and rotation angle of 20 deg.
Demagnetization occurs at the end of the magnet because the orientation of the magnet
and the direction of the flux generated by the rotating magnetic field are opposed.

Fig. 6 shows the induced voltage waveform of the U-phase coil when rotating the rotor
one period of electric angle with no load before and after demagnetization. When theamount of current is set to 20 A, it becomes apparent that demagnetization causes theinduced voltage waveform to change.

Operating Point
The measuring points selected to confirm operating points are displayed in fig. 2, andoperating points for each measuring point in current 20 A are displayed in fig. 3.
Measuring point 1 in fig. 3 is not set on a BH curve based on the operation point where an
irreversible demagnetization occurs. The size of the reverse magnetic field imposed uponeach measuring point is also illustrated on the horizontal axis.

* The demagnetizing ratio is the amount of demagnetization from the standard magnetization that is speciﬁed.

BH curve of magnet

Permeance Coeﬃcient Distribution

Measuring point 1

Me asuring point 2

Fig. 4 shows the permeance coefficient distribution of the magnet when the motor is
stationary with no load. The knee pointʼs permeance is shown in two colors, that are red
and purple. The knee point of the magnet is used for the analysis is 0.699. Fig. 4 showsthat the demagnetization hardly occurs when the motor is stationary with no load.

Magnetic Flux Density (T)

1.6

1.2
1
0.8

Knee pointʼsʼ permeancefor the magnet in use

0.6
0.4
0.2
0
-700000

Demagnetizing Ratio Distribution

Not demagnetized

1.4

Demagnetized-500000-300000-100000

0

Magnetic ﬁeld (A/m)

Fig. 5 uses the no-load static state as a benchmark to show the demagnetizing ratio*
distribution when rotating the rotor one period of electric angle with no load after
demagnetization. As the current increases, both the demagnetizing area anddemagnetizing ratio increases.

Fig. 3. Operating point

Fig. 4. Permeance coefficient distribution

Before Demagnetization
After demagnetization with a 10 A current
After demagnetization with a 20 A current

15

Measuring point 2
Voltage (V)10(Unit: T)
During rotation angleof 20 deg in a full model

Measuring point 1：Orientation of the magnets：Rotating magnetic ﬁeld creating
direction of magnetic ﬂux

Fig. 1 Magnetic flux density distribution

5
0
-5
-10
-15

current 10 A

Fig. 2. Measuring position of operating point

current 20A

Fig. 5 Demagnetizing Ratio Distribution(Unit: %)

0

30

60
90
120
Rotation angle (deg)

150

180

Fig. 6 Induced Voltage Waveform

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Vector Control Analysis of an IPM Motor Using Control
Simulator and the JMAG-RT

Shaft
Coil

Rotor coremodule: DP, RT

Overview
Traditionally, the design of a motor's controls and the design of the motor itself were often performed independently because coordinateddesigns were difficult to carry out. Motor control designs have been getting more advanced, however, so there has been an increasing
demand for simulations that use detailed motor models that exhibit behavior that conforms to that of an actual machine.
With JMAG, it is possible to create a detailed model that conforms to a real machine and accounts for spatial harmonics and magnetic
saturation characteristics that are included in a motor. Importing this motor model, a "JMAG-RT model," to a control/circuit simulator makes it
possible to carry out a linked simulation that accounts for a motor's magnetic saturation and spatial harmonics as well as a motor drive'scontrol characteristics.
The purpose of this Application Note is to demonstrate how to import a JMAG-RT model to a control/circuit simulator after using the JMAG-RT
to obtain the inductance spatial harmonics of the torque and coil. The model is then used to run an analysis that controls the speed of an IPM
motor to its target value.

MagnetStator core

Id and Iq Current Waveforms/ Torque Waveform
Fig. 1 shows the Id and Iq current waveforms from the vector control analysis, and fig. 2
shows the torque waveform. The JMAG-RT motor model accounts for the motor'sgeometry, so it is possible to display the current and torque waveforms, inducing theslot harmonic components. However, the ideal motor model does not include slot
harmonic components. The evaluation for the q-axis current is also smaller than with
the JMAG-RT motor model.

Fig. 1 Id and Iq current waveforms

Case

38

Fig. 2 Torque waveform

Starting Performance Analysis of a Single Phase
Induction Motor

Shaft
Bar

module: DP

Rotor core

Overview
Main winding

Single phase induction motors are widely used as small output motors for the drives in household electrical appliances and office machinery,
like fans and washing machines, because they can use single phase AC, the typical power source for home electronics. Unlike three phase AC,
however, single phase AC cannot create a rotating magnetic field by itself, meaning that it cannot start a motor. For this reason, it needs touse an alternate method to generate a rotating magnetic field to start the motor.
It is important to verify whether or not torque is generated in the intended direction and continues to rotate stably ahead of time in thedesign phase. In order to carry out this verification, the conditions where the rotor follows the equation of motion according to theelectromagnetic force mechanism and starts up need to be analyzed correctly.
The purpose of this Application Note is to introduce an example of a single phase induction motor that uses a capacitor to set up an auxiliary
winding and show its rotation speed versus time, torque versus time, and the magnetic flux density distribution and current density
distribution in the bar just after the motor starts.

Fig. 3 shows the rotation speed versus time graph, and fig. 4 shows the torque versus
time graph.
The time from motor startup until it reaches the synchronous speed is determined by thecapacitance of the capacitor. The slip is very small when the motor reaches synchronous
speed, so the average torque becomes zero.

Rotation speed (rpm)Fig. 1 shows the magnetic flux density distribution right after the motor startup, and fig. 2
shows the eddy current density distribution of the secondary conductor bars.
The magnetic flux generated by the winding causes eddy currents in the secondary
conductor bars, generating a rotating magnetic field.

Starting Performance(Unit: T)

Fig. 1 Magnetic flux density distribution

24

JSOL Corporation(Unit: A/m^2)

Fig. 2 Eddy current density distribution

3500

8.0

3000

6.0

2500

4.0

Torque (Nm)Magnetic Flux Density Distribution, Eddy Current Density
Distribution

Auxiliary winding

2000
1500

2.0
0.0

1000

-2.0

500

-4.0

0
0.00

-6.0
0.05

0.10

0.15

0.20
0.25
Time (sec)Fig. 3 Rotation speed versus time

0.30

0.35

0.40

0.00

0.05

0.10

0.15

0.20
Time (sec)Fig. 4 Torque versus time

0.25

0.30

0.35

0.40

JMAG Application Catalog

Torque Analysis of a Three Phase Induction Motor
Accounting for the Skew

Case

39

Stator core

Rotor coreCage

module: DP, TR

Overview
An induction motor can utilize skew easily because the cage is constructed by metallic casting such as die casting. When skew is applied, itarranges the variations in the magnetic flux that links to the cage in a sinusoidal wave. This makes it possible to eliminate the harmoniccomponents from the induction current that cause negative torque and contain things like the torque variations caused by influence fromthe slots.Applying skew generally affects the flow of magnetic flux in the axial direction, making it complex. This is why an analysis that cancorrectly verify the three dimensional magnetic flux flow is necessary to obtain an advance evaluation of the skew's effects.This Application
Note presents a comparison of the torque waveforms of three phase squirrel cage induction motors with and without torque, and introduces
the effects of using skew to reduce torque variations. Changes in the higher components caused by skew are also displayed by separating thefrequencies of the secondary current, which causes the torque variations.

Torque Variations

3
2.5
Torque (Nm)The torque waveform with and without a cage that has skew is indicated in fig. 1, fig. 2,
and fig. 3, and the size of the torque variations is indicated in Table 1. The results in fig. 1
are obtained with a 3D analysis. The results in fig. 2 are obtained with a 2D analysis. The
results in fig. 3 combine the 2D analysis results with the skew angle. An induction motor
produces torque from the rotating magnetic field of the stator coil and the induction
current of the rotor. The torque variations can be reduced by using a cage that has skew.
Fig. 3 shows that the superimposition of the 2D analysis and the skew angle makes it easy
to display the reductions in torque ripple from skew and the approximate generated
torque.

With/without skew

Torque variations (Nm)

1.5With skew

0.18

1

Without skew

0.41

Accounting for skew withsuperimposition

0.04

2

0.5
0

Current Density Distribution

0

180

360

540

720

Rotation angle (deg)Fig. 4 shows the secondary current density versus frequency, and fig. 5 shows the current
density distribution of the cage. The harmonic components of the current produced by
the harmonic magnetic flux in the stator are reduced by using a cage with skew as
indicated by the secondary current density versus frequency. This is why the torquevariations have been reduced as described above.

3

3

2.5

2.5

Table 1. Torque variations

Fig. 3 Superimposed torque waveform accounting
for the skew angle

With skew

Without skew

1.5
1

2
1.5
1

0.5

0.5

0

0
0

180

360

540

720

Rotation angle (deg)Fig. 1. Torque waveform with skew

Current density (Z)(A/m^2)

2

Torque (Nm)Torque (Nm)

1.6E+07

0

180

360

540

1.2E+078.0E+06

4.0E+06

0.0E+00

720

0

175

350

Rotation angle (deg)Fig. 2 Torque waveform without skew

525

700

875

(Unit: A/m^2)

Frequency (Hz)Fig. 5 Current density distribution (Z component)of the cage at a rotation angle of 630 deg

Fig. 4 Secondary current density versus frequency

Case

40

Stator core
MagnetCogging Torque Analysis of an SPM Motormodule: DP

Overview
The rotor's rotation in a permanent magnet motor can generate positive and negative torque, even when there is no current flow. This torqueis called "cogging torque." While output torque a center of focus in motors used in precision equipment, cogging torque reduction must be
taken into account as well. Skew and fractional slots are means of reducing this cogging torque. Skew is a widely used technique that
attempts to cancel out cogging torque by applying an appropriate amount of twist to the stator or rotor. This generates electromagnetic
force in the thrust direction, however, which presents challenges such as a decrease in performance and an increase in manufacturing cost.
Fractional slots do not have the drawbacks found in skew, but the winding pattern is different from that found with integer slots. This meansthat the torque generation becomes difficult to evaluate because the teeth geometry and the magnet's magnetization distribution are hardto design accurately. In order to carry out these evaluations, an electromagnetic field analysis using the finite element method (FEM) needs tobe carried out because it can perform detailed evaluations of the electromagnetic force distribution produced in the magnetic circuit.
This Application Note presents the use of magnetic field analysis to obtain the cogging torque of an 8-pole, 9-slot SPM motor, which has
relatively small period.

Shaft

Coil

Cogging Torque Waveform
6.00E-05
4.00E-05
Torque (Nm)Fig. 1 shows the cogging torque waveform. As shown in the waveform, the amplitude isabout 5e-5 Nm. The period of the cogging torque is also determined from the number ofslots and poles, which in this analysis are 8 (poles) and 9 (slots). By dividing 360 deg by 72,
the least common multiple of 8 and 9, one arrives at the period of cogging torque, whichin this case is 5 deg.

2.00E-05
0.00E+00-2.00E-05-4.00E-05

Magnetic Flux Lines
Fig. 2 shows the magnetic flux lines at rotation angles of 0 deg, 1 deg, and 4 deg. While
the magnetic flux lines are symmetrical along a symmetry line at 0 deg, they are notsymmetrical at 1 deg and 4 deg. This generates cogging torque.

-6.00E-05
0

1

2

3

4

5

6

7

8

9

10

Rotation angle:
0 deg

Rotation angle:
1 deg

Rotation angle:
4 deg

Rotation angle (deg)Fig. 1 Cogging torque waveform

Fig. 2 Magnetic flux lines (symmetry line)

https://www.jmag-international.com/

25

Positioning Control Analysis of a Permanent Magnet Linear Motor
Using the Control Simulator and the JMAG-RT

Case

41

Mover coreCoilmodule: DP, RT

Overview

Linear motors are widely used for carrier devices and machine tools because of their high-speed performance, high acceleration anddeceleration, and accurate positioning. A type of linear motor called a permanent magnet linear motor has harmonic components such as
cogging and ripples in its thrust because of its structure and how it operates. These components have a large influence on velocity control
and positioning accuracy. The thing that separates linear motors from rotating motors is that linear motors have edges, limiting how far they
can move. This is why linear motors have different characteristics when the mover is positioned in the center of the stator and when it is at
the edge. In order to create a more advanced control design, it is possible to use a control simulation to obtain a linear motor model thatexhibits more detailed behavior that is based on an actual machine.
With JMAG, it is possible to create a linear motor model that is detailed and conforms to a real machine, and that accounts for spatial
harmonics and magnetic saturation characteristics included in a permanent magnet linear motor. Importing this linear motor model, a"JMAG-RT model," to a circuit/control simulator makes it possible to run a linked simulation that accounts for the linear motor's magnetic
saturation characteristics and spatial harmonics as well as the driver's control characteristics.
This Application Note presents how to use the JMAG-RT to obtain the space harmonics and current dependency of the thrust and coilinductance and import them to a circuit/control simulator as a JMAG-RT model. From there, it shows how to perform an analysis that
controls the position of the permanent magnet linear motor to the desired value.

Magnet (4 poles)
Stator core

Translation direction

Response characteristic/Thrust Variation

Fig. 1 shows the mover position versus time, and fig. 2 shows the thrust force versus time.
Fig. 1 shows the time taken for the mover to reach the command value. The JMAG-RT
model accounts for the motor geometry, so the thrust varies according to the positional
relationship between the mover and the slots.

Command0.20

Position of mover yoke

2000
1600

Thrust (N)Position (m)

0.15

0.10

1200
800
400

0.05

0

0.00
0.00

0.05

0.10

0.15

0.20

0.25

0.30

Fig. 1 Position of mover yoke versus time

42

0.05

0.10

0.15

0.20

0.25

0.30

Time(sec)

Time (sec)

Case

-400
0.00

Fig. 2 Thrust force versus time

Displacement Analysis of a Piezoelectric Actuator
module: DS

Overview

Piezoelectric elements are used as actuators and sensors, as well as oscillator circuits and filter circuits in the analog electronic circuits. Whenthe electric potential is applied, the piezoelectric element is deformed. This is called the converse piezoelectric effect. In JMAG, the analysis of
the piezoelectric actuator using the converse piezoelectric effect can be performed by specifying the permittivity matrix and the electric
potential for the material.
This note presents the use of structural analysis to evaluate the displacement of a bimorph piezoelectric actuator, caused by the inverse
piezoelectric effect.

Displacement

Fig. 1 shows the state of displacement and the displacement vector. The part outlined in
red is the piezoelectric actuator at the initial position. In this analysis, one edge of thepiezoelectric actuator is secured, so the other edge can freely be displaced when theelectric potential is applied.

Displacement, μm

1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

26

JSOL Corporation

Electrode 2

Fig. 2 shows the electric potential-displacement characteristics. The displacement
increases linearly with electric potential. The electric potential to obtain the target
displacement can be estimated from the coefficient which is obtained from the slope ofthe line.

1.6

Fig. 1. Displacement

Electrode 1
Direction of polarization

Piezoelectric element 1

Electric Potential-Displacement Characteristics

1.8(Unit:m)Piezoelectric element 2

0

1

2

3

4

5

6

Electric potential, V

Fig. 2. Electric Potential-Displacement CharacteristicsJMAG Application Catalog

Case

Shaft

Torque Analysis of a Coreless Motor

43

module: TR

Overview

As their name implies, coreless motors have a rotor that lacks a core and is made of only a coil. For this reason, there is no core to produce ironloss in the rotor, and its moment of inertia is small. They can be controlled easily because their characteristics are linear and they have small
torque ripples, but they are not versatile enough to produce a large amount of torque. This is why they are often used in small precisionequipment that requires high rotation speeds and good responsiveness. The rotor coil is hard to construct because it is made of only a coil. It is
important to design the coil's twist angle to be able to produce torque. The model needs to be made precisely because coreless motors areused in compact equipment and because the detailed geometry of the parts can affect the characteristics.
In order to carry out these evaluations, the coil's twist needs to be accounted for accurately in three dimensions. An electromagnetic field
analysis using the finite element method (FEM) is necessary to accomplish this because it can evaluate the distribution of the electromagnetic
force produced in the magnetic circuit in detail.
This Application Note presents an evaluation of the torque waveform of a coreless motor when current is running.

Torque Waveform/Current Waveform

The torque waveform and current waveform are shown in fig. 1 and fig. 2. A smoothrotation can be achieved because the rotor doesn't have an iron core, as indicated in fig.
1. The average torque is also indicated at 0.022 mN・m. A ripple has occurred in thetorque waveform calculated by this analysis, but this is caused by the varying contact of
the brush and commutator as indicated in fig. 2.

0.06

Coil 2

Coil

Housing

Current Density Distribution/Lorentz Force Density Distribution

The current density distribution of the coil at a rotation angle of 360 deg is shown in fig. 3,
and the Lorentz force density distribution is shown in fig. 4. A current flows in coil 1 andcoil 3 without flowing in coil 2 when the rotation angle is 360 deg, as indicated in fig. 3.
The currents also flow in opposite directions of one another. The box inside of fig. 4 showsthat coil 3 generates force in the rotation direction, but coil 1 generates force both withand against the rotation direction. This occurs because the direction of the current differs
while the directions of the magnetic fields for each phase coil are the same.

Coil 3
Direction of magnetization

0.06
0.04

0.04
Current (A)Torque (m Nm)

Coil 10.08

0.05

Magnet

0.03
0.02

0.02
0.00
Coil 1

-0.02
-0.04

0.01
0.00
0

60

120

180

240

Rotation angle (deg)Fig. 1. Torque waveform

300

360

Direction of Rotation

Coil 3-0.06

Coil 2

-0.08
0

60

120

180

240

300

360

(Unit: A/m^2)(Unit：N/m^3)

Rotation angle (deg)Fig. 2 Current waveform

Fig. 3. Current density distribution(rotation angle: 360 deg)

Fig. 4. Lorentz force density distribution(rotation angle: 360 deg)https://www.jmag-international.com/

27

Case

44

Resistance Heating Analysis of a Steel Sheet

Electrodemodule: HT, TR

Overview

In treatments like hot formed pressing, a steel sheet needs to be heated uniformly as a part of pre-processing. Resistance heating is a methodof uniform heating that uses a steel sheet's electric resistance. In resistance heating, current is run through electrodes placed on both sides ofa steel sheet. The joule heat produced from the ensuing electric resistance is then used to heat the steel sheet. However, the uniformity of therange of heat generation changes depending on the arrangement of the electrodes, so whether or not the uniformity satisfies the heating
conditions needs to be investigated ahead of time. Unevenness in the current distribution flowing from the electrodes through the steel
sheet is determined from the geometry and the material's electric conductivity. Electric conductivity changes according to the temperature,
though, so both the electromagnetic phenomena and the heat transfer phenomena need to be analyzed at the same time.
This Application Note presents how to obtain the calculate current density distribution, Joule loss distribution and temperature distribution ofsteel sheet.

Work piece (steel sheet)

Current Density Distribution and Joule Loss Density Distribution

Fig. 1 shows the current density distribution and current flux line of the work piece
immediately after heating. Fig. 2 shows the Joule loss density distribution of the workpiece immediately after heating. The electrodes are shorter than the work piece in thewidth direction, so the current density is not uniform at the edges of the work piece, as
can be seen in fig. 1. It is high near the tips of the electrodes and gets lower at the four
corners of the work piece. Joule loss comes from electric resistance in the work piece, so
its density has the same distribution as the current density. Their distribution is almost
uniform near the center of the work piece. This joule loss becomes the heat source.

Section 3
Section 1
At 0 sec

Section 2

Temperature deg C

600

At 5 sec(Unit: deg C)

At 10 sec

Temperature Distribution

The work pieces temperature distribution is shown in fig. 3, and its work pieces
temperature of 10 sec are shown in fig. 4 and fig. 5. Electric resistance generates heat in
the work piece and heats it to about 400 deg C in 10 sec. The areas aside from the workpiece's edges are heated uniformly.

Fig. 3 Temperature distribution with a supply currentof 16 kA

Section 6
Section 8

Temperature deg C

600
500
400
Section 4
Section 5
Section 6
Section 7
Section 8

300
200
100
0

0

20

40

60

Distance mm

Fig. 1 Current density distribution and current flux lineof the immediately after heating

28

JSOL Corporation

Fig. 2 Joule loss density distribution of the immediately
after heating

300
Section 1

200

Section 2

100

Section 3
0

60

120
Distance mm

Section 4(Unit: W/m^3)400

0Section 6
Section 7(Unit: A/m^2)

500

Fig. 5 Temperature of the work piece

80

100

Fig. 4 Temperature of the work piece

180

240

JMAG Application Catalog

Case

45

High-Frequency Induction Heating Analysisof an IH Cooking Heater

Stainless steel sheet

module: FQ, HT

Overview

An IH cooking heater cooks food by heating a pot that acts as a conductive body. It heats this pot with an induction heating method that uses
electromagnetic induction. Eddy currents flow in the iron pot when a high frequency current is applied to the coil. These eddy currents
produce joule loss, which acts as a heat source to raise the temperature of the iron pot. When designing the heating coil, the main points tolook out for are: What kind of magnetic circuit design will raise heating efficiency, and whether it is generating uniform heat in the iron pot.
Another factor is controlling leakage flux to the circuit component in the board box that surrounds the apparatus.
A magnetic field analysis simulation that uses the finite element method (FEM) is best for studying a three dimensional combination of thegeometry, number, and alignment of the magnetic material that adjusts the magnetic circuit, and for quickly obtaining the electric circuit
constant of the high frequency circuit that performs the heating.
This Application Note shows how to obtain the magnetic flux density surrounding an IH cooking heater that uses high frequency induction
heating and the temperature distribution of its iron pot.

Magnetic Flux Density Distribution

Fig. 1 shows the magnetic flux density distribution of the IH cooking heater. The
aluminum plate on the board box shields it from magnetic flux, reducing the amount ofleakage flux that flows to it.

Coil
Core

Iron pot
Top board

Board box
Aluminum plate

Eddy Current Loss Density Distribution and Temperature
Distribution of the Iron Pot

Fig. 2 shows the eddy current loss density distribution of the iron pot. The magnetic field
generated by the coil produces eddy currents on the surface of the iron pot. The iron pot
is then heated because its electric resistance produces joule heat.
Fig. 3 shows the temperature distribution of the iron pot at 10 s, 30 s, and 60 s. The iron
pot is heated by eddy currents. The temperature of the bottom of the iron pot rises overtime, reaching 100 deg C at 60 s. In this analysis, the temperature rise of the bottom of theiron pot is not uniform. To heat the bottom of the iron pot uniformly during heating, thecoil arrangement and the core geometry need to be improved.

At 30 sAt 10 s(Unit: T)
Aluminum plate
Board box

Fig. 1 Magnetic flux density distribution of the IH cooking heater(Unit: deg C)(Unit: W/m^3)

At 60 sFig. 2 Eddy current loss density distribution of the iron pot

Fig. 3 Temperature distribution of the iron pot

https://www.jmag-international.com/

29

Sensitivity Analysis of the Magnetization Patternof an SPM Motor

Case

46

Shaft
Magnetmodule: DP, ST

Overview
The magnet in a surface permanent magnet (SPM) motor is arranged on the rotor's surface, facing the stator. The motor produces torquefrom the interaction between the magnetic field produced by the magnet and the magnetic field produced from the excitation coil. Cogging
torque, which is generated during no-load rotation, depends largely on the magnet's magnetizing state. Adjusting the magnet's
magnetization pattern makes it possible to reduce the cogging torque, which lowers efficiency and causes vibration and noise.
In order to control the magnetization pattern in the magnet of an actual machine precisely, a great number of magnetization devices isrequired. This makes a real machine hard to control, but with a magnetic field analysis simulation that uses the finite element method (FEM),
it is possible to estimate how the cogging torque in the physical phenomenon will change by simply setting the magnetization pattern. Oncethe optimum magnetization pattern has been found, studying the magnetization method can lead to a reduction in development cost.
This Application Note presents the use of a magnetic field analysis to obtain the surface flux density for radial pattern, parallel anisotropic
pattern, and polar anisotropy pattern magnets. It also displays the changes in induced voltage and cogging torque caused by differences in
the magnetization patterns.

Stator core

Surface Magnetic Flux Density

Induced Voltage, Flux Linkage,and Cogging TorqueThe surface magnetic flux density for magnets with a radial pattern, a parallel anisotropic
pattern, and a polar anisotropy pattern are indicated in fig 1. Furthermore, the decay iscompared between the radial pattern and parallel anisotropic pattern at 10 deg, 20 deg,and 30 deg. The polar anisotropy is compared by the distance from the center pole.
When the magnetization directions of the magnets are compared, the radial pattern isclosest to a trapezoid, but the waveform of the parallel anisotropy pattern and polar
anisotropy pattern are the closest to sinusoidal waves. The waveform of the induced
voltage and the flux linkage can be calculated as being closer to a sinusoidal waveform
for the parallel anisotropy and polar anisotropy patterns than the radial pattern.
Furthermore, the radial pattern and parallel anisotropy are closer to a sinusoidal
waveform the larger the angle of the decay.
The waveform of the polar anisotropic pattern is closer to a sinusoidal waveform thelarger the distance is from the center pole. However, the distance from the center pole is
optimum at 15.35 mm because the waveform when r=16.35 has a higher magnetic flux
density around 45 deg.

The induced voltage for the radial, parallel anisotropy, and polar anisotropy are indicatedin 2, the flux linkage in 3, and the cogging torque in 4.
When the induced voltage waveform is compared for each pattern, the waveform for theradial pattern is closer to a trapezoid, while the waveforms for the parallel anisotropy andpolar anisotropy patterns are closer to a sinusoidal waveform, just as predicted by thesurface magnetic flux waveform. The waveforms for the flux linkage are also the same.
Furthermore, as can be seen from the surface magnetic flux density, when the angle ofdecay and the distance from the center pole are compared, the induced voltage
waveform and flux linkage waveform are closer to a sinusoidal waveform the larger theangle of decay, or the larger the distance from the center pole.
Therefore, after comparing the magnetization patterns, the parallel anisotropy or polar
anisotropy patterns have a smoother rotation that produces less cogging torque than theradial pattern. Furthermore, when comparing the angle of decay and the distance fromthe center pole, the cogging torque is the lowest with a radial pattern at 20 deg, a parallelanisotropy pattern at 30 deg, or for the polar anisotropy pattern, a distance when r= 16.35
mm.

0

30

60
90
120
Angle (deg)

150

180

0
-0.2
-0.4
0

30

Radial pattern

60

90
120
Angle (deg)

150

180

11.35mm
14.35mm

12.35mm
15.35mm

13.35mm
16.35mm

0.4
0.2
0
-0.2
-0.4
0

30

Parallel anisotropy pattern

60

90
120
Angle (deg)

150

0
-10
-20
-30

0

30

60
90
120
Angle (deg)150

Radial patternFig. 2 Induced voltag

30

JSOL Corporation

180

-0.08
0

30

60
90
120
Angle (deg)

150

20deg

0.04
0
-0.04
-0.08

180

11.35mm 12.35mm
14.35mm 15.35mm

30deg

0.08

0

Radial pattern

10deg
30
20
10
0
-10
-20
-30

20deg

30deg

0

30

30
20
10
0
-10
-20
-30

11.35mm

12.35mm

13.35mm

14.35mm

15.35mm

16.35mm

30

60
90
120
Angle (deg)150

13.35mm
16.35mm

0.08
0.04
0
-0.04
-0.08

180

0

30

Parallel anisotropy pattern

Angle (deg)

60
90
120
Angle (deg)Polar anisotropy

90

120

150

180

20deg

0

30

150

180

10deg

30deg

0.04
0
-0.04

Parallel anisotropy pattern

60

10deg
Torque (Nm)30deg

Induced voltage (V)

20deg

0
-0.04

180

10deg

30deg

60
90
120
Angle (deg)

150

180

Polar anisotropy

Fig. 3 Flux linkage

Induced voltage (V)Induced voltage (V)

10deg

20deg

0.04

Polar anisotropy

Fig. 1 Surface magnetic flux density

30
20
10

10deg
0.08

Flux linkage (Wb)

-0.4

30deg

0

30

60

90

120

150

180

20deg

30deg

0.04
Torque (Nm)

0

20deg

0.2

Flux linkage (Wb)

0.2

-0.2

10deg
0.4

Torque (Nm)

30deg

Flux linkage (Wb)

20deg

Surface magnetic ﬂux density Br (T)

10deg
0.4

Surface magnetic ﬂux density Br (T)Surface magnetic ﬂux density Br (T)r: Distance from center pole

0
-0.04
0

30

60

90

120

150

Angle (deg)Angle (deg)Radial pattern

Parallel anisotropy patternFig. 4 Cogging torque180

11.35mm 12.35mm
14.35mm 15.35mm

13.35mm
16.35mm

30

150

0.04
0
-0.04
0

60
90
120
Angle (deg)Polar anisotropy

180

JMAG Application Catalog

High-Frequency Induction Heating Analysis
of a Crankshaft

Case

47

Work piece

Coilmodule: FQ, HT

Overview
Machine parts like shafts and gears are made to be resistant to wear and tear. This is accomplished by giving them a certain degree of flexibility
by maintaining their interior toughness while increasing the hardness of their surfaces. By using high-frequency induction heating, which is a
type of surface hardening method, the part is heated rapidly on only its surface by a high frequency power source. This process also has manyother benefits, such as providing a clean working environment because it uses electrical equipment, being very efficient, and providing uniform
results for each product. This is why it is being aggressively implemented in the field.
With induction hardening like the kind used on the work piece in this analysis, the main requirement is to heat a given surface uniformly andincrease rigidity. Eddy currents generated by the high-frequency varying magnetic field occur in the surface of the work piece. Examining these
phenomena in detail requires handling the phenomena that occur in the work piece itself in a numerical analysis based on the finite element
method (FEM).This Application Note shows an example of an evaluation performed by creating a numerical analysis model, analyzing theelevated temperature process, and seeing whether or not the desired temperature distribution is achieved.Use this analysis when obtaining the
optimum coil geometry, current conditions (power supply frequency, current value), and rotation speed.

Temperature Distribution, Temperature Variations, Joule Loss
Density Distribution
This section shows the temperature distribution in fig. 1, temperature variations in fig. 2,
and the joule loss density distribution in figures 3 and 4 as analysis results of the crank
shaft's high-frequency induction heating.

1200
InsideTemperature (deg C)

1000

Surface

800
600
400
200
0

(Units: deg C)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

(Unit: W/m^3)(Unit: W/m^3)

Time (sec)Fig. 1 Temperature distribution at Step 288

Fig. 2 Temperature variations in the surface
and interior of the work piece

Fig. 3 Joule loss density distribution at Step 1

Fig. 4 Joule loss density distribution at Step 288

High-Frequency Induction Heating Analysis
of a Printer Roller

Case

48

Core

module: FQ, HT

Overview
A printer works by running a piece of paper with toner on it between a heated fuser roller and a pressure roller. The heated fuser roller thenapplies heat to fix the toner to the paper. The fuser roller needs to have uniform temperature distribution in order to handle various types of
paper. It also requires the ability to heat up rapidly in order to shorten the standby time, allowing the person using the printer to print
documents quickly. A magnetic field analysis using the finite element method (FEM) is useful in examining several aspects of the process,including: Differences in heating from the heating coil's geometry or placement, what kind of eddy currents are generated in the roller's thinsurface and whether they provide uniform temperature, and how the magnetic flux flow spreads to the roller, air, and core.
This Application Note confirms the non-uniformity in temperature distribution produced by an assumed coil geometry, as well as thetemperature elevation in each part caused by rotation.

CoilWork (Roller)

Eddy Current Loss Density Distribution

Temperature Distribution/Temperature Variation

The eddy current loss density distribution of the roller is shown in fig. 1. The magnetic
field generated by the coil produces eddy currents on the surface of the roller. The eddy
currents are distributed on the surface of the roller due to the skin effect, which isstronger at high frequencies.

The temperature distribution of the roller is indicated in figures 2 and 3, and thetemperature variation of the roller surface is indicated in fig. 4. Fig. 4 indicates thetemperature variation of the roller at the measuring points shown in fig. 2. The measuring
points have been selected in order to be able to confirm the temperature variation in thecircumferential direction and the rotation axis direction. The roller is normally heated to atemperature of approximately 200 deg C, but in this analysis it is only analyzed for one full
revolution. From figures 2 and 3, it is apparent that the rotational motion causes theroller's surface to be heated in a strip. The temperature increase in the area being heatedis fairly even on the axis direction of the roller, as indicated in fig. 4. On the other hand,
the temperature is not distributed evenly in the circumferential direction due to timing
differences in heating, and heat dissipation into the air. The coil's geometry does notallow it to generate heat in the center part, so the temperature elevation is not smooth.

Direction of Rotation

Vertex A1

Vertex A2

Vertex B1

Vertex B2

40
Temperature (deg C)

Vertex A1
Vertex B1
Vertex B2

Vertex A2
Rotation angle 180 deg
Rotatio angle 15 deg
Rotation(Units: deg C)

Fig. 1 Eddy current loss density distribution directly
after starting heating

30

25

20

(Unit: W/m^3)

Roller cross-section

35

Rotation angle 90 deg

Fig. 2 Temperature distribution of the roller(Units: deg C)
Rotation angle 360 deg

Fig. 3 Temperature distribution of the roller

0

60

120

180

240

300

360

Rotation angle (deg)Fig. 4 Temperature variation at the surface of the roller

https://www.jmag-international.com/

31

Case

49

High-Frequency Induction Heating Analysis
of a Steel Sheet

Translational direction

module: FQ, HT

Work piece(Steel sheet)

Overview

Coil

The rolling process of steel sheets changes the strength and properties of the product, so heat treatment is used. High frequency induction
heating is a type of heat treatment that uses a high frequency power source to produce rapid heating, allowing the equipment on the
production line to be smaller. It also has a multitude of benefits, such as being highly efficient and providing a clean working environment.
When the object being heated is a long steel sheet, this process heats it quickly while sending it through a heating coil. For this reason, there
are several factors that need to be studied when assigning a heating amount to correspond to the speed at which the sheet passes throughthe coil. Examples of these are: the arrangement of the heating coil so that it can fulfill the necessary heating amount, and how to adjust thecurrent's frequency and size. This Application Note presents a simulation of the heating conditions of a sufficiently long steel sheet that passes
through a heating coil. The eddy currents produced from the high frequency's varying magnetic fields are uneven on the steel sheet's surface,
so its material properties change due to increases in temperature. This is why it is necessary to approximate the amount of heat generated in anumerical analysis based on the finite element method (FEM) in order to handle the detailed phenomena.
This Application Note explains how to create a numerical analysis model when obtaining the optimum coil geometry, current conditions(power supply frequency, current value), and movement speed. It also shows how to evaluate whether the model fulfills the target heating
speed by analyzing the elevated temperature process.

Eddy Current Loss Density Distribution
Fig. 1 shows the eddy current loss density distribution in the work piece's surface, as well
as a cross section. Eddy currents are generated in the work piece by the magnetic field
produced by the coil. Eddy currents generated by the skin effect are distributed throughthe surface of the work piece.

Translation direction
Translation
Direction
Translation
Direction
At 1.0 s

At 2.0 s

Temperature Distribution

At 4.0 s

Fig. 2 shows the temperature distribution in the work piece. The eddy currents generateheat in the work piece, and after having moved through the coil for 10 s it has beenheated to a temperature of over 800 deg. Moving the work piece through the coil at 0.1
m/s heats it without almost no irregularities in temperature. Heat transmission heats thework piece's interior, as well.

Case

50

Work piece cross-section-(Unit: W/m^3)

At 10.0 s

Fig. 1 Eddy current loss density distribution
in the work piece at 0 s

Cross-section

Cross-section

Cross-section

Cross-section

Fig. 2 Temperature distribution of the work pieceHigh-Frequency Induction Heating Analysis of a Steel Wire
(Translational Induction Hardening)
module: FQ, HT

Work piece(steel wire)

Overview
Steel wires are made to be resistant to wear and tear. This is accomplished by giving them a certain degree of flexibility by maintaining theirinner toughness while increasing the hardness of their surfaces. By using high-frequency induction heating, which is a type of surface
hardening method, the steel wire is heated rapidly on only its surface by a high frequency power source. This process also has many otherbenefits, such as providing a clean working environment because it uses electrical equipment, being very efficient, and providing uniform
results for each product. This is why it is being aggressively implemented in the field. On the other hand, when the heating target is a longsteel wire, it is heated rapidly while passing through the heating coil. For this reason, there are several factors that need to be studied whenassigning a heating amount to correspond to the speed at which the wire passes through the coil. Examples of these are: the arrangement of
the heating coils so that it can fulfill the necessary heating amount, and how to adjust the current's frequency and size. This Application Notepresents a simulation of the heating conditions of a sufficiently long steel wire that passes through a heating coil. The eddy currents
produced from the high frequency's varying magnetic fields are uneven on the steel wire's surface, so its material properties change due toincreases in temperature. This is why it is necessary to approximate the amount of heat generated in a numerical analysis based on the finite
element method (FEM) in order to handle the detailed phenomena.
This Application Note explains how to create a numerical analysis model when obtaining the optimum coil geometry, current conditions(power supply frequency, current value), and movement speed. It also shows how to evaluate whether the model fulfills the target heating
speed by analyzing the elevated temperature process.

Eddy Current Loss Density Distribution

Temperature Distribution of the Steel Wire

Fig. 1 shows the eddy current loss density distribution of the work piece and the coil. The
magnetic field made by the coil changes with time, generating eddy currents in the workpiece. In induction heating these eddy currents become heat sources and heat up thework piece. At high frequencies, the work piece is influenced by the skin effect, making it
so that the eddy currents are distributed in the surface.

Fig. 2 shows the temperature distribution at each position of the work piece.

Fig. 3 shows the temperature variation at the surface and inside of the work piece.
According to the graph, the maximum temperature is around 750 deg C both at thesurface and inside the work piece. It is apparent that the surface is heated first.

Measurement starting point

Measurement ending point
Translation direction

0s

A

B

C

D

E

C

D

E

(Unit: deg C)(Unit: W/m^3)

Fig. 1 Eddy current loss density distribution(cross-section)

32

JSOL Corporation

100 mmFig. 2 Temperature distribution inside the work piece(cross-section)

Steel wire temperature
variation measuring
point (cross-section)

4s
Temperature (deg C)

Translation direction

B

Coil

Temperature Variation of the Steel Wire

Translation direction

A(Unit: deg C)

Interior

800

Surface elements

600
400
200
0

0.0

0.5

1.0

1.5 2.0 2.5
Time (s)3.0

Fig. 3 Temperature variation of the work piece

3.5

4.0

JMAG Application Catalog

Case

51

Coil

High-Frequency Induction Heating Analysis of a Gear

Work piece (gear)

module: FQ, HT

Overview
Gears are created in such a way that the surfaces of their teeth are hard in order to resist the wear and tear that occurs when they come into
contact with the teeth of other gears. However, this has to be accomplished while maintaining the gear's overall toughness. In the case ofhigh-frequency induction heating, which is one of the surface hardening methods, just the surface of the teeth can be hardened by rapidly
heating only the surface using a high-frequency power source. This process also has many other benefits, such as providing a clean working
environment because it uses electrical equipment, being very efficient, and providing uniform results for each product. This is why it is being
aggressively implemented in the field. On the other hand, there are several factors that need to be studied in order to heat the gear's surface
uniformly, such as how to adjust the heating coil's geometry, arrangement, current frequency and size.
The eddy currents generated by high-frequency varying magnetic fields are uneven in the tooth surface, so the material properties change agreat deal as the temperature rises. In order to handle the detailed phenomena, it is necessary to calculate the heat generation amount in anumerical analysis based on the finite element method (FEM).
This Application Note shows how to create a numerical analysis model when obtaining the optimum coil geometry and current conditions(power supply frequency and current value), analyze the elevated temperature process, and evaluate whether or not the model fulfills thetarget temperature distribution.

Coil lead

Eddy Current Loss Density Distribution of the Gear
Fig. 1 shows the eddy current loss density distribution of the surface of the gear and thecross-section of a tooth top. Each cross-section shown in the figure is at the midway
position of the tooth width. The magnetic field generated by the coil produces eddy
currents on the gear tooth surface. This eddy currents are distributed on the surface of thegear due to the skin effect.

Temperature Distribution and Variation of the Gear
Fig. 2 shows the temperature distribution of the gear, and fig. 3 shows the temperature
variation versus time of the tooth tops. Fig. 3 shows the temperature variation of the
tooth top at the measuring points shown in fig. 2. From the temperature distribution, it isapparent that the eddy currents generate heat in the tooth top.

(Unit: W/m^3)

Measuring points

Fig. 1 Eddy current loss density distribution
between 0.0 sec and 0.15 sec(Units: deg C)

Fig. 2 Temperature distribution at 3.0 sec

Maximum Achieving Temperature Distribution
800
Temperature, deg C

Fig. 4 shows the maximum temperature that each part reaches. This figure is a two color
contour plot, where the parts exceeding 770 deg C are shown in red, and those below
770 deg C are shown in purple.

700
600

Quenchingtemperature

500
400
300
200
100
0
0.0

0.5

1.0

1.5

2.0

2.5

(Units: deg C)

3.0

Time, sec

Fig. 3 Temperature versus time at tooth top

Fig. 4 Maximum achieving temperature distribution

Case

52

Inductance Analysis of a Sheet Coil TransformerCore

module: FQ
Primary coil

Overview
Power transformer requires large currents, so their geometry tends to be large. This means that they are particularly hard parts to miniaturize
when electrical product designs get smaller. The sheet coil transformer introduced in this Application Note achieves thinner dimensions bywinding its coil in a thin sheet.
Self-inductance and leakage inductance are critical items in a transformer's design requirements. The amount of inductance is dependent on
the magnetic circuit, but the nonlinear characteristics of the magnetic properties make it so that the magnetic circuit changes when theoperating point changes. The leakage inductance has all of the same properties, but it also has a flux path in non-magnetic regions, making it
easily affected by the geometry and coil arrangement in addition to the core. This is why a magnetic field analysis using the finite element
method (FEM) is necessary when evaluating these types of inductance.
This Application Note explains how to obtain the self-inductance and leakage inductance of a sheet coil transformer.

Secondary coil

Magnet Flux Density Distribution
The magnetic flux density distribution used to calculate the self-inductance is indicated in
fig. 1. The magnetic flux is concentrated at the center of the core, as indicated in the
figure. The magnetic flux produced by the current flowing through the coil is split in two
from the center core because the magnetic flux flows around the outer circumference ofthe core and then back to the center of the core.
The magnetic flux density is larger at the corners inside of the core than the outside of thecore. This is because the magnetic flux flows through the shortest route of the core.

entire core

lower half of the core

Self inductance

30.13 μH

Leakage inductance

0.832 μH

(Unit: T)

Inductance
The inductance of the sheet coil transformer is indicated in table 1. The inductance can beobtained from the relationship between the voltage and current. The self-inductance is30.0 μH, and the leakage inductance is 0.755 μH.

Fig. 1 Magnetic flux density distributionof the core during self-inductance calculation

Table 1 Inductance of the sheet coil transformer

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Case

53

Magnetic Shielding Analysis of a Shield Roommodule: FQ

Overview

Shield rooms are meant to protect precision equipment from the influence of external magnetic fields, so they have to be an enclosed spacethat implements special processing in the walls that blocks magnetic flux. The effects of external magnetic fields inside the shield room
depend on how they are generated, where the precision equipment is located, and the position of the shield room's opening and supply
cable. A magnetic field analysis using the finite element method is necessary to perform an evaluation that deals with three dimensional andtemporal variations to figure out how magnetic flux enters the shield room when several external magnetic fields have been applied.
This Application Note explains how to handle the magnetic shielding phenomena used by the shield room when an external magnetic field isapplied, and from there how to confirm the magnetic flux density distribution.

Magnetic Flux Density Distribution

The shield room and magnetic flux density distribution surrounding air region are shownin fig. 1. The external magnetic field is applied in the direction of the arrows in the figures.
The magnetic flux from this field flows along the wall surface of the shield room, reducing
the magnetic flux density inside of the room itself. The external field that is uniform in theZ-direction in the figures also tries to flow with the wall in the vicinity of the shield room,so the parts outlined in red are where the magnetic flux concentrates and gets more
dense. The parts outlined in blue are where the magnetic flux density is less dense.

Door
Shield Room

Magnetic Flux Density Waveform

The magnetic flux density waveforms for each evaluation axis are shown in figures 2 and3. The shield room is between the positions of 12 m and 15 m, so the magnetic flux
density is lower in that region compared to the outside because of the room's magnetic
shielding qualities. For an external magnetic field of 50 μT the magnetic flux density
inside of the shield room is 0.9 μT, meaning that the magnetic shielding effect blocks out
approximately 98%.

27 m

External magnetic ﬁeld direction

27 m

Magnetic Flux Density (T)External magnetic ﬁeld direction

Magnetic ﬂux density vector54

1.0E-06

1.0E-07

0

3

6

9

12

15

18

21

24

27

Position (m)0m

1.0E-04

Evaluation axis B

1.0E-05

1.0E-06

1.0E-07

0

3

6

9

12

15

18

21

24

27

Position (m)Fig. 2. Magnetic flux density waveformin evaluation axis A

Fig. 3. Magnetic Flux Density Waveform
at Evaluation Axis B

Analysis of the Cogging Force of a Moving CoilLinear Motor

Mover yoke

Translation direction

module: TR

Overview

Linear motors have been widely used in carrier devices and the drive units of machine tools due to their capability for high acceleration anddeceleration, as well as their accurate positioning. In order to improve performance people are trying to obtain a large thrust force in order toenhance responsiveness, but one also needs to fulfill the demand for the trade-off of reducing thrust force variations and the attraction force.
There are also times when skew is added to the magnets because of requirements to reduce thrust force variation.
In order to obtain a large thrust force, the material's nonlinear magnetic properties and the magnet's demagnetization characteristics need tobe accounted for, and they need to be analyzed after modeling a detailed geometry in order to evaluate thrust force variations. This is why
the characteristics need to be studied with a magnetic field analysis simulation based on the finite element method (FEM).
This Analysis Note explains how to obtain the magnetic flux density distribution and cogging in a moving coil linear motor with skew applied
to its magnets.

Magnetic Flux Density Distribution

The magnetic flux density distribution at a translation distance of 36 mm is shown in fig.
1. Applying skew changes the magnetic circuit. It is anticipated that these changes in themagnetic circuit will affect the cogging torque.

Cogging Waveform

With skew

Without skew

6

Cogging (N)4without skew

2
0
-2
-4
-6

with skew(Unit: T)

Fig. 1. Magnetic flux density distribution ata translation distance of 36 mm

JSOL Corporation

MagnetCoil

RailThe cogging waveform is shown in fig. 2. When skew is applied, the cogging torque's
peak value falls by approximately 50%.

Translation Direction

34

B

A

Evaluation axis A

1.0E-05(Unit: T)

Fig. 1. Magnetic Flux Density DistributionCase

1.0E-04

Magnetic Flux Density (T)0mShield room cross-section
Evaluation plane of the magnetic ﬂux density distribution

0

5

10

15

20

Position (mm)Fig. 2. Cogging Waveform

25

30

35

JMAG Application Catalog

Rotor core

Case

Integrated Magnetization Analysis of an IPM Motor

55

Magnetmodule: DP, ST

Overview

CoilInterior permanent magnet (IPM) motors often use strong rare earth magnets. They have poor workability, however, because the magnets areinserted into the rotor's small gaps during the assembly process. After the magnets have been inserted the rotor generates a strong magnetic
field, which means that the workability when embedding it into the stator gets worse, as well. This is why in some cases they assemble themagnets while still in an unmagnetized state and magnetize them after they have been assembled. This construction method is called
integrated magnetization. Using this means of construction can improve the assembly process a great deal, but there is also the possibility
that the magnets will not be completely magnetized. Consequently, first one needs to confirm whether or not integrated magnetization iseven possible, and then from there to estimate the electrical power that the equipment needs for magnetization. Using a magnetic field
analysis simulation with the finite element method (FEM) provides the ability to change the making current amount and yoke geometry asmagnetization conditions, as well as to account for magnetic saturation and evaluate whether or not the magnets are completely magnetized.
This Application Note explains how to determine the changes that occur in a magnetizing field if the making current is changed during
magnetization, as well as how to obtain the induced voltage and cogging torque in the motor using the aforementioned magnets.

Stator core

Magnetization Field Distribution

Induced Voltage Waveform, Cogging Torque Waveform

The magnetization field distributions of the magnet during integrated magnetization at
making currents of 1.0 kA, 1.5 kA, 2.0 kA, and 2.5 kA are shown in fig. 1. The strength ofthe magnetization field correlates to the size of the making current. On the other hand,the orientation of the magnetization field faces almost the same direction, regardless ofthe making current.

Fig. 3 shows the U-phase induced voltage waveform with uniform magnetization in theentire magnet as an ideal magnetization state, as well as when accounting for themagnetization field with making currents of 1.0 kA, 1.5 kA, 2.0 kA, and 2.5 kA. Fig. 4 shows
the cogging torque waveform with uniform magnetization in the entire magnet as an ideal
state, as well as when accounting for the magnetization field with making currents of 1.0
kA, 1.5 kA, 2.0 kA, and 2.5 kA. The magnetization distributions shown in fig. 4.1 affect theinduced voltages and cogging torques. Figures 3 and 4 indicate that magnetization at 2.0
kA or higher provides mechanical characteristics equivalent to those when using a magnet
with an ideal magnetizing state.

Magnetization Distribution inside the Magnet
Fig. 2 shows the magnetization distribution in two cases: An ideal magnetizing state withuniform magnetization through the entire magnet, and the magnetization field that
occurs with making currents of 1.0 kA, 1.5 kA, 2.0 kA, and 2.5 kA. When the making current
is 1.0 kA and 1.5 kA, the magnetization in the entire magnet is small. However, with amaking current of 2.5 kA, there is almost no difference with the magnetization
distribution in an ideal magnetizing state.

Uniform magnetization

1.0 kA

1.5 kA

2.0 kA

0.20
0.15

40

0.10

20
0
-20
-40

Making current Making current Making current (Unit: A/m)
1.5 kA
2.0 kA
2.5 kA

Fig. 1 Magnetization field distribution

Making current Making current Making current Making current
2.5 kA
2.0 kA
1.0 kA
1.5 kA(Unit: A/m)

1.5 KA

2.0 KA

2.5 KA

0.05
0.00
-0.05

-0.15

-80

-0.20
0

30

60

90

120

Rotation angle (deg)Fig. 2 Magnetization distribution in the magnet

1.0 KA-0.10-60
Making current
1.0 kA

Uniform magnetization

2.5 kA

60
Torque (Nm)

Voltage (V)Uniform magnetization
80

Fig. 3 U-phase induced voltage waveform

150

180

0

5

10

15

20

25

30

Rotation angle (deg)Fig. 4 Cogging torque waveform

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Case

56

Torque Characteristics Analysis
of a Self Starting Type Permanent Magnet Motor

Rotor core

Cage
(end ring)Coilmodule: DP
MagnetOverview
A self starting permanent magnet motor combines the characteristics of an induction motor and a permanent magnet motor, so it has higher
efficiency than an induction motor even without a control device like an inverter. It behaves as an induction motor when it starts, generating
torque when the rotor cage first slips against the rotating magnetic field created by the stator and then produces a secondary current.
Consequently, this kind of motor has superior starting ability because there is no need to account for the rotor's start-up position or rotation
speed. When the rotation speed increases and the motor synchronizes, the permanent magnet begins to generate the magnetomotive force
and produce torque instead of the secondary current, so there is no secondary iron loss. This kind of motor has a weak point, however: The
torque falls a great deal when the motor deviates from its synchronicity, and it gets out of step as a magnet motor so the torque variations arelarge. This is why self starting permanent magnet motors can achieve full-voltage starting with household current and are very efficient while
in a synchronous state, but have drawbacks like relatively low starting torque and recovery once they have lost synchronization.These factors
make it so that a magnetic field analysis simulation based on the finite element method is necessary to investigate whether the motor's
characteristics meet the requirements ahead of time.
This Application Note shows how to obtain the current density distribution and slip versus torque curve.

Cage(bars)
Stator core

Current Density Distribution

Slip vs. Torque Curve

The current density distribution at a 0.8 slip is indicated in fig. 1.
If the slip is large, the motor operates as an induction motor. The induced current largely
affects the torque characteristics because the torque is produced by the rotating
magnetic field produced in the coils and the current induced in the rotor and cage.

The slip versus torque curve is indicated in fig. 3.
The torque of the self starting permanent magnet motor gets smaller as the slip gets larger,
reaching a maximum torque at a 0.2 slip.

Flux Lines
The flux lines at a 0 slip are indicated in fig. 2.
If the slip is 0, the motor operates as a synchronous motor. The torque is produced by theattractive force of the permanent magnet and the reluctance magnetic flux because theself starting permanent magnet motor operates as an IPM motor.

12

Torque (Nm)

10
8
6
4
2
0
1

(Unit: A/m^2)

Fig. 1. Current density distribution for a 0.8 slip at 0.667 s

36

JSOL Corporation

0.8

0.6

0.4
Slip

Fig. 2. Flux lines at 0 slip

Fig. 3. Slip versus torque

0.2

0JMAG Application Catalog

Case

Eﬃciency Analysis of an IPM Motor

58

MagnetStator coremodule: DP, LS

Rotor core

Overview

Coil

An IPM motor's features are in its rotor geometry, where its magnets are embedded. When the stator's rotating magnetic field is applied in a
direction that runs perpendicular to the rotor magnets (the q-axis) the motor operates like a normal SPM motor. When the current phase isdisplaced and the d-axis component is applied, however, the motor operates so that the magnetic fields in the rotor magnets are weakened.
This is called field weakening. In an SPM motor the d-axis current operates enough to weaken the magnetic field, so the rotation speed
increases but the torque decreases. However, the rotor geometry in an IPM motor is created so that there is a difference in inductance
between the d-axis and q-axis, so it is possible to generate torque with the d-axis current, which weakens the magnets. This makes it possible
to recover the part weakened by the flux. Consequently, an IPM motor achieves a greater range of operation by incorporating field
weakening controls. For this reason an IPM motor's characteristics depend greatly on its rotor geometry, so studies using the magnetic circuit
method are difficult. In order to perform an advance design study accurately, an electromagnetic field analysis using the finite element
method is necessary. This Application Note presents the use of magnetic field analysis to obtain the efficiency of an IPM motor in each current
phase with a rotation speed of 1800 rpm and the current amplitude of 4.0 A when the motor is driven by sinusoidal wave current.

Eﬃciency characteristics, loss by type, magnetic ﬂux density
distribution, the torque waveform

90.0

Coil/Joule loss

35

Magnet/Joule loss

30
Loss, W

89.5
89.0
88.5

Stator/Joule loss

25
Rotor/Joule loss

20

Stator/Hysteresis loss

15

88.0

10

87.5

5

87.0
0

10

20

30

40

50

60

Current phase β, deg

Fig. 1 Efficiency characteristics per current phase at
a rotation speed of 1800 rpm and a current
amplitude of 4.0 A

Rotor/Hysteresis loss

0
0

10

20

30

40

50

60

Current phase β, deg

Fig. 2 Loss by type at each current phase

Torque Nm

Current phase:20 deg

β=60

1.5
1.0
0.5
0.0

(Unit: T)
Current phase: 40 deg

0

10

20

30

40

50

60

Current phase β, deg

Current phase: 60 deg

Fig. 3 Magnetic flux density distribution

Average torque

40

β=40

2.0

Average torque, Nm

90.5

β=20

2.5

Current phase:0 deg

45

91.0

Eﬃciency, %

β=0
3.0

Fig. 4. Torque Waveform

Torque ripple rate

2.5

100

2.0

90

1.5

80

1.0

70
60

0.5

Torque ripple rate, percent

When the rotation speed is 1800 rpm and the current amplitude is 4.0 A, the efficiency
characteristics per current phase are shown in fig. 1, the loss by type is shown in fig. 2, themagnetic flux density distribution is shown in fig. 3, the torque waveform is shown in fig.
4, and the average torque and torque ripple rate are shown in fig. 5. The results of theefficiency characteristics indicate that a maximum efficiency of 91% can be obtained at β
= 30 deg. Adjusting the current phase can lower the magnetic flux density of the statorand reduce iron loss. Doing this, however, will reduce the average torque and increase
torque ripple, which can lead to noise problems. This means that the current phase needs
to be selected while accounting for the size of torque ripples in addition to the degree of
efficiency.
Obtaining the efficiency by changing the rotation speed, current value, and current phasewithin the motor's range of possible operation makes it possible to create an efficiency
map.

50

0.0
0

10

20

30

40

50

60

Current phase β, deg

Fig. 5 Average torque and torque ripple rate percurrent phase

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37

Iron Loss Analysis of an IPM Motor Accounting for PWM‒Direct Link-

Case

59

Stator core

Magnetmodule: DP, LS

Rotor core

Overview

Shaft

Vector controls using a PWM (Pulse Width Modulation) control are commonly included in the drive circuits of high efficiency motors. A PWM
control makes it possible to adjust the phase or amplitude of a current according to load and rotation speed, so they can achieve highefficiency in a wide operation range. The control frequency of a PWM is called a carrier frequency. Carrier frequencies are often used up toalmost 20 kHz. To form the current waveform supplied by the PWM control, the carrier harmonic current is superimposed on the basic wave
current. This carrier harmonic current applies a high-frequency magnetic field to each part of the motor. As a result, core iron loss and magnet
eddy current loss are generated. The total amount of these losses is not a dominant factor, but they can be a hindrance when trying to raise
efficiency, so they need to be eliminated in the design process. In order to study these problems, both the motor's electromagnetic behavior
and what kinds of controls the drive circuit performs have to be investigated.
In order to run an advance study of these phenomena in CAE, a high fidelity motor model and inverter model need to be coupled. There arethree ways of accomplishing this: Directly linking with a circuit/control simulator, entering a current waveform obtained by using a JMAG-RT
motor model and a circuit/control simulator, and entering actual current measurements.
In this analysis, the iron losses of the IPM motor that accounts for the carrier harmonic are obtained by directly linking to a circuit/control
simulator.

Joule Loss

Comparisons of the iron losses, hysteresis losses, and joule losses when the IPM motor isdriven by a PWM inverter and a sinusoidal current are shown in table 1.
The iron losses in both the rotor core and the stator core increase when the IPM motor is
run using a PWM inverter. The joule losses increase the most severely, increasing 70% in
the rotor core, and 40% in the stator core.

The frequency components of the joule losses are indicated in figures 1 through 4. The
joule loss density distribution is indicated in figures 5 and 6.
The fundamental frequency and harmonic component have almost the same amount ofiron losses in both the rotor core and the stator core if a sinusoidal current or a PWM
inverter is used with a fundamental frequency of 360 Hz for the rotor core and 60 Hz
stator core. However, losses are only generated when the PWM inverter is used with a
PWM fundamental frequency of 10 kHz and its harmonic component. (These values areindicated inside the green square in fig. 1.)
The increased joule losses when the motor is run using a PWM inverter are caused by the
carrier harmonic. The effects of a carrier harmonic need to be accounted for to analyze
the characteristics of an IPM motor in detail.

4.5

4.5

0.45

4.0

4.0

0.40

0.40

3.5

3.5

0.35

0.35

3.0

3.0

0.30
0.25
0.20
0.15

0.30
0.25
0.20
0.15

2.5
2.0
1.5

0.10

0.10

1.0

0.05

0.05

0.5

0.00

0.0

0.00
0

5

10

15

20

25

30

35

40

45

50

0

5

10

Frequency (kHz)

15

20

25

30

35

40

45

50

Joule loss (W)

0.50

0.45

Joule loss (W)0.50

Joule loss (W)Joule loss (W)Iron Losses

0.0
5

10

15

20

25

Rotor core

PWM inverter drive(Unit: W/m^3)

Fig. 5 Joule loss density distribution of rotor core

38

JSOL Corporation

Sinusoidal current

PWM inverter drive

30

35

40

45

50

Fig. 6 Joule loss density distribution of the stator core

Stator core

Iron Loss

1.14 W

1.89 W

14.44 W

16.80 W

Hysteresis Loss

0.18 W

0.20 W

8.28 W

8.35 W

Joule Loss

0.97 W

1.69 W

6.16 W

8.45 W

Table 1. Iron loss resul ts

5

10

15

20

25

30

35

Fig. 4 Joule loss frequency componentof the stator core (PWM inverter drive)

PWM inverter Sinusoidal
PWM inverter
drivecurrent drive
drive(Unit: W/m^3)0Frequency (kHz)Fig. 3 Joule loss frequency componentof the stator core (sinusoidal current drive)

Sinusoidal
current drive

Sinusoidal current drive

1.5

0.5
0

Frequency (kHz)Fig. 2 Joule loss frequency componentof the rotor core (PWM inverter drive)

2.0

1.0

Frequency (kHz)Fig. 1 Joule loss frequency componentof the rotor core (sinusoidal current drive)

2.5

40

45

50

JMAG Application Catalog

Case

60

Superimposed Direct Current Characteristic
Analysis of a ReactorCore
Bobbinmodule: TR

Overview

CoilSpacerA high-frequency reactor, used in equipment such as DC-DC converters, has a high-frequency current accompanying the switching direct
current. The performance of a reactor is evaluated by a stable inductance in a wide direct current region. The gap that is designed to prevent
magnetic saturation from the core largely affects the inductance, so it is a vital parameter of the reactor's design.
The magnetic resistance is determined by the gap when there is a large gap width, which is used as a parameter for the superimposed direct
current of the inductance. This means that the resistance can be evaluated using the magnetic circuit method, but when the gap width issmall, the current is large, and magnetic saturation has a large effect on the inductance, an advance study using a finite element analysis (FEA)
is an effective tool.
This Application Note explains a case example that obtains the superimposed direct current characteristics of a high-frequency reactor when
the gap width is changed.

Superimposed Direct Current Characteristics
Gap 1.00 mm

Gap 0.75 mm

Gap 0.50 mm

2.0

Inductance (mH)

From fig. 1, it is apparent that the inductance decreases more rapidly as the direct current
increases. This is caused by influence from magnetic saturation, as shown in fig. 2. As thegap width grows larger, the gap becomes the dominant factor for magnetic resistance inthe magnetic circuit, so the sensitivity of the inductance toward the current variation
decreases.

1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0

5

10

15

20

25

30

35

40

45

(Unit: T)

50
Superimposed direct Superimposed direct Superimposed direct
current: 20 A
current: 30 A
current: 40 A

Superimposed direct current (A)Fig. 1. Superimposed direct current characteristics

Fig. 2. Magnetic flux density distribution(top: gap width: 1.0 mm; bottom: gap width 0.5 mm)

Case

61

Current Distribution Analysis of a Choke CoilCore

module: DP, TS, TRANSFORMER_TEMPLATE

BobbinOverview
A choke coil is an electric component that is intended to filter high-frequency current.
The current in a choke coil's interior produces local heat generation because of the skin effect, proximity effect, and current offsets caused byleakage flux near the gap. From a heat resistant design standpoint, visual confirmation of detailed current distribution using a finite element
analysis (FEA) is useful because it provides feedback for the design.
This Application Note explains a case example that obtains the current distribution in a choke coil.

Coil

Current Density Distribution
0.65
0.6
Current (A)The current waveform is shown in fig. 1, and the current density distribution in the coils at
each time is shown in fig. 4.2. In order to evaluate in a steady state, fig. 1 shows the timefrom 40 μs to 80 μs.
From the figures, it is apparent that there are offsets in the current at 40 and 60 μs when
the time variations in the current are severe, and that there is not much of an offset at 50
μs, when the time variations are mild. This is because variations in the magnetic fieldnear the gap, which are the source of current offsets, cause variations in the coil current.

0.55
0.5
0.45
0.4
0.35

4040μsec
μseconds
40

50

60

70

5050μsec
μseconds

6060μsec
μseconds(Unit: A/m^2)

80

Time (μs)

Fig. 1. Current waveform

Fig. 2. Current density distribution

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39

Case

Attraction Force Analysis of a Solenoid Valve

62

Stator coreYokemodule: DP

CoilOverview
Solenoid valves move their iron cores in a translational direction, and are used to adjust the inflow and outflow amounts of liquids and gasses.
Running current through the coil forms an electromagnet, which generates an electromagnetic attraction force between the mover andstator. A high level of responsiveness is required to open and close the valve, so the power supply and valve used in the drive need to beevaluated to determine whether they fulfill the required responsiveness and attraction force.
The attraction force is determined from the size of the current, the arrangement of the iron core, and the material properties. However, theactual flow of magnetic flux is complex, and even if the current is increased, phenomena may occur such as the attractive force not being
proportional to the current due to the effects of magnetic saturation in the core. A magnetic field analysis simulation using the finite element
method (FEM) is useful in studying these kinds of behaviors.
This Application Note obtains the attraction force at each position of the movable core.

Movable core

Magnetic Flux Density Distribution

Attraction Force Characteristics

Fig. 1 shows the magnetic flux density distribution at different movable core positions.
The closer the movable core moves toward the stator core, the smaller the air gap
becomes. So, the magnetic flux increases in the moving direction and the magnetic
circuit changes. The magnetic circuit between the movable core and the stator core has alarge effect on the attractive force.

Fig. 2 shows a vector plot of the surface force density produced in the movable core ateach position, and fig. 3 shows the attraction force characteristics of the solenoid valve.
When the movable core is in a position of 1.0 mm to 2.5 mm, a basically constantattraction force is obtained. As the movable core gets closer to the stator core the space
between them shrinks, causing the magnetic resistance to change rapidly. This is why theattraction force generated in the direction of movement changes depending on the
movable core's position when there is direct current excitation.

35

Attractive force (N)30：：Magneticﬂux ﬂow

25
20
15
10
5
0
0.0

Movable core
position:1.0 mm

Movable core
position:3.0 mm

Movable core
position:3.5 mm

Movable core
position:1.0 mm(Unit: T)

Fig. 1. Magnetic flux density distribution

Movable core
position:3.0 mm

Movable core
position:3.5 mm

0.5

1.0

Fig. 2. Surface force density distribution at each position

1.5

2.0

2.5

63

3.5

Fig. 3. Attraction force characteristics

Analysis of Torque Characteristics of a Cage
Induction Motor

Case

3.0

Movable core position (mm)(Unit: N/m^2)

Rotor coreCage

module: FQ

Overview
Induction motors have been widely used for a long time in general industries because they have a simple structure, and are affordable, robust
and highly efficient. When an induction motor rotates at synchronous speed, no torque is produced. However, when proper slip is caused, themaximum torque can be obtained. Losses are generated in a cage induction motor when current flows through the cage, so the pros and consof continuous rotation depending on the amount of the heat generated need to be studied. In an induction motor, the current induced by theauxiliary conductor exerts a large influence on its characteristics. It also causes strong magnetic saturation in the vicinity of the gap, inparticular. This is why a magnetic field analysis based on the finite element method (FEM) is useful when investigating the motor's
characteristics for a design study.
This Application Note introduces a case example that obtains the Slip-Torque curve, Torque-Current curve, Current-Voltage curve at maximum
torque, and the Current-Joule Loss curve for the cage.

Stator core

Torque Characteristics

Eddy Current Density Distribution / Joule Loss

The Slip-Torque curves for each current amplitude are shown in fig. 1, the Torque-Current
curve is shown in fig. 2, and the Current-Voltage curve is shown in fig. 3. As shown in fig.
1, the slip where the torque reaches its maximum value changes depending on the size ofthe current amplitude. The slip where the torque reaches its maximum value isdetermined by the coil's electric resistance, the cage's electric resistance, and the leakage
reactance. Therefore, the changes in the leakage reactance due to an increase in current
cause the changes in the slip where the torque reaches its maximum value. As can beseen from the slope of the Torque-Current curve in fig. 2, the torque constant of thisinduction motor is about 0.8 Nm/A. There is also a limit on the primary voltage that canbe supplied, so the Current-Voltage curve needs to be used to investigate the primaryvoltage during maximum torque ahead of time.

The eddy current density distribution when the current amplitude is 10 A is shown in fig.
4, and the joule loss in the cage for each current value is shown in fig. 5. Both of these areresults at maximum torque. As shown in fig. 4, an excessive current with current densityof over 2x10^7 A/m^2 flows through the bars, and the heat generation is large, so
continuous operation is difficult. As shown in fig. 5, the joule loss increases quadraticallyagainst the current value. By estimating the heat generation from the loss, it is possible tounderstand the period of time when continuous operation is possible at the maximum
torque. In this way, it is necessary to understand the current and losses in the cage whenstudying the maximum torque in a cage induction motor.

600
2A
7A

3A
8A

4A
9A

5A
10A

10

150

8

120

4

6

90

Voltage (V)Torque (Nm)Torque(Nm)

6

4

500
400
Loss (W)

1A
6A
8

60

300
200

2

2

30

0
1.0

0

0

100
0.8

0.6

0.4

0.2

Slip

0.0

0

2

4

6

8

Current (A)

10

0

2

4

6

8

10

Current (A)0(Unit: A/m^2)

0

2

4

6
Current (A)Fig. 1. Slip-Torque curve

40

JSOL Corporation

Fig. 2. Torque-Current curve

Fig. 3. Current-Voltage curve

Fig. 4. Eddy current density distribution (rotor)

Fig. 5. Joule loss of the cage

8

10

JMAG Application Catalog

Case

Thrust Force Analysis of a Coreless Linear Motor

64

Yoke

module: TR

MagnetOverview
Translation direction

Linear motors are widely used for carrier devices and machine tools because of their high-speed performance, high acceleration anddeceleration, and accurate positioning. Among them there is a type of motor called a coreless linear motor. Coreless linear motors generally
have a smaller thrust force than core linear motors, but they do not produce cogging, so they only have a small amount of thrust variation.
They are used for linear motor stages and electronic packaging machines to make use of this property.
Because the thrust variations in linear motors are small, they can be hard to predict and measure at the design stage. With the finite element
method (FEM), it is possible to obtain thrust variations with accuracy even when they are small, as is the case with a coreless linear motor.
In this example, explains how to obtain the thrust variations in a coreless linear motor when it is driven with a three-phase alternating current.

Lorentz Force Density Distribution

Translation Direction

U-phase coil50
V-phase coil40

W-phase coil

Thrust, N

The Lorentz force density distribution at a translation distance of 21 mm is shown in Fig.
1. The Lorentz force is generated by the magnet's magnetic field when electricity flows
through the coil. The strength of the Lorentz force is determined by the current value andthe strength of the magnetic field. The direction of the Lorentz force is determined fromthe direction of both the current flow and the magnetic field.
When the translation distance is 21 mm current only flows through the U-phase andW-phase coils, so the Lorentz force density gets higher in the places in those coils thatface the magnets. The Lorentz force occurs in the translation direction, becoming thethrust force.

30

20

10

(Unit: N/m^3)00

top: positions of the magnet and coilbottom: Lorentz force density distribution

Fig. 1. Lorentz Force Density Distribution in the
Coils at a Translation Distance of 21 mm

Thrust Force

Coil

7

14

21

28
35
Distance, mm

42

49

56

Fig. 2. Thrust Force Waveform

The thrust force waveform is shown in fig. 2. Because there is no cogging the thrustvariations are small, comprising about 6% of the average thrust force.

Case

Static Thrust Analysis of a Voice Coil Motor

65

Yoke
Magnetmodule: TR

Overview

Translation
Direction

Linear actuators are used in machine tools because of their high-speed performance, high acceleration and deceleration, and accurate
positioning. There are coreless types of linear actuators, as well. They generally have a smaller thrust force than core linear actuators, but they
do not produce cogging, so they only have a small amount of thrust variation. Because of this property, they are used in fields where
high-accuracy positioning is necessary, like with head drives of packaging machines or the slight movements of precision stages.
Static thrust variations at the translation position of the actuator have an effect on determining the position accurately. The static thrust isdetermined by the amount of current, so its current characteristics need to be obtained.
This Application Note explains how to obtain the current characteristics and the translation position characteristics of the static thrust in avoice coil motor, which is a type of coreless linear actuator.

Coil

Translation Position Characteristics of Static Thrust

Current Characteristics of Static Thrust

The translation position characteristics of the static thrust at a supply current of 0.05 A are
shown in fig. 1. The Lorentz force density distribution and magnetic field distribution of
the coil at translation positions of 0 mm and 1 mm are shown in figures 2 and 3. From fig.
1, it is apparent that the static thrust reaches its peak at a translation position of 1 mm,
when the coil is directly beneath each magnetic pole. From fig. 2, it is apparent that theLorentz force density inside the circle at the translation position of 0 mm is small
compared to the translation position of 1 mm. From fig. 3, it is apparent that the magnetic
field in the circle at the translation position of 0 mm is weak, and its direction is not
perpendicular to the translation direction. Because the direction and size of the current in
the coil is constant and the static thrust (Lorentz force) is determined from the strengthand direction of the magnetic field, the static thrust is reduced by the magnetic field
distribution in the coil.

The current characteristics of the static thrust are shown in fig. 4. The figure shows thelinearity of the static thrust against the supply current. According to the slope of this line,
the thrust constant is 6 N/A. This means that it is possible to control the static thrust withthe supply current value.

Translation
Direction

Magnetization Direction
N
N
S
S

S
N
N

S
S
N
N

N
N
S
S

Translation position: 1mm

S
S
N
N

0.8
Current Direction

0.350
Static thrust (N)1.0Translation position: 2mm

0.325

Static thrust (N)

N
N
S
S

Current ﬂow direction
Translation position: 0 mm

Translation position 0 mm

Translation position 0mm

0.300

0.6
0.4
0.2

0.275
0.250
0.0

0.0
0.5

1.0

1.5

Translation position (mm)Fig. 1 Translation position characteristicsof the static thrust(Unit: A/m)

(Unit:N/m^3)

2.0
Translation position 1mm

Fig. 2 Lorentz force density distribution(translation direction component)

Translation position 1 mm

Fig. 3 Magnetic Field Distribution

0

0.05

0.1

0.15

0.2

Supply current value (A)Fig. 4 Current characteristics of the static thrust

https://www.jmag-international.com/

41

Case

Core

Operating Time Analysis of an Injector

66

module: DP

Coil
Yoke

Overview
A solenoid type injector used in engines opens a valve and injects fuel by moving a plunger with magnetic force created by anelectromagnet. Injectors in engines need to respond quickly for applied voltage to improve fuel consumption by maintaining the amount offuel flow. In solenoid injectors, one of the reasons that the response is delayed is eddy currents, which are produced when the magnetic flux
generated by current flow undergoes time variations. The eddy currents are generated in a direction that inhibits changes in the magnetic
flux, causing a delay in the initial rise of the attraction force when the current begins to flow. This reduces the injector's responsiveness. JMAG
makes it possible to account for the effects from eddy currents and obtain an injector's responsiveness by running a transient response
analysis. Identifying the places where eddy currents are generated enables a designer to study whether or not responsiveness can be
improved. This Application Note explains how to apply direct current voltage to a solenoid injector and obtain its response characteristics byaccounting for effects from eddy currents.

Plunger

Response Characteristics

Current characteristics

Fig. 1 shows the response characteristics of the plunger versus time. The plunger's
displacement assumes the opening direction to be positive, and the valve is fully open
when the displacement is 0.1 mm.
Fig. 1 demonstrates the difference in responsiveness of the plunger depending if eddy
current is accounted for or not. Responsiveness is lower when eddy current is accounted
for.

Fig. 3 shows response characteristics for current versus time.
Fig. 3 demonstrates the valve is open at 0.003(s) when eddy current is accounted for.
However, when eddy current isn't accounted for, the valve is open at 0.0025(s). A slower
responsiveness can be verified from the current flowing in the coil when eddy current isaccounted for. Furthermore, when eddy current is accounted for, there is a lot of current
flowing through the coil even though attractive force is lowered, compared to when it
isn't accounted for.

Attraction force characteristics
Fig. 2 shows the attraction force characteristics.
Fig. 2 shows attraction force is lower when eddy current is accounted for compared towhen it isn't accounted for. Responsiveness decreases when eddy current is raised due toa decrease in attraction force.

Excitation begins

Excitation begins

Plunger begins moving

Fig. 4 shows the current density distribution when the current starts flowing 1 ms, when
the plunger starts moving 2.3 ms and when the plunger moves one stroke 4.2 ms. The
plunger moves in the direction that the valve opens in. When current flows in the coil,eddy currents are produced in the core, yoke and plunger. Magnetic flux flows rapidly
when the current starts running, which causes eddy currents to concentrate in the surfaceof each part. However, eddy currents begin flowing in the interior of each part as time
passes because the power supply is a direct current. These eddy currents lower the
injector's responsiveness.

1.2

Plunger begins moving

50

120

1

80
60
40

Accounting
for eddy currents

20

Not accounting
for eddy currents

0
0

1

2

3

4

5

Time (ms)Fig. 1 Response characteristics

Case

68

40
0.8
Current (A)Attraction force (N)100
Position (μm)

Current Density Distribution

30
20

Accounting
for eddy currents

10

Not accounting
for eddy currents

0
0

1

2

3

4

5

0.6
0.4

Accounting
for eddy currents

0.2

Not accounting
for eddy currents
0

0.001

0.002

0.003

0.004

0.005

1.0 ms

2.3 ms

4.2 ms(Unit: A/m^2)

Time (s)Time (ms)Fig. 2 Attraction f orce characteristics

Translation
direction

0

Fig. 3 Current characteristics

Fig. 4 Current density distribution

Speed Versus Torque Characteristic Analysis
of a Three-Phase Induction Motor

Rotor core
Cage
Bolt holemodule: DP

Overview
An induction motor is a motor in which a rotating magnetic field in the stator coils causes induced current to flow in an auxiliary conductor.
This current and magnetic field exert force on the auxiliary conductor in the rotation direction and cause the motor's rotor to rotate. Induction
motors are widely used in everything from industrial machines to home appliances because they have a simple construction and are small,light, affordable, and maintenance-free.
In an induction motor, the current induced by the auxiliary conductor exerts a large influence on its characteristics. It also causes strong
magnetic saturation in the vicinity of the gap, in particular. This is why a magnetic field analysis based on the finite element method (FEM) is
useful when investigating the motor's characteristics for a design study.
This Application Note explains an analysis that confirms the Speed-Torque curve and current density distribution of an induction motor.

Stator core

Current Density Distribution
The current density distribution at a rotation speed of 1050 r/min is shown in fig. 1. The
rotating magnetic field created by the stator coils generates induced current in the rotor
cage. Torque is generated from the magnetic flux and rotating magnetic field created bythis induced current, so the current has a large effect on the torque characteristics.

4

Torque (Nm)

3

2

1

Speed-Torque Curve
The Speed-Torque curve is shown in fig. 2. For this induction motor, the maximum torquecan be obtained at the lower speed range. This means the induction motor has highresistance. In this case, changing the thickness of bars can reduce the electric resistance,
and then the maximum torque can be obtained at the higher speed range.

42

JSOL Corporation

0.033 s0.057 s00rotation speed: 1050 r/min

Fig. 1 Current density distribution(Unit:A/m^2)

300

600

900

1200

Rotation speed (rpm)Fig. 2. Speed-Torque Curve

1500

1800

JMAG Application Catalog

Case

69

Iron Loss Analysis of an IPM Motor

Stator core

Magnetmodule: DP, LS

Rotor core

Overview
Coil

Demand for higher efficiency and smaller size in motors has grown from the need to accommodate devices that incorporate miniaturizationand energy efficiency in their designs. In order to meet this demand, motors have to improve their output density and reduce their losses.
One type of loss commonly found in motors is iron loss, which increases drastically at high rotation speeds and high magnetic flux densities.
This increase can lead to a rise in temperature and a reduction in efficiency. Consequently, it is growing more important to predict iron loss
levels at the motor design stage.
Unfortunately, it is not possible to obtain iron losses accurately in studies that use the magnetic circuit method or rules of thumb. In order toobtain them accurately, one needs to find the distribution and time variations of the magnetic flux density in each part of the motor after
accounting for a fine geometry and the material's nonlinear magnetic properties. Using the finite element method (FEM) is essential in orderto carry out this kind of a detailed analysis.
This Application Note explains a case example that obtains the iron loss and its distribution in a permanent magnet motor.

Magnetic Flux Density Distribution
The magnetic flux density distribution of the stator core and rotor core is shown in fig. 1,
and the magnetic flux density waveforms of the magnetic flux density r-component at
measuring points 1 and 2 is shown in fig. 2.
While both the value and amount of variation in the magnetic flux density at measuring
point 1 are large, at measuring point 2 the magnetic flux density has a large value but asmall amount of variation. These differences in magnetic flux density affect the iron loss.

Measuring point (1)Measuring point (2)2.0Magnetic ﬂux density (T)Measuring point 1

Joule Loss Density Distribution/Hysteresis Loss Density
Distribution/Iron Loss Density Distribution
The joule loss density distributions for the stator core and rotor core are shown in fig. 3.
The hysteresis loss density distributions for the stator core and rotor core are shown in fig.
4. The iron loss density distributions for the stator core and rotor core are shown in fig. 5.
The ratios of loss for the stator core and rotor core are shown in fig. 6.
As shown in the magnetic flux density waveforms, joule losses are greater in places where
the amount of variation in the magnetic flux density is large, compared to places where
the amount of variation in the magnetic flux density is small. The same trend can be seenin the hysteresis losses as well, as shown in fig. 6 by the fact that the ratio of the joule andhysteresis losses taken up by the stator is large.

1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5

Measuring point 2-2.0(Unit: T)

0

60

120

180

240

300

360

Electric angle (deg)Fig. 1 Magnetic flux density distribution (1 rotation)

Measuring point 2

Fig. 2 Magnetic flux density waveforms (r-component)

18.0
Measuring point 1

16.0

Joule loss (W)14.0

Hysteresis loss (W)Loss, W

12.0
10.0
8.0
6.0
4.0
rotor corestator core

(Unit: W/m^3)

rotor corestator core

(Unit: W/m^3)

rotor corestator core

(Unit: W/m^3)

2.0
0.0
Rotor coreFig. 3 Joule loss density distribution

Fig. 4 Hysteresis loss density distribution

Fig. 5 Iron loss density distribution

Stator coreFig. 6 Ratio of loss

https://www.jmag-international.com/

43

Analysis of Impedance-Frequency Characteristics
of a Cable

Case

70

module: FQ

SheathOverview
Twisted pair cables are used in situations that require strict noise reduction like with signal lines and speaker cables because they are notaffected by external noise and do not emit much noise of their own. The cable's performance relies on its electric properties, but thesechange depending on the state of the current that is flowing. For example, when the current frequency rises, the current is offset in thecopper wires because of the skin effect and proximity effect. As a result, the apparent cross-sectional area of the current flow is reduced,
causing the alternating current resistance to increase and the inductance to change. An increase in the resistance leads to an increase inlosses, and changes in inductance result in distortions in the signal. This is why these frequency characteristics need to be understood in
advance. The above phenomena occur in the interior of the copper wires, so an evaluation using a magnetic field analysis based on the finite
element method (FEM) is useful because they can be difficult to predict with manual calculations.
This Application Note explains how to obtain the frequency characteristics of the resistance and inductance in twisted pair cables.

Insulating ﬁlm
Copper wire

Current Density Distribution

Resistance/Inductance Frequency Characteristics

The current density distribution of the copper wire is shown in fig. 1. At a frequency of 10
kHz the proximity effect causes offsets in the current, and at 100 kHz both the skin andproximity effects cause offsets in the current. It is anticipated that these offsets in current
have an effect on the resistance and inductance.

The frequency characteristic of the resistance in the copper wire is shown in fig. 2, and thefrequency characteristic of the inductance is shown in fig. 3. The resistance andinductance begin to change gradually at around 10 kHz due to the proximity effect, andthe skin effect begins to increase as the frequency grows larger, so both resistance andinductance change rapidly. In order to reduce the skin and proximity effects, the structureneeds to be studied so that the current in the frequency range being used can flow
through the conductor uniformly.

0.30

30

Inductance (Hμ)

Resistance (m ohm)

25
20
15
10

0.25

0.20

0.15
5
0

0.10
0.01

20 Hz

10 kHz

100 kHz

0.1

1

10

100

0.01

Fig. 1 Current density distribution in the copper wire

71

1

10

100

Frequency (kHz)Fig. 2 Frequency characteristic of the resistanceFig. 3 Frequency characteristic of thn e inductance

Basic Characteristic Analysis of a Motor with 2 Brushes,
6 Poles, and 19 Slots

Case

0.1

Frequency (kHz)(Unit: A/m^2)

CoilRotor coremodule: DP

Overview
Small brush motors generally have a structure containing 2 poles and 3 slots, but there are times when a multi-pole structure is adopted inorder to produce higher torque. The reason for this is because achieving a higher torque makes it possible to omit deceleration systems.
Brush motors have a construction where the number of poles and number of slots are not divisible, with the objective of raising therectification effect or limiting torque variations. In exchange for reducing torque variations, however, there is a drawback when it comes totorque output. This is why selecting the number of poles and slots have become a design theme, especially when it comes to small motors,
which have a small number of slots. This makes the selection process difficult because the difference in distribution becomes large. The
model for this analysis has 6 poles and 19 slots, so there are 3.16 slots per pole. They cannot be divided into decimals however, so there haveto be either 3 or 4 slots for each magnetic pole. As a result, the induced voltage in each coil and the torque generated are unbalanced.
These evaluations need to be able to account for an accurate circuit geometry, and the current flowing through coil connected via acommutator needs to be handled accurately, as well. This is why an electromagnetic field analysis using the finite element method (FEM) is
necessary to account for everything. This Application Note presents an analysis to obtain the speed versus torque and torque versus current
for a motor that has 2 brushes, 6 poles, and 19 slots.

MagnetStator core

Magentic Flux Density Distribution

Fig. 1 shows the speed versus torque curve and fig. 2 shows the current versus torquecurve.
As shown in fig. 1, the torque in the DC brush motor decreases as the rotation speed
increases. This is because the current flowing in the coil is reduced by the increase ofreverse electromagnetic force produced as the rotation speed increases. As shown in fig.
2, the relationship between the current and torque is mostly linear, and the torquecoefficient of this brush motor is 0.05 Nm/A.

Fig. 3 shows the magnetic flux density distribution at rotation speeds of 2000 r/min and4000 r/min.
The magnetic flux density increases at lower a lower rotation speed, as indicated in the
figure. This is because the current flowing in circuit is larger, as mentioned earlier. A motorand control design that does not allow magnetic saturation to occur at low rotation
speeds is desirable because the performance of the motor worsens with magnetic
saturation.

2.5

2.5

2.0

2.0
Torque (Nm)Torque (Nm)Speed-Torque Curve and Torque-Current Curve

1.5
1.0

1.0
0.5

0.5

0.0

0.0
0

1000

2000

3000

Rotation speed (r/min)Fig. 1. Speed-Torque curve

44

1.5

JSOL Corporation

4000

5000

0

10

20

30

40

50

2000 r/min

4000 r/min

Current (A)Fig. 2. Torque-Current curve

Fig. 3. Magnetic flux density distribution(Unit: T)

JMAG Application Catalog

Analysis of Attractive Force between Steel Plates
and a Magnet

Case

72

Top steel plate
MagnetBottom steel plate

module: ST

CASE1:Magnet and steel sheets are detached

Overview
Many electronic products that have movers use the attractive force generated between magnets and magnetic materials. Even magnetic
materials that are not magnetized themselves take on magnetic properties when in the vicinity of a magnet, generating an attractive force
between the two bodies.
In order to understand the phenomena where the magnet and magnetic material attract each other, it is helpful to visualize the magnetic flux
flowing through the air. Using a magnetic field analysis makes it easy to visualize and understand these phenomena.
This Application Note explains how to obtain the magnetic flux density distribution and attractive force generated between a steel plate anda magnet.

Top steel plate
MagnetBottom steel plate

CASE2：Magnet is attached to the bottom steel sheet

Attractive Force Characteristics of CASE 1

Attractive Force Characteristics of CASE 2

The magnetic flux density distribution for CASE 1 is shown in fig. 4.1, and its attractive
force characteristics are shown in fig. 4.2. The attractive force acting in the positive
direction on the Y-axis is set to positive.
The relative positions of the magnet and steel plate allow the magnetic flux to bevertically symmetrical, meaning that the forces acting on the top steel plate and thebottom steel plate are the same, and almost no force is acting on the magnet.

The magnetic flux density distribution for CASE 2 is shown in fig. 3, and its attractive force
characteristics are shown in fig. 4. In fig. 4, (4) is a value that represents the sum of theattractive forces of (2) Magnet and (3) Bottom steel plate. The attractive force acting in thepositive direction on the Y-axis is set to positive.
In CASE 2 the magnet is touching the bottom steel plate, so the magnetic flux density ishigher in the bottom steel plate than in the top steel plate, and the attractive force
between the magnet and the bottom steel plate is larger than between the magnet andthe top steel plate. This is why the magnet's attractive force in fig. 4 is shown acting in thenegative direction, and the sum of (1) and (2) is equal to (3). (1) and (4) are also equal, so it
is possible to confirm that the attractive force that the attractive force acting on themagnet and the bottom steel plate has been obtained correctly.

Top steel plate

MagnetBottom steel plate(1) Top steel plate
100
80

34.3

30

Attractive force (N)Attractive force (N)

40
20
10

0.0

0
-10
-20

Flux lines near the steel plates and magnet(Unit: T)

Fig. 1 Magnetic Flux Density Distribution of CASE 1

0
-20
-40

-34.3
Calculation Region

Attractive Force (N)Top steel plate

-34.3
-

Magnet0.0

Bottom steel plate

34.3(Unit:T)

Fig. 2. Attractive Force Characteristics for CASE 1

Fig. 3 Attractive Force Characteristics of CASE 1(3) Bottom steel plate(4) (2)+(3)

77.8

24.8

-24.8

-60
-80

-30
-40

(2) Magnet

60
40
20

-53.0
Calculation Region

Attractive Force (N)(1)

Top steel plate--24.8(2)

Magnet-53.0(3)

Bottom steel plate

77.8

(4)

(2)+(3)24.8

Fig. 4 Attractive Force Characteristics of CASE 2

Case

Analysis of the Capacitance of a Parallel Plate Capacitor

73

Electrode plate

module: EL

Overview
Capacitors are widely used as basic passive components in a variety of electric circuits. Parallel plate capacitors are taken up as the subject
matter for how to obtain capacitance, a basic electric property, because they follow simple phenomena. Two electrode plates are placed in
the air to face each other. When a constant electric potential difference is applied to them, they store an electric charge. The capacity of thecharge stored in the electrode plates changes depending on the permittivity of the dielectric material between them.
This Application Note explains how to use JMAG to calculate the capacitance when the dielectric material is air and when it is a highpermittivity material, while obtaining the amount of charge in each.

Dielectric material

Electric Field Distribution

Capacitance Calculation

The magnetic field distributions generated between the two electrode plates when they
are in parallel and an electric potential difference is applied to them are shown in fig. 1.
When the dielectric material is air, there is a large magnetic field that extends from the
opposing faces of the electrode plates to the air, so there is a large magnetic field
generated locally at the corner of the electrode.

The electric charge distribution generated on the surfaces of the electrode plates isshown in fig. 2. When the high permittivity material is used, the amount of surface charge
density is larger than that of air. The capacitance is calculated with the following formulaby using the amount of surface charge density and the electric potential difference
applied in the analysis. The result is shown in table 1.

CQ
VDielectric material (air):Surface charge density 2.34e-12 (C)C : Capacitance, F,
Q : Electric charge, C,
V : Electric potential diﬀerence, V(Unit: C/m^2)

Dielectric material (air)(Unit:V/m)
Dielectric material (high permittivity material)

Fig. 1 Electric Field Distribution

Dielectric material (air)Dielectric material(high permittivity material)

0.47 pF

443 pF

CapacitanceDielectric material (high permittivity material):
Surface charge density 2.21e-9 (C)(Unit: C/m^2)

Fig. 2 Electric Charge Distribution of the Electrode
Plate Surface

Table1

Capacitancehttps://www.jmag-international.com/

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74

Speed Versus Torque Analysis of a Single-Phase
Induction Motor

Stator coreCage(end ring)
Cage (bar)module: DP

Overview

Rotor coreSingle-phase induction motors are widely used as small output motors for the drives in household electrical appliances and office machinery,
like fans and washing machines, because they can use single-phase AC, the typical power source for home electronics. Unlike three-phase AC,
however, single-phase AC cannot create a rotating magnetic field by itself, meaning that it cannot start a motor. For this reason, it needs touse an alternate method to generate a rotating magnetic field to start the motor.
The induced current flowing in the secondary conductor largely affect the performance of the motor because the motor rotates by using theinteraction between this current and the magnetic field of the stator coils. Strong magnetic saturation distribution is also generated near thegap, so the nonlinear characteristics of the magnetic properties have a big influence on performance, as well. At the step before the designphase, it is helpful to run an analysis and evaluation using the finite element method (FEM) to understand a single-phase induction motor's
characteristics by accounting for induced current and magnetic saturation characteristics.
This Application Note explains how to obtain the current density distribution and Speed-Torque curve created by auxiliary winding that uses acapacitor.

Shaft

Current Density Distribution

Speed-Torque Curve

The current density distributions for maximum torques at rotation speeds of 2,100 rpm
are shown in fig. 1. The difference in speed between the rotor and rotating magnetic field
becomes smaller as the rotor reaches high speed. This makes the variations in themagnetic field acting on the cage grow smaller, which in turn reduces the eddy currents
flowing in the bars accordingly. The eddy current distribution in the bars largely affects
the torque characteristics.

The Speed-Torque curve is shown in fig. 2. The maximum torque can also be obtained ataround 2000 rpm. The rotation region that provides maximum or high torque varies
depending on the electric resistance, so changing the cage materials or geometry makes
it possible to study the motor characteristics that most closely meet the goals of thedesign.

2.50E+00

Torque, Nm

2.00E+00

1.50E+00
1.00E+00

5.00E-01
0.00E+00
0

Rotation speed: 2100 rpm(Unit:A/m^2)

300

600

900 1200 1500 1800 2100 2400 2700 3000
Rotation speed, rpm

Fig. 2. Speed-Torque curve

Fig. 1. Current density distribution

Case

75

Core

Iron Loss Analysis of a Reactor

Bobbinmodule: FQ, LS

CoilOverview
Reactors are installed on the input or output side of inverter circuits. Because they are required for long-term operation, the ability to control
running costs from losses is an important challenge for their design. Iron loss is one of the major types of losses in a reactor. It consumes
electric power as heat in a magnetic body, so it causes heat to increase and efficiency to decrease in the reactor.
Using a finite element analysis (FEA) to confirm the distribution of iron loss density makes it possible to study a reactor's local geometry
during its design, so it is useful in providing feedback about the design itself.
This Application Note analyzes the iron loss of a reactor.

SpacerMagnetic Flux Density Distribution/
Iron Loss Density Distribution

Joule Loss Density Distribution/
Hysteresis Loss Density Distribution

The core's magnetic flux density distribution and iron loss density distribution are shownin figures 1 and 2, respectively. The magnetic flux concentrates on the surface because it
cannot flow easily in the direction of the laminations, increasing the iron losses
accordingly. The magnetic flux also tries to flow through the shortest path in themagnetic circuit, so the magnetic flux density and iron loss are large at the top andbottom of the core, which are not wrapped by the coil.

The core's joule loss density distribution and hysteresis loss density distribution are
shown in fig. 3, the ratio of losses is shown in fig. 4, and the losses are shown in table 1. As
can be seen from the figures, the contribution of the joule loss toward the iron loss islarge. The amount of core loss may be effectively decreased by reducing the amount ofJoule losses.

Hysteresis loss
7(%)

Joule loss
93(%)

Iron loss (W)Joule loss (W)Hysteresis loss (W)

21.1
19.7
1.4

(Unit:W/m^3)(Unit:T)(Unit:W/m^3)
Joule loss density distribution Hysteresis loss density distribution

Fig. 1. Magnetic flux density distribution

46

JSOL Corporation

Fig. 2. Iron loss density distribution

Fig. 3 Joule loss density distribution
and hysteresis loss density distribution

Fig. 4 Ratio of loss

Table 1 Losses

JMAG Application Catalog

Case

77

Inductance Analysis of an RFID Tag
module: FQ

CoilOverview

Covering material

An RFID tag uses electromagnetic induction to communicate information by supplying electrical power to an IC chip from a reading device. In
order to relay information in specific frequencies with a good degree of sensitivity, the RFID tag uses resonance between its internal coilantenna and capacitor.
The coil antenna's inductance and the capacitor's resonance frequency need to be estimated for resonance to be produced accurately at the
specified frequency. When the capacitor is external, the inductance of the coil antenna needs to be obtained accurately and the capacitor'scapacity needs to be identified. Some RFID tags have magnetic sheets or metallic films on them. The sheet's magnetic properties and eddy
currents generated in the film can affect the inductance.
This Application Note performs a magnetic field analysis of an RFID tag that has a metallic film and a magnetic sheet with a resonancefrequency of 13.56 MHz, and obtains the magnetic field distribution and the RFID tag's inductance.

Substrate
Magnetic sheet
Metal plate

Magnetic Field Distribution

Inductance

The magnetic field distribution around the RFID tag is shown in fig. 1.
The magnetic field is shielded toward the metal plate, as shown in fig. 1. This magnetic
field distribution has a large effect on the inductance.

The inductance of the RFID tag and the capacity of the resonant capacitor are shown intable 1.
The inductance of this RFID tag can be obtained from the relationship between thevoltage and current. In this analysis, it is 942 nH. The capacity of the external capacitor,
calculated from the resonance frequency and inductance, is 146 pF.

Resonance frequency

13.56 MHz

Inductance

942 nHCapacitance

146 pF

Metal plate(Unit: A/m)Fig. 1 Magnetic field distribution

Table 1. Each value of the RFID tag

Case

78

Loss Analysis of a Sheet Coil TransformerCore

module: FQ, LS

Primary coil

Overview

Secondary coilPower transformers need to be able to operate over the long term, so they are always required to control running costs from losses. Iron loss
is one of the main losses in a transformer, and it can lead to temperature increases and efficiency decreases because it consumes electric
power as heat in a magnetic material.
Using a finite element analysis (FEA) to display the iron loss density distribution and obtain the iron loss values in a transformer makes it
possible to study the local geometry and get design feedback at the design stage.
This Application Note shows how to obtain the iron loss of a sheet coil transformer.

Magnetic Flux Density Distribution/
Iron Loss Density Distribution

Joule Loss Density Distribution/
Hysteresis Loss Density Distribution

The core's magnetic flux density distribution and iron loss density distribution are shownin figures 1 and 2, respectively. The magnetic flux density distribution and iron loss
density distribution are similar, as indicated in the figures. This is because the frequency ofthe magnetic flux density variations is constant, so the size of the iron losses is
proportional to the size of the magnetic flux density variations.

The core's joule loss density distribution and hysteresis loss density distribution are
shown in fig. 3, the ratio of losses is shown in fig. 4, and the losses are shown in table 1.

Hysteresis loss
45 (%)

Joule loss density distribution

Joule loss
55 (%)

0.24 W

Hysteresis loss

0.20 W

Iron Loss

0.44 W(Unit: W/m^3)(Unit:W/m^3)(Unit:T)

Joule loss

Hysteresis loss density distribution

Fig. 1. Magnetic flux density distribution

Fig. 2. Iron loss density distribution

Fig. 3. Joule loss density distribution andhysteresis loss density distribution

Fig. 4. Loss ratio

Table 1. Losses

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Case

79

Analysis of an Eﬀect of Magnetic Field Orientation
on Magnetization

Nonmangetic material

Coil

Magnet(to be magnetized)
Molding die

module: ST

Overview

Coil
Magnetizing yoke

A magnet's magnetizing state largely affects the characteristics of a device that uses permanent magnets. The manufacturing process of amagnet is divided into two parts: the orientation process and the magnetization process. In the orientation process, an orientation device isused at the stage where magnetic power is formed to align the growth direction of the magnetization. After this, the magnet is temporarily
demagnetized. Next, in the magnetization process, a magnetization device achieves the desired state of magnetization in the magnet.
Both the orientation process and the magnetization process need to be accounted for in order to evaluate a magnet's magnetization state
accurately. Consequently, it is extremely difficult to study a magnetizing state by obtaining the behaviors of the molding die, magnetization
device, and magnetization current with manual calculations. Running a magnetic field analysis simulation that uses the finite element
method (FEM) it is possible to calculate the magnetic field orientation and magnetization field from the material properties, equipment
geometry, and applied current. One can also run an analysis that uses the magnetization state inside the magnet based on these results.
This Application Note explains how to obtain a magnet's surface magnetic flux density distribution while accounting for the orientation andmagnetization processes.

Non-magnetic material

Magnetic Field Orientation Distribution/
Magnetization Field Distribution

Magnetization Distribution/
Surface Magnetic Flux Density Distribution

The magnetic field orientation distribution is shown in fig. 1, and the magnetization field
distribution is shown in fig. 2. As can be seen in fig. 1, the magnetic field orientation
distribution is almost entirely radial distribution. For this orientation, the magnetization
state of the magnet is determined by applying the magnetization field shown in fig. 2.

Fig. 3 shows the magnetization distribution in Magnets A and B. Magnet A is the magnet
that was magnetized by accounting for the magnetic field orientation shown in fig. 1.
Magnet B is a magnet that was magnetized with an ideal radial orientation. The magnetic
flux density distribution at a distance of 0.5 mm from the surface of a magnet that has
been placed in air is shown in fig. 4.
From fig. 3, it can be seen that there is greater magnetization at the ends of Magnet A
than of Magnet B. This is because the orientation at the ends of Magnet A shown in fig. 1
is more similar to the magnetization field distribution shown in fig. 2 than with Magnet B.
From fig. 4, it can be seen that the surface magnetic flux density is higher in Magnet B
than Magnet A at the center of the magnetic poles. This is because the orientation at thecenter of the poles in Magnet A shown in fig. 1 differs more from the direction of themagnetization field shown in fig. 2 than it does with Magnet B. As can be seen from theresults in fig. 3, the surface magnetic flux density distribution at the ends of the magnets
differs greatly.
This is why in order to evaluate the magnetization state of a magnet accurately, themagnetic field orientation and magnetization field need to be taken into account.

0.30

45 degMagnetic Flux Density, T

0.25
0.20
0.15

0 deg

0.10

Magnet A

0.05

Magnet B

0.5 mm

0.00
-45

-30

-15

0
Angle, deg

15

30

45

--45 deg(Unit: A/m)
Magnet A

Fig. 1 Magnetic field orientation distribution(red lines: magnetic flux)

48

JSOL Corporation

Fig. 2. Magnetization Field Distribution

Magnet B(Unit: A/m)Fig. 3 Magnetization distribution in the magnet

Fig. 4 Surface magnetic flux density of the magnet

JMAG Application Catalog

Case

80

Cogging Torque Analysis of an SPM Motorwith Skewed Magnetization

Shaft
Magnetmodule: TR

Overview
Reductions in vibration and noise are being sought after because they are a cause of torque variations in motors, and demands for reduction
are particularly strong with motors that are used in machine tools and power steering. Cogging torque, which is a torque variation that is
produced when there is no current, is generated because the electromagnetic force produced in the gap changes according to the rotor's
rotation. This makes it necessary to apply skew to the stator and rotor and come up with innovative geometry for the magnet and stator inorder to reduce torque variations by limiting variations in the electromagnetic force. Applying skew reduces the cogging torque, but it also
brings disadvantages such as producing force in the thrust direction and generating eddy currents from the magnetic flux that links in thelamination direction.
Consequently, in order to accurately evaluate a motor that has skew applied, one needs a magnetic field analysis simulation that uses thefinite element method (FEM), which can account for a detailed 3D geometry, instead of studies that use the magnetic circuit method or a 2D
magnetic field analysis.
This Application Note presents the use of a magnetic field analysis to obtain the flux density distribution, cogging torque, and induced
voltage of an SPM motor that has skewed magnetization applied to its magnet.

Stator core

Magnetic Flux Density Distribution

Induced Voltage Waveform

Fig. 1 shows the magnetic flux density distribution at the rotation angle of 90 deg. From
fig. 1, it is apparent that the magnetic circuit changes when the magnet is skewed. Thesevariations in the magnetic circuit have an effect on the cogging torque waveform.

The induced voltage waveforms of the U-phase at skew angles of 0 deg and 30 deg areshown in fig. 3, and the frequency components of the U-phase's induced voltage
waveforms are shown in fig. 4.
The high frequency components are reduced when skew is applied to the magnet. The
ratio taken up by the fundamental frequency of 60 Hz out of the total frequency
components is 72% with a skew angle of 0 deg and 82% with a skew angle of 30 deg.
According to these results, the induced voltage waveform is smoother at a skew angle of
30 deg than at an angle of 0 deg.

Cogging Torque Waveform
Fig. 2 shows cogging torque waveforms at skew angles of 0 deg and 30 deg.
The peak value of the cogging torque is reduced by approximately 90% when skew has
been applied to the magnet.

Skew angle 0 deg

Skew angle 30 deg

Skew angle 0 deg

Skew angle 30 deg

Skew angle 0 deg

8

0.015

6

0
-0.005(Unit: T)

Fig. 1 Magnetic flux density distribution

0-6-0.015skew angle of 30 deg

2

-2
-4

-0.01

skew angle of 0 deg

Voltage (V)

4

0.005

Voltage (V)Torque (Nm)

0.01

-8
0

5

10

15

20

25

30

Rotation angle (deg)

1

31

61

91

121

151

181

60 120 180 240 300 360 420 480 540 600 660 720 780
Frequency (Hz)

Rotation angle (deg)Fig. 2 Cogging torque waveform

Skew angle 30 deg

10
9
8
7
6
5
4
3
2
1
0

Fig. 4 Frequency componentsof the U-phase induced voltage waveform

Fig. 3 U-phase induced voltage waveform

Case

81

AL-value Analysis of a Choke CoilCore

module: ST, TRANSFORMER_TEMPLATE

Overview

BobbinCoil

A choke coil is an electric component that is intended to filter high-frequency current. The AL-value is a vital parameter in a choke coil designthat determines the cutoff frequency of a high-frequency current.
The AL-value varies greatly depending on gap width, so it is effective for the advance study of a design to accurately obtain the AL-value
dependency on the gap width (air gap versus AL characteristics) for the geometry of a choke coil by using a finite element analysis (FEA).
This Application Note explains a case example that obtains the air gap versus AL characteristics of a choke coil.

Air Gap-AL Characteristics
The air gap versus AL curve of the choke coil is shown in fig. 1.
Fig. 1 demonstrates the relationship between the gap width and the AL-value. This makes
it possible to obtain the AL-value from the gap width and estimate the number of turns toachieve the specified inductance.

AL-value (nH/N^2)

1000

100

10

1
0.1

1

10

Width of air gap (mm)Fig. 1. Air gap versus AL-value

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Case

82

Analysis of a Synchronous Reluctance Motor
Rotor coremodule: DP

Overview
Skyrocketing prices of rare earth magnets have led to rising expectations for synchronous reluctance motors (referred to below as SynRMs),
which do not use permanent magnets. SynRMs have a simple structure that can achieve solid performance at a low price. However, torque isgenerated only by the rotor's saliency and the coil's magnetomotive force, so raising the torque density depends greatly on the core's
nonlinear magnetic properties and the rotor geometry. This is why they have a different format than a typical motor. On the other hand, theaforementioned rising prices of rare earth magnets, improvements in current control technology, and the ability of optimization designs using
magnetic field analysis have raised the possibility of lowering these barriers, giving SynRMs the chance to be reexamined.
SynRMs operate using the nonlinear region of a magnetic steel sheet, so the inductance expresses nonlinear behavior as well. This behavior
distorts the excitation current waveform a great deal, making it impossible to run advanced projections that are accurate with calculation
methods that follow linear formulas. Consequently, it becomes necessary to use the finite element method (FEM), which can handle nonlinear
magnetic properties in material, detailed motor geometry, and transient currents.
This Application Note presents an evaluation of torque variations that occur when the phase of a sinusoidal wave current is changed.

Stator core

Magnetic Flux Density Distribution and Average Torque per
Current Phase
The magnetic flux density distribution at a rotation speed of 600 r/min and a current
amplitude of 3.0 A is shown in fig. 1. The average torque at each current phase is shown infig. 2. The magnetic flux density distribution changes with the current phase. When thecurrent phase is 0 deg and 90 deg the average torque is 0, and when the current phase is
45 deg the torque reaches its peak.

1.2

Average torque (Nm)

1.0
0.8
0.6
0.4
0.2
0.0
0
(Unit: T)

Fig. 1 Magnetic flux density (top: β= 0, bottom: β= 45)

15

30

45

60

75

90

Current phase β (deg)Fig. 2 Average torque per current phase

Case

83

Lining
Molten metal

Magnetic Shielding Analysis of an Induction Furnace
module: FQ

Overview
An induction furnace is an apparatus that uses high-frequency induction heating to melt metal. Running current through the coil surrounding
the crucible starts electromagnetic induction phenomena, which generate current in the metal in the crucible. This current produces joule losses
in the metal, which are used to heat and melt it. Magnetic yokes are arranged around the coil. The yokes are used as strong components thatprevent the Lorentz force generated by the coil from damaging and deforming it. The magnetic yokes also reduce the leakage flux that flows
out of the appliance, preventing unintended heating in surrounding structures. Keeping the amount of material used in the magnetic yokes toa minimum makes it possible to reduce the cost of the apparatus.
To understand the magnetic flux that spreads from the induction furnace, it is necessary to use the eddy current distribution and magnetic flux
flow in the metal in the crucible, as well as the concentrations in magnetic flux caused by the positions of the yokes.
This Application Note displays magnetic flux density distribution to evaluate the differences in magnetic flux with and without yokes.

Yoke

Coil

Magnetic Flux Density Distribution
The magnetic flux density distribution generated in the yokes and the air region is shownin fig. 1. When the yokes are present, there is less leakage flux than when they are notpresent. This means that it is possible to prevent a steel plate outside of the yokes frombeing heated.

80
75

Joule loss (kW)70

without yokes

Cut plane

Molten Metal Joule Loss
The joule loss values of the molten metal are shown in fig. 2. There is 5% more joule loss inthe furnace with the yokes than without them, so it is applying energy to the molten metal
efficiently.

JSOL Corporation

60
55
50
45

Cut plane(Unit: T)

40
Without yokeswith yokes

Fig. 1 Magnetic flux density distribution

50

65

Fig.2.Joule Loss of the Molten Metal

With yokes

JMAG Application Catalog

Case

84

Current direction

Electromagnetic Forming Analysis of a Tube
module: DP

Radial force generation

Overview
Electromagnetic forming is a process in which eddy currents are generated in a tube when a large, instantaneous current is run through a coil,creating a strong magnetic field. The tube is formed by using the Lorentz force produced by the interaction between these magnetic fields.
This method is different from forming that uses plastic deformation with a press mold because it deforms with the power of the tube itself.
This makes it possible to perform difficult forming, like with an object that will not fit into a normal press mold. However, the force generated
in the tube is determined from induced eddy currents, which means that it is dynamic deformation, so the behavior of its deformation is
known for being difficult to analyze. Using JMAG to properly analyze the Lorentz force distribution generated in the tube can assist withpredicting the deformation behavior of the electromagnetic forming process.
This Application Note explains how to obtain the Lorentz force density distribution generated in a tube when current is run through a coil.

Tube
Coil

Eddy Current Density Distribution

Lorentz Force Density Waveform

The eddy current density distribution generated in the tube at 125 μs is shown in fig. 1.
Running high current through the coil creates a strong magnetic field around it, inducing
eddy currents in the tube. More eddy currents are generated in the ends of the tube, as
shown in fig. 1. This is because the magnetic flux concentrates near the ends of the tube.

The Lorentz force densities in the radial direction and the axial direction at each time areshown in figures 2 and 3. The tube expands with the positive values of Lorentz force
density in the radial direction, and the tube shrinks with negative values of Lorentz force
density in the tube. The Lorentz force that occurs in the radial direction at 65 μs and 125
μs is produced consistently around the center of the tube, and it becomes drasticallylarger at the ends, as indicated in fig. 2. This is because the eddy currents concentrate in
the ends of the tube. When the current has declined to 187.5 μs, the Lorentz force that
occurs shrinks the ends of the tube.
The Lorentz force density in the axial direction only occurs within a 10 mm range at eachend of the tube, and it is distributed symmetrically across the center position of 0.
The Lorentz force generated works to shrink the tube as well as to expand it while current
is flowing in the coil, so the Lorentz force density in the tube needs to be evaluated using
a simulation at the design stage.

62.5μs0mm

Center of tube

-75mm

End of tube

40
0
-40

85

187.5μs400-400.0E+00

1.0E+112.0E+113.0E+11-2.0E+11-1.0E+11

Fig. 2 Lorentz force density
in the radial direction

High-Frequency Induction Heating Analysis
of a Constant Velocity Joint

0.0E+00

1.0E+112.0E+11

Lorentz force density (N/m^3)

Lorentz force density (N/m^3)

Fig. 1 Eddy current density distribution in the tube at 125 μs

125μs

-80

-80

(Unit: A/m^2)Case62.5μs

187.5μs
80-1.0E+11

Position for the tube

125μs

80
Position (mm)End of tube
Position (mm)75mm

Fig. 3 Lorentz force density
in the axial direction

Work piece(outer race)

module: FQ, HT

Overview
Constant velocity joints are used in the connections at each end of a drive shaft in a vehicular drive system. The inside of the outer race of aconstant velocity joint makes direct contact with steel balls or rollers on the inner race, so it needs to have increased hardness to improve its
ability to resist wear and tear. On the other hand, its inside needs to retain its toughness in order to maintain its flexibility as a part.
High-frequency induction hardening is used as a heat treatment method that hardens only the surface of a product. With this method, using
a high-frequency power supply makes it possible to heat the surface locally and rapidly. This process also has many other benefits, such asproviding a clean working environment because it uses electrical equipment, being very efficient, and providing uniform results for eachproduct. This is why it is being aggressively implemented in the field. It is difficult to predict the hardening of the inside of a constant velocity
joint because its uneven geometry makes eddy currents and magnetic flux flow in a complex manner.
With interior hardening like the kind used in this example, the heating coil design must follow spatial constraints. The eddy currents that are
generated from a high-frequency varying magnetic field are offset on the surface of the part's interior, so the material properties change agreat deal as the temperature rises. This is why it is necessary to predict the amount of heat generated in a numerical analysis based on thefinite element method (FEM) in order to handle the detailed phenomena.
This Application Note shows how to run an analysis of the elevated temperature process by using the geometry of a coil to evaluate whether
or not the target temperature conditions are fulfilled.

Coil

Joule Loss Density Distribution
The joule loss density distribution produced in the outer race is shown in fig. 1. Eddy
currents are generated in the outer race by time variations in the magnetic field createdby the coil. In induction heating, these eddy currents become a heat source for the workpiece. At high frequencies the work piece is strongly affected by the skin effect, so theeddy currents are distributed near its surface.

0.12 mm

Temperature Distribution
The temperature distribution of the outer race is shown in fig. 2. The eddy currents cause
heat to generate in the outer race, so the temperature rises as time passes. At 2 s most of
the inside of the outer race reaches the Curie point of 800 deg C, but there is variation in
the temperature distribution. Improvements need to be made to the coil geometry inorder to achieve uniform heating on the inside of the outer race.

After 0.5 s

After 1.0 s

After 2.0 s(Unit: deg C)(Unit: W/m^3)

Fig. 1. Joule loss density distribution

Fig. 2. Temperature distribution

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51

Case

86

Power Transmission Analysis
of a Wireless Power Transfer SystemCoremodule: FQ

Secondary Coil

Overview
A wireless power transfer system is a device which uses electromagnetic induction to provide electric power without physical contact. They
can be used for various applications, such as moving or rotating devices, or devices sealed inside enclosed spaces. Because the primary andsecondary sides do not touch, the power transmission efficiency and leakage flux vary depending on their positions relative to each other.
Therefore, it is important at the design stage to understand how properties change according to their placement.
When evaluating the properties of a transformer whose primary and secondary sides are separated by a gap, it is helpful to use magnetic field
analysis based on the finite element method (FEM), which allows precise modeling of the geometry of the parts and their relative positions,and makes it possible to visualize the leakage in the magnetic flux that is generated in the primary side and transmitted to the secondary side.
This Application Note presents how to obtain the power transmission efficiency when the feeder wire's position is moved in both thehorizontal and vertical directions from a reference position, and how to display the flow of magnetic flux.

Feeder Wire

Vertical Distance vs. Power Transmission Eﬃciency

Horizontal Distance vs. Power Transmission Eﬃciency

Diagrams of the magnetic flux lines with the feeder wire moved in the vertical direction
are shown in fig. 1, and the power transmission efficiency is shown in fig. 2. As the
distance between the feeder wire and the core gets larger, the amount of leakage flux
increases and the linkage flux in the secondary coil decreases. In this way, the powertransmission efficiency is reduced.

Diagrams of the magnetic flux lines with the feeder wire moved in the horizontal
direction are shown in fig. 3, and the power transmission efficiency is shown in fig. 4.
When the feeder wire is moved horizontally from the reference position, the balance ofmagnetic flux in the core becomes distorted to the left and right. Variations in the linkage
flux in the secondary coil are small, and variations in power transmission efficiency arealso small.

87.0Power transmission eﬃciency (%)

Power transmission eﬃciency (%)

95
90
85
80
75
70
65
-15

-10

-5

0

5

10

15

52

JSOL Corporation

Fig. 2 Vertical distance vs. power transmission efficiency

86.0
85.5
85.0
84.5
84.0
0

Displacement in vertical direction (mm)Fig. 1 Magnetic flux lines (left: primary coil moved
downward vertically 15 mm; right: reference position)

86.5

5

10

15

Displacement in horizontal direction (mm)Fig. 3 Magnetic flux lines (left: reference position;right: primary coil moved horizontally 15 mm)

Fig.4 Horizontal distance vs. power transmission efficiency

JMAG Application Catalog

Case

87

Iron Loss Analysisof an IPM Motor Including the Eﬀect of Shrink Fitting

Frame
Magnetmodule: DP, DS, LS

CoilOverview
Magnetic steel sheet is used for the cores of drive motors for HEVs and EVs. This is to make them more compact, lighter, and more efficient.
The main point for improving efficiency in an IPM motor's high rotation speed region is how to reduce iron loss. However, shrink fitting is usedin order to strengthen the joint between frames and stator cores with laminated structure. The compressive stress generated during shrink
fitting is known to increase iron loss. Therefore, it is important to account for the effects of this stress when evaluating iron loss.
Iron loss is generated when there are magnetic-field variations in steel sheet. Also, the amount of iron loss depends on the steel sheet's ironloss properties. These iron loss properties of steel sheet become worse when it is subjected to stresses such as shrink fitting. The stress causedby shrink fitting has its own distribution, and is particularly large in the back yoke. So, in order to evaluate the iron loss with good accuracy, it isnecessary to correctly obtain the stress distribution for the magnetic flux, time variation, and steel sheet. In recent research, it has been
mentioned that in order to increase the precision of stress dependency for iron loss, stress distribution is not the only requirement; evaluations
accounting for changes in magnetic flux density over time for each primary axis of principle stress is also a requirement.
Based on the results of iron loss with no stress, this document will compare the results of iron loss accounting for only the size of stress, andiron loss accounting for changes in magnetic flux density over time for each principle stress. (hereafter referred to as iron loss accounting forstress tensor)

Rotor core

Stator corePrincipal stress distribution

Iron Loss Density Distribution

Fig. 1 demonstrates compressive stress distribution and stress tensor for principle stress. It
shows that compressive stress occurs in the back yoke and the direction is thecircumferential direction. We learn that compressive stress caused by the frame
compressing in the radial direction becomes bigger in the slot area. This is because thestress is damped towards the tips of the teeth, while it cannot be damped andconcentrates at the slots.

Fig. 3 shows the iron loss density distribution. In order from the left: results of notaccounting for stress, accounting for only stress size, and accounting for stress tensor. We
learn that depending on if stress is accounted for or not, iron loss density distribution
varies, and iron loss values increase approximately 60 percent. This is believed to be due
to loss becoming greater from stress, and in order to match iron loss with actual
measurements, an analysis accounting for stress due to shrink fitting is required. In thecomparison regarding stress dependency methods, differences become visible with ananalysis accounting for stress size only and an analysis accounting for stress tensors
around the base of the teeth. This is due to magnetic flux density distribution around thebase of the teeth greatly affects the radial direction components, and an analysis
accounting only for compressive stress can be overestimated compared to an analysis
accounting for variations of magnetic flux density over time for each principle stress.

Magnetic ﬂux density (basic wave components) distribution
Fig. 2 demonstrates effective value distribution for the basic wave components (60Hz) ofmagnetic flux density when accounting for stress. The radial direction components of themagnetic flux density are the main components in the teeth, on the other hand, thecircumferential direction components are the main components in the back yolk. Upon
evaluating iron loss for stress dependencies, these differences in magnetic flux density
distribution effect on the methods: if only the stress size is accounted for, or if changes inmagnetic flux density over time for each principle stress are accounted for.

Low

High
compressive stress distribution

Fig. 1. Principal stressstress tensor(Units: MPa)

all direction
components

radial direction
components

circumferential direction (Units: T)
components

Fig. 2. Effective value distribution of magnetic flux
density for basic wave (60Hz)

Iron loss：3.7 W
not accounting
for stress

Iron loss：6.1 W
accounting for only
the size of stress

Iron loss：5.6 W
accounting
for stress tensor(Unit: W/m^3)

Fig. 3. Iron loss density distribution

https://www.jmag-international.com/

53

Case

Stiﬀness Torque Analysis of a PM Stepper Motor

89

Magnetmodule: TR

Teeth

Yoke

Coil

Overview
PM stepper motors are commonly used for positioning of moving parts in small devices such as printers and video equipment. In order for its
drive to function with an open loop, the most important characteristics for a stepper motor are controllability and holding torque, and not the
motor's output. Therefore, the desired characteristics are detent torque, which is a non-excitation holding torque, and stiffness torque, which
is an excitation holding torque. A PM stepper motor is made up of a multi-pole magnetized rotor and offset inductors for each phase. In orderto reduce their size and number of parts, claw pole inductors are made from folded steel sheet. Because of this, the flow of magnetic flux isthree dimensional, so it is necessary to carry out a 3D electromagnetic field analysis using the finite element method (FEM) to proceed with anaccurate preliminary study.
This Application Note describes how the stiffness torque at 0.5 A of current can be calculated for a PM stepper motor.

Stiﬀness Torque

0.3

0.3

0.2

0.2

0.1

0.1

Torque (Nm)Torque (Nm)Fig. 1 shows the stiffness torque with one-phase excitation and fig. 2 shows the stiffness
torque with two-phase excitation. Fig. 3 shows the magnetic flux density distribution of
the yoke with one-phase excitation at a rotation angle of 5.5 deg. Fig. 4 shows themagnetic flux density distribution of the yoke with two-phase excitation at a rotation
angle of 5.0 deg. A maximum torque of 0.26 Nm is obtained at 5.5 deg for one-phase
excitation, and 0.27 Nm is obtained at 5.0 deg for two-phase excitation. The magnetic flux
density is concentrated at the inductor regardless of the excitation.

0
-0.1

-0.2

-0.2

-0.3

-0.3
0

5

10

15

20

25

Rotation angle (deg)Fig. 1 Stiffness torque with one-phase excitation

54

0
-0.1

JSOL Corporation

30

0

5

10

15

20

25

30

(Unit: T)(Unit: T)

Rotation angle (deg)Fig. 2 Stiffness torque with two-phase excitation

Fig. 3 Magnetic flux density distribution (at 5.5 deg)for one-phase excitation

Fig. 4 Magnetic flux density distribution (at 5.0 deg)
for two-phase excitationJMAG Application Catalog

Analysis of the Eﬀect of PWM on the Iron Loss ofan IPM Motor

Case

90

Coil

module: DP, LS

Overview
Current vector controls are generally used in interior permanent magnet synchronous motors (hereinafter referred to as IPMs), and among
them PWM inverters are widely utilized to create a command current. It is vital to get a good understanding of iron losses in order to raise theefficiency of an IPM motor. However, iron losses increase when power is converted by the PWM inverter because the carrier harmonic createdby the PWM becomes superimposed on the current and the magnetic flux density waveform in the IPM motor's core.
There are three methods to gain an understanding of iron loss that accounts for PWM carrier harmonics: There is the method of linking acontrol / circuit simulator that contains the PWM inverter with a magnetic field analysis by inputting the current waveform obtained from thesimulation into the analysis, the method of inputting the actual measurements of a current into a magnetic field analysis, and the method of
inputting the estimated harmonic current.
This Application Note demonstrates a case study where the effects of a carrier harmonic against an IPM motorʼ s iron loss is confirmed using
estimated harmonic current waveform as input.

Rotor core

MagnetStator core

Iron Loss

Joule Loss/ Joule Loss Density Distribution

A comparison of the hysteresis losses, and joule losses when the IPM motor is driven by a
PWM inverter and a sinusoidal current are shown in fig 1 and Fig. 2.
The iron losses in both the rotor core and the stator core increase when the IPM motor is
run using a PWM inverter. The joule losses increase the most severely, increasing 70% in
the rotor core, and 20% in the stator core.

The joule loss frequency components are shown in Fig. 3 and Fig. 4, and the joule loss
density distribution is shown in Fig. 5 and Fig. 6.
The fundamental frequencies (360 Hz in the rotor core and 60 Hz in the stator core) andharmonic components have almost the same amounts of iron loss in the rotor core andstator core whether a sinusoidal current drive or a PWM inverter drive is used. However,iron loss is only produced during PWM inverter drive for the PWMʻs fundamental
frequency (10 kHz). (These values are indicated inside the green square in the Fig. 3 andFig. 4.)The increased joule losses when the motor is run using a PWM inverter are caused by the
carrier harmonic.
The effects of a carrier harmonic need to be evaluated to analysis the characteristics of an
IPM motor in detail.

1.20

18.00
16.00

1.00

14.00
12.00
Hysteresis loss
Joule loss

0.60

Loss W

Loss W

0.80

10.00

Hysteresis loss
Joule loss

8.00
6.00

0.40

4.00
0.20

2.00
0.00

0.00
Sinusoidal current drive

Snusoidal current drive

PWM inverter drive

Fig. 2 Ratio of loss (Stator core)

0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00

3.00
2.50

Joule loss W

Joule loss W

Fig. 1 Ratio of loss (Rotor core)

0

5

10

15

20

25

2.00
1.50
1.00
0.50
0.00

30

0

Frequency kHz

5

20

25

30
Joule loss density
Hysteresis loss density
Iron loss density（sinusoidal current drive） （sinusoidal current drive） （sinusoidal current drive）

Joule loss density（sinusoidal current drive）

Hysteresis loss density（sinusoidal current drive）

Iron loss density（sinusoidal current drive）

Joule loss density(PWM inverter drive)

Hysteresis loss density(PWM inverter drive)

Iron loss density(PWM inverter drive)

3.00
2.50

Joule loss W

Joule loss W

15

sinusoidal current drive

0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0

5

10

15

20

25

1.50
1.00

（Unit：W/m^3）

0.50
0

5

10

15

20

25

Joule loss density(PWM inverter drive)

30

Hysteresis loss density(PWM inverter drive)（Unit：W/m^3）

Iron loss density(PWM inverter drive)

Frequency kHz

PWM inverter drive

Fig. 3. Joule Loss Frequency Component of Rotor Core

2.00

0.00

30

Frequency kHz

91

10Frequency kHz

sinusoidal current drive

Case

PWM inverter drive

PWM inverter drive

Fig. 4. Joule Loss Frequency Component of Stator Core

Fig. 5. Loss Density Distribution of Rotor Core

Fig. 6. Loss Density Distribution of Stator Core

Iron Loss Analysis of an IPM Motor Including the Eﬀects
of Press Fitting Stress

Frame
Magnetmodule: DP, DS, LS

CoilOverview
One of the demands for IPM motors is higher efficiency over a wide range of rotation speeds in combination with motor drives, as reluctancetorque can be used in addition to magnet torque. Iron loss makes up a particularly large proportion of total loss in the high rotation region,and how to make this smaller is a major design issue. Generally, IPM motor cores have laminated structures, and methods such as press fitting
or shrink fitting are used to maintain them.For motors using magnetic steel sheet for their cores, the stress generated by press fitting canincrease iron loss, so it is important to take this stress into account when evaluating iron loss. Iron loss is generated when there aremagnetic-field variations in steel sheet. Also, the amount of iron loss depends on the steel sheet's iron loss properties. These iron loss
properties of steel sheet become worse when it is subjected to stresses such as press fitting. The stress caused by press fitting has its own
distribution, and is particularly large in the back yoke. So, in order to evaluate the iron loss with good accuracy, it is necessary to correctly
obtain the stress distribution for the magnetic flux, time variation, and steel sheet.
This Application Note presents modeling the press fitting of a core and frame with the Press Fit condition and then obtaining the iron loss
density of an IPM motor with and without accounting for the stress generated at that time.

Rotor core
Stator core

Minimum Principal Stress Distribution
The minimum principal stress distribution is indicated in fig.1. Tensile stress occurs in theframe and teeth, but compressive stress that especially affects the iron losses is produced
largely in the yoke. The compressive stress is especially large in the area of the model
within the square in fig.1, but this is because of the slots' geometry.

Low

Iron Loss Density Distribution
Fig.2 shows the iron loss density distribution. The iron loss density distribution varies
depending on whether or not the stress is accounted for. The iron losses are increased by
10 % due to the stress. An analysis that accounts for the compressive stress caused by thepress-fitting needs to be performed to evaluate the iron losses more accurately because
the iron losses increase when the compressive stress is applied.

(Units: MPa)

Fig. 1Minimum principal stress distribution

High

Iron loss: 3.7 W
without stress

Iron loss: 4.1 W
with stress(Unit: W/m^3)

Fig. 2 Iron loss density distribution

https://www.jmag-international.com/

55

Case

Analysis of Detent Torque of a PM Stepper Motor

94

Magnetmodule: TR

Inductor

Yoke

Overview

Coil

PM stepper motors are commonly used for positioning of moving parts in small devices such as printers and video equipment. In order for its
drive to function with an open loop, the most important characteristics for a stepper motor are controllability and holding torque, and not the
motor's output. Therefore, the desired characteristics are detent torque, which is a non-excitation holding torque, and stiffness torque, which
is an excitation holding torque.
A PM stepper motor is made up of a multi-pole magnetized rotor and offset inductors for each phase. In order to reduce their size andnumber of parts, claw pole inductors are made from folded steel sheet. Because of this, the flow of magnetic flux is three dimensional, so it isnecessary to carry out a 3D electromagnetic field analysis using the finite element method (FEM) to proceed with an accurate preliminary
study.
This Application Note describes how the detent torque can be calculated for a PM stepper motor.

Detent Torque/ Magnetic Flux Density Distribution
The detent torque is shown in fig. 1, and the magnetic flux density distribution in the yoke
at a rotation of 7.5 deg is shown in fig. 2.
From the geometry, the detent torque seems to have a period of 7.5 deg, but due to thethree dimensional flow of magnetic flux between the upper inductor and the lower
inductor, the period is 15 deg.

0.04
0.03

Torque (Nm)

0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
0

3

6

9

12

15

(Unit:T)

Rotation angle (deg)Fig.1. Detent TorqueFig. 2 Magnetic flux density distribution in the yoke
(at rotation of 7.5 deg)

Case

95

Analysis of a Universal Motor

Rotor coremodule: DP

Stator core

Overview
Motors in electrical industrial equipment are often used for cutting and machining, so it is desirable for them to be usable at high rotation
speeds. Induction motors cannot achieve high rotation speeds with commercial power supplies because their rotation speed is determinedby alternating current frequency. DC brush motors can achieve high rotation speeds, but it is necessary to provide DC power for them, so theycannot be used with commercial power supplies. Universal motors, on the other hand, are designed to be connected to commercial power
supplies, and because they can rotate at high rotation speeds with either direct or alternating current, they can fulfill purposes needing
thousands of r/min, such as electric drills. Because of their high degree of versatility, they are widely used in everyday-life applications. Usersof industrial tools often hold them in their hands, making reduced weight desirable in addition to low vibration and noise levels.
Development to solve these issues is being carried out in universal motor design.
The basic structure is the same as in a brush motor, but when the supplied power is alternating current, the output itself varies periodically.
Further, they are used under difficult conditions such as when inrush current occurs when a commercial power supply is directly connectedfrom a stopped state. Therefore, electromagnetic field analysis using the finite element method (FEM), which can account for nonlinear
magnetization properties of the materials, is useful in order to evaluate magnetic saturation during overload.
This note presents how the characteristics of a universal motor can be obtained, including torque versus current (T-I), torque versus speed(T-N), and magnetic flux density distribution.

Speed-Torque and Torque-Current Curves

Fig. 1 shows the magnetic flux density distribution at rotation speeds of 4000 r/min and8000 r/min.
More current flows in a universal motor during low rotation speeds than during highrotation speeds, so magnetic flux and torque are both higher. However, a magnetic circuitand motor design that does not allow magnetic saturation to occur at low rotation
speeds is desirable because performance may be worse for a motor in which magnetic
saturation occurs easily.

The speed-torque curve is shown in fig. 2, and the torque-current curve is shown in fig. 3.
As can be seen from the two graphs, in a universal motor, the torque gets smaller as
rotation speed rises, and the torque gets larger as current increases.

4000 r/min

8000 r/min(Unit: T)

24000

3.0

20000

2.5

16000

2.0

Current (A)Rotational Velocity (r/min)

Magnetic Flux Density Distribution

12000
8000

0.0

0
20

40

60

Torque (mNm)

56

JSOL Corporation

1.0
0.5

4000

0

Fig. 1. Magnetic flux density distribution at a rotation angle of 1060 deg

1.5

Fig. 2. Speed-Torque Curve

80

100

0

20

40

60

Torque (mNm)

Fig. 3. Torque-Current Curve

80

100

JMAG Application Catalog

Case

97

Core

Sound Pressure Analysis of a Transformer
Bobbinmodule: DS, FQ

Overview

Secondary CoilIn recent years, the demand to reduce vibration and noise is growing while the requirements for higher efficiency and smaller and lightertransformers grow with environmental conservation trends. The primary cause of noise for transformers is the electromagnetic vibrations andthe resonance phenomena at the eigenfrequency of the structure. A sound pressure analysis can be performed with a coupled magnetic fieldand structural analysis that uses the electromagnetic force as excitation force.
This example presents the use of a coupled magnetic field and structural analysis to obtain the sound pressure distribution accounting for theelectromagnetic force of the core when the transformer is operating on a power supply frequency of 6 kHz.

Primary CoilElectromagnetic Force Distribution

Acceleration Distribution

Fig. 1 shows the electromagnetic force distribution of the core while driven.
From the figure, it can be seen that a large force is generated in the Z-axis direction in thegap. Also, because this electromagnetic force works as an attraction force, the frequency
is 12 kHz, double the power supply frequency.

The acceleration distribution at 12 kHz is shown in fig. 3.
The deformation modes are larger at 47 kHz and 66 kHz, as indicated in this figure. This isbecause the electromagnetic force generated near the core's gap and the deformation
modes at 47 kHz and 66 kHz are in the same Z-axis direction. The figure also shows thatthe deformation in the core's sides increases with the attraction force in the gap.
However, there are no resonance phenomena because the frequency of theelectromagnetic force is sufficiently smaller than the Eigen frequency.

Eigenmode
The Eigenmode of the transformer are indicated in fig. 2.
This shows that there is not much deformation in the Z-axis direction in the low-order
Eigenmode, relatively close to the electromagnetic force's frequency of 12 kHz. Modes
where the area around the gap deforms in the Z-axis direction include 47 kHz and 66 kHz.

27 kHz28 kHz47 kHz66 kHzSound Pressure Distribution
The sound level distribution at 12 kHz is shown in fig. 4.
The sound pressure is higher in the Z-axis direction, in which the electromagnetic force
works, and in the X-axis direction, in which the core's deformation is larger. The reason
why the sound pressure levels are not too high is that the electromagnetic frequency of
12 kHz and the Eigen frequencies of the deformation modes that greatly influence theelectromagnetic vibration are far apart.

(Unit: dB)(Unit: m/s^2)(Unit: N)

Fig.1. Electromagnetic force distribution

Case

98

Fig.2. Eigenmode (red outline: original model)

Fig.3. Acceleration distribution at 12 kHz

Fig.4. Sound pressure level distribution at 12 kHz

Response Analysis of a Solenoid Valve Usinga Control Simulator and the JMAG-RT

Stator coreYokemodule: DP, RT

CoilOverview
A solenoid valve opens and closes valves using the force of electromagnets. A valve that controlｓ the amount of fluid flowing has various
states of open and closed. For this reason, this analysis evaluates the response, such as the speed and position control, of a valve tocommands. An analysis that accounts for both the magnetic circuit characteristics of solenoid valve and control characteristics are necessary
to measure the response time of the solenoid valve or position controllability because both two characteristics have influences.
JMAG enables you to create a detailed and actual-machine faithful solenoid valve model that accounts for attraction force contained in asolenoid valve, inductance harmonic, or current dependency. A link simulation that accounts for both the solenoid valve magnetic saturation,or spatial harmonic and the control characteristics of the motor drive can be run by importing this solenoid valve model "JMAG-RT model" ina circuit/control simulator.
This note presents how JMAG-RT can be used to calculate attraction forces and coil inductance that varies with harmonic and current. The
result is the JMAG-RT model used as a reference for a circuit / control simulator that runs a transient analysis to run an analysis that controls
the solenoid valve position within the desired Value.

Movable core

The position of the movable core and the excitation current
versus time
Desired level

1.0

Actual level

4.0

0.9

3.5

0.8

3.0

0.7
Current (A)Position (mm)The position of the movable core versus time is indicated in Fig.1. , and the excitation
current versus time is indicated in Fig.2. The position of the movable core is moved from 0
mm to 1.5 mm, and then moved from 1.5 mm to 3.0 mm, as indicated in Fig.1. Even
though the movable core moves 1.5 meters in both instances, the method to move thecore differs due to the electromagnetic force, current dependency of the inductance,
position, and the spring constant. A surge occurs the moment the movable core begins
moving, as indicated in Fig.2. This surge should be investigated further if it is fairly large,
because electrical equipment near the surge could be damaged.

2.5
2.0
1.5
1.0

0.6
0.5
0.4
0.3
0.2

0.5

0.1

0.0

0.0
0

50

100

150

Time (msec)

Fig. 1. Position of the movable core versus time

200

0

50

100

150

200

Time (msec)

Fig. 2. Excitation current versus timehttps://www.jmag-international.com/

57

Case

99

Superimposed Direct Current Characteristics Analysis
of a High Current Reactor
module: TR

Overview
High-frequency, high-current reactors have a current composed of direct current with high-frequency ripple superimposed on it. In terms ofreactor performance, stable inductance over a wide range of direct current is desirable. Also, the gap installed to prevent magnetic saturationin the core has a large effect on the inductance, and thus is a vital parameter in reactor design. Finite element method (FEM) analysis is usefulfor accurately estimating inductance over a wide current range, while also accounting for the effects of the gap's inductance and nonlinearity
in magnetization properties, and applying this during design. This Application Note presents how to obtain a high-current reactor's
superimposed direct current characteristics when the gap length is varied.

Superimposed Direct Current Characteristics
the relationship between the length of the flux path, the gap length, and the inductancewhen the direct current is 25 A is indicated in table 1, the superimposed direct current
characteristics are indicated in fig. 1, and the magnetic flux density distribution isindicated in fig. 2. From table 1, it is apparent that the inductance decreases as the gap
length becomes larger. This is because the magnetic resistance of the entire magnetic
circuit is made up of the magnetic resistances in the core and the gap connected in series,
and the magnetic resistance of the gap increases with gap length. From fig. 1, it can beseen that the direct current at which the inductance starts decreasing is smaller with ashorter gap. This is because the core becomes magnetically saturated more easily as thegap becomes shorter, as shown in fig. 2.

Gap length 2mm

40

Gap length 3mm

Gap length 4mm

Flux path length (mm)440

Gap length (mm)Inductance (μH)

2

36.06

3

26.96

4

22.07

Inductance (μH)

35
30
25
20
15
10
5
(Unit: T)
Superimposed direct
current: 50 A

Table 1 Values at current of 25 A

58

JSOL Corporation

Superimposed direct Superimposed direct
current: 75A
current: 100 A

Fig. 2. Magnetic flux density distribution(top: gap length 2 mm ; bottom: gap length 4 mm)

0
0

25

50

75

100

125

150

175

Superimposed direct current (A)Fig. 1 Superimposed direct current characteristics

Core
BobbinCoil
Gap

JMAG Application Catalog

Case

100 Surface Heating Analysis of a Steel Plate

Coilmodule: FQ, HT

Overview
High-frequency induction heating is one heating method used when heat treating the surface of a steel plate. With induction heating, theheating depth can be adjusted because it is possible to localize heating by modifying the coil's geometry and electrical power. The coilgeometry, heating conditions, etc. must be designed correctly in order to achieve heating as desired, but the cost and time needed for
prototyping can be a problem. To make accurate predictions, it is necessary to account for the temperature dependency of the thermal
conductivity, the electrical conductivity, and the detailed coil geometry in order to find the heat generation distribution. Electromagnetic field
simulation using the finite element method (FEM) is needed for this type of prediction.
This Application Note presents how to confirm the uniformity of the surface temperature distribution in the steel plate facing the coil andcheck for an eddy current loss density distribution that causes the temperature distribution to be uneven, when a mosquito-coil-shaped coilis in position.

Work piece (steel plate)

Eddy Current Loss Density Distribution

Temperature Distribution

The eddy current loss density distribution generated in the steel plate and the eddy
current loss density distribution in a cross-section of the steel plate are shown in Fig .1
and Fig .2. The eddy currents generated by the magnetic field variations created by thecoil are distributed to the surface of the steel plate by the skin effect.

The temperature distribution of the steel plate at 0.1 s, 1.0 s, and 2.0 s is indicated in fig. 3.
The temperature versus time is indicated in fig. 4, and the temperature distribution in the
cross-section of the steel plate is indicated in fig. 5. The temperature of the surface of thesteel plate rises over time, reaching about 100 deg C in the steel plate surface below thecoil at 2.0 s. Heating occurs first in the surface, with the heat then transferred gradually tothe inside of the steel plate, because the eddy currents are concentrated on the steel
plate surface.
The temperature distribution of the steel plate is uneven because of the coil's geometry.
Improvements need to be made to the coil geometry in order to achieve uniform
temperatures in the steel plate.

(Unit: W/m^3)

Fig. 1 Eddy current loss density distribution
in the steel plate between 1.9 and 2.0 s

0.1s

1.0 s

(Unit: Celsius)

Steel plate cross-section
Steel plate cross-section enlarged

100

Steel plate cross-section

80

0.1 sec

2.0 s

Fig. 3 Temperature distribution of the steel plate

Steel plate cross-section enlarged(Unit: Celsius)

60
40
Evaluation point: surface

20
0(Unit: W/m^3)

Fig. 2 Eddy current loss density distribution
in the steel plate cross section between 1.9 and 2.0 s

Temperature (deg C)

120
0.2mm

0.0

0.5

1.0

1.5

2.0

Steel plate cross-section

Steel plate cross-section

1.0 sec

2.0 sec

Steel plate cross-section enlarged

Steel plate cross-section enlarged

Time (s)Fig. 4 Temperature variation of the steel plate surface

Fig. 5 Temperature distribution in the steel plate crosssectionhttps://www.jmag-international.com/

59

Case

Core

101 AL-Value Current Characteristics Analysis of a Choke Coilmodule: ST, TRANSFORMER_TEMPLATE

Overview

Coil

A choke coil is an electric component that is intended to filter high-frequency current. The AL-value is a vital parameter in a choke coil designthat determines the cutoff frequency of a high-frequency current.
Because the AL-value is often set as a design specification and AL-value is a nonlinear magnetic property of the core, it varies widely
according to the making current. Finite element analysis (FEA) enables accurate reflection of magnetic properties, and so can obtain AL-value
current properties and provide feedback for design.
This note presents the use of a magnetic field analysis to obtain the AL-value current properties of a choke coil.

BobbinMagnetic Flux Density, Flux Linkage, and AL-value
The magnetic flux density distributions of the core for currents of 0.5 A and 0.7 A are
indicated in fig. 1, the current characteristics of the flux linkage are indicated in fig. 2, andthe current characteristics for the AL-value are indicated in fig. 3. If the current flowing in
the coil increases, the flux linkage increases only minimally because of magnetic
saturation in the core. Furthermore, the AL-value is smaller due to magnetic saturation inareas where the current is large because the AL-value is the inductance per turn in the
coil.

700

0.004

AL-value (nH/N^2)600
Magnetic ﬂux (Wb)

0.003

0.002

500
400
300
200

0.001

100
0

(Unit:T)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Direct current (A)Direct current (A)Fig. 1. Magnetic flux density distribution(left: current 0.5 A; right: current 0.7 A)

Fig. 2. Direct current versus flux linkage

Fig. 3. Direct current versus AL-value

Case

102

Magnetization
direction

Magnetic Field Analysis of a Magnetic Sensor

Magnetmodule: ST

Overview
With recent improvements in the functionality of electric devices and home appliances, magnetic sensors are being used more for contactlesssensing of whether device doors, etc. are open or closed. An open/closed switch using a magnetic sensor switches between open and closed
by sensing distance according to the size of the magnets' magnetic field. At the design stage, it is necessary to evaluate magnet type andposition, sensor sensitivity, and other issues.
Magnetic field analysis simulation using the finite element (FEM) method is effective for accounting for differences in magnetic field strength
due to three-dimensional positioning and interference from other magnetic parts.
This Application Note presents how to obtain the magnetic flux density distribution at points in the horizontal and vertical directions awayfrom the magnet.

Sensing
element

Magnetic Flux Density Distribution
The magnetic flux density distribution of the Z component produced around the magnet
is indicated in fig. 1. The horizontal distance versus Z-component magnetic flux is
indicated in fig. 2. The rotation angle versus θ component magnetic flux is indicated in
fig. 3. Positive values are indicated in the positive direction of the Z-axis, and negativevalues are indicated in the negative direction of the Z-axis. The magnetic flux density
value at a spatial coordinate is analyzed ignoring the volume of the sensing element.
As indicated in fig. 1, the magnetization direction of the magnet has negative values on
the Z-axis in the vertical direction, and works in the opposite direction in the horizontal
direction. For this reason, if the sensing element is moved in a horizontal direction, themagnetic flux density that has a negative value in its initial position weakens toward a
positive value when the sensing element is moved away from the magnet, as indicated in
fig. 2. The direction of the magnetic flux density does not change within a rotation angle
range of 15 deg, and the θ-component magnetic flux density decreases when thesensing element is moved away from the magnet, as indicated in fig. 3.
The magnetic flux density distribution can be evaluated in detail, and the detection rangeof the magnetic flux density can be estimated from these results.

Z-axis positive

Z-axis negative(Unit: T)

Fig. 1. Magnetic flux density distribution of the Z component

MagnetMagnetSensing element
15(deg)

20 mm

Sensing element

Rotation angle

Horizontal Distance X
0 mm

0

-0.05
Z=1.0mm
Z=4.0mm-0.1

Z=2.0mm
Z=5.0mm

Z=3.0mm0.20

15 deg

0.15
0.10
0.05
0.00
-0.05

-0.15
0

5

10

15

20

Horizontal distance (mm)Fig. 2. Horizontal distance versus Z-component
magnetic flux density

JSOL Corporation

0.25
Magnetic ﬂux density (T)Magnetic ﬂux density (T)

0.05

60

0 deg

Length: 100 mm

Vertical distance
Z=1.0 to 5.0 mm

0

2

4

6
8
10
Rotation angle (deg)1214Fig. 3. θ-component magnetic flux density versus
rotation angle (rotation radius: 100 mm)

JMAG Application Catalog

Eﬃciency Analysis of a Permanent Magnet
Synchronous Motor

Case

103

Shaft

module: DP, LS

MagnetOverview
PM motors can achieve high efficiency over a wide range of drive regions compared to inductance and brush motors because they use themagnetic field from their magnets. Thinking of the motor as an energy converter, "efficiency" here means the degree of efficiency with whichelectric energy is converted to kinetic energy. The main indicators are the power factor, which can show whether electric power is being usedeffectively, and efficiency, which shows the ratio of input power to output power. Because a motor's characteristics vary widely according to
its geometry and control, it is necessary to do an evaluation considering a number of parameters in order to improve efficiency.
It is difficult to accurately evaluate a motor's characteristics in advance using the magnetic circuit method or other methods, therefore it isnecessary to run an electromagnetic field analysis using the finite element method (FEM).
In this example, how to obtain the efficiency with a sinusoidal wave voltage drive at a rotation speed of 1,800 r/min and voltage amplitude of
40 V.

Stator core

Voltage Waveform/Current Waveform

Active Power Waveform/Power Vector Plot

The voltage waveform and current waveform for the U-phase are indicated in Fig. 1.
The Fig. 1 shows that there is displacement between the phase voltage and phasecurrent. This phenomenon occurs due to inductance. In addition, the apparent power is78.9 VA when calculated from each effective value and converted to the three phases.

The U-phase active power waveform is shown in Fig. 2, and the power vector plot isshown in Fig. 3.
When the U-phase average active power is calculated according to Fig. 2 and converted
to the three phases, the value is 76.6 W. The power factor and power factor angle arecalculated from the ratio of the active power and apparent power, which is 0.97 and 13.8
deg for this motor. The reactive power can be obtained from the apparent power andpower factor angle and displayed in a vector plot as in Fig. 3.

Current (U-phase)
2.0

30

1.5

20

1.0

10

0.5

0

0.0

-10

-0.5

-20

-1.0

-30
-40
180

Output Characteristics
The torque waveform is shown in Fig. 4 and the output characteristics value is shown in
Table 1.
The average torque 0.30 Nm can be obtained from Fig. 4. The motor output 55.7 W can becalculated from the obtained torque and rotation speed. Note that the copper loss
consumed by the coil is 20.0 W. The efficiency of this motor, 70.0 % is calculated from theoutput, iron loss and copper loss.

A
Current,

Voltage,V

Voltage (U-phase)
40

-1.5
210

240

270

300

330

-2.0
360

Rotation angle, deg

Fig. 1. U-phase Voltage Waveform and the Current Waveform

50

0.34

40

Apparent power 78.9 VA
20Reactive power 18.9 VAR
Power factor angle θ=13.8 deg

10

Average Torque, Nm

0.30

Output, W

55.7

0.3

Iron loss, W

3.9

Copper Loss, W

20.0

Eﬃciency, %

70.0

0.28

Active power 76.6 W
0.26

0
-10
180

Torque, Nm

Active power,W

0.32
30

210

240

270
300
Rotation angle,deg

Fig. 2. U-phase Active Power Waveform

330

0.24
180

360

210

240

270

300

330

360

Rotation angle,deg

Fig. 3. Power Vector Plot

Table 1. Output Characteristics Value

Fig. 4. Torque Waveform

Case

104 Thrust Force Analysis of a Linear Induction Motor

Mover yoke
Direction of motion

module: DP

Overview

Aluminum plate

Coil

Linear motors are widely used for carrier devices and machine tools because of their high-speed performance, high acceleration anddeceleration, and accurate positioning. One type of linear motor, the linear induction motor, can be constructed at low cost because it canuse a primary side with coils, and a secondary side made of a conductor that is not magnetized, such as aluminum or copper.
Linear induction motors have unique phenomena called end effects that cause reduced performance at low slip, so it is important to grasp
thrust force characteristics, including end effects, using magnetic field analysis when evaluating these motors.
This Application Note presents how to obtain thrust force characteristics for a linear inductance motor.

Stator yoke

Eddy Current Density Distribution

Thrust Force
The thrust force characteristics are shown in fig. 2. When this linear induction motor starts
from a stationary state and accelerated to a synchronous speed, the thrust force reaches amaximum of 360 N at a translation speed of 1.2375 m/s and then falls thereafter. This isbecause the amount of variation in the magnetic flux in the aluminum sheet is reduced as
the variation in the magnetic field generated by the coil reaches a synchronous speed.

Direction of motion

400
350
300

Translation speed
3.7125 m/s

Thrust force (N)The eddy current density distribution and magnetic flux lines are shown in fig. 1. Thrustforce is produced through the interaction between the eddy currents and magnetic field
when eddy currents occur in the aluminum sheet due to variations in the magnetic fieldcreated by the coils.
More eddy currents remain behind the motor at a translation speed of 3.7125 m/s than at
a translation speed of 1.2375 m/s, as indicated in fig. 1. This is because variations inmagnetic flux to the rear of the motor increase as the translation speed is increased.

A large amount of eddy currents remain

Direction of motion

250
200
150
100
50
0
-50

Translation speed
1.2375 m/s

0(Unit:A/m^2)

1

2

3

4

Translation speed (m/s)

A small amount of eddy currents remain

Fig. 1. Eddy current density distribution and magnetic
flux lines at 0.16 s

Fig. 2. Thrust force

https://www.jmag-international.com/

61

Case

105

Primary winding

Leakage Inductance Analysis of a Transformer

Insulation
Secondary winding
Winding cross-section

module: FQ, TS, TRANSFORMER_TEMPLATE

Overview
Inductance is an important physical quantity that determines a transformer's response characteristics against electric signals. Inductance isgenerally categorized into self-inductance and leakage inductance. Self-inductance is an indicator of to what extent the transformer can
produce magnetic flux, and leakage inductance is an indicator of how much magnetic flux the transformer can send from the primary coil tothe secondary coil without leaking. This is why self-inductance and leakage inductance are important items for transformer design
requirements. The amount of inductance is dependent on the magnetic circuit, but the nonlinear characteristics of the magnetic properties
make it so that the magnetic circuit changes when the operating point changes. The leakage inductance has all of the same properties, but it
also has a flux path in non-magnetic regions, making it easily affected by the arrangement and geometry of the winding in addition to thecore. This is why a magnetic field analysis using the finite element method (FEM) is necessary when evaluating these types of inductance.
This Application Note explains how to obtain self-inductance and leakage inductance for two types of secondary coiling in a transformer:
uniform coiling and close coiling.

WindingBobbinCore

Inductance
The inductance of the transformer when the secondary coil has a uniform coiling andclose coiling is indicated in table 1. The model that has a secondary coil with a closed
coiling has a larger leakage inductance.

Evaluating Leakage Inductance
The leakage inductance is obtained from the flux linkage of the primary coil when thesecondary coil has a short circuit.
The flux linkage of the primary coil is evaluated as the flux leakage because the magnetic
flux produced by the primary coil eliminates the flux linkage of the secondary coil. The
variations of the flux leakage versus time can be evaluated from the voltage terminal of
the primary circuit because they are induced voltage.

62

JSOL Corporation

Uniform coiling

Close coilingSelf-inductance

104 μH105 μH

Leakage inductance

20 μH

25 μH

Table 1. Inductance of the transformer

Uniform coiling

3.7 mV

Close coiling

3.8 mV

Table 2. Voltage terminal connect to the current source

JMAG Application Catalog

Case

106

Stator core

Iron Loss Analysis of a Brush Motor

Magnetmodule: DP, LS

Rotor core

Overview

Coil

Brush motors are used in many devices, particularly smaller-sized ones. With demands for energy-saving in recent years, higher efficiency isdesired not only in high performance and large-scale motors used in HEVs and large appliances, but also in small-scale brush motors. To
respond to these demands, it is important to reduce loss. Loss reduces efficiency directly, and also causes further reductions by increasing adevice's temperature through heat generation, so it is necessary to know the amount and distribution of loss in order to create improved
designs that suppress this loss. Motor loss is dominated by copper loss and iron loss, and copper loss can be more or less known from thecurrent flowing in the coils. Iron loss, however, depends on material properties, drive conditions, and geometry, and is therefore difficult toevaluate through desktop calculation.
Magnetic analysis using the finite element method (FEM) is useful at the design stage because it can consider all electromagnetic behavior
and motor geometry, and therefore makes estimation of the distribution and total amount of iron loss possible.
This note presents how to obtain the iron loss in the stator core and rotor core of a brush motor.

Magnetic Flux Density Distribution
The magnetic flux density distribution is shown in fig. 1, the r-component and theta
component of the magnetic flux density waveform for measuring point 1 are shown infig. 2, and the r-component and theta component of the magnetic flux density waveform
for measuring point 2 are shown in fig. 3.
The magnetic flux density of the rotor core is greater than that of the stator core, as
indicated in fig. 1. Figures 2 and 3 show that the amount of variation in the magnetic flux
density at measuring point 1 is large, while the amount of variation in the magnetic flux
density at measuring point 2 is small. This difference in the amount of variation has aneffect on the iron loss.

R-component

Magnetic ﬂux density (T)Measuring
point 2

Measuring
point 1

(Unit: T)

Fig. 1 Magnetic flux density distribution

R-component

2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
360

420

480

540

Joule Loss Density Distribution/Hysteresis Loss Density
Distribution/Iron Loss Density Distribution
The joule loss density distribution is shown in fig. 4, the hysteresis loss density distribution
is shown in fig. 5, and the iron loss density distribution is shown in fig. 6. The proportionsof the loss in the stator core and the rotor core are shown in fig. 7 and fig. 8.
The joule loss density is greater in areas that have larger variations in magnetic flux
density compared to areas with smaller variations in magnetic flux density, as confirmedby the magnetic flux density waveform. The same trend can be seen with the hysteresis
loss. Fig. 7 and fig. 8 confirms that the rotor core accounts for a greater proportion of thejoule loss and hysteresis loss.

Theta-component

600

660

720

Fig. 2. Magnetic flux density waveform (measuring point 1)(Unit: W/m^3)(Unit: W/m^3)

Rotation angle (deg)Fig. 5. Hysteresis loss density distribution

Fig. 6. Iron loss density distribution

6

Theta-component
0.2

1.6

0.18
0.14

1

0.12

0.8
0.6

0.1
0.08

0.4

0.06

0.2

0.04

0
360

Hysteresis loss
Joule loss

4
Loss W

1.2

5

Hysteresis loss
Joule loss

0.16

Loss W

Magnetic ﬂux density (T)

1.4

3
2
1

0.02
420

480

540

600

660

720

(Unit: W/m^3)

Rotation angle (deg)Fig. 3. Magnetic flux density waveform(measuring point 2)

Fig. 4. Joule loss density distribution

0
Stator coreFig. 7 Ratio of loss(Stator core)0
Rotor coreFig. 8 Ratio of loss(Rotor core)

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63

Case

107

Current Voltage Characteristics Analysis
of Flyback Converters

Core
Bobbinmodule: DP

Overview
DC-DC converters such as flyback converters control current of primary / secondary winding with switch operation, and transmit electrical
power on the load side using diodes and condensers. Depending on its application, there is a need to satisfy current voltage characteristics
but it is affected by magnetic saturation and skin effect. To accurately evaluate current voltage characteristics, it will be important tospecifically account for the impact of skin effects and proximity effects, as well as leakage flux from the core gap. Especially for high switchingfrequencies, it will be necessary to control alternating resistance occurring in the winding due to switch period, and to accurately evaluate
alternating resistance to design converters with low loss.
Magnetic field analysis using the finite element method is effective for accurate analyses of magnetic saturation and alternating resistance. By
coupling external circuits and magnetic field analysis running switching, the current voltage characteristics of the converter can be obtained.
In this example, this case study obtains the current waveform of the primary / secondary winding and the core magnetic flux density
distribution, current density distribution inside the winding, joule loss density distribution at each time when the voltage is applied.

Primary coil

Secondary coil

Current Waveform, Magnetic Flux Density Distribution

Voltage Characteristics

Primary current and secondary current waveform obtained in the time of one cycle in thesteady state is shown in Fig. 1, and the magnetic flux density distribution of each time isshown in Fig. 2. While voltage is applied to the primary side when the switch is on, it canbe confirmed that primary current is increasing. It can be confirmed that when the switch
is off at 12.5 μs, the current of the primary side will be 0 and that current is flowing in thesecondary side. The magnetic energy in the core accumulated by flowing current can beassumed by the magnetic flux density distribution on Fig. 2. Switching to the secondaryside at 13 μs, it can be confirmed that the core magnetic flux density hits the maximum,and it is slowly discharged, reducing the magnetic flux density.

Primary coil terminal voltage and load voltage waveform obtained in the time of one
cycle in the steady state is shown in Fig. 3. It can be confirmed that while the switch is on,
the input voltage of 70 V is applied to the coil. It can be confirmed that the load voltage isa constant value of 21 V and has stepped down following the winding ratio.

Primary current

Joule Loss Density Distribution
The joule loss density when current is applied to the secondary winding is displayed inFig. 4. The Fig. 4 shows that joule loss inside secondary winding is concentrated on theoutside of the winding. This may be caused by the skin effect due to current variations
inside the winding and leakage flux from the core gap.

Secondary current

Primary voltage

3.0

Secondary coil Primary coil

70.0
60.0

2.0

Voltage, V

Current, A

Load voltage

80.0

2.5

1.5

50.0
40.0
30.0

1.0
13 μsec

0.5

18 μsec

23 μsec

20.0(Unit: T)

0.0
0.0

5.0

10.0

15.0
Time,µs

Fig. 1. Current Waveform

20.0

25.0

10.0
0.0

30.0

0.0

Fig. 2. Magnetic Flux Density Distribution

5.0

10.0

15.0
Time,µs

20.0

25.0

(Unit: W/m^3)

30.0

Fig. 3. Voltage Characteristics

Fig. 4. Joule Loss Density Distribution

Case

108 Centrifugal Force Rupture Analysis of a Ring Magnet

module: DS

Overview

Stator core

Shaft
MagnetAs the applications for permanent magnet synchronous motors expand in the manufacturing sector, the development of high-speed-capable
motors is continuing apace. One problem during high-speed operation is centrifugal force produced in the rotor, because, in an SPM using
ring magnets, a magnet can rupture when the stress acting on it surpasses its mechanical strength. Analyzing the maximum rotation speed ofa motor in advance to evaluate methods to prevent the magnet from rupturing, such as designing reinforcing rings, is highly advantageous
during the design stage. When an SPM motor rotates, centrifugal force is generated, producing stress on the magnets. The stress distribution
inside the magnets is not uniform. In addition to evaluating their mechanical strength, it is necessary to thoroughly investigate areas of stressconcentration using the finite element method (FEM).
This Application Note presents how to obtain the stress distribution of a ring magnet in an SPM motor rotating at high speeds.

Reinforcement ringTensile Stress Distribution by Rotation Speed
The tensile stress distributions of the magnet without a reinforcing ring under different
rotation speeds are shown in fig. 1. It can be seen from the figure that, without areinforcing ring, the tensile stress in the magnet becomes higher than its mechanicalstrength between 2,750 and 3,000 r/s, and the magnet will rupture.

Tensile strength limit of magnet

Tensile strength limit of magnet

Reinforcing ringTensile Stress Distribution with and without a Reinforcement
Ring
The tensile stress distribution with and without a reinforcing ring at a rotation speed of
3,000 r/s is shown in fig. 2. It can be seen from the figure that, with a reinforcing ring, themagnet does not exceed its tensile strength limit. This is because the reinforcing ring
suppresses the deformation of the magnet. In this way, the motor can be driven at fasterrotation speeds by using measures to prevent the ring magnet from rupturing.

64

JSOL Corporation

2500 r/s

2750 r/s

3000 r/s(Unit: MPa)

Fig. 1. Tensile stress distribution of the magnet

Without reinforcing ringWith reinforcing ring

(Unit: MPa)

Fig. 2. Tensile stress distribution with and without
the reinforcement ringJMAG Application Catalog

Case

109

Operating Time Analysis of an Electromagnetic
Relay Accounting for Eddy Currents

Movable core
Stator coremodule: TR

CoilOverview
Electromagnetic relays are devices that use an electromagnet to physically connect and disconnect contact points. Magnetic flux is generated
from the magnetomotive force, which is expressed as the product of the number of turns in the coil and the current that is applied to the coil.
This flux produces an attraction force in the movable core, making the relay close.
To put it simply, the attractive force is determined from the area of the gap between the movable core and the stator core and the size of themagnetic flux density produced in said gap. With a relay whose movable core does not move linearly, however, it is a difficult problem topredict the magnetic flux density in the gap because it does not become parallel. The nonlinear magnetic properties of the iron core and yoke
also affect the magnetic flux density in the gap. With a JMAG magnetic field analysis, it is possible to obtain the attraction force of the
movable core while accounting for these factors. One of the reasons that the response is delayed in electromagnetic relays is eddy currents,
which are produced when the magnetic flux generated by current flow undergoes time variations. The eddy currents are generated in a
direction that inhibits changes in the magnetic flux, causing a delay in the initial rise of the attraction force when the current begins to flow.
This reduces the injector's responsiveness. JMAG makes it possible to account for the effects from eddy currents and obtain anelectromagnetic relay's responsiveness by running a transient response analysis.
This Application Note presents the use of the motion equation function to evaluate the operating time of an electromagnetic relay with DC
voltage drive. Eddy currents generated in the core are considered for this purpose.

Yoke

Operating Time

Magnetic Flux Density Distribution

The displacement versus time is indicated in fig. 1, the current versus time is indicated in
fig. 2, and the attraction force versus time is indicated in fig. 3.
The current gradually increases after excitation starts, but it does not increase easily foraround 2 ms because of the reverse electromotive force that comes with an increase inmagnetic flux. Even after the excitation stops at 5 ms, the movable core still contacts thestator core because of the electromagnetic energy stored in the coil flowing through thediode, but when the elastic force of the spring becomes stronger than the attractive force
at around 7 ms, the movable core returns to its initial position.

The magnetic flux density distributions just before excitation stops, just after excitation
stops, and just after the movable core begins to return to its initial position are indicatedin fig. 4.
After excitation stops, the magnetic flux density decreases gradually because current isstill flowing into the coil through the diode due to the effect of the electromagnetic
energy in the coil as indicated in fig. 2.

Time of excitation

70

10

60
Current (mA)

Displacement (deg)Time of excitation

80

Attraction force / Elastic force (mNm)

12

8
6
4

50
40
30
20

2

10
0

0
0

2

4

6

8

10

0

2

6
Time (ms)Time (ms)Fig. 1. Displacement versus time

4

Fig. 2 Current versus time

8

10

Time of excitation

4.0
3.5

Attraction force

3.0

Elastic force of spring

2.5
2.0
1.5
1.0
0.5
0.0
0

2

4

6
Time (ms)Fig. 3 Attraction force versus time

8

10
Just before
excitation stops

Just after
excitation stops

Just after the core
begins to return(Unit: T)

Fig. 4 Magnetic flux density distribution

https://www.jmag-international.com/

65

Case

BobbinLoss Analysis of a Choke Coil

110

Core

module: FQ, LS, TS, TRANSFORMER_TEMPLATE

Overview
A choke coil is an electric component that is intended to filter high-frequency current. Measures to evaluate the heat source as well as thecore iron losses that occur within the choke coil and the copper losses of the coil that decrease efficiency need to be used for this analysis. The
current generated in the choke coil has offsets caused by the skin effect, proximity effect, and leakage flux near the gap, so it is distributed
both inside of and between the wires. Iron loss generated in the core is also distributed by offsets in the core's magnetic flux density. It is
helpful to get tips for the design quantitatively and visually studying these detailed distributions, and an effective way of doing this is amagnetic field analysis that uses the finite element method (FEM).
This Application Note shows how to obtain the iron loss and copper loss in a choke coil.

Coil

Iron Loss Density Distribution

Copper Loss Density Distribution

The iron loss density distribution is shown in fig. 1, the joule loss density distribution in fig.
2, and the hysteresis loss density distribution in fig. 3.
From fig. 1, it is apparent that iron losses increase in the inside corners. This is caused by
the flow of magnet flux concentrating on the shortest path through the magnetic circuit.
The hysteresis losses are the largest contributor to the iron losses, as indicated in figures 2
and 3.

Fig. 4 shows the joule loss density distribution in the coil. The joule losses are larger
around the gap, as indicated in this figure. These joule losses are produced by the leakage
flux from the gap that causes the current distribution in the wires to be uneven.

Percentage of Loss
The percentage of the iron and copper losses is indicated in fig. 5. As the table indicates,
the hysteresis losses dominate the losses of the choke coil. However, when looking at theloss densities it becomes apparent that there is a region near the coil where they are high.
In a heating design it is possible to resolve this issue and lower the heat generation
density in the coil by changing the gap width or the coil arrangement.

(Unit:W/m^3)

Fig. 1. Iron loss density distribution

0.035
0.030

Loss W

0.025
0.020

Copper Loss
Hysteresis loss

0.015

Joule loss

0.010
0.005
(Unit: W/m^3)(Unit:W/m^3)(Unit:W/m^3)

Fig. 2. Joule loss density distribution

Fig. 3. Hysteresis loss density distribution

0.000

Fig. 4. Joule loss density distribution of the coilFig. 5. Ratio of loss

Case

111 Starting Performance Analysis of a Universal Motor

Stator coremodule: DP

Overview
A universal motor is a motor that rotates on both direct and alternating currents. A universal motor is used in home appliances and industrial
machines because these motors are robust, compact, light, and fast with a simple construction, and have a large starting torque. However,problems such as vibration and a reduction in starting torque caused by the cogging torque occur as the size of the motor becomes smaller.
Evaluating the starting performance of a universal motor at the design stage is necessary to resolve these problems.
This document introduces the use of a magnetic field analysis to obtain the speed versus time graph, the current waveform, and the torqueversus time graph for a universal motor.

Starting Performance
The speed versus time graph from when the motor starts to 0.4 seconds is indicated inFig. 1, the current waveform of the stator coil is indicated in Fig. 2, and the torque versus
time graph is indicated in Fig. 3.
The universal motor reaches a rotation speed of 6,000 rpm at approximately 0.35 seconds
as indicated in Fig. 1. Furthermore, the larger the current flowing through the stator coilis, the larger the torque becomes when the motor is started. As the rotation speed
increases, the current and torque decrease.

4000

2000

0.5

8

0.4

6
4
2

0
0.1

0.2
Time (sec)Fig.1. Speed versus time graph

JSOL Corporation

0.3

0.4

0.3
0.2
0.1

0
0

66

10

Torque (N・m)

6000
Current (A)Rotation speed (rpm)

8000

0
0

0.1

0.2

0.3

Time (sec)Fig.2. Current waveform of the stator coil

0.4

0

0.1

0.2
Time (sec)Fig.3.Torque versus time graph

0.3

0.4

Rotor core

JMAG Application Catalog

Case

112

Starting Thrust Force Analysis of a Linear
Induction Motor

Mover yoke

module: FQ

Direction of motion

Overview

Eddy Current Density Distribution/
Frequency of the Aluminum Sheet vs. the Joule Losses

The starting thrust force for power supply frequencies of 1 to 15 Hz is indicated in fig. 3.
As the figure shows, thrust force increases as the power supply frequency gets higher.
This is because eddy currents in the aluminum sheet increase with higher power supply
frequencies as shown in fig. 1. The maximum starting thrust force of this linear induction
motor is approximately 180 N.

200

700
600

160
Thrust force (N)Joule loss (W)

500
400
300
200

Power supply frequency 7 Hz

0

Case

113

80

0
0

Fig.1 Eddy current density distribution

120

40

100

Power supply frequency 15 Hz

Stator yoke

Starting Thrust ForceThe eddy current density distribution for each power supply frequency is indicated in fig.
1, and the aluminum sheet's frequency vs. joule loss is indicated in fig. 2.
As indicated in fig. 1, eddy currents in the aluminum sheet increase as the power supply
frequency becomes higher. This is because time variations in the magnetic flux acting on
the aluminum sheet increase along with power supply frequency. As indicated in fig. 2,
joule loss in the aluminum sheet increases as eddy currents increase.

Power supply frequency 1 Hz

Aluminum plate

Coil

Linear motors are widely used for carrier devices and machine tools because of their high-speed performance, high acceleration anddeceleration, and accurate positioning. One type of linear motor, the linear induction motor, can be constructed at low cost because it canuse a primary side with coils, and a secondary side made of a conductor that is not magnetized, such as aluminum or copper.
There are some problems when building a linear inductance motor, such as complex eddy currents flowing in the secondary conductor sheet,
and a large amount of leakage flux between the mover and stator. In order to improve linear inductance motor efficiency, therefore, it isimportant to gain an understanding of the paths of eddy currents and the leakage flux. Evaluation using the finite element method (FEM) is
useful for this.
This Application Note presents how to obtain the starting thrust force for a linear inductance motor.

2(Unit: A/m^2)

4

6

8

10

12

14

16

0

2

Power supply frequency (Hz)

4

6

8

10

12

14

16

Power supply frequency (Hz)Fig. 2 Aluminum sheet frequency vs. joule loss

Fig. 3 Starting thrust force

Power Transmission Analysis of a WirelessPower Transfer System with Opposing Cores

Secondary coilCoremodule: FQ

Overview
A wireless power transfer system is a device which uses electromagnetic induction to provide electric power without physical contact. They
can be used for various applications, such as supplying power to moving or rotating devices, or devices sealed inside enclosed spaces.
Because the primary and secondary sides do not touch, the power transmission efficiency and leakage flux vary depending on their positions
relative to each other. Therefore, it is important at the design stage to understand how properties change according to their placement.
When evaluating the properties of a transformer whose primary and secondary sides are separated by a gap, it is helpful to use magnetic field
analysis based on the finite element method (FEM), which allows precise modeling of the geometry of the parts and their relative positions,and makes it possible to visualize the leakage in the magnetic flux that is generated in the primary side and transmitted to the secondary side.
In this example, how to obtain the power transmission efficiency when the feeder wire's position is moved in both the horizontal and verticaldirections from a reference position, and how to display the flow of magnetic flux.

Primary coil

Vertical Distance vs. Power Transmission Eﬃciency

Horizontal Distance vs. Power Transmission Eﬃciency

MMagnetic flux density distributions with varied vertical distance between the primary andsecondary sides are shown in Fig. 1, and the vertical distance versus power transmission
efficiency curve is shown in Fig. 2. The leakage flux increases and the magnetic flux inside thesecondary side core falls with greater vertical distance, as indicated in Fig. 1. Approximately 80%
of maximum efficiency can be achieved when the vertical distance is 2 mm, as indicated in Fig. 2.
The efficiency falls basically linearly because the reduction in flux linkage in the secondary coil iscaused simply by the increase of the vertical distance.

Magnetic flux density distributions with a vertical distance of 2 mm and varied horizontal
distance between the primary and secondary sides are shown in Fig. 3, and the horizontal
distance versus power transmission efficiency curve is shown in Fig. 4. As indicated in Fig.
3, magnetic flux in the left side of the secondary side core falls with horizontal distance,
and a flow channel that does not link the secondary coil is formed when the horizontal
distance is 10 mm. The efficiency is reduced nonlinearly, as shown in Fig. 4. This isbecause the reduction in linkage flux in the secondary coil is due not only to distance, butalso to changes in the way the magnetic flux flows.

90
80

80

Horizontal distance 2 mm

70
60
50
Horizontal distance 6 mm

40
30
20
10

Vertical distance 10mm

Horizontal distance 10 mm

The ﬂux linkage is notproduced in thesecondary coil

0
0

2

4

6

8

10

Fig. 2. Vertical distance vs. power transmission efficiency

70
60
50
40
30
20
10
0
0

12

2

4

6

8

10

12

Horizontal
distance,mm

Verticaldistance,mmFig. 1. Magnetic flux lines

Power transmission
eﬃciency,%

Vertical distance 6 mm

Power transmission eﬃciency, %90Vertical distance 2 mm

Fig. 3. Magnetic flux lines

Fig. 4. Horizontal distance vs. power transmission
efficiency

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67

Case

114

Rotor core

Stator core

Vibration Analysis of an Outer Rotor Motor

Shaft

module: DP, DS

Holes

Overview
An outer rotor motor has a magnetic rotor that rotates around a stator. The rotor radius of an outer rotor motor is large, so it can produce alarger amount of drive torque than an inner rotor motor with the same diameter, giving it a superior constant velocity. On the other hand,countermeasures for vibration and noise that occur during rotation are vital as well. The electromotive force is a cause of the vibration that
occurs when a motor rotates. Additionally, when this electromotive force resonates with the motor's eigenmodes, it causes even larger
vibrations and noise. Countermeasures such as changing the motor's eigenfrequency through processes like setting a hole in the rotor core
have been taken with the objective of preventing resonance. In order to carry out these kinds of studies, it is necessary to get a precise,definite grasp of the electromotive force's spatial distribution, frequency analysis, and natural frequency.
This note presents the use of a magnetic field analysis and structural analysis to obtain the sound pressure caused by electromagnetic
vibrations in an outer rotor motor with holes fabricated in the rotor core.

Electromagnetic Force Distribution
Fig. 1 shows the electromagnetic force distribution where the electromagnetic force isgreatest among the frequency measurement component points, and Fig. 2 shows thefrequency components of the electromagnetic force displayed at the points in Fig. 1. The
frequency components of the electromagnetic force are concentrated at 180 Hz and theelectromagnetic force is larger between the rotor core and poles.

Eigenmode

Electromagnetic force（amplitude) (N)The eigenmode for a rotor core with and without fabricated holes are indicated in Fig. 3.
This analysis confirms the minimum frequency for the mode that is deformed in the radial
direction, which was 649 Hz with holes and was 677 Hz without them. A large amount ofvibration is generated by the resonance phenomenon of the eigenfrequency and theelectromagnetic force.

The frequency component
measurement point for theelectromagnetic force(Unit: N)

Fig.1. Electromagnetic force distribution

With holes：649 Hz

Without holes：677 Hz

Fig. 3. Eigenmode deformation in the radial direction

68

JSOL Corporation

Magnets

Acceleration (RMS) Distribution and Sound Pressure Level
Distribution
The acceleration (R-RMS) distribution at 720 Hz is indicated in Fig. 4.4 and the soundpressure level distribution at 720 Hz is indicated in Fig. 4.5. The electromagnetic force
takes effect, and it is possible to confirm a large acceleration and a high sound pressurelevel distribution due to the resonance phenomenon at an eigenfrequency of 720 Hz, afrequency close to the eigenfrequency shown in 4.2 "Eigenmode". Fig. 4.4 and Fig. 4.5
show that opening holes shown below can suppress acceleration in the R direction andreduce noise in this motor.

25
20
15
10
5
0
0

360

720

1080

1440

1800

2160

2520

2880

With holes

Without holes(Unit：m/s^2)Frequency(Hz)Maximum value
with holes: 42 dB

Fig. 2. Frequency components of the electromagnetic force

Fig. 4. Acceleration (intensity) distribution at 720 Hz

Maximum value
without holes: 44 dB(Unit：dB)

Fig. 5. Sound pressure level distribution at 720 Hz

JMAG Application Catalog

Case
Shaft

Eccentricity Analysis of an SPM Motor

115

module: DP

MagnetOverview
Motors have many parts, which must be assembled correctly for a usable product. Even if each part is made within acceptable limits for
manufacturing errors, when various parts with small errors are put together, the errors can have a cumulative effect. In particular, if eccentricity(deviation, deflection) occurs in the cylindrical axis of the rotor and stator, the magnetic flux density distribution and electromagnetic force canbecome unbalanced, causing vibration and noise. Ideally, parts would be manufactured without any errors, but in reality, error reduction
requires precise mechanical manufacturing, which means a huge increase in costs. This is why it is necessary to figure out the tolerance zone oftrade-off between settings and performance for each part at the design stage.
In order to grasp these at the design stage, highly precise evaluation sensitive to parts' manufacturing errors is needed, so electromagnetic field
analysis using the finite element method (FEM) is effective.
This Application Note presents how to evaluate the cogging torque waveform and effects on the electromagnetic force acting on the stator inan SPM motor with and without eccentricity.

Stator core

Torque Waveform/ Magnetic Flux Density Distribution

Electromagnetic force acting on the stator

The torque waveform with and without eccentricity is indicated in fig. 1. The magnetic flux
density distribution with a rotation angle of 180 deg at the largest difference in the positionof the rotor is indicated in fig. 2. Torque variations occur with each one rotation of one
period when eccentricity is applied, as indicated in fig. 1. The magnetic flux density
increases in the direction of the eccentricity in the stator core, as indicated in fig. 2. This is
how variations in the magnetic flux density distribution caused by eccentricity affect thetorque waveform.

The eccentricity direction component of the electromagnetic force acting on the stator isindicated in fig. 3. The perpendicular direction component of the eccentricity is indicatedin fig. 4. Each of the components has an electromagnetic force waveform with one periodof one rotation due to eccentricity. Vibration and noise can be caused by electromagnetic
resonance if the frequency component of the electromagnetic force acting on each part is
close to the eigenmode of the motor.

Without eccentricity

With eccentricity

Electromagnetic force, N

Torque, Nm

0.36

0.32
Eccentricity direction
0.28

0.24
Without eccentricity(Unit: T)
0

60

120

180

240

300

0

40

-20
-40
-60

60

Rotation angle, deg

Fig. 2. Magnetic Flux Density Distribution(Rotation Angle: 180 deg)

20
0
-20

120
180
240
Rotation angle, deg

300

-60

360

0

60

116

120

180

240

300

360

Rotation angle

Fig. 3. Electromagnetic Force Acting on the Stator(Eccentricity Direction Component)

Fig. 4. Electromagnetic Force Acting on the Stator(Perpendicular Direction Component)

Operating Time Analysis of an Injector
by Evaluating the Reduction in Eddy Currents

Case

With eccentricity

-40

-100
0

360

Fig. 1. Torque Waveform

Without eccentricity
60-80With eccentricity

0.20

With eccentricity

20

Electromagnetic force

Without eccentricity
0.40

Core

Coil

module: TR

Overview

Yoke

A solenoid type injector used in engines opens a valve and injects fuel by moving a plunger with magnetic force created by anelectromagnet. Injectors in engines need to respond quickly to applied voltage in order to control the amount of fuel flow and improve fuelefficiency. In solenoid injectors, one of the reasons for slow response is eddy currents, which are produced when the magnetic flux generated
by current flow undergoes time variations. The eddy currents are generated in a direction that inhibits changes in the magnetic flux, causing adelay in the initial rise of the attraction force when the current begins to flow. This reduces the injector's responsiveness. JMAG makes it
possible to account for effects from eddy currents and obtain an injector's responsiveness by running a transient response analysis.
Identifying the places where eddy currents are generated enables a designer to study how responsiveness can be improved. This Application
Note explains how to apply direct current voltage to a solenoid injector and obtain its response characteristics by accounting for effects fromeddy currents. The effectiveness of slots added to reduce eddy currents are evaluated by comparing the analysis results with a model withoutslots added.

Translation direction
Plunger

Joule Loss / Current Density Distribution

Response Characteristics

The joule loss for each part is shown in fig. 1, and current density distributions at 1.3 ms,
when the core's joule loss is at its approximate peak, are shown in fig 2.
Fig. 1 shows that the joule loss is particularly different in the core, and the peak joule loss
is reduced by 20%. Eddy currents are reduced by using a core with narrow slots installed,
as shown in figures 2 and 3.

Fig. 3 shows the response characteristics, and fig. 4 shows the attraction force
characteristics of the plunger versus time.
The displacement of the plunger is in the direction the valve opens, and the valve is fully
open at a position of 0.1 mm. There is a constant fuel pressure of 5 N against the directionof the displacement. The plunger does not move right after the current is applied, butrather begins moving when the attraction force exceeds the fuel pressure, as indicated in
the figures. The response characteristics and attraction force are improved by adding slots
to reduce eddy currents.

Plunger without slots
Yoke without slots
Core without slots

Plunger with slots
Yoke with slots
Core with slots

Excitation begins

1.0
0.5

0.08
0.06
0.04
Without slots

0.02
without slots
0

1

2

3

4

with slots

20
15
10
Without slots
With slots

012Time (msec)

Fig.2. Current Density Distribution

25

5

With slots
0.00
0

5

Time (ms)Fig.1. Joule losses(Unit: A/m^2)

Plunger begins moving

30
Attractive force (N)Position (mm)Joule loss (W)Excitation begins
35

0.10

1.5

0.0

Plunger begins moving

0.12

2.0

Fig.3. Response characteristics

3

4

5

0

1

2

3

4

5

Time (msec)

Fig.4. Attraction force characteristics

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69

Case

117 Iron Loss Analysis of a Transformer

Cor emodule: FQ, LS

Primary winding

Overview
Mid- and large-sized power transformers need to be able to operate over the long term, so they are always required to control running costs
from losses. Iron loss is one of the main losses in a transformer, and it can lead to temperature increases and efficiency decreases because it
consumes electric power as heat in a magnetic material.
Using finite element analysis (FEA) to confirm the distribution of iron loss density makes it possible to study a transformer's local geometry
during design. Further, iron loss is divided into hysteresis loss caused by hysteresis in the core and joule loss caused by eddy currents, andanalysis makes it possible to compare the relative contributions of each of these.
This Application Note presents how to obtain the iron loss and the ratio of hysteresis loss and joule loss within that iron loss for a three-phasetransformer.

Secondary winding

Magnetic Flux Density Distribution/
Iron Loss Density Distribution

Joule Loss Density Distribution/
Hysteresis Loss Density Distribution

The core's magnetic flux density distribution and iron loss density distribution are shownin figures 1 and 2, respectively. The magnetic flux density distribution and iron loss
distribution are similar, as indicated in the figures. This is because the frequency of themagnetic flux density variations is constant, so the size of the iron losses is proportional to
the size of the magnetic flux density variations.

The core's joule loss density distribution and hysteresis loss density distribution are
shown in fig. 3, the ratio of losses is shown in fig. 4, and the losses are shown in table 1. As
can be seen from the figures, the contribution of the hysteresis loss toward the iron loss islarge. From this, it seems that investigating ways to reduce hysteresis loss would be
effective in reducing iron loss.

Joule loss
37(%)
Hysteresis loss
63(%)

Joule loss
density denstribution(Unit:T)

Hysteresis loss
density distribution(Unit:W/m^3)(Unit:W/m^3)

Fig. 2 Iron loss density distribution(winding is hidden)

Fig. 1 Magnetic flux density distribution(winding is hidden)

Fig. 3 Joule loss density distribution and hysteresis
loss density distribution (winding is hidden)

Iron loss (W)324

Joule loss (W)120

Hysteresis loss (W)204

Table 1. Iron loss

Fig. 4 Ratio of loss

Current

Case

118 Thermal Analysis of a Busbar

Busbarmodule: FQ, HT

Overview

Bonding wire

Current is supplied via busbars or wire bonding in power supply lines for power electronics devices such as inverters. Because inverters andsimilar devices operate with PWM carrier frequencies of several kHz, high-frequency current flows in their busbars. Influences from the skineffect cannot be ignored in this kind of high-frequency current, so increased resistance and loss become problems. A design that accounts forheat and temperature distributions at each frequency is vital because excess heat can cause a reduction in efficiency or even damage thedevice. Because a busbar's geometry is complicated, it is difficult to predict in advance where there will be unevenness in the current's flow
while current is running, and whether this will cause heat generation. With magnetic field analysis using the Finite Element Method (FEM), it ispossible to correctly obtain the unevenness in current distribution and the joule loss, and then predict the temperature distribution withthese as causes of heat generation.
This Application Note presents how to obtain the temperature distribution in a busbar or the like with changes in the power supply
frequency.

Thin copper sheet

Current

Current Density Distribution/
Joule Loss Frequency Characteristics

Temperature Distribution/
Average Tempera ture Frequency Characteristics

The current density distribution at each frequency is indicated in fig. 1, and the frequency
characteristics of the joule loss are indicated in fig. 2. The current density in the bonding
wires and the thin copper sheet increase as indicated in fig. 1. Joule loss increases as thefrequency increases, as indicated in fig. 2. This is because the cross-sectional area where
current is flowing decreases and the resistance increases due to the skin effect.

The temperature distribution of the busbar is indicated in fig. 3, and the frequency
characteristics of the average temperature are indicated in fig. 4. The temperature
increases in the bonding wires and thin copper sheet along with the current density at allof the frequencies. Also, the temperature increases drastically at 1 MHz compared to 10
kHz and 100 kHz. Caution needs to be taken because the increase in temperature can not
only cause the busbar to burn out, but can also damage the device as thermal stressincreases.

BusbarThin copper sheet

Wire 1

Wire 2

BusbarWire 3Thin copper sheet

Wire 1

Wire 2

Wire 360

0.5
Temperature (deg C)

55
Joule loss (W)0.4

Wire 3
Wire 2
Wire 1

0.3
0.2

Wire 3
Wire 2
Wire 1

0.1
10 kHz

100 kHz

1 MHz(Unit: A/m^2)

10 kHz

100 kHz0.0
100 (kHz)1 (MHz)

JSOL Corporation

40
35
30
25
10 (kHz)

Frequency

70(Unit: Celsius)

45

20
10 (kHz)

Fig. 1. Current Density Distribution

1 MHz

50

Fig. 2. Joule loss frequency characteristics

100 (kHz)1 (MHz)Frequency (Hz)Fig. 3. Temperature distribution

Fig. 4. Average temperature frequency characteristicsJMAG Application Catalog

Case

119

Torque Characteristic Analysis
of a Three Phase Wound Rotor Induction Motor

Insulation
Rotor coremodule: DP

Stator coil
Rotor coil

Overview
A wound-rotor induction motor is a motor that causes induced current to flow in the secondary coil using a rotating magnetic field in thestator coil, and thus produces turning force through the interaction of the current and the rotating magnetic field. Because induced current
flows through the coil, electric power can be extracted and regenerated via slip rings.
In an induction motor, the current induced by the secondary conductor exerts a large influence on its characteristics. It also causes strong
magnetic saturation in the vicinity of the gap, in particular. For these reasons, analysis based on the finite element method (FEM) canaccurately handle induced current and magnetic saturation, and is useful for understanding detailed characteristics in advance.
This Application Note shows how to obtain the current density distribution and slip-torque curve.

Stator core

Current Density Distribution
1200
1000
Torque (Nm)The current density distribution at a 0.5 slip is indicated in fig. 1.
Induced current is generated in the rotor coil by the rotating magnetic field of the statorcoil. Torque is generated from the magnetic flux and rotating magnetic field created bythis induced current, so the current has a large effect on the torque characteristics.

0.124 sec

600
400
200

Slip-Torque Curve
A graph of the slip-torque curve is shown in fig. 2.
The amount of torque that can be produced by this induction motor increases as the slip
gets larger. It is necessary to investigate measures such as reducing the rotor's external
resistance when attempting to obtain higher torque with a small slip, because the torqueat each slip is determined by the induced current in the secondary side.

800

0.131 sec

(Unit:A/m^2)

0
1.00

0.75

0.50

0.25

0.00

Slip

Fig. 1. Current density distribution at 0.5 slip

Fig. 2. Slip-torque curve

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71

Case

120 Thermal Demagnetization Analysis of an SPM Motormodule: DP

Shaft
MagnetOverview
Exactly how to resolve the problem of rising temperatures is a vital issue when trying to achieve an improvement in a motor's efficiency andoutput. Among the materials used in a motor, the magnet experiences the greatest variations in properties in relation to temperature. In thecase of rare-earth magnets, demagnetization can occur within tens of degrees above 100 deg C. Whether they demagnetize or not depends
on the reverse magnetic field applied and the temperature. They still have some resistance if either the temperature is raised only or if areverse magnetic field is applied only, but the combination of the two causes a great reduction in resistance. A large current flows in the coils
when the motor is overloaded and is producing a lot of torque, which leads to a large reverse magnetic field and heat, increasing thepossibility of demagnetization. Solutions to this problem include heat-resistant magnets and increased motor size, but these lead to trade-off
issues during the design stage because of the larger size and higher cost.
In order to carry out a precise evaluation of demagnetization, it is necessary to get a definite grasp of areas where a reverse magnetic fieldoccurs and the materials' demagnetization properties. With magnetic field analysis simulation using the finite analysis method (FEM), it ispossible to calculate the reverse magnetic field and determine whether magnets and other parts demagnetize due to reverse magnetic field,
taking material demagnetization properties into account.
This Application Note presents how to change the temperature of permanent magnets in an analysis, and then evaluate the effects on thetorque waveform, magnetic flux density distribution, etc.

Stator core

Torque Waveform
TorqueAverage torque0.35
60(deg C)

140(deg C)

60(deg C)

0.30
Torque (Nm)The torque waveform when the temperature of the magnet is varied from 60 deg C to
140 deg C and then back to 60 deg C for one period of electrical angle (180 deg of
mechanical angle) is indicated in fig. 1.
The average torque is reduced due to thermal demagnetization when the magnet is at atemperature of 140 deg C, as indicated in fig. 1. Also, the irreversible demagnetizationthat occurred at 140 deg C is apparent in that the average torque is 17% lower when thetemperature of the SPM motor is returned to 60 deg C.

17% reduction

Temperature of Magnet:
60 deg Celsius(rotation angle: 180 deg)

0.25

Temperature of Magnet:
140 deg Celsius(rotation angle: 360 deg)

0.20

0.15
0

60

120

180

240

300

360

420

480

Temperature of Magnet:
60 deg Celsius(rotation angle: 540 deg)

540

Rotation angle (deg)Magnetic Flux Density Distribution
The magnetic flux density distributions under the different conditions are shown in fig. 2.
From the figure, it can be seen that the overall magnetic flux density is lower at 140 deg C
compared to before the temperature is raised. It can also be seen that it does not return
to its original state even after the temperature is returned to 60 deg C.

Fig.1. Torque waveform

Fig.2. Magnetic flux density distribution

Measuring point 1

Demagnetizing Ratio Distribution
Fig. 3 shows the demagnetization ratio (*1) distribution after the temperature is returnedto 60 deg C, with magnetization conditions prior to the raised temperature as a standard.
The demagnetization shown here is irreversible demagnetization from going over theknee point, and indicates that the magnet properties have worsened.
It can be seen from the figure that even when the magnet is returned to 60 deg C, there isdemagnetization across a wide range compared to before.
*The demagnetizing ratio is the amount of emagnetizationwith reference tomagnetization that is specified.

Measuring point 2

Temperature of magnet:
60 deg C(rotation angle: 540 deg)

Temperature of magnet:
60 deg C(rotation angle: 180 deg)

(Unit: %)Fig.3. Demagnetizing ratio distribution

Fig. 4. Measuring points for operating point

Operating Point
1.2
Magnetic ﬂux density (T)The measuring points selected to confirm operating points are shown in fig. 4, and theoperating points are shown in fig. 5.
In fig. 5, it can be seen that irreversible demagnetization has occurred because theoperating point exceeds the knee point at measuring point 1 after heat is added and theoperating point does not return to the original B-H curve after the temperature is reduced
to 60 deg C. The size of the reverse magnetic field acting on each element can also beseen on the horizontal axis.

○ : Point 1
× : Point 2

60 deg Celsius (befor)

60 deg Celsius (befor)
140 deg Celsius

0.8

60 deg Celsius
0.6

B-H curve of magnet: 60 deg Celsius: 140 deg Celsius

140 deg Celsius
0.4-400000-300000-200000
Magnetic ﬁeld (A/m)

Fig.5. Operating point

72

JSOL Corporation

60 deg Celsius

1.0-100000

0(Unit:T)JMAG Application Catalog

Output Analysis
of a Salient-Pole Synchronous Generator

Case

121

Stator core
Rotor coilmodule: DP

Overview

Rotor core

Salient-pole synchronous generators are used in hydroelectric generators and the like. Power is generated in the stator coils (armature) by afield current flowing in the rotor coils and the rotor rotating.
Reactions occur between the field current and the armature current that either strengthen or weaken the magnetic flux depending on the
power factor of the connected load of a salient-pole synchronous generator. This causes the operating point of the magnetic circuit inside thegenerator to change, which affects the output. The core normally has nonlinear magnetic properties, so an evaluation of the magnetic circuitwith magnetic field analysis, which can handle nonlinear magnetic properties, is useful.
This Application Note presents the use of a magnetic field analysis to obtain the magnetic flux density distribution, no-load saturation curve,
and output of a salient-pole synchronous generator.

Stator coilNo-load Saturation Curve/ Magnetic Flux Density Distribution

Output

The no-load saturation curve is shown in fig. 1, and the magnetic flux density distributions
for excitation currents of 75 A and 100 A at 0.005 s are shown in fig. 2. The slope of the
no-load saturation curve begins to decrease after exceeding an excitation current of 75 A.
The magnetic flux density increases in the teeth of the rotor core and stator core as theexcitation current gets larger, and the slope of the no-load saturation curve decreases as
magnetic saturation increases. The effective value of the induced voltage is obtainedusing the root mean square.

The time versus current of the shaft resistance at an excitation current of 75 A is indicatedin fig. 3. The output current of this salient-pole synchronous generator is 880 A.

1000

1000
800
800
Current (A)U-phase induced voltage(eﬀective value) (V)

1200

600
400
200

600
400
200

0
0

50

100

150

200

250

300

Excitation current 75 A

Excitation current 100 A

0(Unit: T)0

Excitation current (A)Fig.1. No-load saturation curve

Case

122

0.005

0.01

0.015

0.02

Time (s)Fig. 2. Magnetic flux density distribution at 0.005 s

Fig. 3. Time versus current at excitation current 75 A

Inductance Analysis of an IPM Motor- d/q-axis Inductance Obtained by Actual Measurement module: DP

Overview
Evaluating the inductance characteristics along the d/q-axis is important when analyzing the saliency of a rotor in an IPM motor. With actual
measurements, it is possible to calculate the inductance in the d-axis and q-axis by measuring the no-load magnetic flux or the voltage andcurrent with a three-phase current flowing when the motor is in actual operation. If it cannot be measured while the motor is operating, an
LCR meter can be entered in two phases when the rotor is in a stationary state. However, current conditions are different between threephases and two phases, so the motor will express different characteristics during actual drive, especially when it is strongly affected bymagnetic saturation. For this reason, the analysis contents need to be determined according to the situation of the actual measurements
when comparing the measurements with an analysis.
This Application Note presents an analysis that obtains d/q-axis inductance in an IPM motor while assuming actual measurements in astationary rotor.

MagnetRotor core
Coil
Stator core

Ld/Lq versus current phase angle /Inductance
0.040
Ld

0.035

Lq

0.030
Inductance (H)From fig. 1, it is apparent that the d-axis inductance hardly changes depending on thecurrent phase angle. The reason for this is that the inductance is small because the d-axis
magnetic flux passes through a magnet with low permeability, meaning that it is not very
affected by the magnetic saturation of the iron core because the magnet's magnetic
resistance makes a big contribution. On the other hand, the magnetic flux from the q-axis
passes through the iron core, so the permeability is large compared with the d-axis,
meaning that the inductance is large as well. The current phase also progresses, so whenthe magnetic flux is weaker the iron core's magnetic saturation is alleviated, meaning thatthe q-axis inductance increases.
However, a three-phase current is not run with the method of calculation using theline-to-line inductance, so it is not possible to obtain the variations in d-axis/q-axis
inductance caused by the current phase angle. The magnetic saturation conditions of theiron core are also different when running a three-phase AC and a line-to-line current, so
the q-axis inductance has been generally overestimated. In contrast, the contribution of
the aforementioned magnet's magnetic resistance to the d-axis inductance is large, so
there is not a big gap between the results of the two methods.

0.025

Parameter

Value

0.015

d-axis inductance Ld

0.010 H

0.010

q-axis inductance Lq

0.030 H

0.020

0.005
0.000
0

10

20

30

40

50

60

70

80

90

Current phase angle (deg)Fig. 1 Ld/Lq versus current phase angle(LdLq analysis model)

Table 1 Ld/Lq (line-to-line inductance model)

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73

Case

123

BobbinThermal Analysis of a Choke CoilCore

module: HT, TRANSFORMER_TEMPLATE

Overview
A choke coil is an electric component that is intended to filter high-frequency current. The current generated in the choke coil has offsets
caused by the skin effect, proximity effect, and leakage flux near the gap, so it is distributed both inside of and between the wires. Iron loss
generated in the core is also distributed by offsets in the core's magnetic flux density. Iron loss in the choke coil's core and copper loss in its
coil become a heat source in addition to reducing efficiency, so they need to be understood and reduced from a heating design standpoint.
An analysis using the finite element method (FEM) is effective in getting more information about the design by quantitatively evaluating theheat generation phenomena with copper and iron losses as the heat sources.
This Application Note shows the use of a thermal analysis to obtain the temperature distribution using the iron losses and copper losses in
the choke coil as the heat source.

Coil

Temperature Distribution
Fig. 1 shows its temperature distribution, and fig. 2 shows the maximum temperature ineach part.

26.6
26.4

Temperature degC

26.2
26.0
25.8
25.6
25.4
25.2
25.0
(Unit: deg C)

Fig. 1. Temperature distribution in the core

Core

Coil

Fig. 2. Maximum temperature in each part

Magnet(before magnetization)

Case

126 Magnetization Analysis Accounting for Eddy Currents

module: ST, TR

Coil
Non-magnetic
material

Overview
A magnet's magnetizing state largely affects the characteristics of a device that uses permanent magnets. A magnetization device performs
magnetization by applying an extremely strong magnetic field to a magnet. The capacity of the magnetization power supply is also
determined by multiplying the magnetization current by time. The production process requires that magnetization be performed by runninga large current through a magnetization coil in an extremely short period of time. However, eddy currents are generated in the magnetic
material to be magnetized when the currents time variations are too severe, opening the possibility that the intended magnetization
distribution cannot be obtained. On the other hand, when the current changes too slowly the magnetizing device's capacity needs to be
expanded, leading to an increase in production cost. This is why the trade-off between production costs and whether sufficientmagnetization can be carried out needs to be studied.
A magnetic field analysis using the finite element method (FEM) can handle the nonlinear magnetic properties of materials and eddy currents
that are generated in the magnetic material to be magnetized when current flows through the coil. This makes it possible to discover how themagnetizing field will be generated and what effect it will have on the magnetization distribution.
This Application Note presents how to obtain the magnetization field distribution accounting for eddy currents in the magnetic material to
be magnetized, the eddy current density distribution in the magnetic material to be magnetized, and the surface magnetic flux density of themagnet.

Magnetization Field/ Eddy Current Density Distribution/
Surface Magnetic Flux Density
This section shows the magnetization field in fig. 1, the eddy current density distribution
in the magnetic material to be magnetized in fig. 2 and the magnet's surface magnetic
flux density in fig. 3.

z=18.045 deg0 deg(Unit: A/m^2)(Unit: A/m)Fig. 1. Magnetization field distribution

74

JSOL Corporation

Fig. 2. Eddy current density distributionof the magnetic material to be magnetized

Magnetic ﬂux density (T)

0.5 mm0.26-45 deg

0.24
0.22
0.20
0.18
0.16
-45

-30

-15
0
15
Angle (deg)30

45Fig. 3. Surface magnetic flux density waveform of the magnet(at 0.5 mm above the magnet surface)

Magnetization yoke(laminated steel)JMAG Application Catalog

Case

127 Resistance Heating Analysis of Steelmodule: FQ, HT

Work piece (steel)

Overview

Electrode

Deformation occurs when large metal parts such as shafts undergo machining, causing their material properties to worsen. Because of this,these material properties are restored by eliminating machining deformations using thermal processing, which returns the metal structure toits standard condition. It is necessary to heat the whole part to the same temperature in order to restore the metal's overall properties withthermal heating, and ohmic heating is often used for this purpose. It is helpful to measure the temperature distribution in advance with aninvestigation of heating conditions.
Evaluation with analysis based on the finite element method (FEM) is necessary to find whether a product with a 3D geometry is heateduniformly by a given electrode configuration.
This Application Note presents how to obtain the temperature distribution, temperature variations, and heat flux distribution in a bodyheated by ohmic heating.

Temperature Distribution

Heat Flux Distribution

The temperature distribution of the heated body is shown in fig. 1. From the figure, it isapparent that the central portion of the heated body has the highest temperature at all
times.

The heat flux distribution at 700 s is shown in fig. 3. From the figure, it is clear that heat is
conducted from the heated body to the electrodes, and that this is the cause of thetemperature variation along the lengthwise direction of the heated body.
Some means of reducing this temperature variation in the lengthwise direction isnecessary in order to uniformly heat the body and improve its properties.

Temperature Variations
Temperature variations at each measurement point A, A', B, B' are shown in fig. 2. In the
figure, it can be seen that there is no temperature variation between the surface and theinside (A-A', B-B'). Also, there is a temperature variation between the ends and the central
portion (A-B, A'-B') as seen in fig. 1, and this temperature variation is 200 deg C after 700 sof heating.

AA'
B

after 300 s heating

after 500 s heating

after 700 s heating

Temperature (deg C)

B'(Unit: Celsius)

A

700

A'

B

B'

about 200 deg C

300
400
Time (s)

500

600
500
400
300
200
100
0
0

100

200

600

700

Fig.2. Temperature variations

Fig.1. Temperature distribution(Unit: W/m^2)

Fig.3. Heat flux distribution

Case

128 Structural Analysis of a Cantilever

Fixed end

module: DS

Concentrated load

Overview
As the importance of robust design in today's smaller and thinner electric devices continues to rise, so does the demand for ways of reducing
vibration. In order to precisely evaluate an electric device's vibration characteristics and robustness, it is important to first evaluate these
phenomena in the device's parts individually, because overall characteristics can often be understood by understanding the characteristics ofeach of its component parts. This Application Note presents how to evaluate the vibration characteristics and robustness of a basic cantileverbeam, rather than of an actual electric device.
Electric devices and their individual component parts have eigenmodes that are determined by their mechanical properties and theirgeometry. Vibration increases when the excitation force resonates with the electromagnetic device's eigenmode. Also, electric devices andtheir component parts deform when a load is applied to them. Even in a simple form like a cantilever beam, the displacement direction differs
depending on the direction of the load, its position, and the frequency. For this reason, it is necessary to evaluate the eigenmode anddisplacement for an electric device and its component parts in order to understand these phenomena.
In this Application Note, the displacement is obtained for three simple cantilever beam models with different mesh conditions when theireigenmode or a concentrated load is applied to them, and the displacement is then compared with theoretical values.

BeamFree end

Eigenmode

Displacement

The first-order eigenmodes for Models A (coarse mesh, first-order elements ), B (finemesh, first-order elements ), and C (coarse mesh, second-order elements ) are shown infig. 1 and their first- through third-order natural frequencies are shown in table 1. Theseshow that the deformation direction of the first-order eigenmode can be found even witha coarse mesh, but in order to keep the natural frequency including bending withinaround 5% off from the theoretical value, it is necessary to either use a finer mesh or use
second-order elements.

The Z-direction displacements for Models A, B, and C are shown in fig. 2, and theirmaximum Z-direction displacements are shown in table 2. These show that the
deformation can be found approximately with a coarse mesh, but in order to keep the
maximum displacement including bending within around 5% off from the theoreticalvalue, it is necessary to either use a finer mesh or use second-order elements.

Fixed end

Fixed end

Free end

Model A

Model B

1st-ordermode

2st-ordermode

3st-order
mode
Model C

Fig. 1 First-order eigenmodes (red lines: original model)

Theoretic
Model Aal value

Model B

Model C
Concentrated load

Natural frequency (Hz)130.92

160.7

134.9

129.9

Diﬀerence fromtheoretical value (%)

--

22.7

3.0

0.8

Natural frequency (Hz)163.65

184.5

171.7

168.7

Diﬀerence fromtheoretical value (%)

--

12.7

4.9

3.1

Natural frequency (Hz)818.27

1008

839.7

808.1

Diﬀerence fromtheoretical value (%)

--

23.2

2.6

1.2

Table 1. Natural frequency

Theoretic Model A Model B Model C
al value

Model AFree end

Z-direction
maximum
Zdisplacement (mm)139.23116.38134.28139.14Diﬀerence fromtheoretical value (%)

--

16.4

3.6

0.1

Model
M d lB(Unit: m)Model C

Fig.2. Displacement (red lines: original model)

Table 2. Maximum displacement

https://www.jmag-international.com/

75

Case

129

Coil

Analysis of a PM Stepper Motor Accounting
for Magnetization

Magnet(to be magnetized)

Yokemodule: ST, TR
Magnetization Device

Overview
Stepper motors are commonly used for positioning in printers and digital cameras. In PM stepper motors, the motor's characteristics are
determined through design of the magnetization state of the permanent magnet used. It is necessary to accurately evaluate thismagnetization state in order to improve the precision of estimations of a PM stepper motor's characteristics.
Detailed measurements of the magnetization distribution inside a magnet are difficult to make. However, it is possible to accurately find themagnet's magnetization because it can be obtained through analysis of the magnetization field from the magnetization device, using thefinite element method (FEM).
In this analysis, a magnetization device model is created and a magnet is magnetized. The magnetization distribution and surface magnetic
flux density of the magnetized magnet, and the induced voltage of a PM stepper motor with the magnetized magnet integrated into it, arethen obtained.

Magnetization Distribution and Surface Magnetic Flux Densityof the Magnet
The magnetization distribution of the magnet magnetized by the magnetization device isshown in fig. 1, and fig. 2 shows the radial direction component of a two-pole span of thesurface magnetic flux density at 0.2 mm from the magnet surface.
It can be seen from fig. 1 that magnetization is strong in all parts of the magnet undercomplete magnetization, and magnetization is weak toward the inner side of the magnet
under incomplete magnetization. This is why in fig. 2, the maximum surface magnetic flux
density value under complete magnetization is around 0.8 T, and is smaller at around 0.3
T under incomplete magnetization. This shows that it is necessary to consider themagnetization ratio and recoil relative permeability in order to grasp the magnetization
state.

Yoke

MagnetCoil
Conductor
Stepping motor

Induced Voltage Waveform
The induced voltage waveform of the stepper motor rotating at 1,000 r/min is shown infig. 3. The figure makes it clear that the maximum induced voltage value goes down fromaround 7.5 V under complete magnetization to around 5.0 V under incomplete
magnetization due to the difference in the strength of the magnet's magnetization, and
the PM stepper motor's characteristics are different depending on whether themagnetization ratio and recoil relative permeability are accounted for or not.

30 deg4.35 mm(Unit: A/m)
Incomplete magnetization

Magnetic ﬂux density (T)Surface magnetic ﬂux density measurement area
Complete magnetization

Complete magnetization

Incomplete magnetization

0.8
0.4

Incomplete magnetization

6
4
2
0
-2
-4
-6

0.0

-8

-0.4

0

-0.8
0

Fig. 1. Magnetization distribution in the magnet

Complete magnetization

8
4.35 mm

0.2 mmInduced voltage (V)

0 deg

5

10

15
Angle (deg)

20

25

30

Fig. 2. Radial direction component of the magnet's
surface magnetic flux density

5

10
15
20
Rotation angle (deg)2530

Fig. 3. Induced voltage waveform

Cas e

130

Thermal Conductivity Analysis of Basic Geometry

Part 1(assumed rotor core)module: HT

Overview
With improvements in the miniaturization and increased output power of electric devices in recent years, demands for ways of dealing withheat are also increasing. It is of course important to reduce loss in order to prevent heat, but it is also important to find ways to expel any heatthat is generated. The actual heat conduction and heat transfer phenomena in each part need to be accurately modeled in order to precisely
evaluate an electric device's heat dissipation characteristics. In other words, if the thermal behavior of each part is accurately modeled, it
becomes possible to grasp the thermal characteristics of the entire electric device.
Here, to make thermal analysis easier to understand, two basic cube-shaped objects are modeled as stand-ins for a rotating machine's rotor
core and stator core, and a thermal conductivity analysis is carried out between them.
In this Application Note, the analysis accuracy is investigated by comparing the temperatures obtained through analysis in each part withtheoretical values.

Part 2(assumed stator core)

Temperature Distribution

Inner side temperature of each part

The temperature distribution of part 1 and part 2 is shown in fig. 1. The temperaturechanges linearly from the outer to the inner side of each part.

Analysis results and theoretical calculations for the inner side temperatures of part 1 andpart 2 are shown in table 1. The analysis results are consistent with the theoretical
calculations for both parts. This indicates that these analysis results are accurate.

Part 1

60.7

Temperature fromtheoretical calculation(deg C)
60.7

Part 289.3

89.3

Temperature fromanalysis results (deg C)(Units: deg C)

Fig. 1. Temperature distribution

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JSOL Corporation

Table 1. Inner side temperature of each partJMAG Application Catalog

Case

131

Rotor core

Coil

MagnetStray Capacitance Analysis of a Motor

Stator coremodule: EL

Overview
Motors driven by PWM inverters are widely used in household appliances. It is known that when a PWM inverter is used for drive, common
mode current flows and axial voltage is produced in the motor's bearings. Electric corrosion is caused in the bearings by this axial voltage,
which can reduce the motor's reliability and life. It is advantageous to use analysis to grasp the stray capacitance in advance, because thecommon mode current is produced by stray capacitance acting parasitically on the motor's parts. This note presents the use of analysis toobtain the stray capacitance between an IPM motor's coil and stator core, between its rotor core and stator core, and between the inner andouter rings of its bearings.

Motor
Inner ring

Outer ring

Balls

Bearing

Electric Field Distribution

Stray Capacitance

The electric field distribution when the electric potential difference is applied between
the motor's coil and stator core, the motor's rotor core and stator core, and the inner andouter rings of the bearing are indicated in figures 1, 2, and 3. A large electric field is
produced: in the air in the slots near the coil and stator core when the electric potential
difference is applied between the coil and stator core; in the air in the gap when theelectric potential difference is applied between the rotor core and the stator core; andthrough the balls, which are conductors, when the electric potential difference is applied
between the inner and outer rings.

The stray capacitance between the motor's coil and stator core, between the motor'srotor core and stator core, and between the inner and outer rings of the bearings isindicated in table 1. Axial voltage is produced by the stray capacitance.

Between the motor's
coil and stator coreStraycapacitance(pF)(Unit: V/m)electric potential difference applied between
the motor's rotor core and stator coreFig. 1. Electric field distribution

Fig. 2. Electric field distribution

78

8.2

(Unit: V/m)

(Unit: V/m)electric potential difference applied between
the motor's coil and stator core

Between the motor's
Between therotor core and stator bearing's inner andcoreouter rings

40

Table 1. Stray capacitanceFig. 3. Electric field distribution(electric potential difference applied between
the inner and outer rings of the bearing)Case

132 Loss Analysis of a Three-phase TransformerCore

module: FQ, LS
Primary winding

Overview
Mid- and large-sized power transformers need to be able to operate over the long term, so they are always required to control running costs
from losses. Iron loss is one of the main losses in a transformer, and it can lead to temperature increases and efficiency decreases because it
consumes electric power as heat in a magnetic material. Using finite element analysis (FEA) to confirm the distribution of iron loss density
makes it possible to study a transformer's local geometry during design. Further, evaluating the ratio and distribution of the iron and copper
losses through FEA becomes advantageous when designing a transformer.
This note presents how to obtain the iron and copper losses of a three-phase transformer.

Secondary winding

Iron loss Density Distribution

Percentage of Loss

The iron loss density distribution of the core is shown in fig. 1, the joule loss density
distribution of the core in fig. 2, and the hysteresis loss density distribution of the core in
fig. 3. From fig. 1, it is apparent that iron losses increase in the inside corners. This isbecause the magnetic flux flows through the shortest route of the core. As can be seenfrom figures 2 and 3, a large percentage of the iron loss is hysteresis loss.

The percentages of the iron and copper losses are indicated in table 1. Table 1 shows thatthe ratio of iron loss in the total loss is large. Therefore, investigating ways to reduce ironloss would be effective in reducing the total amount of loss. Furthermore, finding ways toreduce hysteresis loss would be effective in reducing the iron loss.

Current Density Distribution and Copper Loss Density
Distribution
The current density distribution of the coil is shown in fig. 4, and the joule loss density
distribution in fig. 5. From fig. 4, it is apparent that the current density distribution isapproximately the same in each phase. Joule loss is higher in the secondary coil than the
primary coil, as indicated in fig. 5. This is because the current is larger in the secondary
coil.
(Unit:W/m^3)

Fig.3. Hysteresis loss density distribution(Unit:A/m^2)

Fig. 4. Eddy current density distribution

Type of loss
Joule loss
Hysteresis
loss
Copper Loss

Iron loss(Unit:W/m^3)

Fig.1. Iron loss density distribution(Unit:W/m^3)

Fig.2. Joule loss density distribution

Iron losses (W)120

Percentage (%)
32

204

54

52

14

(Unit:W/m^3)

Fig.5. Joule loss density distribution

Table 1. Ratio of loss

https://www.jmag-international.com/

77

Case

133 Thermal Analysis of a Three-phase TransformerCore

module:HT

Overview

Primary coil
Secondary coil

One of the most important principles in transformer design is how low loss can be kept to during operation. Iron loss in the transformer's coreand copper loss in its coil become heat sources in addition to reducing efficiency, so they need to be understood and reduced from thestandpoint of heat resistant design for all parts, particularly the coil's insulation.
An analysis using the finite element method (FEM) is effective in getting more information about thermal design by quantitatively evaluating
the heat generation phenomena with copper and iron losses as heat sources, including the temperature distribution.
This Application Note presents an example of how to confirm a temperature distribution using the iron and copper losses of a three-phasetransformer as heat sources.

Heat Generation/Temperature Distribution
From fig. 1, it is apparent that the heat generation density is higher on the inside corners of
the core. This is because the flow of magnet flux is concentrated in the shortest paththrough the magnetic circuit. The temperature around the bases of the core's center leg ishigher because the heat generation density is higher there, as indicated in fig. 2(a). For thisreason, the temperature of the winding at the center leg is also higher.

(a) Core(Unit: deg C)(b) Winding(Unit: deg C)(Unit: W/m^3)

Fig. 1. Heat generation density distribution of the core

Case

134

Fig. 2. Temperature distribution

Speed Control Analysis of a Permanent Magnet
Linear Motor Using the Control Simulator and the JMAG-RT

Mover coreCoilmodule: DP, RT

Overview

Magnet (4poles)

Linear motors are widely used for carrier devices and machine tools because of their high-speed performance, high acceleration anddeceleration, and accurate positioning. A type of linear motor called a permanent magnet linear motor has harmonic components such as
cogging and ripples in its thrust because of its structure and how it operates. These components have a large influence on velocity control
and positioning accuracy. The thing that separates linear motors from rotating motors is that linear motors have edges, limiting how far they
can move. This is why linear motors have different characteristics when the mover is positioned in the center of the stator and when it is at
the edge. In order to create a more advanced control design, it is possible to use a control simulation to obtain a linear motor model thatexhibits more detailed behavior that is based on an actual machine.
With JMAG, it is possible to create a linear motor model that is detailed and conforms to a real machine, and that accounts for spatial
harmonics and magnetic saturation characteristics included in a permanent magnet linear motor. Importing this linear motor model, a"JMAG-RT model," to a circuit/control simulator makes it possible to run a linked simulation that accounts for the linear motor's magnetic
saturation characteristics and spatial harmonics as well as the driver's control characteristics.
This Application Note presents how to use JMAG-RT to obtain the spatial harmonics and current dependency of the thrust and coilinductance, and import them to a circuit/control simulator as a JMAG-RT model. From there, it shows how to perform an analysis that controls
the speed of the permanent magnet linear motor to the desired value. The system that creates a JMAG-RT model using magnetic field
analysis is referred to as JMAG-RT.

Translation direction

Stator core

Speed/ Mover position
Command

Speed

0.1
Mover position (m)10.8
Speed (m/sec)

The time variations in speed are shown in fig. 1 and the time variations in mover position
are shown in fig. 2 for when there is a constant command value of 0.8 m/s. The fact that
this linear motor is running at its command value of 0.8 m/s can be confirmed in that the
speed oscillates around 0.8 m/s in fig. 1, and that the slope in fig. 2 is about 0.8.

0.6
0.4

0.04

0

0
0

0.02

0.04

0.06
Time (s)Fig. 1. Speed time variations

JSOL Corporation

0.06

0.02

0.2

78

0.08

0.08

0.1

0

0.02

0.04

0.06
Time (s)Fig. 2. Mover position time variations

0.08

0.1

JMAG Application Catalog

Case

Case

138 Vibration Analysis of an SR Motor

Shaft
Rotor coreCoilmodule: DP, DS

Overview

Stator core

There are high hopes for SR motors to provide robustness and low cost, thanks to their relatively simple construction without permanent
magnets. However, the large electromagnetic force produced by the saliency of their stator and rotor causes vibration and noise.
The electromagnetic force working in an SR motor causes vibration and noise as an electromagnetic excitation force. Vibration and noise arecaused when this electromagnetic excitation force resonates with the motor's eigenmode. In order to evaluate this phenomenon withacceptable precision, it is necessary to accurately ascertain the distribution of the electromagnetic force acting on the stator core, which is thesource of radiated sound, and to obtain the eigenmode of the entire motor including its connected case.
In this example, how to obtain the electromagnetic force generated in the stator core of an SR motor and evaluate the sound pressure bylinking it to the motor's eigenmode.

Current Density Distribution and Electromagnetic ForceDistribution
The contour plot of the current density distribution when the SR motor is at 45 deg ofrotation and the vector plot of the electromagnetic force distribution of the stator coreare indicated in Fig. 1. The frequency components of the electromagnetic force produced
at the ends of the stator core's teeth are indicated in Fig. 2.
The electromagnetic force at the stator core teeth tips gets very large from the timing of
the current flow, as seen in Fig. 1. Fig. 2 shows that the 66.67 Hz frequency component isdominant in the electromagnetic force.

Acceleration Distribution and Sound Pressure Distribution
Fig. 4 shows the acceleration distribution at 2,067 Hz, the frequency at which the mode
occurs with elliptical deformation confirmed by eigenmode analysis. Fig. 5 shows thesound pressure level distribution. It can be seen that at 2,067 Hz, electromagnetic
resonance occurs between the electromagnetic force and the eigenmode shown in Fig. 3.

Eigenmode
The eigenmode at around 2,100 Hz is shown in Fig. 3. It can be seen that there is a mode
at which the stator core is deformed to an ellipse at around 2,100 Hz. A great deal ofvibration is likely to be caused by resonance phenomena between this eigenmode andthe electromagnetic force.

Total

Stator core onlyaround 2,100 Hz, red lines: original modelFig. 3. Eigenmode with Deformation in the Radial Direction(Unit: A/m^2)

Current density
contour plot

Electromagnetic force
vector plot

(Unit: N)

Electromagnetic force(amplitude)(N)

Measuring point for electromagnetic
force frequency components

180
160
140
120
100
80
60
40
20
0

(Unit: dB)(Unit: m/s^2)
0

200 400 600 800 1000 1200 1400 1600 1800 2000

Total

Stator core only

Frequency (Hz)Fig. 1. Electromagnetic Force Distribution
at a Rotation of 45 deg

Fig. 2. Electromagnetic Force Frequency Components

Fig. 4. Acceleration Distribution at 2,067 Hz

Fig. 5. Sound Pressure Level Distribution at 2,067 Hz

https://www.jmag-international.com/

79Power Transmission Analysis Using Magnetic
Resonance Phenomena

Case

139

Transmission coilmodule: FQ

Overview
Recently, magnetic resonance is gaining attention as a new type of wireless transmission technology. Magnetic resonance differs fromconventional types of electromagnetic induction transmission that are widely used today in that the axes of the transmission and receiving
coils do not need to be aligned and in that it allows efficient transmission at a distance of several meters. A design for the coil geometry andcircuit that is optimized for the frequency being used is necessary to make the transmitting and receiving sides resonate and thus transmit
power. It is very difficult to use measurement to visualize how magnetic field is being generated, and is therefore transmitting power, in thespace between the transmitting side and the receiving side. Reproducing the power transmission state using analysis can help with designing
optimized coils.
This Application Note presents how to confirm the power transmission efficiency and the magnetic flux density distribution.

Receiving coil

18

19

20

15

17

18

19

Frequency (MHz)250 (mm)

100
80
60
40
20
0
15

16

200 (mm)

16

17

18

19

15

16

17

18

19

Frequency (MHz)Frequency (MHz)275 (mm)

300 (mm)20

Fig. 1 Power transmission efficiency

140

16

17

18

19

100
80
60
40
20
0

20

17.0(MHz)17.5(MHz)
Transmission coil

15

16

17

18

19

18.0(MHz)

Receiving coil
15

16

17

18

19

20

Frequency (MHz)325 (mm)distance between transmission and receiving coils: 200 - 300 mm

Case

15

100
80
60
40
20
0

Frequency (MHz)

100
80
60
40
20
0

20

100
80
60
40
20
0

20

Power transmission
eﬃciency (%)

17

Frequency (MHz)

100
80
60
40
20
0

Power transmission
eﬃciency (%)

16Power transmission
eﬃciency (%)

Power transmission
eﬃciency (%)

15

100
80
60
40
20
0

Power transmission
eﬃciency (%)

100
80
60
40
20
0

Power transmission
eﬃciency (%)

Fig. 3 shows magnetic flux density distributions (amplitude) for 17.0 MHz, 17.5 MHz, and18.0 MHz with distances between transmission and receiving coils of 200 mm, 300 mm,
and 400 mm.
The figure makes it clear that, for each distance between transmission and receiving coils,
magnetic flux density is larger near the receiving coil at a frequency with good powertransmission efficiency. The size of the magnetic field can also be displayed in JMAG.

Power transmission
eﬃciency (%)

Magnetic Flux Density Distribution

Graphs of power transmission efficiency for varying distances between the transmission
and receiving coils are shown in figures 1 and 2. From fig. 1, it is apparent that with
distances between the coils of 200 to 275 mm, there are peaks to each side of theresonance frequency of 17.5 MHz, at which a transmission frequency of nearly 100% isachieved. With a distance between the coils of 300 mm, there is one consolidated peakequal to the resonance frequency. Fig. 2 shows that with distances between the coils of
325 to 400 mm, power transmission efficiency does not reach 100% even at its peak,meaning reduced efficiency. Therefore, it can be said that the optimal distance for powertransmission at a resonance frequency of 17.5 MHz is around 300 mm.

Power transmission
eﬃciency (%)

Power Transmission Eﬃciency

Distance between transmission and receiving coils: 200 mm

350 (mm)20

Distance between transmission and receiving coils: 300 mm
15

16

17

18

19

Frequency (MHz)Frequency (MHz)375 (mm)

400 (mm)20(Unit: T)
Distance between transmission and receiving coils: 400 mm

distance between transmission and receiving coils: 325 - 400 mm

Fig. 2 Power transmission efficiency

Fig. 3 Magnetic flux density distribution (amplitude)

Analysis of Electromagnetic Repulsion Producedin Small Contact Bridges
module: TR

Overview
Electromagnetic repulsion is generated in the contact points of switches used in electrical equipment and current inflow points duringresistance heating when contact terminals bridge through a small point. It is advantageous to investigate the size of the electromagnetic
repulsion when designing a device, or to understand the phenomena that are actually occurring.
This Application Note presents the use of a magnetic field analysis to obtain the electromagnetic repulsion when a switch is closed, creating acontact bridge.

Current Density Distribution, Lorentz Force Density
Distribution, Electromagnetic Repulsion

Electromagnetic repulsion (N)This section shows the current density distribution near the contact terminal in fig. 1, theLorentz force density distribution in fig. 2, and the electromagnetic repulsion in fig. 3.

100
80
60
40
20
0

(Unit: A/m^3)(Unit: A/m^3)

Fig. 1 Current density distribution

80

JSOL Corporation

Fig. 2 Lorentz force density distribution

0

5

10

15

20

25

Supply current (kA)

Fig. 3 Current characteristics of the electromagnetic repulsion

Contact partJMAG Application Catalog

Case

142

Frame

Press Fit Analysis of a Divided Core

MagnetCoilmodule: DS, DP, LS

Overview
Divided core

Smaller size and higher output are being demanded of the motors used for applications such as air conditioning compressors. One
production technique for achieving this is a higher lamination factor in divided cores. The stress caused by press-fitting a divided stator core
into a frame is known to increase iron loss in a motor if magnetic steel sheet is used for the core.
Iron loss is affected by magnetic flux density and stress. Specifically, it increases in areas of high magnetic flux density with high frequency,
and in areas of high stress. Further, the stress caused by press fitting has its own distribution, and is particularly large in the core and backyoke. So, in order to evaluate the iron loss with good accuracy, it is necessary to correctly obtain the magnetic flux density distribution, time
variations, and stress distribution.
This Application Note presents how to use the Press Fit condition to model an analysis of the stress from fitting a core to a frame, and thenobtain the iron loss density of an IPM motor under no load, with and without accounting for the stress.

Rotor core
Stator core

Minimum Principal Stress Distribution

Iron Loss Density Distribution

The minimum principal stress distribution is indicated in fig.1. Tensile stress occurs in theframe and teeth, but compressive stress that especially effects iron loss is produced
largely in the contact faces between the divisions of the core. The compressive stress is
especially large in the area of the model within the square in fig.1, but this is because of
the slots' geometry.

Fig.2 shows the iron loss density distribution, and Fig.3 shows the comparison of iron loss
values. The iron loss density distribution varies depending on whether or not the stress isaccounted for, with iron loss increased by around 10% due to the stress. An analysis that
accounts for the compressive stress caused by the press-fitting needs to be performed toevaluate the iron losses more accurately because the iron losses increase when thecompressive stress is applied.

5
4.5
Low

4
Iron loss W

3.5
3
2.5
2
1.5
1
0.5
High

without stresswith stress(Unit: W/m^3)0
without stress(Units: MPa)

Fig. 1. Minimum principal stress distribution

Fig. 2. Iron loss density distribution

with stressFig. 3. Iron loss

Case

143 Inductance Analysis of an Air Core Coilmodule: ST

Overview
Cross-section

Electric components such as choke coils and chip inductors are used for purposes such as reducing electromagnetic noise by taking
advantage of their own inductance. Air core coils have smaller inductance than those with cores, but because they do not use cores withnonlinear magnetization properties, they maintain linearity in their inductance and can therefore be used in high-frequency filters andoscillators.
In a coil with a core, most of the magnetic flux flows in the core, but because the magnetic circuit in an air core coil is not as clearly structured
as in a core, the inductance is easily influenced by the coil's dimensions and geometry. Because of this, it is necessary to accurately understandthe inductance by analyzing the flow of magnetic flux in the air region.
This analysis shows how to obtain an air core coil's inductance through analysis and compare it with theoretical values.

Coil

Magnetic Flux Density Distribution/ Inductance
The magnetic flux density distribution in the coil and the air region around it is shown infig. 1 and the coil's inductance is shown in table 1. The inductance from the analysis
results is compared to theoretical values.
Fig. 1 shows that the density of the coil's flux linkage is higher in the center of the coil. The
difference between the inductance obtained from the flux linkage and theoretical values
is 0.28%. This confirms that essentially accurate inductance can be obtained using
analysis.

Analysis (H)Theoretical (H)Diﬀerence (%)

6.77e-03

6.79e-03

0.28(Unit: T)

Fig.1. Magnetic flux density distribution

Table 1. Inductancehttps://www.jmag-international.com/

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146

Core

Analysis of Stray Loss in a Transformer
module: FQ, HT, LS

Overview
Transformers are made to be used long-term, so it has become an important design policy to control running costs from losses. These losses
include copper loss in the coil and iron loss in the core. In high-capacity transformers, however, there is also stray loss in the tank from flux
leakage from the core. From a safety standpoint, companies want to contain the heat produced from stray losses in the tank to well below thestandards required for heat resistant design because they anticipate injuries from people touching the tank itself.
Predicting these losses and the heat that they generate is a vital component of transformer design, but it is difficult to estimate them fromdesktop calculations, so evaluations and detailed analyses using the finite element method (FEM) are indispensible.
This Application Note explains how to obtain losses in a transformer tank and use them to evaluate the temperature distribution in each part.

Tank
Secondary winding
Primary winding

Joule Loss Density Distribution

LossesFig. 1 shows the magnetic flux density distribution, fig. 2 shows the current density
distribution, and fig. 3 shows the joule loss density distribution in the tank. From fig. 1, it isapparent that magnetic flux is leaking from the transformer to the tank. This leakage flux
generates eddy currents in the tank, as shown in fig. 2. Fig. 3 shows that joule losses getbigger in places with high current density.

Fig. 5 and Fig. 6 shows the losses in each part. Use these loss values as heat sources to run
the thermal analysis.

Iron Loss Density Distribution
Fig. 4 shows the iron loss density distribution in the core. From fig. 4, it is apparent thatiron losses increase in the inside corners. This is caused by the flow of magnet flux
concentrating on the shortest path through the magnetic circuit.

Temperature Distribution
Fig. 7 shows the core's temperature distribution, fig. 8 shows the winding's temperature
distribution, and fig. 9 shows the tank's temperature distribution. From fig. 7, it isapparent that the temperature in the outside of the core is lower than in the inside of thecore. This is because the core is cooled by insulating oil. In the same way, from fig. 8 it isapparent that the primary winding on the outside has a lower temperature than thesecondary coil on the inside. From fig. 9, it is apparent that the temperature of the tankhas not risen higher than the temperature of the insulating oil.

16.0
14.0
Iron losses W

12.0
10.0
8.0
6.0
4.0
2.0

Fig. 6. Losses (Other than core)

Secondarywinding W

Secondarywinding V

Secondarywinding U

Primary
winding W

Fig. 3 Joule loss density
distribution in the tankPrimary
winding V

Fig. 2 Current density
distribution in the tank

(Unit: W/m^3)

Tank

Fig. 1 Magnetic flux density
distribution in the tank

(Unit: A/m^2)

Primary
winding U

0.0(Unit: T)

full model

half model（Unit: deg C）

Fig. 7. Temperature distribution of the core

350.0

Iron losses W

300.0
250.0
200.0
150.0
100.0
50.0
0.0
Core

(Unit: deg C)（Unit: deg C)(Unit: W/m^3)

Fig. 4 Iron loss density distribution

82

JSOL Corporation

Fig. 5. Losses(Core)

Fig. 8 Temperature distribution in the winding

Fig. 9. Temperature distribution of the tank

JMAG Application Catalog

Case

148

Loss Analysis of a Power Transformer (Flyback Converter)
module: TS, DP, ST, LSCore
BobbinPrimary coil

Overview
A flyback converter is a well-known system for small capacity power supplies in the several-dozen W class. They are cheap and have a simple
structure, so they are widely used as converters for pressurization in home appliances. In recent years there has been a trend toward making
small-scale switching transformers even smaller and higher-frequency, so it is not rare to see converters using the flyback system drive 100
kHz or more. Because of the higher frequencies and smaller scales of transformers, an important challenge of how to control their heatgeneration has emerged in the design process. The losses that produce heat can be separated into copper loss in the coil and iron loss in thecore. Copper loss is distributed inside of the coils because of the proximity effect, which is caused by influence from the skin effect andleakage flux. This means that local heat generation in the coils becomes a problem.
Iron loss also has a complex distribution because it depends on the magnetic flux density distribution that accounts for the core's magnetic
saturation, so the core's local heat generation becomes a problem as well.
A magnetic field analysis simulation based on the finite element method (FEM) can precisely evaluate the complex loss distributions of thecoil and core, so it is optimal for an advance study of a switching transformer's thermal design.

Secondary coil

Iron Loss Density Distribution

Copper Loss Density Distribution

Fig. 1 shows the iron loss density distribution.
From the figure, it is apparent that iron losses increase in the inside corners. This is causedby the flow of magnet flux concentrating on the shortest path through the magnetic
circuit.

Fig. 2 shows the coil's joule loss density distribution when the maximum current isflowing through the primary coil.
The joule losses are larger around the gap, as indicated in this figure. These joule losses
are produced by the leakage flux from the gap that causes the current distribution in thewires to be uneven.

Percentage of Loss
The percentages of the iron and copper losses are indicated in table 1. As the tableindicates, the iron losses are dominant in this transformer.

Type of loss

Loss value (W)Percentage (%)

Iron loss

1.56e-01

76.7

Copper loss

4.73e-02

23.3(Unit: W/m^3)(Unit: W/m^3)

Fig. 1. Iron loss density distribution

Case

149

Table 1. Ratio of loss

Fig. 2. Joule loss density distribution of the coil

Analysis of Magnetic Force Acting on the Arcof a Vacuum Circuit BreakerContact cup

Contact cup(with slits)

module: TR

Overview
Vacuum circuit breakers have the function of breaking the current when a large current flows due to a short circuit or other reason in
distribution boards, etc. When the vacuum circuit breaker breaks the current, metal vapor is discharged from the contacts and becomes aplasma, producing an arc. Vacuum evaporation may occur due to this arc.
To avoid vacuum evaporation caused by an arc, a magnetic field is created by the arrangement of the contacts, and the arc is diffused withLorentz force. Because the construction of the contacts is complex, it is important to analyze the magnetic field using the finite element
method (FEM), which can handle complex geometries, to estimate whether the arc will be adequately diffused.
This Application Note presents how to calculate the current density and Lorentz force of a vacuum circuit breaker, and obtain its arc-diffusing
force.

Lorentz Force
The direction of the Lorentz force generated in the arc is shown in fig. 1. A force of 6 N
works in a direction -51 deg from an X-axis passing through the center of the arc.

Ring contact
Arc

Current Density Distribution and Magnetic Flux Density
Distribution
The current density distribution is shown as a vector plot in fig. 2, and the magnetic flux
density distribution of a cross-section of the arc is shown as a contour plot in fig. 3.
Current flowing in the contact cup as shown in fig. 2 causes magnetic flux to begenerated as shown in fig. 3. The magnetic flux density to the upper left of the center of
the arc is larger than the magnetic flux density to the lower right. Therefore, force isproduced from the upper left to the lower right.

51 deg6N
Force direction(Unit: T)(Unit: A/m^2)

Fig. 1. Direction of Lorentz force

Fig. 2. Current density distribution

Fig. 3. Magnetic flux density distribution (arc cross-section)

https://www.jmag-international.com/

83

Case

150

Electromagnetic Repulsion Analysis of a Vacuum
Circuit BreakerSupporting
conductor
Coil

Upper
electrodes

Contact

module: TR

Overview
Vacuum circuit breakers have the function of breaking the current when a large current flows due to a short circuit or other reason in
distribution boards, etc. Contact pressure is applied to the contacts in a vacuum circuit breaker because it is necessary for the vacuum circuit
breaker's contacts to be in contact and the electric circuit to be closed when normal current is flowing, unlike the large current from a shortcircuit.
Electromagnetic repulsion is produced by the current flowing in a vacuum circuit breaker's contacts when current flows through the vacuum
circuit breaker. Therefore, it is necessary to evaluate the electromagnetic repulsion and set the contact pressure higher than theelectromagnetic repulsion in order to maintain a closed state during normal current.
This Application Note presents how to obtain the electromagnetic repulsion in a circuit breaker from the Lorentz force when a 100 kA current
is flowing.

Contact points
Lowerelectrode

Assembly
diagram

Expanded diagram

Current Density Distribution

Lorentz Force Density Distribution and Electromagnetic Force

Current density distributions are shown in figures 1, 2, and 3. The current that flows into
the supporting conductor of the lower electrode divides in the coil and flows to thecontact via the contact points as shown in (a). Then, the current that has flowed into thecontact is concentrated at the contact point as seen in (b). When current flows into thecontact of the upper electrode from the narrow contact points, it tries to spread out asshown in fig. 3.

The Lorentz force density distribution is shown in fig. 4. Vertically repulsive force isgenerated in the contacts by the current flowing as shown in fig. 3.
Electromagnetic repulsion of approximately 700 N is generated in the contacts.

(b)
(d)

(a)

(c)

(Unit: A/m^2)(Unit: N/m^3)(Unit: A/m^2)(Unit: A/m^2)

Fig. 1 Current density distribution (upper)

Fig. 2 Current density distribution (lower)

Fig. 3 Current density distribution(ZX plane around contacts)

Fig. 4 Lorentz force density distribution(ZX plane around contacts)

Case

151

Pressboard

Insulation Evaluation Analysis of a Power Transformer

Board
Insulation oil

module: EL

Overview
A transformer is an electrical device that uses electromagnetic induction to convert the voltage level of alternating current power. Amongtransformers, power transformers that are used in power conversion have extremely high electric fields applied to their coils, so insulation
technology like the winding structure, insulating materials, and insulation structure are vital in attaining miniaturization and higher capacity.
In addition to normal power conversion, a transformer's insulation structure has to be designed to be able to withstand lightning strikes andexcess voltage during short circuits.
A transformer's dielectric strength is dependent on the electric field intensity applied to the insulating material, so the electric field intensity
needs to be examined before the safety factor for the insulation breakdown can be estimated.
This Application Note shows how to obtain the electric field intensity distribution when a lightning strike or short circuit occurs and theassumed maximum voltage is applied to the winding.

Electric Filed Distribution
Fig.1 shows the electric field distribution when a voltage difference is applied between
the primary coil's insulating material and the secondary coil's insulating material.
The maximum electric field intensity is about 8 kV/mm, which does not exceed thedielectric strength. From the distribution, it is also possible to confirm which parts large
electric fields are generated in.

(Unit: V/m)Fig. 1 Electric field distribution

84

JSOL Corporation

Insulation of primary coil
Insulation of secondary coil
Cross-section

Enlarged viewof the power transformer

JMAG Application Catalog

Electromagnetic Force Analysis of Short-circuited
Power Transformer Coils

Case

152

Core

module: FQ

Overview
A transformer is an electrical device that uses electromagnetic induction to convert the voltage level of alternating current power.
Electromagnetic force is produced by the current and magnetic field in a transformer's coils while it is converting power. There is a risk that alarge electromagnetic force can cause the coils to deform or rupture, particularly when something goes wrong and current flows in a shortcircuit. Electromagnetic force is produced by the current and magnetic field in a coil, but the coil is exposed to not only the magnetic field it
produces by itself but also the magnetic field from other coils and leakage flux from the core. Because of this, it is important to evaluate inadvance what kinds of forces are produced in what areas due to the arrangement of the coils and the positional relationship of the coils andcore using magnetic field analysis.
This analysis uses different coil positions to evaluate the Lorentz force density and electromagnetic force produced in the coils during a shortcircuit in order to confirm the effects that the primary and secondary coils have on each other.

Secondary coilPrimary coil

1000

Model C

Model D

Primary coil (Z-component)

Model B

Model A

Model C2

Model D
Model D

Model B

0

Secondary coil (Y-component)

Model A

2000
Model D

(Unit: N/m^3)

Model D (Unit: N/m^3)

Model B

Model C

4000

Model A

Model B

6000

Model C2

Model A

8000

Model C1

Base model

Model B

Base model

Model AElectromagnetic force (N)Primary coil (Y-component)

Model C1

0
Base model

Model B

8000
7000
6000
5000
4000
3000
2000
1000
0

2000

Model D

Model A

Model C2

Model C1

0

3000

Model C2

1000

4000

Model C1

2000

Base model

3000

Base model

Base model

4000

Electromagnetic force (N)The electromagnetic force of the V-phase coil is shown in fig. 3. The Z-component graphshows that, in all of the models, repulsion force is produced that mutually separates theprimary and secondary coils. In an actual machine, the Z-component force is canceled out
by symmetry, but this force can be seen in analysis because only a half model is used.
From the Y-component graph, it can be seen that a greater force is working in models A
and B compared to the other models. When the balance of the coil's winding method isasymmetrical vertically, a large force is produced in the vertical direction in addition to
the repulsion in the radial direction, as indicated by the Lorentz force density vectors in
fig. 1. The overall force in model D is canceled out so its Y-component force is zero, but bylooking at the Lorentz force density vector distribution, it can be seen that force isproduced a direction that compresses the primary coil as a whole.

Electromagnetic force (N)Electromagnetic ForceThe Lorentz force density distribution vectors of the V-phase coil as seen from the X-axis
are shown in fig. 1, and the Lorentz force density distribution vectors of the three phases
as seen from the Z-axis are shown in fig. 2. From fig. 1, it can be seen that the direction of
the force working on the coil varies according to how the coils are wound. Fig. 2 indicatesthat in models A and D, in which the deviation of the Lorentz force is large, the Lorentz
force density to the inside grows large because it is influenced by the adjacent coils.

Electromagnetic force (N)Lorentz Force Density

Secondary coil (Z-component)* Model C1 indicates the upper coil, and model C2 indicates the lower coil.

Fig. 1. Coil Lorentz force density distribution(V-phase, YZ plane)

Fig. 2. Coil Lorentz force density distribution (XY plane)

Fig. 3. V-phase coil electromagnetic force graphs

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85

Case

154

Calculation of Equivalent Circuit Parameters
in a Three-Phase Induction Motor

Shaft
Cage
(end rings)

module: DP

Stator core

Overview

Rotor core

An induction motor is a motor in which the rotating magnetic field of the stator coils causes induced current to flow in an auxiliary conductor,
exerting force on the rotor in the rotational direction and causing it to spin. Induction motors are widely used in everything from industrial
machines to home appliances because they have a simple construction and are small, light, affordable, and maintenance-free. An induction
motor's characteristics are influenced by leakage reactance and resistance, including resistance on the secondary side. These are referred to as
equivalent circuit parameters, and they are important because they characterize a device's properties. Equivalent circuit parameters are greatly
affected by both the current distribution induced in the auxiliary conductor and the magnetic saturation near the gap, so a finite element
analysis (FEA) needs to be run in order to investigate these characteristics with precision.
This Application Note explains how to obtain the secondary resistance, leakage inductance, and excitation inductance of an induction motor
when its power supply frequency has been changed with regard to its voltage and current controls.

Cage (bars)

Equivalent Circuit Parameter Frequency Characteristics(Voltage Control)

Equivalent Circuit Parameter Frequency Characteristics(Current Control)

The secondary resistance when the power supply frequency in the voltage control has
been changed is shown in fig. 1, the leakage inductance is shown in fig. 2, the magnetic
flux density distribution during the lock test is shown in fig. 3, the current density
distribution is shown in fig. 4, the excitation inductance is shown in fig. 5, and themagnetic flux density distribution during the no-load test is shown in fig. 6. The leakage
inductance is the sum of the primary and secondary leakage inductances.
The equivalent circuit parameters change according to the frequency in each result. Thisis because the primary and secondary currents as well as the current distribution in thebars change according to the frequency.

The secondary resistance when the power supply frequency in the current control has
been changed is shown in fig. 7, the leakage inductance is shown in fig. 8, the magnetic
flux density distribution during the lock test is shown in fig. 9, the current density
distribution is shown in fig. 10, the excitation inductance is shown in fig. 11, and themagnetic flux density distribution during the no-load test is shown in fig. 12. The leakage
inductance is the sum of the primary and secondary leakage inductances.
The equivalent circuit parameters change according to the frequency in each result. The
excitation inductance also has a constant value. For current control, the excitation current
during the no-load test does not change with the frequency and the excitationinductance is constant because induction current does not flow through the cage.

3.0
2.5
2.0
1.5
1.0
0.5
0.0

0.07

4.5

0.08

0.06

4.0

0.07

0.05
0.04
0.03
0.02
0.01

10

20

30

40

50

60

0

3.0
2.5
2.0
1.5
1.0
0.5

10

20

30

40

50

Fig. 1 Frequency characteristics of the secondary
resistance

50 Hz(Unit: T)

50 Hz

10

20

30

40

50

60

0.04
0.03
0.02
0.01
0

10

20 Hz

20

30

40

50

60

Frequency (Hz)Fig. 7 Frequency characteristics of the secondary
resistance(Unit: A/m^2)

Fig. 4 Current density distribution during lock test

Fig.3 Magnetic flux density distribution during
lock test

0.05

Frequency (Hz)Fig. 2 Frequency characteristics of the leakage
inductance

20 Hz

0.06

0.00
0

60

Frequency (Hz)Frequency (Hz)

20 Hz

3.5

0.0

0.00
0

Leakage inductance (H)3.5

Leakage inductance (H)Secondary resistance (ohm)

4.0Secondary resistance (ohm)

4.5

50 Hz

Fig. 8 Frequency characteristics of the leakage
inductance(Unit: T)

20 Hz

50 Hz

(Unit: A/m^2)

Fig. 10 Current density distribution during lock test

Fig. 9 Magnetic flux density distribution during
lock test

0.20

2.5
Excitation inductance (H)Excitation inductance (H)

0.18
2.0
1.5
1.0
0.5
0.0

0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00

0

10

20

30

40

50

60

20 Hz

50 Hz

(Unit: T)

Frequency (Hz)Fig. 5 Frequency characteristics of the excitationinductance

86

0.16

JSOL Corporation

0

10

20

30

40

50

60

20 Hz

50 Hz

(Unit: T)

Frequency (Hz)Fig. 6 Magnetic flux density distribution during
no-load test

Fig. 11 Frequency characteristics of the excitationinductanceFig. 12 Magnetic flux density distribution during
no-load test

JMAG Application Catalog

Case

156

Segregation Analysis of Torque Components
for an IPM Motor

Stator core
MagnetCoilmodule: DP

Overview
IPM motors are often used as high performance motors because they are highly efficient and their structure makes it possible to achieve awide range of operation. They are able to achieve high efficiency because they obtain maximum total torque by using their controls to adjust
their magnet and reluctance torques. For this reason, it is important to find out the distribution of both of these torques during operation
when the IPM motor is being designed. The motorʼ s detailed geometry and the material's nonlinear magnetic properties need to be taken
into account to obtain the torque characteristics, and it is even more difficult to segregate the torque into two components by using manual
calculations.
In order to proceed with the design while looking into how much each one contributes, it needs to be studied with an electromagnetic field
analysis that uses the finite element method (FEM).
In this Application Note, the torque components are separated and the magnetic flux density distributions created by each magnetomotive
force are confirmed.

Rotor core

Torque Versus Current Phase Angle
Fig. 1 shows a graph of the torque versus current phase angle. As can be seen in thegraph, the maximum torque is obtained when the current phase angle is near 20 deg. The
magnet torque also takes up a bigger portion of the total torque.

Fig. 2 shows a contour plot of the magnetic flux density distribution and magnetic flux
lines.
As can be understood from these figures, it is possible to visualize the condition of themagnetic circuit in addition to segregating the torque values. From these, it is possible toconfirm the effects on the torque of each magneto motive force.

Reluctance torque

Magnet torque

2.5
Torque (Nm)Magnetic Flux Density Distribution, Magnetic Flux Lines

Total torque
3.0

2.0(a) During load

1.5
1.0
0.5

(Unit: T)

0.0
0

10

20

30

40

50

60

70

80

(b) Magneto motive force
with current only90

Current phase angle (deg)Current phase: 20 deg step 1

Fig. 1 Torque versus current phase angle

Case

157

(c) Magneto motive force
with the magnets only

Fig. 2 Magnetic flux density distribution, magnetic flux lines

Analysis of Eddy Currents in an IPM Motor Using
the Gap Flux Boundary

Stator core
MagnetCoilmodule: DP, FQ

Overview
It is becoming increasingly common for permanent magnet motors to use rare earth magnets in order to achieve higher output density
because they have a high energy product. Neodymium rare earth magnets have a high electric conductivity because they contain a great
deal of iron, so when a varying magnetic field is applied to them they produce joule loss from eddy currents. IPM structure adoption and field
weakening controls have become prevalent in recent years in order to allow faster rotation. This has led to an increase in the frequencies andfluctuation ranges of the varying fields applied to magnets, resulting in a corresponding increase in joule losses. By dividing the magnet likeone would a laminated core to control eddy currents, one can obtain a method of raising the apparent electric conductivity while lowering
the eddy currents. Armature reactions in the stator occur before the eddy currents produced in the magnet, so the eddy currents are
determined by: The slot geometry of the stator core, the geometry of the rotor, the nonlinear magnetic properties of the core material, andthe current waveform that flows through the coil. In order to examine these kinds of magnet eddy currents ahead of time, one has to beprecise when accounting for things like these various geometries and material properties. This is why a magnetic field simulation using thefinite element method (FEM), which can account for them, would be the most effective.
This Application Note explains how to use the gap flux boundary condition to evaluate the eddy current loss in the magnet by changing thenumber of magnet divisions. This will make it possible to obtain effective results in a shorter period of time than with a normal transient
response analysis.

Rotor core

Change in the Losses of the Divided Magnet

Eddy Current Density Distribution in the Magnet

Fig. 1 shows the magnet's eddy current losses and fig. 2 shows the eddy current loss
density distribution.
From fig. 1, it is apparent that the losses are largest at 2880 Hz. This is caused by theeffects from the slot harmonic components, which are decided by multiplying therotation speed by the number of slots. From fig. 1 and 2, it is also apparent that the eddy
current losses decrease as the number of magnet divisions increases.
By increasing the number of divisions in the magnet, the magnetic flux that links theindividual magnets is reduced. Because of this, the eddy current density in each divided
magnet is reduced as well, making it so that the total amount of eddy current loss
decreases.

Fig. 3 shows the eddy current density distribution in the magnet at 2880 Hz. It is apparent
that dividing the magnet causes the eddy currents to decrease.

0.05
No divisions
2 divisions
4 divisions
8 divisions

0.04

Loss (W)

0.03
0.02
0.01
0.00
1440

2880

5760

8640

Frequency (Hz)14400
No division

2 divisions

4 divisions

8 divisions(Unit: W/m^3)

frequency: 2880 Hz

Fig. 1 Eddy current loss of the magnet

Fig. 2 Eddy current loss density distribution in the magnet

No division

2 divisions

4 divisions

8 divisions(Unit:A/m^2)

frequency: 2880 Hz

Fig. 3 Eddy current density distribution in the magnet

https://www.jmag-international.com/

87Superimposed Direct Current Characteristic Analysis
of a Reactor That Accounts for Minor Hysteresis Loops

Case

158

Core
Bobbinmodule: FQ, ST

Coil

Overview
High-frequency reactors used in equipment like DC-DC converters have a high-frequency current accompanying the switching direct current.
The reactor's performance requires a stable inductance in a wide direct current region that is superimposed by alternating current
components. If there is only a direct current, the magnetic flux is generated against the external magnetic field, following the magnetic steel
sheet's DC magnetization curve. However, when there is a current waveform whose high-frequency components are superimposed on the
direct current component, the response displays a minor loop against the external magnetic field. The values of the inductance in the reactor
can have significant differences depending on the method used to measure them. This can make it difficult to carry out a performance
prediction during an actual state of operation.
In order to handle the responsiveness of a magnetic field against a current waveform that is superimposed by a higher harmonic with a small
amplitude for the direct current component, a magnetic field analysis that accounts for material modeling needs to be carried out. With amagnetic field analysis, it is possible to analyze the machine characteristics from the magnetic flux density distribution.
This Application Note presents the use of the frozen permeability condition to obtain the superimposed direct current characteristic that
includes minor hysteresis loops of a high-frequency reactor.

SpacerSuperimposed direct current characteristics, Magnetic ﬂux
density distribution
This section shows the superimposed direct current characteristics in fig. 1 and the core's
magnetic flux density distribution in fig. 2 as analysis results of the reactor described.
From fig. 1, it is apparent that the inductance decreases more rapidly as the direct current
increases. This is caused by influence from magnetic saturation, as shown in fig. 2.

1
0.9
0.8
Inductance (mH)

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

10

20

30

40

50

60

Superimposed direct Superimposed direct Superimposed direct
current 5 A
current 30 A
current 45 A

(Unit: T)

Superimposed direct current (A)Fig. 1 Superimposed direct current characteristics

Fig. 2 Magnetic flux density distribution (during DC flow)

Sensitivity Analysis of Dimensional Tolerance
in an SPM Motor

Case

159

Shaft
Magnetmodule: DP

Stator core

Overview
The corners of magnets used in surface permanent magnet (SPM) motors can be filleted, chamfered, etc. However, it is difficult to maintainexactly the same production in the manufacturing process, and some variation among finished products will occur. Dimensional tolerance isset so as to eliminate the effects of these variations on motor performance.
There are tradeoffs between dimensional tolerance, performance, and cost, so it is important to investigate these at the design stage. With
numerical analysis using the finite element method (FEM), it is possible to evaluate the sensitivity of motor performance, such as torque, bysimply changing the dimensions.
This Application Note presents how to assume a dimensional tolerance of ±0.4 mm for a chamfer, and find out whether changing
dimensions within the tolerance range has an effect on motor performance by comparing cogging torque and induced voltage.

45 deg1.3mm
Geometry of magnet chamfer

Cogging torque

Induced voltage

The cogging torque waveform is shown in fig. 1, and the frequency components of thecogging torque are shown in fig. 2. Although the location of the peak value is different
when the size of the chamfer is changed, the peak value is almost the same, as indicatedin fig. 1. Differences at the fundamental frequency of 360 Hz are small when comparedwith higher harmonic components, as confirmed by the frequency components in fig. 2.
From this, it can be determined that variations in the chamfer occurring within thetolerance range have little effect on the cogging torque.

The induced voltage waveform is shown in fig. 3, and the frequency components of theinduced voltage are shown in fig. 4. Even when the size of the chamfer is changed, theinduced voltage value is almost the same, as indicated in fig. 3. Differences at thefundamental frequency of 60 Hz are small when compared with higher harmoniccomponents, as confirmed by the frequency components in fig. 4.
From this, it can be determined that there are few effects from the chamfer tolerance.

0.00
0

5

10

15

20

-0.04
-0.08

25

30

0.09
0.08

-0.4mm-0.3mm-0.2mm-0.1mm±0.0mm+0.1mm+0.2mm+0.3mm+0.4mm

0.07
0.06
0.05
0.04
0.03
0.02
0.01

40

20

Voltage (V)0.04-0.4mm-0.3mm-0.2mm-0.1mm±0.0mm+0.1mm+0.2mm+0.3mm+0.4mmAmplitude (Nm)Torque (Nm)

30

0.10

0.08

10
0
0

30

60

90

-10
-20

0.00
Angle (deg)Fig. 1. Cogging torque88

JSOL Corporation

150

180

-0.4mm-0.3mm-0.2mm-0.1mm±0.0mm+0.1mm+0.2mm+0.3mm+0.4mm

35
30
25
20
15
10
5
0
0

0

-0.12

120

-0.4mm-0.3mm-0.2mm-0.1mm±0.0mm+0.1mm+0.2mm+0.3mm+0.4mmAmplitude (V)

0.12

360

720

1080

1440

1800

2160

Frequency (Hz)Fig. 2. Frequency components of cogging torque60 120 180 240 300 360 420 480 540 600 660-30
Angle (deg)Fig. 3. Induced voltage

Frequency (Hz)Fig. 4. Frequency components of the induced voltage

JMAG Application Catalog

Case

Shaft

161 Line Start Analysis of a Three-phase Induction Motor

Cage(end rings)
Stator coremodule: DP

Overview

Rotor core

The simplest method for starting an induction motor is a line start that connects the motor to a direct power supply. For a line start, the static
impedance is small compared to impedance during rated operation, so a large current flows during the initial start-up.
The large current flowing through both the primary and secondary sides during start-up causes intense magnetic saturation near the
induction motor's gap. This magnetic saturation results in reduced impedance, so the starting current grows even larger. The size of thestarting current affects the voltage source capacity connected to the induction motor, as well as both the electromagnetic force and heat
capacity that operate on the motor's coils. This is why it is beneficial to investigate the starting performance of an induction motor with thefinite element method (FEM), which can account for local magnetic saturation.
This Application Note presents an analysis that simulates the line start of an induction motor and obtains the starting performance of its
rotation speed variations.

Cage (bars)

Rotation Speed Waveform

Current waveform

The rotation speed waveform when starting the induction motor is indicted in fig. 1.
Fig. 1 shows that the motor accelerates rapidly at the initial start-up, and reaches a steadystate at approximately 0.5 s.

The current waveform of the stator coil when the induction motor starts is indicated in fig.
3. Fig. 3 shows that an extremely large current is produced during the initial start-up. Thisis because the reverse voltage at initial start up is small and almost all of the power supply
voltage is applied to the stator coils. At a steady state, the three-phase AC has anamplitude of less that 5 A.

Torque waveform

1400

7.0

1200

6.0

Drive torque

1000

5.0

Load torque

800
600

15
U-phase
10

4.0
3.0
2.0

-5

200

1.0

-10

0.0
0.10

0.20

0.30

0.40

0.00

0.50

0.10

0.20

Time (sec)

0.30

0.40

-15
0.00

0.50

W-phase

0

400

0
0.00

V-phase5

Current (A)Torque (Nm)

Rotation speed (r/min)

The waveforms of the drive torque and load torque when starting the induction motor
are indicated in fig. 2. Fig. 2 shows that an extremely large drive torque is produced at theinitial start-up. In a steady state, a drive torque that is almost constant is generated inorder to achieve balance with the load torque.

0.10

0.20
0.30
Time (sec)Time (sec)Fig. 1. Rotation speed waveform

Fig. 2. Torque waveform

0.40

0.50

Fig. 3. Current waveform

Drive Simulation of an SR Motor using a Control Simulator
and JMAG-RT

Case

162

module: DP, RT

Overview
SR motors are gaining attention as motors that do not use permanent magnets. However, torque pulsation is greater in SR motors because of
the way their torque is generated, so it is seen as important to suppress torque pulsation not only with detailed design of magnetic circuits,
but also using SR motor control. In order to evaluate suppression of torque pulsation with control simulation, it is necessary to have an SR
motor model that shows behavior conforming to a real motor.
With JMAG, it is possible to create a detailed model that conforms to a real machine and accounts for the torque's dependency on rotor angle
and the magnetic saturation characteristics of the magnetic circuit in an SR motor. Importing this motor model, a "JMAG-RT model," to acontrol/circuit simulator makes it possible to carry out a linked simulation that accounts for an SR motor's detailed characteristics as well as a
motor drive's control characteristics. This kind of operation makes control simulation to suppress an SR motor's torque pulsation possible.
In this Application Note, after using JMAG-RT to obtain an SR motor's torque and inductance characteristics, the JMAG-RT model is imported
into a circuit/control simulator, voltage is applied to drive the SR motor, and the torque pulsation is analyzed. By evaluating the obtained
torque pulsation in detail, investigations can be carried out to reduce the torque pulsation (this Application Note does not incorporate actual
attempts to reduce torque pulsation).

Shaft

Coil

Rotor core

Stator core

Rotation Speed Waveform/Torque Waveform/Current Waveform
Fig. 1 shows the rotation speed waveform, fig. 2 shows the torque waveform, and fig. 3
shows the current waveform when the SR motor is driven. A large phase current flows
immediately after starting because the rotation speed is slow and the starting voltage issmall, but this is suppressed to an upper limit by the current limit value. At this time, a lotof torque is generated, and the rotation speed increases rapidly.
Once the rotation speed increases a certain amount, the rotation speed, torque, andcurrent settle into a steady state because the starting voltage rises and the current limit
becomes irrelevant. The rotation speed also oscillates because the torque ripple in thesteady state is large. Evaluation of excitation timing and control of the current waveform
itself are possibilities for ways to reduce this.

18

12
A-phase

16
14

800

12

600
400

10

C-phase

10
8
6
4

200

0.25

0.50

0.75
Time (s)Fig. 1. Rotation speed waveform

1.00

1.25

1.50

6
4
2

2

0
0.00

B-phase8
Current (A)1000
Torque (Nm)

Rotation speed (r/min)

1200

0

0
0.00

0.25

0.50

0.75

1.00

1.25

1.50

0.00

0.25

0.50

Fig. 2. Torque waveform

0.75

1.00

1.25

1.50

Time (s)Time (s)Fig. 3. Current waveform

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89

Case

163

Shaft

Torque-Current Curve Analysis of an SPM Motor

Magnetmodule: DP

Rotor core

Overview
One of the fundamental properties of a permanent magnet synchronous motor is the relationship between its current and torque(torque-current curve). The torque generated at each current value is uniform with increases in current up to a certain point, so the torqueincreases in a linear fashion. However, magnetic saturation effects occur with further current increases, and the torque generated with eachincrease in current begins to drop off.
Because a permanent magnet synchronous motor's torque-current curve is highly susceptible to saturation effects in the motor's magnetic
circuit, it is helpful to obtain the torque-current curve with a magnetic field analysis taking saturation into account in order to evaluate themotor's design and drive characteristics.
This Application Note presents how to obtain the torque-current curve as a basic property of one type of permanent magnet synchronous
motor, the surface permanent magnet synchronous (SPM) motor.

Stator core

Magnetic Flux Density Distribution

Torque-Current Curve, Torque Constant-Current Curve

Figures 1 and 2 show the magnetic flux density distribution at current amplitudes of 10 A
and 200 A.
From the figures, it can be seen that there are differences in the flow of magnetic flux due
to larger and smaller current amplitude, such as changes in the locations of teeth withhigh magnetic flux density. This is due to magnetic saturation in the teeth, and thesechanges in the flow of magnetic flux affect the torque.

Figures 3 and 4 show the torque-current curve and torque constant-current curves.
It can be seen from figures 3 and 4 that the ratio of torque increase declines withincreased current. This is because the flow of magnetic flux changes, as seen in figures 1
and 2.

0.12

18

Torque coeﬃcient (Nm/A)

16
14
Torque (Nm)

12
10
8
6
4
2

0.08
0.06
0.04
0.02
0.00

0

(Unit: T)(Unit: T)

0.10

0

50

100

150

0

200

Fig. 1. Magnetic flux density distribution(current amplitude: 10 A)

Fig. 2. Magnetic flux density distribution(current amplitude: 200 A)

Fig. 3. Torque-current curve

100

150

200

Fig. 4. Torque constant-current curve

Case

165

50

Current (A)Current (A)Stator core

Creating IPM Motor Eﬃciency Maps

MagnetCoilmodule: DP, LS

Overview
IPM motors use rare earth sintered permanent magnets because they have strong magnetic energy. They can use the magnetic torque fromthe magnet's field and the rotating magnetic field in addition to the reluctance torque that originates from the difference in inductance
between the d-axis and q-axis, so they have a wide drive range and are highly efficient.
Their efficiency changes with their rotation speed and their load, so it is beneficial to create an efficiency map when designing the motor andits controls. However, the calculations required to create an efficiency map are typically huge, so it takes time to organize the results as well.
Though is possible to estimate the efficiency by using the motor's voltage equation and torque formula to calculate the torque, voltage, andcurrent, one cannot use this method to estimate the iron loss or account for the effects of the nonlinear magnetic properties of the motor'siron core. The main problem is the difficulty of correctly calculating the efficiency. To help with this problem, an efficiency map that accounts
for influence from iron loss and nonlinear magnetic properties can be easily obtained by creating a JMAG-RT model of the target and using
JMAG-RT Viewer's efficiency map calculation function.
This Application Note presents the use of the JMAG-RT Viewer to create an efficiency map for an IPM motor.

Rotor core

Speed-Torque curve / Eﬃciency map
141210Torque (Nm)Fig. 1 shows the Speed-Torque curve, and fig. 2 shows the efficiency map.
As seen in fig. 1, in the low speed region the torque becomes constant. This is due to thecurrent controls. The torque also decreases as speed increases. This is because the current
flowing to the motor coil decreases due to back electromotive voltage.
In fig. 2 it is possible to get a grasp of the efficiency at each drive condition. The rangethat maps the torque's positive value and efficiency also becomes the possible range of
output for this motor.

8
6
4
2
0
0

1000

2000

3000

4000

5000

6000

Rotation speed (r/min)Fig. 1. Speed-Torque curve

90

JSOL Corporation

Fig. 2. Efficiency map

JMAG Application Catalog

Line Start Simulation of an Induction Machine
Using a Control Simulator and the JMAG-RT

Case

166

Shaft
Cage (end rings)

module: FQ, LS, RT

Overview

Stator core
Rotor core

Collaborative design is difficult because the controls and motor are designed independently. However, it has become necessary to resolve
challenges through high-accuracy simulations at the beginning of the development process in order meet demands for more advanced
motors. An effective way of achieving this is for the simulations to be performed while collaborating on the motor design circuit/control
designs.
An induction motor's characteristics are influenced by leakage reactance and resistance, including resistance on the secondary side. The
resistance on the secondary side is affected by the skin effect, so the finite element method (FEM) needs to be used to obtain the distributionof the secondary induced current. With JMAG, it is possible to use a magnetic field analysis to obtain the resistance and leakage reactance,
and to create a model of an induction motor, as well. Incorporating this motor model, called a "JMAG-RT model," to a circuit/control simulator
makes it possible to use JMAG-RT to run a linked simulation with it.
This Application Note explains how to use the JMAG-RT to create a JMAG-RT model of an induction motor, import it to a circuit/control
simulator, and run an induction motor line start simulation.

Cage (bars)

Rotation Speed Waveform

Current Waveform

The rotation speed waveform when starting the induction motor is indicated in fig. 1.
From fig. 1, it is apparent that the motor accelerates rapidly at the initial start-up andmoves to a more steady acceleration while transitioning to a steady state.

The current waveform of the stator coil when the induction motor starts is indicated in fig.
3.
Fig. 3 shows that an extremely large current is produced during the initial start-up. This isbecause the back electromotive voltage at initial start-up is small and almost all of thepower supply voltage is applied to the stator coils. At a steady state, the three-phase AC
has an amplitude of less than 5 A.

Torque waveform

1600

9.0

1400

8.0

Drive torque

1200

7.0

Load torque

1000

6.0

800

5.0

600
400
200
0
-200
0.00

0.30

0.60

0.90

1.20

1.50

20

Fig. 1. Rotation speed waveform

V-phase
10

4.0
3.0
2.0

0
-5
-10

0.0

-15

-1.0
0.00

0.30

0.60

0.90

1.20

1.50

W-phase5

1.0Time (s)U-phase15

Current (A)Torque (Nm)

Rotation speed (r/min)

The waveforms of the drive torque and load torque when starting the induction motor
are indicated in fig. 2.
From fig. 2, it is apparent that a large drive torque is generated at initial start-up.
Furthermore, the drive torque becomes almost constant as the load torque balances out
in the steady state.

-20
0.00

0.30

0.60

Time (s)Fig. 2 Torque waveform

0.90

1.20

1.50

Time (s)Fig. 3. Current waveform

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91

Case

167

Bolt hole

Rotor core

Iron Loss Analysis of a Three Phase Induction Motor

Cage(end rings)

module: DP, LS

Overview
An induction motor is a motor in which the rotating magnetic field of the stator coils causes induced current to flow in an auxiliary conductor,
which produces force in the rotational direction. Induction motors are widely used in everything from industrial machines to home
appliances because they have a simple construction without parts that experience wear from abrasion, and can be used simply by connecting
them to a power source.
Improved efficiency in induction motors is an important theme. Iron loss, a cause of lower efficiency along with primary and secondary
copper loss, must be reduced in order to improve efficiency. The relative importance of iron loss tends to grow especially with higher
rotations due to the inverter drive, so it is helpful to estimate the complex iron loss distribution inside the core.
This Application Note presents an example of how to find the iron loss in the stator core and rotor core at a rotation speed of 3,300 r/min.

Magnetic Flux Density Distribution
The magnetic flux density distribution of the stator core and rotor core is shown in fig. 1,
the magnetic flux density waveforms of the magnetic flux density R-component at
measuring points 1, 2, 3, and 4 are shown in fig. 2, and a graph of the results frommeasuring point 2 after FFT is shown in fig. 3. Also, the steady-state torque and primary
current amplitude are shown in table 1, and the secondary current flowing in the cage isshown in fig. 4.
Comparing measuring points 1 and 2, at measuring point 2 the magnetic flux density has
a large value and amount of variation, but at measuring point 1 the magnetic flux density
has a small value and amount of variation. For measuring points 3 and 4, it can be
confirmed that the value is large at measuring point 3. Further, when measuring points 2
and 3 are compared, the amount of variation is seen to be larger at measuring point 2.
These differences in magnetic flux density have an effect on iron loss.
120 Hz, the frequency of the power supply, is dominant at measuring point 2, as seen in
fig. 3.
The frequency of the secondary current affects the magnetic flux density on the rotor
side.

Measuring point 1

Measuring point 2

Measuring point 3

Measuring point 4

Cage (bars)

Stator core

Joule Loss Density Distribution/Hysteresis Loss Density
Distribution/Iron Loss Density Distribution
The joule loss density distributions for the stator core and rotor core are shown in fig. 5.
The hysteresis loss density distributions for the stator core and rotor core are shown in fig.
6. The iron loss density distributions for the stator core and rotor core are shown in fig. 7.
The ratios of loss for the stator core and rotor core are shown in fig. 8.
As shown in the magnetic flux density waveforms, joule losses are greater in places where
the amount of variation in the magnetic flux density is large, compared to places where
the amount of variation in the magnetic flux density is small. The same trend can be seenin the hysteresis losses as well, as shown in table 2 by the fact that the ratio of the joule
and hysteresis losses taken up by the stator is large.

Magnetic ﬂux density T

0.8
Measuring point 1
Measuring point 2
Measuring point 3
Measuring point 4

0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8

(Unit: T)

0

60

120

180

240

300

360

rotor corestator core

(Unit: W/m^3)

rotor corestator core

(Unit: W/m^3)

Electric angle deg

Fig. 1. Magnetic flux density distribution at a rotationof 1,800 deg

Fig. 2. Magnetic flux density waveforms (R-component)

Fig. 5. Joule loss density distribution

Fig. 6. Hysteresis loss density distribution

1.0E+00

3
2.5

Rotation speed 3,300 r/min

Results

Torque (Nm)0.37

Primary current (A)1.5

2Iron loss W

Amplitude T

1.0E-01

1.0E-02

1.0E-03

120

600

1080

1560

2040

2520

3000

stator core

3480

Bar3

Bar2

Bar3
r3

Current A

100
80
60
40
20
0
-20
-40
-60
-80
-100
360

720

1080

1440

1800

Rotation angle deg

Fig. 4. Secondary curren

92

JSOL Corporation

0.5

Rotor coreFig. 3. Frequency components at measuring point 20(Unit: W/m^3)0Frequency Hz

Bar1Joule loss
1

rotor core

1.0E-04

Hysteresis loss

1.5

2160

2520

Bar2
Bar1

Table 1. Reference results

Fig. 7. Iron loss density distribution

Fig. 8. Ratio of loss

Stator core

JMAG Application Catalog

Case

172

High-Frequency Induction Heating Analysis of a Test Piece(Rotational Induction Hardening)
module: FQ, HT

CoilOverview
Machine parts like shafts and gears are made to be resistant to wear and tear. This is accomplished by giving them a certain degree of
flexibility by maintaining their interior toughness while increasing the hardness of their surfaces. By using high-frequency induction heating,which is a type of surface hardening method, the part is heated rapidly on only its surface by a high frequency power source. This process also
has many other benefits, such as providing a clean working environment because it uses electrical equipment, being very efficient, andproviding uniform results for each product. This is why it is being aggressively implemented in the field.
With induction hardening like the kind used on the work piece in this analysis, the main requirement is to heat a given surface uniformly andincrease rigidity. The high-frequency's varying magnetic field produces eddy currents with an offset in the surface of the work piece, so
handling the phenomena inside the work piece with a numerical analysis based on the finite element method (FEM) is the most effectivemeans analyzing the process in detail.
This Application Note explains how to create a numerical analysis model when obtaining the optimum coil geometry, current conditions(power supply frequency, current value), and rotation speed. It also shows how to evaluate whether the target temperature distribution is
being achieved by analyzing the elevated temperature process.

Work piece

Joule Loss Density Distribution

Temperature Distribution and Variation vs. Time

Figures 1 and 2 show the joule loss density distribution generated in the work piece andthe coil. The magnetic field generated by the coil produces eddy currents in the workpiece, which become heat sources in induction heating. The places where the heated partexceeds the Curie temperature lose their magnetism, so the eddy current loss density
decreases, as well. The skin effect produced by the high frequency also causes the eddy
current to be distributed near the work piece's surface.

Fig. 3 shows the work piece's temperature distribution, and fig. 4 shows its temperature
variation versus time. In the surface facing the heating coil, the temperature is higher
because more eddy currents are generated. As shown in fig. 4, induction heating causes
the temperature in the surface of the work piece to rise first, and the temperature of thework piece's interior rises gradually due to thermal conduction.

Evaluation point: surface

Work piece cross-section

Work piece cross-section

Work piece cross-section(Unit: W/m^3)(Units: deg C)(Unit: W/m^3)

Temperature (deg C)

900

Evaluation point: interior

800

Surface elements

700

Interior

600
500
400
300
200
100
0
0.0

1.0

2.0

3.0

4.0

Time (s)Fig. 1 Joule loss density distribution at 0.0 s

Fig. 2 Joule loss density distribution at 4.0 s

Fig. 3 Temperature distribution at 4.0 s

Fig. 4 Temperature variation of the work piece

https://www.jmag-international.com/

93

Case

Stator core

173 Basic Characteristic Analysis of an IPM Motor

MagnetCoilmodule: DP, LS

Overview
Demand for higher efficiency and smaller size in motors has grown from the need to accommodate devices that incorporate miniaturizationand energy efficiency in their designs. In order to meet this demand, motors have to improve their output density and reduce their losses.
One type of loss commonly found in motors is iron loss, which increases drastically at high rotation speeds and high magnetic flux densities.
This increase can lead to a rise in temperature and a reduction in efficiency. Consequently, it is growing more important to predict iron loss at
the motor design stage.
Unfortunately, it is not possible to obtain iron losses accurately in studies that use the magnetic circuit method or rules of thumb. In order toobtain them accurately, one needs to find the distribution and time variations of the magnetic flux density in each part of the motor after
accounting for a fine geometry and the material's nonlinear magnetic properties. Using the finite element method (FEM) is essential in orderto carry out this kind of a detailed analysis.
This Application Note demonstrates an analysis in which an IPM motor's cogging torque, torque, magnetic flux density distribution, and ironloss in the stator core are obtained.

Rotor core

Torque Analysis (No-load)

Torque Analysis (Load)

The cogging torque waveform is shown in fig. 1, the magnetic flux lines where thecogging torque is 0 Nm at a rotation angle of 0 deg are shown in fig. 2, the induced
voltage waveform is shown in fig. 3, the iron loss density distribution is shown in fig. 4,
and the ratios of loss for the stator core are shown in fig. 5.

The current phase versus torque is shown in fig. 6, the magnetic flux density distribution
and magnetic flux lines at current phase 10 deg are shown in fig. 7, the current phaseversus loss is shown in fig. 8, and the iron loss density distribution at current phase 10 deg
is shown in fig. 9.

3.0

0.20

2.5
Torque (Nm)Torque (Nm)

0.16
0.12
0.08

2.0
1.5
1.0

0.04
0.5
0.00
0

1.5

3

4.5

6

0.0

7.5

10

Rotation angle (deg)Fig. 1 Cogging torque waveform

Fig. 2 Magnetic flux density distribution
and magnetic flux lines (rotation angle: 0 deg)V-phase

30

40

50

60

70

80
(Unit:T)

Current Phase (deg)Fig. 6 Current phase versus torqueW-phaseIron loss

80

8

60

7

Joule loss

Fig. 7 Magnetic flux density distribution
and magnetic flux lines (current phase: 10 deg)

Hysteresis loss

40
6

20

Loss (W)

Voltage (V)U-phase20(Unit:T)

0
-20

5
4

-40
-60

3

-80
0

15

30

45

60

75

90

2
(Unit:W/m^2)

Rotation angle (deg)Fig. 3 Induced voltage waveform

Joule loss

Hysteresis loss

6
5

Loss (W)43210Stator coreFig. 5 Ratio of loss

94

JSOL Corporation

10

20

30

40

50

60

70

80

(Unit: W/m^2)

Current Phase (deg)Fig. 4 Iron loss density distribution

Fig. 8 Current phase versus loss

Fig. 9 Iron loss density distribution (current phase: 10 deg)JMAG Application Catalog

Drive Characteristic Analysis of a Three-Phase
Induction Motor

Case

176

Shaft
Cage
(end rings)

module: DP, LS

Stator core

Overview

Rotor core

An induction motor is a motor in which the rotating magnetic field of the stator coils causes induced current to flow in an auxiliary conductor,
which exerts force on the rotor in the rotational direction and causes it to spin. Induction motors are widely used in everything from industrial
machines to home appliances because they have a simple construction and are small, light, affordable, and maintenance-free.
In an induction motor, the current induced by the auxiliary conductor exerts a large influence on its characteristics. It also causes strong
magnetic saturation in the vicinity of the gap, in particular. This is why a magnetic field analysis based on the finite element method (FEM) is
useful when investigating the motor's characteristics for a design study.
This Application Note explains how to confirm drive characteristics such as torque, loss, and efficiency in an induction motor when its rotation
speed changes.

Cage (bars)

Drive Characteristics (Current Control)

The primary current, primary copper loss, secondary copper loss, iron loss, torque, andefficiency with changing rotation speeds during voltage control are shown in figures 1
through 6. The eddy current density distribution, joule loss density distribution, hysteresis
loss density distribution, and iron loss distribution at maximum efficiency are shown infigures 7 through 10.

The primary voltage, primary copper loss, secondary copper loss, iron loss, torque, andefficiency with changing rotation speeds during current control are shown in figures 11
through 16. The eddy current distribution, joule loss density distribution, hysteresis loss
density distribution, and iron loss distribution at maximum efficiency are shown in figures
17 through 20

7.0E+04

100
80
60
40

6.0E+04
5.0E+04
4.0E+04
3.0E+04
2.0E+0420

1.0E+0400.0E+00
300

600

900

1200

300

900

1200

1500

1.0E+03

8.0E+04
6.0E+04
4.0E+041200

8.0E+02
6.0E+02
4.0E+02

300

600

900

1200

Eﬃciency (%)

1.0E+03
8.0E+02
6.0E+02
4.0E+02

900

300

600

900

1200

1500

stator corerotor core

2.0E+03
1.5E+03

Fig. 9 Hysteresis loss density distribution

1200

1500

1200

1500

70
60
50
40
30
20
10
0

0

300

600

900

1200

1500

0

300

600

900

Rotational Velocity (r/min)Rotational Velocity (r/min)Fig. 15 Rotation speed versus torqueFig. 16 Rotation speed versus efficiency(Unit: W/m^3)

stator core

(Unit: A/m^2)

rotor core

(Unit: W/m^3)(Unit: A/m^2)

Fig. 8 Joule loss density distribution(Unit: W/m^3)

900

80

2.5E+03

Rotational Veｌ
ocity (r/min)Fig. 6 Rotation speed versus efficiency

600

Fig. 14 Rotation speed versus iron loss

90

0

300

Rotatioal Velocity (r/min)

100

Rotational Veｌ
ocity (r/min)rotor core0

3.0E+03

0.0E+00

stator core1500

3.5E+030Fig. 7 Eddy current density distribution

1200

4.0E+03

0.0E+00

Fig. 5 Rotation speed versus torque

600

70

30

1500

0.0E+00
300

80

40

1200

1.0E+03

90

50

900

5.0E+02

Fig. 13 Rotation speed versus secondary copper loss

60

600

1.5E+03

Rotational Velocity (r/min)

5.0E+021500

300

2.0E+03

1.0E+04

10
1200

0

2.0E+0420900

5.0E+03

Fig. 12 Rotation speed versus primary copper loss

3.0E+04

2.0E+02600

1.0E+04

Rotational Velocity (r/min)

4.0E+04

1.0E+03300

1.5E+04

2.5E+030Torque (Nm)

1.2E+03

2.0E+041500

5.0E+04

1500

Fig. 4 Rotation speed versus iron loss

1.4E+031200

6.0E+04

Rotational Veｌ
ocity (r/min)Fig. 3 Rotation speed versus secondary copper loss

900

0.0E+00
0

1500

600

7.0E+04

Rotational Veｌ
ocity (r/min)

0300

Fig. 11 Rotation speed versus primary voltage

0.0E+00900

2.5E+04

Rotational Veｌ
ocity (r/min)

2.0E+02

2.0E+04

3.0E+04

0.0E+00
0

Secondary Copper Loss (W)

1.2E+03

1.0E+05

Iron Loss (W)Secondary Copper Loss (W)

1.4E+03

1.2E+05

0.0E+00

Torque (Nm)600

Fig. 2 Rotation speed versus primary copper loss

1.4E+05

600

2000

Rotational Veｌ
ocity (r/min)Fig. 1 Rotation speed versus primary current

300

3000

0

Rotational Veｌ
ocity (r/min)

03.5E+04

4000

1000

0

1500

4.0E+04

5000

Iron Loss (W)

0

6000

Eﬃciency (%)

Primary Current (A)120

Primary Copper Loss (W)8.0E+04140

Primary Copper Loss (W)160

Primary Voltage (V)Drive Characteristics (Voltage Control)

stator corerotor coreFig. 10 Iron loss density distribution

Fig. 17 Eddy current density distribution(Unit: W/m^3)

stator corerotor coreFig. 19 Hysteresis loss density distribution

Fig. 18 Joule loss density distribution(Unit: W/m^3)

stator corerotor core

(Unit: W/m^3)

Fig. 20 Iron loss density distribution

https://www.jmag-international.com/

95

Case

177

Torque Characteristic Analysis of a Three Phase
Induction Motor

Shaft
Cage
(end rings)

module: DP, LS

Stator core

Overview

Rotor core

An induction motor is a motor in which the rotating magnetic field of the stator coils causes induced current to flow in an auxiliary conductor,
which exerts force on the rotor in the rotational direction and causes it to spin. Induction motors are widely used in everything from industrial
machines to home appliances because they have a simple construction and are small, light, affordable, and maintenance-free.
It is possible to drive an induction motor so that its slip is constant by adjusting the voltage or current against load variations. When thishappens, each characteristic changes with influence from magnetic saturation and leakage flux because of the excitation variations in a
specific slip.
This Application Note explains how to obtain the torque characteristics in an induction motor when the current amplitude has been changed
in a specific slip.

Torque-Current Curve / Current-Voltage Curve
The Torque-Current curve and Current-Voltage curve when the current has been changed
with a constant slip are shown in figures 1 and 2.
The figures show that the torque and voltage both rise when the current increases.

Eddy Current Density Distribution
The eddy current density when the current has been changed with a constant slip isshown in fig. 3. The figure shows that the eddy currents in the cage increase when thecurrent value rises.

Joule Loss Density Distribution/Hysteresis Loss Density
Distribution
The joule loss density distribution and hysteresis loss distribution in the stator core when
the current has been changed with a constant slip are shown in figures 4 and 5, and thejoule loss density distribution and hysteresis loss density distribution in the rotor core areshown in figures 6 and 7.
The figures show that all of the losses increase when the current value rises.

5000

3500
3000

4000

2500
Voltage (V)Torque (Nm)Cage (bars)

2000
1500

3000
2000

1000
1000

500
0

0
0

20

40

60

80

100

120

140

Current (A)a current of 125 A

20

40

60

80

100

120

140

a current of 50 Aa current of 125 A(Unit: W/m^3)

a current of 50 Aa current of 125 A(Unit: W/m^3)

Current (A)Fig. 1 Torque-Current curve

a current of 50 A

0

Fig. 2 Current-Voltage curve

Fig. 4 Joule density distribution of the stator core

(Unit: A/m^2)

Fig. 3 Eddy current density distribution

a current of 50 Aa current of 125 A

Fig. 5 Hysteresis loss density distribution of the stator core

(Unit: W/m^3)

Fig. 6 Joule density distribution of the rotor corea current of 50 Aa current of 125 A(Unit: W/m^3)

Fig. 7 Hysteresis loss density distribution of the rotor coreCase

178 Analysis of SR Motor I-Psi Characteristics

Shaft

module: DP

Overview

Coil

Rotor core

Stator core

With the skyrocketing prices of rare earth magnets, expectations have been rising for SR (switched reluctance) motors because they have a
motor format that does not use permanent magnets. SR motors have a simple structure that can achieve solid performance at a low price.
However, torque generation depends only upon the saliency between the stator and rotor, so torque variations are extremely large and cause
vibration and noise, meaning that the use applications are limited. On the other hand, because of the skyrocketing prices of rare earth metals,
the improvement in current control technology, the possibility of optimized designs thanks to magnetic field analysis, and the rising ability to
reduce challenges, SR motors are being re-examined.
SR motors operate using the nonlinear region of a magnetic steel sheet, so because the inductance displays nonlinear behavior, it is
impossible to carry out advanced projections that are accurate with calculation methods that follow linear formulas. Consequently, it
becomes necessary to use the finite element method (FEM), which can handle nonlinear magnetic properties in material and minute
geometry.
This example presents an evaluation of flux linkage, inductance and torque for each rotor position when the flowing current value is changed.

I-Psi Characteristics
0.000(deg)5.625(deg)11.250(deg)16.875(deg)28.125(deg)33.750(deg)39.375(deg)45.000(deg)

6

8

1.0
0.8
0.6
0.4
0.2
0.0
0

2

JSOL Corporation

4

10

2(A)

10(A)

Current (A)Fig. 1. I-Psi characteristics

96

22.500(deg)

1.2
Magnetic Flux (Wb)Fig. 1 shows the SR motor's I-Psi characteristics, fig. 2 shows the current value 2 A and themagnetic flux density distribution for 10 A.
When the tooth is not facing to each other and is at 0 deg, because magnetic resistance isdominant in the main circuit, there is an almost linear increase in magnetic flux linkage in
relation to the current. Also, we understand that particularly when the tooth is facing toeach other, the more the current is increased, the more the amount of magnetic flux
decreases. This is caused by influence from magnetic saturation of the core, as shown infig. 2. In addition, the bigger the difference in the flux leakage (inductance) when thetooth is facing to each other and when it is not, increases torque so it is also possible toconsider how this could be used.

Fig. 2. Magnetic flux density distribution(Unit: T)

JMAG Application Catalog

Case

179 Analysis of SR Motor Static Characteristics

Shaft

Coil

Rotor coremodule: DP

Overview

Stator core

With the skyrocketing prices of rare earth magnets, expectations have been rising for SR (switched reluctance) motors because they have a
motor format that does not use permanent magnets. SR motors have a simple structure that can achieve solid performance at a low price.
However, torque generation depends only upon the saliency between the stator and rotor, so torque variations are extremely large and cause
vibration and noise, meaning that the use applications are limited. On the other hand, because of the skyrocketing prices of rare earth metals,
the improvement in current control technology, the possibility of optimized designs thanks to magnetic field analysis, and the rising ability to
reduce challenges, SR motors are being re-examined.
SR motors operate using the nonlinear region of a magnetic steel sheet, so because the inductance displays nonlinear behavior, it is
impossible to carry out advanced projections that are accurate with calculation methods that follow linear formulas. Consequently, it
becomes necessary to use the finite element method (FEM), which can handle nonlinear magnetic properties in material and minute
geometry.
This example presents an evaluation for each rotor position of the effect on flux linkage (shown as I-Psi characteristics below) when excitation
current is changed.

Flux linkage waveform/Inductance waveform/
Torque waveform
4(A)

6(A)

8(A)

10(A)

2(A)

Inductance (H)

0.8

0.4

4(A)

6(A)

8(A)

2(A)

10(A)

0.40

20

0.30

15

Torque (Nm)

2(A)
1.2
Magnetic Flux (Wb)For each rotor's position change, the SR motor's flux linage waveform is indicated in fig. 1,
the inductance waveform in fig. 2, and the torque waveform in fig. 3.
From fig. 1 we understand that in the range of 45 deg when the tooth is facing to eachother, the more the current is increased, the more the amount of magnetic flux decreases.
As a result, as the currency value increases, inductance decreases, which we also
understand from fig. 2. These are due to the nonlinear properties of the core material.
In fig. 3 it is possible to estimate the approximate average torque for each current value.
Also, it is possible to consider geometry in accordance with setting specifications, such as
peaks and reductions in the torque.

0.20
0.10
0.00

0.0
0

5

10

15

20

25

30

Angle (deg)Fig. 1. Flux linkage waveform

35

40

45

4(A)

6(A)

8(A)

10(A)

10
5
0

0

5

10

15

20

25

30

35

40

Angle (deg)Fig. 2. Inductance waveform

45

0

5

10

15

20

25

30

35

40

45

Angle (deg)Fig. 3. Torque waveform

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97

Case

180 Analysis of SR Motor Dynamic Characteristics

Shaft

Coil

Rotor coremodule: DP

Overview

Stator core

With the skyrocketing prices of rare earth magnets, expectations have been rising for SR (switched reluctance) motors because they have a
motor format that does not use permanent magnets. SR motors have a simple structure that can achieve solid performance at a low price.
However, torque generation depends only upon the saliency between the stator and rotor, so torque variations are extremely large and cause
vibration and noise, meaning that the use applications are limited. On the other hand, because of the skyrocketing prices of rare earth metals,
the improvement in current control technology, the possibility of optimized designs thanks to magnetic field analysis, and the rising ability to
reduce challenges, SR motors are being re-examined.
SR motors create their excitation state by alternating between opening and closing switches in accordance with the position of the rotor's
rotation, however the timing of the alternating causes major changes in the torque properties. Also, it is important not only to increase thetorque average and torque constant, but also to consider the optimum switch timing to control vibration and noise.
This example presents how to carry out an analysis with different switch timings to evaluate torque and current in SR motors.

Torque Waveform

Copper Loss Waveform

The torque waveform for each voltage application width θw with changing the voltage
application start angle θs are shown in fig .1 and fig .2.
By changing θs and θw, we understand that the torque waveform has changed.

The copper loss waveform for each voltage application width θw with changing thevoltage application start angle θs are shown in fig. 5 and fig. 6.
It is apparent that depending on the voltage application width, there is a large differencein the waveform. We can understand that if θw is 30 deg, copper loss is bigger in therange of multiple phase currents flowing simultaneously.

The current waveform of the A-phase coil for each voltage application width θw withchanging the voltage application start angle θs are shown in fig. 3 and fig. 4.
It is apparent that if the θw is 30 deg, it takes a longer time for the current to decrease to
0A. This is because the rotor tooth is nearing the stator and inductance has increased.

θs=-5(deg)θs=0(deg)θs=5(deg)θs=-5(deg)20

The average torque Tave(θs), torque ripple Tr(θs), torque ripple rate Tr(θs)/Tave(θs),
and torque constant Kt for each voltage application width θw with changing the voltage
application start angles are shown in table 1 and table 2.
In this case, average torque and torque constant are at a maximum and the torque ripple
rate is at a minimum at θw = 30 deg, θs = 5 deg. It is possible to consider optimum
switch timing in accordance with required specifications.

θs=-5(deg)θs=5(deg)

50Copper loss (W)Torque (Nm)

15

10

10
5
0

θs=0(deg)θs=5(deg)
180

160

160

140

140

120
100
80
60
40

-5
90

105

120

135

150

165

90

180

105

120

Fig. 1. Torque waveform (θw: 25 deg)θs=-5(deg)θs=5(deg)

12

12

10

10

8

8

6
4
2
0

180

θs=0(deg)

90

105

120

135

150

Rotation angle (deg)Fig. 3. Current waveform (θw: 25 deg)

JSOL Corporation

165

180

100
80
60
40
20
0

0
90

105

120

135

150

165

180

90

105

120

135

150

Fig. 5. Copper loss waveform (θw: 25 deg)2

Fig. 6. Copper loss waveform (θw: 30 deg)

θs
(deg)

Tave(θs)(Nm)

Tr(θs)(Nm)

Tr(θs)/Tave(θs)(%)

Kt(Nm/A)θs(deg)

Tr(θs)(Nm)

Tr(θs)/Tave(θs)
(%)

-5

5.2

16.2

313

1.01

-5

7.9

15.2

193

1.40

0

7.9

15.4

195

1.54

0

10.5

13.9

132

1.87

5

10.3

14.8

144

2.04

5

12.6

12.4

98

2.25

Tave(θs)
(Nm)

-2
120

135

150

165

180

Rotation angle (deg)Fig. 4. Current waveform (θw: 30 deg)180

θs=5(deg)

4

105

165

Rotation angle (deg)Rotation angle (deg)690θs=5(deg)

120

0

-2

98

165

Fig. 2. Torque waveform (θw: 30 deg)

Current (A)Current (A)θs=0(deg)

150

Rotation angle (deg)Rotation angle (deg)θs=-5(deg)135θs=0(deg)

200

180

20
-5

θs=-5(deg)

200

20

15
Torque (Nm)

θs=0(deg)

Switching Properties

Copper loss (W)Current Waveform

Table 1. Switching properties (θw: 25 deg)

Table 2. Switching properties (θw: 30 deg)

Kt
(Nm/A)JMAG Application Catalog

Case

181 Analysis of SR Motor Drive Characteristics

Shaft

Coil

Rotor coremodule: DP, LS

Overview

Stator core

With the skyrocketing prices of rare earth magnets, expectations have been rising for SR (switched reluctance) motors because they have a
motor format that does not use permanent magnets. SR motors have a simple structure that can achieve solid performance at a low price.
However, torque generation depends only upon the saliency between the stator and rotor, so torque variations are extremely large and cause
vibration and noise, meaning that the use applications are limited. On the other hand, because of the skyrocketing prices of rare earth metals,
the improvement in current control technology, the possibility of optimized designs thanks to magnetic field analysis, and the rising ability to
reduce challenges, SR motors are being re-examined.
SR motors sometimes drive while changing switch timing in accordance with rotation speed so it is useful to understand properties such as
torque, current and iron loss in accordance with revolution speed.
This example presents how to confirm drive characteristics such as torque, loss, and efficiency in a motor when its switch timing changes foreach rotation speed.

Drive Characteristics
The average torque, current, iron loss and efficiency with altered rotation speed andexcitation specifications are shown in figures 1 through 4.
From fig. 1 we understand that whether iron loss is taken into account or not, there isalmost no effect on torque. This is because the ratio of iron loss to output is small. From
fig. 2 we understand that current does not drop gradually in relation to rotation speed.
We understand that this is because switch timing has been changed in accordance withrotation speed, and switch timing exerts a large influence on its characteristics. In
addition, we understand from fig. 4 that as rotation speed increases, efficiency rises andpeaks at around 90%.

6
Iron loss not accounted for
Current (Arms)

Torque (Nm)

8
6

100%

10

90%

9

5

Iron loss accounted for

80%

8

3

4

2

2

1

70%

7

4
Iron loss (W)10Eﬀciency (%)

12

6
5
4
3

0
0

500

1000

1500

2000

2500

0

3000

Rotation speed (r/min)Fig. 1. Rotation speed versus torque

500

1000

1500

2000

2500

3000

50%
40%
30%
20%

2

10%

1
0

60%

0%

0
0

500

1000

1500

2000

2500

0

3000

500

1000

1500

2000

2500

Rotetion spwwd (r/min)Rotetion spwwd (r/min)Rotetion spwwd (r/min)Fig. 2. Rotation speed versus current

Fig. 3. Rotation speed versus iron loss

Fig. 4. Rotation speed versus efficiency

3000

Case

183 Agitation Force Analysis of an Induction Furnace

Molten metal

module: FQ

LiningCoil
Yoke

Overview
An induction furnace is an apparatus that uses high-frequency induction heating to melt metal. Running current through the coil surrounding
the crucible starts electromagnetic induction phenomena, which generate current in the metal in the crucible. This current produces joulelosses in the metal, which are used to heat and melt it. Magnetic yokes are arranged around the coil. The yokes are used as strong
components that prevent the Lorentz force generated by the coil from damaging and deforming it. The magnetic yokes also reduce theleakage flux that flows out of the appliance, preventing unintended heating in surrounding structures. Keeping the amount of material usedin the magnetic yokes to a minimum makes it possible to reduce the cost of the apparatus.
When evaluating to determine at what temperature does the molten metal in the crucible become uniform, the issue becomes one of whatcaused the molten metal to become agitated. As parts where heat was generated through induction heating are heated locally a gap intemperature with surrounding metals arises and sparks a convection phenomenon. Applying a large current and using the significant Lorentz
force generated in the molten metal enables agitation of the molten metal.
This analysis will evaluate the size of Lorentz force required to agitate and where this is generated.

Magnetic Flux Density Distribution

Magnetic Flux Density Distribution

The magnetic flux density distribution generated in the yokes and the air region is shownin fig. 1. The figure shows that Lorentz force is generated toward the inside of the molten
metal. This enables confirmation of the size and direction of the agitation force.

The magnetic flux density distribution generated in the molten metal is shown in fig. 3.

Lorentz Force of Molten Metal
The Lorentz force of the molten metal is shown in Table 1. We learned the force of the axis
direction component Z is small and radial direction component R is large.

Current Density Distribution
The current density distribution generated in the molten metal is shown in fig. 2. You cancheck the depth and size of the eddy current generated inside the molten metal. The
Lorentz force distribution generated in the molten metal is shown in fig. 2.

(Unit: A/m^2)(Unit: N/m^3)

Fig. 1. Lorentz force density distribution

Fig. 2. Current density distribution

Lorentz Force (R -component)-21.7(N)Lorentz Force (Z -component)

0.318(N)(Unit:T)

Fig. 3. Magnetic flux density distribution

Table 1. Magnetic flux density distribution

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99

Case

184

Reactor Sound Pressure Analysis taking Magnetostriction
into Accountmodule: DS, TR

Core

Coil

Overview
Reactors are used in all sorts of systems related to power systems. For example, they fill the role of making the current pulsation between an
inverter and a motor smoother. Reactor vibration noise can also be an issue and countermeasures are sought from analysis.
Among target reactors, vibration is not generated solely by electromagnetic force but also by magnetrostriction force caused bymagnetrostriction. To evaluate this phenomenon with decent accuracy requires an accurate grasp of resonance phenomena with an eigen
frequency by adding both electromagnetic force and magnetostrictive force to the vibratory force.
These notes will conduct an analysis of changed vibratory force and show a case study of confirming the impact magnetrostriction has on
vibration noise.

SpacerElectromagnetic Force Distribution
Fig. 1 shows the distribution of the electromagnetic forces. It is apparent that theelectromagnetic force is concentrated in the gap. This is because the air permeability inthe core is different from the gap.

Magnetostrictive force distribution
Magnetostrictive force distribution dependent on magnetic flux density is shown in Fig. 2.
As magnetostrictive force is larger than electromagnetic force, it needs to be added as avibratory force, which is not just electromagnetic force alone.

Eigenmode(Unit: N)

Fig. 1 Electromagnetic force distribution（Unit：N）

Fig. 2 Magnetrostriction force distribution

The eigenmode at around 10 kHz is shown in fig. 3. It was learned that the reactor has amode in which it vibrates horizontally in the vicinity of 100 kHz. A great deal of vibration is
likely to be caused by resonance phenomena between this eigenmode and theelectromagnetic force.

Acceleration
Fig. 4 shows acceleration distribution. In this analysis data, confirm the results of analysis
setting magnetic force only or magnetostrictive force only for the excitation force, andanalysis when setting both of them for the excitation force. Looking at Fig. 4 shows thatthe degree of acceleration generated is greater in magnetostrictive force than inelectromagnetic force. Taking both electromagnetic and magnetostrictive force into
account with excitation force sees each cancel out the other and vibration reduced.

Magnetostrictive
force only

Fig. 3 The eigenmode of approximately 10 kHz

Electromagnetic
force only

(Unit: m/s^2)
Magnetostrictiv
force andElectromagnetic force

Fig. 4 Acceleration distribution

Sound Pressure Distribution
The distribution of the sound pressure level is shown in fig. 5. In this analysis data, confirm
the results of analysis setting magnetic force only or magnetrostrictive force only for theexcitation force, as well as for analyses when setting both as the excitation force. Looking
at Fig. 5 shows that sound pressure is greater in electromagnetic force than inmagnetostrictive force as in the acceleration. Taking both electromagnetic andmagnetostrictive force into account with excitation force sees each cancel out the otherand sound pressure reduced.
Fig. 4 shows acceleration distribution. In this analysis data, confirm the results of analysis
setting magnetic force only or magnetrostrictive force only for the excitation force, andanalysis when setting both of them for the excitation force. Looking at Fig. 4 shows thatthe degree of acceleration generated is greater in magnetostrictive force than inelectromagnetic force. Taking both electromagnetic and magnetrostrictive force into
account with excitation force sees each cancel out the other and vibration reduced.

100

JSOL Corporation

Magnetostrictive
force only

Electromagnetic
force only

Magnetostrictiv
force andElectromagnetic force

Fig. 5 Sound pressure distribution(Unit: dB)

JMAG Application Catalog

Case

Tank

185 Analysis of Stray Loss in a Power Transformer

Clamp

Connector

module: FQ, LS

Overview
Transformers are made to be used long-term, so it has become an important design policy to control running costs from losses.
These losses include copper loss in the coil and iron loss in the core. In high-capacity transformers, however, there is also stray loss
in the tank from flux leakage from the core. Stray loss is not only the total loss value, it also poses the risk of generating heat locallyso there is a need to check loss distribution.
Predicting these losses and the heat that they generate is a vital component of transformer design, but it is difficult to estimate
them from desktop calculations, so evaluations and detailed analyses using the finite element method (FEM) are indispensable.
This note obtains loss distribution for each component in drive condition and checks local overheating.

Core
Secondary winding
Primary winding

ShieldJoule Loss Density Distribution

Iron Loss Density Distribution

Fig. 1 shows the magnetic flux density distribution, fig. 2 shows the current density
distribution, and fig. 3 shows the joule loss density distribution in the tank. From fig. 1, it isapparent that magnetic flux is leaking from the transformer to the tank. This leakage flux
generates eddy currents in the tank, as shown in fig. 2. Fig. 3 shows that joule losses getbigger in places with high current density.
Joule loss distribution or the shield, clamp and connector are displayed in figures 4
through 6 Check the danger of localized heating from joule loss distribution in each part.

Fig. 7 shows the iron loss density distribution in the core. From fig. 7, it is apparent thatiron losses increase in the inside corners. This is caused by the flow of magnet flux
concentrating on the shortest path through the magnetic circuit.

(Unit: T)

Fig. 1 Magnetic flux density distribution in the tank

LossesTable 1 shows the losses in each part.

(Unit: A/m^2)

Fig. 2 Current density distribution in the tank

(Unit: A/m^2)(Unit: W/m^3)

Fig. 3 Joule loss density distribution in the tankFig. 4 Joule loss density distribution of the shield

Part
Tank

(Unit: A/m^2)

Fig. 5 Joule loss density distribution of the clamp

7558.8Core80364.9

Secondary Winding

445566.9

Primary Winding

1111014.0

Shield14190.7

Clamp5444.7

Connector

5265.5(Unit: W/m^3)(Unit: A/m^2)

Fig. 6 Joule loss density distribution of the connector

Iron losses (W)Fig. 7 Iron loss density distribution

Table 1. Losses

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101

Case

186

Gear Induction Hardening Analysis
Using Two Frequencies

CoilWork piece(gear)

module: FQ, HT

Overview
Gears are created in such a way that the surfaces of their teeth are hard in order to resist the wear and tear that occurs when they come into
contact with the teeth of other gears. However, this has to be accomplished while maintaining the gear's overall toughness. By using
high-frequency induction heating, which is a type of surface hardening method, the teeth are heated rapidly on only their surface by a highfrequency power source. This process also has many other benefits, such as providing a clean working environment because it uses electricalequipment, being very efficient, and providing uniform results for each product. This is why it is being aggressively implemented in the field.
On the other hand, there are several factors that need to be studied in order to heat the gear's surface uniformly, such as how to adjust theheating coil's geometry, arrangement, current frequency and size.
The eddy currents generated by high-frequency varying magnetic fields are uneven in the tooth surface, so the material properties change agreat deal as the temperature rises. In order to handle the detailed phenomena, it is necessary to calculate the heat generation amount in anumerical analysis based on the finite element method (FEM).
In this material, the validity of coil geometry used for quenching is evaluated by heating the gear top and bottom at high and low
frequencies, respectively, to confirm uniform temperature distribution on the surface of the gear.

Coil leadCore

Eddy Current Loss Density Distribution of the Gear

Temperature Distribution and Variation of the Gear

Fig. 1 shows the eddy current loss density distribution of the surface of the gear and thecross-section of a tooth top. Each cross-section shown in the figure is the XY-plane at the
midway position of the tooth width. The magnetic field generated by the coil produces
eddy currents on the gear tooth surface. This eddy currents are distributed on the surfaceof the gear due to the skin effect.
Itʼ s also possible to confirm the changes in the eddy currents due to different frequencies.
Eddy currents are distributed on the gear bottom at low frequency and on the gearsurface and tip at high frequency due to the skin effect.

Temperature distribution in the gears is shown in Fig. 2. Fig. 3 shows the time
characteristics of temperature variations at each temperature measurement point. From
the temperature distribution, it is apparent that the eddy currents generate heat in thetooth top.
The difference in heat generation from the coil geometry can also be verified in Fig. 3. It isalso possible to confirm at each measurement point that the temperature exceeds theCurie point of 800 deg C at around 2.0 sec.

Magnetic Flux Lines
The Magnetic flux lines are shown in fig. 4. It can be confirmed that magnetic flux is
flowing from the core to the top of the gear.

1200

Temperature (deg C)

1000

model A (thick coil)

800
600
400

bottom
top

200

edge

(Unit: W/m^3)0Temperature
measurement points

model B (thin coil)Fig. 1 Eddy current loss density distribution
between 0.0 s and 0.2 s

Case

188

(Units: deg C)

Fig. 2 Temperature distribution at 2.4 s

0.0

1.0

2.0

3.0

4.0

low-frequency (5kHz)

high-frequency(100kHz)

Time (sec)Fig. 3 Temperature versus time at tooth top

Fig. 4 Magnetic Flux Lines

Thermal Demagnetization Analysis of IPM Motors
Accounting for Coercive Force Distribution of Magnets

Stator Core

module: DP

Rotor Magnet

Overview
The purpose of this Application Note is to help JMAG users understand the steps and settings used in a JMAG analysis. It is intended to help
those who are working with a new analysis target better understand the analysis steps and the setting contents.
The actual JMAG model described in this Application Note is available for download from the JMAG Application Catalog. This model will allow
you to view the model, settings and results. You can also create a template from the model to use as a starting point when analyzing yourown geometry. They can be downloaded using the same method as the analysis model data. Please refer to the JMAG manual for more
information on templates.
This example presents whether or not the demagnetization region can be smaller by assigning distribution to the coercive force of
permanent magnets.

Rotor Core
Coil

TorqueFig. 1 indicates the reduction of torque when the temperature has changed. It can be
confirmed that the reduction of torque can be controlled more than the original magnet
by assigning coercive force distribution.

9.00
8.50
8.00

Fig. 2 shows the demagnetization ratio (*1) distribution after temperature increase withmagnetization conditions prior to the raised temperature as a standard. By distributing
high coercive force on the outer magnet surface, reduction in demagnetization ratio of
the magnet can be confimed.
(*1) The demagnetization ratio shows whether demagnetization has occurred from thespecified standard magnetization.

102

JSOL Corporation

Torque (Nm)

7.50

Demagnetizing Ratio Distribution

7.00
Without coercive force distribution

6.50
6.00
5.50

Without coercive force distribution

5.00

With coercive force distribution

4.50
4.00
50

70

90

110

Temparature (deg C)

Fig. 1 Temperature versus Torque

130

150

（Units: %）With coercive force distribution

Fig. 2 Demagnetizing ratio distribution

JMAG Application Catalog

Case

189

Response Characteristics Analysis of Injectors Accounting
for CollisionCore

module: DP

Coil
Yoke

Overview
A solenoid type injector used in engines opens a valve and injects fuel by moving a plunger with magnetic force created by anelectromagnet. Injectors in engines need to respond quickly for applied voltage to improve fuel consumption by maintaining the amount offuel flow. In solenoid injectors, one of the reasons that the response is delayed is eddy currents, which are produced when the magnetic flux
generated by current flow undergoes time variations. The eddy currents are generated in a direction that inhibits changes in the magnetic
flux, causing a delay in the initial rise of the attraction force when the current begins to flow. This reduces the injector's responsiveness. Also,
the reflection of the plunger at collision during open/close inhibits the open/close motion. By running a transient response analysis in JMAG,
the responsiveness of the injector accounting for the effect of the collision of the plunger and the eddy current can be obtained. Identifying
the places where eddy currents are generated enables a designer to study whether or not responsiveness can be improved.
For this case, direct current voltage is applied to the solenoid type injector and the response characteristics is obtained after accounting forthe effect of the collision and eddy current.

Plunger

Position of Plunger and Coil Current
Position of plunger (μm)

0.5

80
60
0.25
40
20
0

0
0

4

6

8

10

Time (ms)Fig. 1 Position of plunger and coil current

Eddy current and collision are accounted for
Eddy current and collision are not accounted forTranslation
direction

Current starts
to ﬂow

Comparison of Response Characteristics

Plunger moves
in the direction
valve opens

Plunger moves
in the direction
valve closes

100
80
60
40
20
0

(Unit：A/m^2)

0

2

4

6

8

10

Time (ms)Fig. 2 Current density distribution

Fig. 3 Comparison of response characteristics

Eddy current and collision are accounted for

Eddy current and collision are accounted for

Eddy current and collision are not accounted for

Eddy current and collision are not accounted for
0.5

25

0.45
Attraction Force(N)Depending on whether you account for eddy current and collision, the comparison of
response characteristics of the plunger for each time is shown in Fig. 3 The attraction
force characteristics is shown in Fig. 4, and current characteristics is in Fig. 5. The positionof the plunger is positive in the direction the valve opens and the valvle is full open at
position 100(μm) and the valve is closed at position 0 (μm).
Fig. 3 shows that the plunger is bouncing near 3.2 (ms) due to the collision, when eddy
current and collision are accounted for. Also, the responsiveness is slower compared towhen they are not accounted for.
The slow responsiveness is caused by the reduced attraction force. In Fig. 4, the lower
attraction force can be verified when eddy current and collision are considered compared
to when they are not considered.
As for the current characteristics, when the eddy current and collision are considered, thelarger current value can be seen compared to when they are not considered. For more
information about the cause, see the Application Note "66 - Operating Time Analysis of an
Injector ".
When the plunger moves in the direction to open the valve, there will be a delay inresponse of approximately 1.5(ms) due to the fuel pressure, spring and the weight of theplunger in the case that eddy current and collision are not accounted for. In the casewhen eddy current and collision are accounted for, the delay in response is approximately
2 (ms) and it can be confirmed that the cause of delay in response is the decrease of theattractive force due to eddy current. The plunger will move to the direction of closing thevalve from 7 (ms). Also, the eddy current caused magnetic flux and result in a delay of theplunger displacement. In addition, the plunger will collide and reflect, and it can be
confirmed that more time is required for the open/close of the valve.

2

Position of plunger (μm)

Fig. 2 shows the current density distribution when current starts to flow at 0.05 (ms),when the plunger moves in the direction the valve opens at 2.2 (ms), when the plunger
moves in the direction valve closes at 7.1 (ms). The translation direction of the plunger isthe direction the valve opens. By running current through the coil, eddy current will occur
in the core, york and plunger. Eddy current will focus on the surface of each part sincemagnetic flux flow will suddenly increase at the start of current flow and because thepower supply is direct current, eddy current will flow inside the parts as time passes.
These eddy currents become the cause of the decrease in responsiveness.

Coil current

0.4

20
Current (A)Current Density Distribution

Position of plunger
100

Coil current(A)When accounting for collision and eddy current, the comparison of coil current andposition of the plunger for each time is shown in Fig. 1. The position of the plunger ispositive in the direction the valve opens, and the valve is fully open at position 100(μm)
and closed at position 0(μm). The coil current value continues to rise to keep applying
supply voltage to the coil until the time is 4(ms). To keep the current at a minimum whilethe valve is fully open during 4ms to 6ms, the voltage is controlled to maintain the coilcurrent at 0.25 (A). The coil current is controlled so that it does not flow from 6(ms) andthe valve will start to close due to fuel pressure, spring, and plunger weight.

15
10

0.35
0.3
0.25
0.2
0.15

5

0.1
0.05

0
0

2

4

6
Time (ms)Fig. 4 Attraction Force Characteristics

8

10

0
0

2

4

6

8

10

Time (ms)Fig. 5 Current Characteristics

https://www.jmag-international.com/

103

Case

190

Arrangement Optimization of Wireless Power Transfer
SystemReceiving Coilmodule: FQ

Overview
Originally there were only two types of wireless power transfer systems: proximity and long distance. However, recently mid-distance devices
have been developed and has it made it possible for highly-efficient power transmission even when the transmission and receiving end are
disconnected. Although there are charging devices for cellular phones that have already been released as products, research of charging
large products such as automobiles and machine tools is currently in the process.
This document introduces the procedure in obtaining the optimum solution for geometries of transmission coils (angle that defines position)
for simple magnetic resonance wireless power transfer system. Even if you are a user of applications excluding wireless power transfer
systems, please use this document as a function reference material to get a grasp of the optimization procedure and the flow of result
evaluation.

Transmission CoilEﬃciency
Fig. 1 shows the efficiency relative to the tilt calculated in optimization.
This shows that efficiency is decreasing as the tilt gets bigger.

80

0.16

70

0.14

Coupling Coeﬃcient

If the coupling coefficient, which is the coupling level of the transmission coil and thereceiving coil, becomes poor, it will also decrease efficiency. The coupling coefficient inrelation to the tilt is shown in fig. 2. The coupling coefficient will get smaller as the tilt getslarger. Fig. 3 shows the mutual inductance in relation to the tilt. It can also be confirmedthat as the tilt gets larger, mutual inductance drops as well. This is because flux linkage tothe receiving coil decreases, as shown in fig. 4.

Eﬃciency (%)

Coupling Coeﬃcient

0.18

90

60
50
40
30
20
10
0

0.12
0.1
0.08
0.06
0.04
0.02

0

5

10

15

20

25

0

30

0

5

10

15

Tilt (deg)

20

25

30

25

30

Tilt (deg)Fig. 1 Efficiency

Fig. 2 Coupling Coefficient

5.0E-06
4.5E-06
Mutual Inductance (H)Fig. 5 shows the magnetic flux line at 0 deg and 30 deg as the tilts of receiving coil. The
magnetic flux line is displayed at 90 deg for phase settings, where transmission coilcurrent will be the maximum value.
The receiving coil where tilt occurs has been marked with a circle. When comparing 0
degrees and 30 deg, the coil at 30 deg is outside the flux line. For this reason, themagnetic flux that links to the receiving coil drops. It can be said that adjusting
parameters such as capacitance is required when creating models with tilts.

4.0E-06
3.5E-06
3.0E-06
2.5E-06
2.0E-06
1.5E-06
1.0E-06
5.0E-07
0.0E+00

0

5

10

15

20

25

30

Flux Linkage of the Receiving Coil (Wb)

Magnetic Flux Lines
5.0E-06
4.5E-06
4.0E-06
3.5E-06
3.0E-06
2.5E-06
2.0E-06
1.5E-06
1.0E-06
5.0E-07
0.0E+00
0

5

10

Tilt (deg)Fig. 3 Mutual Inductance15

Fig. 4 Flux Linkage of the Receiving Coil(a) 0 deg tilt(b) 30 deg tilt

Fig. 5 Magnetic Flux Lines

104

JSOL Corporation

20

Tilt (deg)JMAG Application Catalog

Torque Analysis of Three-Phase Induction Motors
Accounting for the Skew Using the Multi-slice Method

Case

191

Rotor core

Stator coreCage

module: DP

Overview
An induction motor can utilize skew easily because the cage is constructed by metallic casting such as die casting. When skew is applied, itarranges the variations in the magnetic flux that links to the cage in a sinusoidal wave. This makes it possible to eliminate the harmoniccomponents from the induction current that cause negative torque and constrain things like the torque variations caused by influence fromthe slots. Generally when skew is applied, magnetic flux also flows in the axis direction, and this requires that a 3D analysis is performed inorder to thoroughly evaluate skew effect. However, 3D analyses tend to cause an inordinate amount of calculation costs. The multi-slice
method, representing skew approximately with a 2D cross-section superimposition, is effective.
In this document, the multi-slice method is used to compare torque waveforms of a cage three-phase induction motor with and withoutskew, and is then used to verify the effects of using skew to reduce torque variations.

Torque Characteristic

Current in the Secondary Conductor

The harmonics of torque are significantly deleted due to skewing, showing that this isconnected to a decrease in vibration and noise.

Average current density distribution in one period of electric angle in the secondary
conductor is shown in fig. 2 and 3, a section graph is shown in fig. 4, and current
waveform and frequency components for the secondary conductor are shown in fig. 5
and 6.
As shown in fig. 2, 3 and 4, bar current without skewing is concentrated on the rotor
surface, and with skewing, bar current enters almost to the inside. As shown in fig. 5 and6, it is believed that this is due to the harmonic components in the secondary current
decreases due to skewing, and the skin depth then becomes thicker. Slot harmoniccomponents around 420(Hz), determined by the number of revolutions and slots, aregreatly reduced.

2.5

Torque (Nm)

2.3
2.1
With/without skew

Tave (Nm)

Tr (Nm)Tr/Tave (%)

With skew

2.2

0.4

18

Without skew

2.2

0.1

3

1.9
no skew

1.7

with skew
1.5
234

252

270

288

306

324

342

360

(Unit: A/m^2)(Unit: A/m^2)

Angle (deg)Fig. 1 Torque wave

Table 1 Torque characteristic

Fig. 2 Average current density distribution (no skew)

50-50
with skew

8.4E+06
8.3E+06-100
Current (A)no skew

8.5E+06

Current (A)Current Density (Average) ( A/m^2)

08.6E+06

Fig. 3 Average current density distribution (with skew)

-150
-200

45

no skew

40

with skew

35
30
25
20
15

8.2E+06

no skew-250
8.1E+06

10

with skew

8.0E+06
0

1

2

3

4

5

6

7

-300
234

252

270

288

306

324

342

5
360

Angle (deg)Distance (mm)Fig. 4 Section graph of average current density

Fig. 5 Current waveform in the secondary conductor

0
50

150

250

350

450 550 650 750
Frequency (Hz)850

950 1,050

Fig. 6 Current in the secondary conductor(frequency component)

https://www.jmag-international.com/

105Response Characteristics Analysis of Solenoid Valves
Accounting for Residual Magnetization

Case

192

Core

Coil

module: DP

Yoke

Overv iew

Plunger

A solenoid valve becomes an electromagnet by applying current through the coil and moves the plunger by using electromagnetic force(attractive force) generated in the magnetic materials. Especially in products that require a quick response in the plunger, the time it takes forthe valve to open and close is an evaluation indicator, and it is necessary to investigate in advance causes for decreases in responsivenesssuch as eddy current.
In this example. we will discuss the steps to investigate the effect on responsiveness caused by residual magnetization generated in themagnetic materials, and look at case examples.

ResponsivenessFig. 1 shows a complete response graph comparing residual magnetization being
accounted for and not accounted for, fig. 2 shows responsiveness when the valve opens,and fig. 3 shows responsiveness when the valve closes.
Fig. 1 demonstrates that responsiveness for the valve opening and closing differs
depending if residual magnetization is accounted for or not. Fig. 2 and 3 demonstrates
that responsiveness is lowered when residual magnetization is accounted for.

Responsiveness When Valve Opens
The cause behind a slower responsiveness when accounting for residual magnetization
will be examined. Attraction force in the Y-axis direction when the valve opens is shownin Fig 4, and current values are shown in Fig. 5. We can verify that attractive force
decreases in the Y-axis direction, and current values increase.
The cause behind a decrease in attractive force in this case is due to the difference in the
how magnetic flux flows. Magnetic flux during the valve opening is shown in Fig. 6. It
becomes clear that magnetic flux decreases when residual magnetization is accounted
when compared to when it isn't accounted for. The root cause is not due to the effect ofresidual magnetization, but due to the handling of the magnetization properties whenthe valve opens.
Magnetization properties used when the valve opens are shown in Fig. 7. During ananalysis, an initial magnetization curve is used when residual magnetization is accounted
for, and an average magnetization curve is used when residual magnetization is not

With residual magnetization

No residual magnetization

100

80
60
40

No residual magnetization

With residual magnetization

0

2

4

6

80
60
40

0.5

Time, msec

1

1.5

0.4
0
-0.4

No residual magnetization
No residual magnetization

With residual magnetization

With residual magnetization

300

500

0
1.2

1.4

1.6

1.8

Fig. 8. Back Electromotive Force When the Valve Opens

No residual magnetization

With residual magnetization

0.025

15.0

10.0
40

0.020
0.015
0.010

5.0
0.005

0
5

5.5

6

6.5

1

7

1.2

1.6

1.8

(Unit: A/m)

0.000

2

1.0

1.2

Time, msec

Time, msec

Fig. 3. Responsiveness When the Valve Closes

No residual magnetization

1.4

No residual magnetization

With residual magnetization

With residual magnetization

No residual magnetization
6

0.8

0.020

5

0.010

0.2

0.005

1.8

2.0

With residual magnetization

3
2
1

0.0
1

1.2

1.4

1.6

1.8

Time, msec

Fig. 5. Current Values When the Valve Opens

106

JSOL Corporation

2

1.0

1.2

1.4

1.6

1.8

Time, msec

Fig. 6. Magnetic Flux When the Valve Opens

2.0

0
5.0

5.5

6.0
Time, msec

6.5

Fig. 11. Attractive Force When the Valve Closes

Without residual
magnetization

With residual
magnetization

Fig. 10. Magnetization When the Valve Closes

4

0.015

Force, N

Flux, Wb

0.025

0.4

1.6

Fig. 9. Amount of Magnetic Flux Per Ampere

1.0

0.6

1.4
Time, msec

Fig. 4. Attractive Force When the Valve Opens(Y-axis Direction)

2.0

Time, msec

0.030

Flux/Current, Wb/A

Force, N

60

100

Fig. 7. B-H Curve

20.0

80

-100

1.0

20

Current, A-300

Magnetic Field, A/m

25.0

100

4

2

-1.6
-500

2

Fig. 2. Responsiveness When the Valve Opens

120

6

-0.8

Time, msec

Fig. 1. ResponsivenessWith residual magnetization

8

0.8

-1.2

0

8

No residual magnetization
10

1.2

0

0

With residual magnetization

1.6

20

20

Position of plunger, μm

Magnetization contour plot when the valve closes at 0.005s is shown in Fig. 10. It showsthat the yolk is magnetized while valve closes if residual magnetization is accounted for.
Fig. 11 indicates that attractive force is maintained due to residual magnetization of yolk.
This applied attractive force inhibits valve movement, and responsiveness decreases.

EMF, V

100

Responsiveness When Valve Closes

Magnetic ﬂux density, T

120
Position of plunger, μm

Position of plunger, μm

No residual magnetization120

accounted for, and then analysis is performed. In cases when average magnetization
curves (not accounting for residual magnetization) are used, magnetic flux is generated
easily when current flows, as shown in Fig. 7. As a result, attractive force in the Y-axis
direction increases, and valve responsiveness improves compared to the initial
magnetization curve (accounting for residual magnetization).
Furthermore, when residual magnetization is not accounted for, attractive force becomeslarger than when residual magnetization is accounted for, but current values become
smaller. For this case study, as shown in Fig. 8, using an average magnetization curve (notaccounting for residual magnetization) results in a larger back electromotive force due toa lot of magnetic flux flow. Because of this, as shown in Fig. 9, as looking at the amount ofmagnetic flux generated per Ampere when the valve opens, the case where residual
magnetization is not accounted for generates greater amount of magnetic flux per
Ampere. Therefore, attractive force increases even if current values are small.
The cause for a decrease in responsiveness when the valve opens, as shown above, is notdue to the effects of residual magnetization, but due to the differences in handlingmagnetization properties.

7.0

JMAG Application Catalog

Case

193

Calculation of Equivalent Circuit Parameters
in a Single-Phase Induction Motor

Stator Core
Cage
(end-rings)

module: DP

Overview

Cage (bars)

Rotor Core

Since single phase induction motors can be used with single-phase AC, which is common in general household power supply, it is widelyused for electrical home and office appliances such as washers and fans as small power motors. However, unlike three-phase AC, single phaseAC cannot independently create a rotational magnetic field, and since it cannot run a motor at this state, there is a need to develop arotational magnetic field somehow and make it run.
Characteristics of single phase induction motors are affected by factors such as the secondary side and leakage inductance. These are called
equivalent circuit parameters and they are important because they characterize the device's properties. Equivalent circuit parameters are
greatly affected by current distribution induced by secondary conductors and magnetic saturation near the gap, and accurately
understanding these characteristics using finite element analysis will be required.
In this example, We will be obtaining the secondary resistance, leakage inductance and excitation inductance of main coil winding of
capacitator-start single-phase induction motor that has auxiliary winding.

Shaft

Equivalent Circuit Parameter
The equivalent circuit parameter of 100V voltage and frequency 50Hz will be shown in fig.
1. In addition, the magnetic flux density (intensity) distribution of lock test analysis isshown in fig. 1, and the magnetic flux density (intensity) distribution of no-load test
analysis in fig.2. Fig. 3 will show the current density (intensity) distribution of the lock test
analysis bar.

Primary resistance

0.80 ohm

Secondary resistance

4.35 ohm

Primary, secondary leakage
inductance

2.19 mH

excitation inductance

204 mH(Unit: T)

Table 1 Each value of equivalent circuit parameters

Fig.1 Magnetic flux density distribution (intensity)
during lock test

(Unit: T)

Fig. 2 Magnetic flux density distribution (intensity)
during no-load test

(Unit: A/m^2)

Fig. 3 Current density distribution during lock test

https://www.jmag-international.com/

107

Case

194

Analysis of High-Speed Rotation Motors Accounting
for Eddy Current

Stator coreCoilmodule: DP, LS

Overview
PM synchronous motors are used in HEV drives, air conditioner compressors, and various other industries. Especially in case of PM
synchronous motors for HEV drives, high-speed rotation is advancing faster than ever to achieve a small size with high output density. The
iron core of a PM synchronous motor is composed of magnetic steel sheets, and the iron loss generated in the magnetic steel sheets havegenerally been evaluated in the post processing of a magnetic field analysis, In this case, the effects of iron loss on the magnetic field analysis
were not accounted for. However, the effects of iron loss on the magnetic field can no longer be ignored as rotation speeds are further
increased, which causes the risk of not being able to accurately analyze the torque output by the motor when evaluating iron loss inpost-processing.
Here we discuss the steps for accounting for eddy current loss occurring in the magnetic steel sheets during a magnetic field analysis, andthen analyze the effects on torque.

Rotor core

MagnetTorque characteristics
5.0

2,000

4.5

1,500

No eddy current

1,000

Accounting for eddy current

4.0Line Voltage V

3.5
Torque Nm

Fig. 1 compares torques between one that does not account for eddy current inside themagnetic steel sheet during magnetic field analysis, and one that does. It shows thattransient phenomena occur in the early stages of analysis when accounting for eddy
current. When the revolution speed is high like in this motor, the torque is overestimatedwhen eddy current is not accounted for in the magnetic steel sheet.

3.0
2.5
2.0
1.5
1.0

No eddy current

Fig. 2 shows the comparison between U-V line voltage when the eddy current inside themagnetic steel sheet is accounted for in the magnetic field analysis and when it is not.
Like it is with torque, if eddy current inside the magnetic steel sheet is not accounted for,voltage will be overestimated. Fig. 2 shows that accounting for eddy current controls the
harmonic component of voltage. This is because changes in the magnetic flux of magnets
and current is controlled by the eddy current.

0-500-1,000-1,500Accounting for eddy current

0.5-2,0000.0Line voltage

500

0

45

90

135

180

225

270

315

0

360

45

90

135

180

225

270

315

360

Angle deg

Angle deg

Fig. 1 Torque Wave

Fig. 2 U-V Line Voltage Waveform

4000
3500

Iron Loss, Eﬃciency

損失 (W)

Fig.3 shows the eddy current distribution (average value between electric angle of 180 to360deg) obtained in the analysis accounting for eddy current of magnetic steel sheets.
Large eddy currents occur in the tip of the teeth part and rotor surface of the stator. Next,
the iron loss of analysis accounting for eddy current of magnetic steel sheets and one thatis not will be summarized in fig 4. Iron losses are sum of the hysteresis loss and eddy
current loss of magnetic steel sheet and eddy current loss of the magnet. However, evenin an analysis accounting for eddy current, the hysteresis loss is calculated with
post-processing of the magnetic field analysis.
The efficiency calculated from copper loss, iron loss and the output will be shown in fig 5.
This table shows that there is when eddy current of magnetic steel sheets is notaccounted for in the magnetic field analysis, efficiency is also overestimated by about 1%.

3000

電磁鋼板
渦電流損 (W)
電磁鋼板
ヒステリシス損 (W)

1000
500
0
(Unit：A/m^2)

Fig. 3 Eddy current distribution

90.0
89.0

Eﬃciency %

87.0
86.0
85.0
84.0
83.0
82.0
81.0
80.0
No eddy current

Fig. 5 Efficiency

JSOL Corporation

2000
1500

88.0

108

磁石
渦電流損 (W)2500

Accounting for eddy current

渦電流なし

Fig. 4 Iron loss

渦電流考慮

JMAG Application Catalog

Case

195

Torque characteristics analysis of three-phaseinduction motors using 3D correction function

Stator core
Coil
Coil-end

module: DP, TR

Cage(end-rings)
Rotor core

Overview
An induction motor is a motor in which the rotating magnetic field of the stator winding causes induced current to flow in an secondary
conductor, which exerts force on the rotor in the rotational direction and causes it to spin.
In an induction motor, magnetic flux in the axis direction, such as leakage flux in the coil end, can have a large effect on motor characteristics.
However, transient phenomena are strong in the inductance motor, and analysis time is needed in order to achieve a steady state solution.
Due to this, being able to evaluate characteristics in a 2D analysis is desirable, but 3D effects such as leakage flux in the coil ends are ignored
in a 2D analysis, and there are cases when an accurate analysis cannot be obtained. In JMAG, highly accurate results can be quickly obtainedby performing corrections on 3D effects.
This document introduces introduces a case study which obtains inductance correction values, and then obtains N-T characteristics(revolution speed-torque characteristics) using the 3D correction function.

Shaft
Cage (bars)

Self Inductance

N-T Characteristics

Fig.1 shows the results of self inductance at lock tests.
Fig.1 shows that inductance has decreased by approximately 25% in a 2D analysis whencompared to a 3D analysis. This is because a 2D analysis cannot account for leakage
inductance in the coil end.

Fig.2 shows those N-T characteristics of 2D, 3D and 2D analysis with 3D correction andfig.3 shows N-I characteristics.
Fig. 2 shows that N-T characteristics becomes the same as 3D by using 3D correction. As
shown in fig.3, this is because current flowing in the coil has come closer by accounting
for leakage inductance.

120

160

0.006

140

100

0.005
0.004
2D

0.003

3D
0.002

100
80
60
2D̲no correction40

2D̲corrected

20

0.001

Current (Arms)

Torque (Nm)

Self Inductance (H)120

V Phase40

2D̲no correction
2D̲corrected

20

3D

0
0

U Phase

60

3D

0
0

80

300

W Phase

600

900

1200

0

1500

300

Fig. 2 N-T characteristics

Fig. 1 Self Inductance

600

900

1200

1500

Rotation speed (r/min)Rotation speed (r/min)Fig. 3 N-I characteristics

Case

196 Analysis of coil stray capacitanceCoilmodule: EL

Overview
In a high frequency application, not only the coil's inductance component but its capacitance component cannot be ignored and results indegradation of the circuit impedance characteristics and inconsistencies in resonance frequencies. Stray capacitance that occurs between thecoil windings is one of them. For example, with a helical coil, electrical potential varies depending on its rotation so difference in electricalpotential occurs between the neighboring windings. As a result, capacitance occurs between the windings and operates as a condenser
component.
The stray capacitance that occurs in the coil depends on its geometry. To obtain the stray capacitance that occurs in various types of coils, amagnetic analysis using FEM is effective. By modeling the entire coil, stray capacitance accounting for the distance between windings anddifferences in winding pattern can be obtained.
This document introduces target on the helical coil, changes the distance between windings, and obtains stray capacitance that occurs ineach coil.

Stray capacitanceElectric potential distribution, Current density distribution

Stray capacitance obtained from the analysis results are shown in Table 1. The stray
capacitance is larger when the coil distance is shorter.

Each electric potential (intensity) distribution of when electric potential of 1(V) is supplied
between terminals of the coil with coil distance of 1.5 (mm) is shown in Fig.1, and current
density distribution (intensity) is shown in Fig.2. It can be confirmed that the electric
potential distribution inside the coil is solved from the supplied electric potential
boundary condition (amplitude: 1V, 0V). It can be confirmed that current flowing in thecoil depending on the electric potential difference is solved, and current density
distribution is obtained.

coil distance mm

15Stray capacitance pF

1.42

0.89

1 mm

Table 1 Stray capacitance

5 mm

Fig. 1 Electric potential distribution (RMS)(Unit：V)

1 mm

5 mm

(Unit：A/m^2)

Fig. 2 Current density distribution (RMS)

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109

Case

197 Electric ﬁeld distribution Analysis of a Electroplating

Electrolyte solution

module: EL

Positive pole

Overview

Negative pole

Electroplating is a technology that forms a metal film on the surface of an object and is currently used in various fields from electric parts toautomobiles.
When electroplating, use a solution that includes the component you wish to use. Within that solvent, set the board including thecomponent you wish to electroplate at a positive pole, and set the board you with to electroplate at the negative pole, and run voltage. Then,current will flow, and the component of the positive pole will melt in the solvent, and electroplating for the negative pole side will start.
This example will introduce the electric field distribution in a model with the negative pole geometry facing the positive pole that is notparallel but slanted.

Electric ﬁeld distribution

Current density distribution

Fig.1 shows the flat model and the slanted model in the electric field graph, and Fig.2
shows the contour of electric field distribution. Also, Fig.1 is measured along the electricpole.
For the flat model shown with the red line, Fig.1 shows that it is symmetrical at 15mm as
the center. On the other hand, for the slanted model as shown with the blue line, it can beseen that the electric field gets larger as the distance between positive and negative poles
becomes smaller.
Fig.2 shows that biases are occurring in the electric field for slanted model on thenegative pole.

Fig.3 shows the current density graph of the flat model and the slanted model, and Fig.4
shows the contour of the current density distribution.
For the flat model shown with the red line, Fig.3 shows that current density is symmetricalat 15mm as the center. On the other hand, for a slanted model as shown with the blueline, the current density close to the positive pole will be higher. In this case study, since
the material property inside the solution is constant, the electric field intensity is in
proportion with the amount of the current. For this reason, current density would havethe same results.

600000Flat model
Slanted model

10000

Current density (A/m^2)

Electric ﬁeld (V/mm)

12000

8000
(a) Flat model

6000
4000
2000

（Unit：V/m）0

Flat model
Slanted model

500000400000300000((a)) Flat model

200000100000
0

0

5

10

15
20
Distance (mm)

25

30

0

(b) Slanted model

5

10

15

20

25

Fig.1 Electric field(b) Slanted model

Fig.3 Current density

Fig.2 Electric field distribution（Unit：A/m^2）

30

Distance (mm)Fig.4 Current density distribution

BusbarCase

198 Busbar Thermal Stress Analysis

module: FQ, DS, HT

Bonding wire

Overview
Current is supplied via busbars or wire bonding in power supply lines for power electronics devices such as inverters. Because inverters andsimilar devices operate with PWM carrier frequencies of several kHz, high-frequency current flows in their busbars. Influences from the skineffect cannot be ignored in this kind of high-frequency current, so increased resistance and loss become problems.Furthermore, whenthermal strain is caused due to excessive heat generation, disconnections may occur in a worse case scenario of stress accumulating in thebonding wire connection region. This type of stress is called thermal stress. When thermal stress exceeds yield stress, irreversible changes togeometry occur, so verifying thermal stress distribution due to heat generation is important.
Because a busbar's geometry is complicated, it is difficult to predict in advance where there will be unevenness in the current's flow whilecurrent is running, and whether this will cause heat generation. With magnetic field analysis using the Finite Element Method (FEM), it ispossible to correctly obtain the unevenness in current distribution and the joule loss, and then predict the temperature distribution withthese as causes of heat generation.
Here, temperature distributions for busbars are obtained when the power supply frequency is changed.

Thin copper sheet
Silicon resin

Current Density Distribution

Thermal Stress Distribution

Fig.1 shows current density distribution. Distribution is occurring in current density for thebus bar and thin copper sheet due to the skin effect as indicated in Fig.1. As the section of
the bonding wire where current flows is small, it is clear that current density is particularly
getting higher, when compared to others.

Fig.3 shows thermal stress distribution of the bonding wire. There is distribution inthermal stress, and it also shows that large stress is put on the connection joint between
the busbar and the copper thin sheet, as well as the top of the bending part. Caution
needs to be taken because temperature increase will not only cause burn out, but it may
also damage the wire due to local thermal stress.

Temperature Distribution
The temperature distribution of the bus bar is shown in Fig.2. Temperature is rising withthe bonding wire and the copper thin sheets. The temperature of the bonding wire has
increased especially at over 200 deg C.

(Unit: A/m^2)

Fig. 1 Current density distribution

110

JSOL Corporation(Unit: MPa)

(Unit: deg C)

Fig.2 Temperature density distribution

Fig. 3 Thermal stress distribution

JMAG Application Catalog

Eﬃciency Analysis of Wireless Power Transfer via
Electromagnetic Induction Method using Resonance Circuits

Case

199

module: FQ

Ferrite Core（Receiving）

Overview
In recent years, electromagnetic induction type transmission is considered for applied use in wireless power supply systems for electricvehicles (hereinafter, referred to as EV) and plug-in hybrid cars (hereinafter, referred to as PHEV), which are demanded for practical use. As for
wireless power supply systems for EV / PHEV, even if the car-equipped coil of the receiving end and the ground buried coil of the transmission
end slip out of place in the horizontal position, electric power transmission needs to be retained with high efficiency. Recently, circuits withexcitation condensers, which retain a large power supply of electric power is also considered.
Efficiency changes, which is triggered by change of positioning, is primarily because of the change in magnetic flux in the receiving end
where the transmission coil is slightly shifted in relation to the receiving coil. Furthermore, run magnetic field analysis when evaluating power
supply of electric power or capturing the variations in spatial magnetic flux.
This example presents, an electromagnetic induction-type wireless power supply system using an excitation circuit positioned with a parallel
resonance condenser on the receiving end, and a series resonance condenser on the transmission end will be analyzed. An efficiency change
in relation to the disposition in the horizontal direction is evaluated from both power transmission efficiency and the coupling coefficient.

Coil（Receiving）

Coil（Transmission）
Ferrite Core（Transmission）

Power Value / Power Transmission Eﬃciency

Coupling Coeﬃcient

Graphs of power value and power transmission efficiency when changing the horizontal
distance between the transmission and receiving coils are shown in Fig. 1.
When there are no displacements in the horizontal direction, power close to 4.7kW can beobtained with a maximum efficiency of 99%. The state where both the transmission andreceiving coil centers match up is considered 0, and the distance that it moves in thehorizontal direction (horizontal distance) is defined as displacement in the horizontal
direction. As the horizontal distance between the transmission and receiving coil getslarger and displaced, both power supply and power transmission efficiency will decrease.
Power transmission efficiency / power value will both be 0 at 180mm.
Since voltage of the transmission coil is constant in this model, the input voltage willchange. Reduction of power value is thought to be because of the circuit design. With
displacement in the horizontal direction, self-inductance of the coil will not change, butthere will be a change in mutual inductance between coils. This causes the resonancefrequency determined by the inductance and capacitor to be displaced, and powertransmission efficiency will drop.

Fig. 2 shows the coupling coefficient when the horizontal distance between transmission
and receiving coils is changed.
As with coupling coefficients, it will drop due to displacements in the horizontal direction.

Power transmission eﬃciency

0

50

100
150
Horizontal direction (mm)Fig.3 shows the magnetic flux density distribution when the horizontal distance ischanged between the transmission and receiving coils. As horizontal distance changes,
magnetic flux linked to the receiving coil (above) decreases. It is possible to confirm thatthese changes in flow of magnetic flux are the cause of decrease in power supply,
efficiency, and coupling coefficient.

0.300
Coupling coeﬃcients K

100.0
90.0
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.0
200

Power transmission eﬃciency (%)

Power supply (W)Power supply
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0

Magnetic Flux Density

0.250
0.200

Horizontal distance 0mm

Horizontal distance 30mm

Horizontal distance 60mm

Horizontal distance 120mm

0.150
0.100
0.050

(Unit：T)

0.000
0

50

100

150

200

Horizontal direction (mm)Horizontal distance 180mm

Fig. 1 Power supply and power transmission
efficiency relative to the horizontal direction

Fig. 2 Coupling coefficients relative to the horizontal
direction

Fig. 3 Magnetic Flux Density

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Case

200

Iron Loss Analysis of IPM Motor using Hysteresis Model
module: DP, LS

Overview

Recently, PM motors are being applied in home appliances and electric automobiles, and the demand for compactness and high power isincreasing. On the other hand, small and high-power motors tend to be affected by high-frequency components due to magnetic saturationand high rotations, which results in increased loss in the motor; therefore, they are also expected to have increased efficiency. In therealization of highly-efficient motors, it will become increasingly important to accurately estimate loss.
In the drive circuit of a PM motor, there are times when current vector control using PWM control (pulse width modulation control) is run.
Current waveform that is being supplied by PWM control have carrier harmonic current superimposed on the basic wave control, and withthis carrier harmonic current, high-frequency of the magnetic field is applied to the motor core and iron loss occurs. With harmoniccomponents and magnetic saturation of the iron core due to permanent magnets, it will be necessary to evaluate iron loss with magnetic flux
density waveform that is different from the conventional sinusoidal alternating current. Also, in a high rotation PM motor, the variablefrequency of magnetic flux density becomes increasingly high and the eddy current distribution of the depth direction of the laminated steel
sheet due to skin effect becomes difficult to ignore. With these conditions, evaluation using the highly-accurate iron loss calculation function
becomes required.
In this example, a hysteresis model accounting for minor loops of the direct current bias magnetism and the lamination analysis function
accounting for eddy current distribution in the lamination direction are run simultaneously to evaluate iron loss due to high carrier harmoniccomponents in the IPM motor.

Iron Loss

Iron loss is indicated in Fig. 1. The result of the hysteresis model and lamination analysis iscompared with the loss analysis result of the conventional iron loss properties.
The eddy current loss obtained by conventional iron loss characteristics isunderestimated. The magnetic flux density due to the carrier component during actual
driving fluctuates more, so that the magnetic flux density peak is higher than the peakattained for the magnetic flux density used to obtain iron loss characteristics. It is thoughtthat the loss increased because the magnetic permeability decreased, the skin effect was
less, and eddy currents were greater. The hysteresis loss is larger than the iron loss in therotor core. In the rotor core, the positions and areas of the minor loops change mainly
due to DC superposition due to magnetic flux. It is thought that this is because thehysteresis model captured these effects more accurately.

10.00

Magnetic Flux Density Waveform

Point B
Point A

Hysteresis loss density(Iron loss properties)

2

1.5

4.00
3.00

Eddy current loss

2.00
0.00

(1)Hysteresis model + lamination model (2)Iron loss characteristics

Fig. 1. Iron Loss

JSOL Corporation

1

0.5

1.00

112

measurement point(Unit：W/m^3)

5.00

Joule loss density(Lamination Analysis)

Joule loss density(Iron loss properties)

Fig. 2. Hysteresis Loss Density andJoule Loss Density Distribution(Unit：W/m^3)

0

0

0.005
0.01
Time, sec

0.015

Magnetic ﬂux density waveform in point A

Magnetic ﬂux density, T

Hysteresis loss density(Hysteresis model)

Coil

The waveform of magnetic flux density in the rotor core and stator core are displayed inFig. 3.
Super imposed direct current components that affect the minor loop appear in themagnetic flux density waveform of the rotor core.

Magnetic ﬂux density, T

Iron loss, W

Hysteresis Loss

6.00

Magnet

The distribution of joule loss density and hysteresis loss density is displayed in Fig. 2.
It can be confirmed that hysteresis loss is higher in the rotor core surface. This is thought
to be the effects of the minor loop. Furthermore, as with the results shown in Fig. 1, theJoule loss density is higher in the lamination analysis than the iron loss density.

9.00
7.00

Rotor core

Joule Loss / Hysteresis Loss Density Distribution

Stator Core Hysteresis loss(1)(2)
Rotor Core Hysteresis loss(1)(2)
Stator Core Eddy current loss (2)Rotor Core Eddy current loss (2)Stator Core Anomalous eddy current(1)Rotor Core Anomalous eddy current(1)Stator Core Classical eddy current loss density(1)Rotor Core Classical eddy current loss density(1)8.00

Stator core

2
1.5
1
0.5
0
-0.5
-1
-1.5
-2

0

0.005
0.01
Time, sec

0.015

Magnetic ﬂux density waveform in point B

Fig. 3. Magnetic Flux Density Waveformof the Rotor Core and Stator CoreJMAG Application Catalog

Case

201 Iron Loss Analysis of an SR Motor

Shaft

module: DP, LS

Overview
With the skyrocketing prices of rare earth magnets, expectations have been rising for SR (switched reluctance) motors because they have a
motor format that does not use permanent magnets. SR motors have a simple structure that can achieve solid performance at a low price.
However, since they do not use permanent magnets, it is difficult to obtain the same efficiency as a permanent magnet motor, and efficiency
increase is becoming an issue. Reduction of iron loss that occurs during drive is said to be a possible solution to this problem. This is where it
becomes important to accurately obtain iron loss of SR motors.
The magnetic flux density waveform in each part of the SR motor is in a state where direct current magnetic flux is superimposed on thealternating current magnetic flux. Direct current bias magnetism, like in this case, has confirmed that the iron loss analysis method using theconventional Steinmetz empirical law produces underestimated evaluations of loss.
This example, to accurately capture iron loss that occurs in the SR motor, we will introduce hysteresis loss accounting for the hysteresis loopof the materials and the iron loss analysis methods accounting for eddy current loss on the inside of the lamination cross section.

Coil

Rotor core

Stator core

Iron loss

Iron Loss Density Distribution

Fig.1 shows the obtained iron loss value, and Fig. 2 shows the breakdown of hysteresis
loss and joule loss of the rotor and stator. It is comparing with the analysis using theoriginal iron loss properties. For more information about setting iron loss calculation
using iron loss properties, see "JAC181 Analysis of SR Motor Drive Characteristics." It is
confirmed that iron loss increases as power supply voltage gets larger. By using thehysteresis model and lamination analysis, larger iron loss is obtained from the iron loss
properties, and as supply voltage gets large, the difference also increases. At 200V, a value
higher by approximately 30% than the original iron loss properties is estimated. It can be
confirmed that the hysteresis loss is largely increased in a hysteresis model although jouleloss drops in the lamination analysis, when confirming the breakdown of iron loss atvoltage of 100V to 200V.

Fig.3 shows the iron loss distribution at supply voltage of 100V, and Fig.4 shows the ironloss distribution at 200V. It can be confirmed that hysteresis loss density is high at thecorner of the salient pole part of the rotor and stator in a hysteresis model. The density ofjoule loss is also high at the corner of the salient pole part, but the joule loss of iron loss
properties is high.

Joule Loss (Stator core)
Hysteresis Loss (Stator core)
Hysteresis model+Lamination analysis

30.0

Iron loss properties

25.0
20.0
15.0
10.0
5.0
0.0
0

50

100

150
Voltage V

200

250

10.0
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
Hysteresis model+ Lamination analysis

Joule Loss (Rotor core)
Hysteresis Loss (Rotor core)
40.0
35.0
Iron loss W

35.0

Iron loss W

Ironn loss W

40.0

25.0

Hysteresis Loss Density(Hysteresis model)

Hysteresis Loss Density(Iron loss properties)(Unit: W/m^3)

Joule Loss Density(Lamination Analysis)

Joule Loss Density(Iron loss properties)(Unit: W/m^3)

20.0

Hysteresis Loss Density(Hysteresis model)

Hysteresis Loss Density(Iron loss properties)(Unit: W/m^3)

Joule Loss Density(Lamination Analysis)

Joule Loss Density(Iron loss properties)(Unit: W/m^3)

15.0
10.0
5.0
Iron loss
properties

Power supply voltage:100 V

Fig. 1 Iron loss

30.0

Fig. 2 Breakdown of iron loss

0.0
Hysteresis model+ Lamination analysis

Iron loss
properties

Power supply voltage:200 V

Fig. 3 Iron Loss Density Distribution (100V)

Fig. 4 Iron Loss Density Distribution (200V)

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113

Case

202 Core Stray Loss Analysis of Power Transformer

Secondary Winding

module: FQ, LS

Primary Winding

Overview
Stray loss is mainly caused by leakage flux from winding, but structures closer to the winding will tend to be affected easier by leakage flux.
The ratio of stray loss relative to all losses may not necessarily be large but heat is generated in certain areas and may become an issue in theoperation of transformers. In particular, stray loss occurring in the core tends to have effect on not only the core, but insulated items that
construct the core, and may result in degradation of insulating oil due to burnout.
Stray loss of the core is physically iron loss, but the cause is not the main magnetic flux in the core. Categorically, it is considered stray loss as it
is evaluated from short-circuit tests in the actual machine test and because it is caused by leakage flux from the winding.
This document introduces runs analysis by modeling the short-circuit test that is conducted when evaluating stray loss and obtains stray loss
distribution in the core close to the winding.

Eddy Current Loss Density Distribution,
Hysteresis Loss Density Distribution

Core

Eddy Current Loss Density Waveform

Fig.1 shows the eddy current loss density distribution of the core, and Fig.2 shows
hysteresis loss density distribution. It can be seen that the effects of leakage flux from thewinding ends are large

Fig.6 shows eddy current loss density waveform equivalent to 1 block of the lamination
direction near the winding end part of the center foot. With the effects of leakage flux, it
can be seen that there is strong heat generation on the core surface.

Magnetic Flux Lines
Fig.3 shows the magnetic flux lines from the center foot section direction. As this is a
short-circuit test, it can assume that all occurring magnetic flux are leakage flux. It can beseen that magnetic flux that is leaking from the winding ends are almost perpendicular tothe core. This occurs because the permeability inside the core differs greatly from thepermeability on the outside of the core. This leakage flux causes local eddy current loss inthe core surface.

Magnetic Flux Density Distribution
Core(Unit: T)

Fig. 3 Magnetic Flux Lines

Fig. 4 Magnetic flux density distribution of the core

(Unit: W/m^3)(Unit: W/m^3)

Fig.1 Eddy current loss density distribution

114

JSOL Corporation

Fig. 2 Hysteresis loss density distribution

Magnetic ﬂux density (T)0(mm)

10(mm)

1.2
0.8
0.4
0
0

2

4
6
Distance (mm)Fig. 5 Magnetic flux density waveform

8

10

Eddy current loss density (W/m3)

Fig.4 shows magnetic flux density distribution of the core, and Fig.5 shows the current
density waveform of 1 block in the lamination direction of the center foot. The block on
the outermost layer of the core accounts for eddy current; therefore, with secondary
magnetic flux occurring due to eddy current, magnetic flux density has become large on
the steel sheet near the surface. On the contrary, as it is a short-circuit test, it can be seen
the magnetic flux density is low as magnetic flux does not flow inside the core.

0(mm)10(mm)

4

6

80000
60000
40000
20000
0
0

2

8

Distance (mm)Fig. 6 Eddy current loss density waveform

10

JMAG Application Catalog

Case

Core(leg)

No-Load Test Analysis of Power Transformer

203

Core(yoke)

Secondary Coilmodule: FQ, LS
Primary Coil

Overview
Large-scale power transformers are expected to be highly-reliable as machinery that will support the industry infrastructure, but
unfortunately, it is impossible to conduct tests connecting to actual load. For this reason, tests of special circuit structure including no-load
tests and contradiction tests are run.
No-load tests are tests that apply rated voltage in the primary side at the no-load state of the secondary side. Current only flows to theprimary side in the no-load test, but this current causes excitation and iron loss of the iron core.
If it is an iron core that has a tripod structure, the magnetic circuit of each phase is not completely equivalent, and thus causes unbalanced
current.
This example focuses on modeling in the analysis of no-load tests in JMAG and obtains the magnetic flux line, current, excitation
conductance, and excitation susceptance that can be acquired from there.

Magnetic ﬂux line

Excitation conductance, Excitation usceptance

The magnetic flux line of the core is shown in fig. 1. According to the material propertiesof grain oriented steel, you can see that the magnetic flux line of the core is oriented
along the easy axis of magnetization.

Excitation conductance and excitation susceptance are shown in table. 2, and the
T-shaped equivalent circuit diagram in the transformer is shown in fig. 2. In the no-load
test, the circuit constant in the dotted lines are obtained in the equivalent circuit shownin fig. 2. Voltage decreases due to the primary side coil resistance and the primary side
leakage reactance are considered to be sufficiently small, so V'=V is assumed.

Iron Loss Current, Magnetization Current
Effective values for iron loss current and magnetization current are shown in table. 1. In
the three-legged iron core, the center leg and the left/right legs are not equivalent in themagnetic circuit, so unbalanced current is generated between the three phases. In table.
1, the average values are taken for the current running through the three phases.

Primary sidecoil resistanceIron loss current A

Excitation conductance μS

Magnetization current A

Primary side
leakage reactance

Secondary side
Leakage reactance

Excitation susceptance μS
Vʼ

V

0.18

31.2

Secondary sidecoil resistance

6.1

1042.3

Excitation
conductanceFig. 1 Magnetic flux line of core

Table. 1 Iron loss current, magnetization current

Table. 2 Excitation conductance, Excitation susceptance

Copper Loss Analysis of Switching Transformers using
Litz Wire

Case

204

ExcitationsusceptanceFig. 2 T-shaped equivalent circuit diagram

Core

BobbinWinding cross-section

module: TS

Overview
Flyback converters are used for various applications such as power supply circuits, which is suitable for small electrical power that realizes
switching power supply. Since there are not many parts and it is a very simple structure, it has realized power supply circuits that limit
manufacturing cost. Power supply circuits require designs that limit the cost of operation but as portability becomes a focus and harmonics ofswitch frequencies become common, heating of the winding becomes an issue. To increase pressure resistance, there are times when gap isnecessary for the core; however, with magnetic flux fringing near the gap, local heat generation may occur due to proximity effects in thewinding. Local heat generation will not only drop the conversion efficiency of electrical power but it will also cause degradation in theinsulating film so it is important to control it as much as possible. The Litz wire is one wire separated into multiple thin wires, painted withenamel film, and twisted together. Separating one wire into multiple thin wires controls the concentration of current due to the skin effect. By
adding twisted structure to the bundle wire, it is expected that the current flows more evenly, and local heat generation is controlled.
This document introduces confirms the effects on copper loss due to the presence or absence of a twisted structure in the winding regarding
the copper loss that occurs in the small switching transformer structured with PQ cores.

Primary coil
Secondary coil

Current Waveform, Copper Loss Waveform

Joule Loss Density Distribution

Fig. 1 shows primary and secondary current waveform depending on whether twisted
wire is used or not, and Fig. 2 shows primary copper loss waveform. Fig. 1 shows that it
has reached the steady state after 0.0006(sec) and whether or not it has twists, there areresponses at approximately the same times. Also, Fig. 2 confirms that the average value ofcopper loss in winding for the primary winding has dropped by approximately 26(%) forthe case with twists compared to the one without. It is assumed that with local flux
linkage inside the winding dropping due to twist effects, local biases of eddy current
caused by time variations have been resolved.

Fig. 3 shows the time average of joule loss density distribution of winding at the steadystate. In the results without winding, joule loss has increased as eddy current is inducedby the reaction of the current in secondary winding, which occurs when the switch is off
in the wires of the primary winding that is close to the secondary winding because ofproximity effects. As opposed to this, the results with twists show that the impact ofproximity effects from the secondary winding is controlled.

Flux Linkage Waveform
Fig. 4 shows the winding flux linkage waveform. It can be confirmed that the transformer
is operating regardless of whether there are twists or not, and without losing properties
as a magnetic circuit.

No twists

With twists

3
With twists secondary

25
20
15
10

0

2
1.5
no twists

1

with twists

0.5

5(Unit:W/m^3)00.00020.00040.0006Time (sec)Fig. 1 Current Waveform

0.0008No twists primary

No twists secondary

With twists primaryWith twists secondary

5.0E-042.5

00.00070.00072

0.000740.000760.00078

4.0E-04
3.0E-04
2.0E-04
1.0E-04
0.0E+00
00.0008Time (sec)Fig. 2 Copper loss waveform

Winding ﬂux linkage (Wb)

No twists secondary
Copper Loss (W)Current (A)30

No twists primaryWith twists primary

0.00020.00040.00060.0008Time (sec)Fig. 3 Joule loss density distribution

Fig. 4 Flux Linkage Waveform

https://www.jmag-international.com/

115

Case

205

Analyzing the Torque Characteristics of IPM Motors
Using a Thermal Equivalent Circuit

Stator core

Rotor coremodule: DP
MagnetOverview
In order to realize the high efficiency and high output of motors, it is necessary to understand the temperature increases that occur in eachpart of a motor. This is because coil resistance and magnet characteristics change with increases in temperature, and this may result in motor
characteristics being largely affected.
Although it is difficult to understand the temperatures of each part in a real machine, the temperatures of each of these parts can be obtainedwith ease through simulations, and changes to characteristics due to increases in temperature can also be taken into account. As methods ofanalysis that account for the increases in temperature for each part, there exists the method of coupled analysis via magnetic field analysis
and thermal analysis, as well as the method of performing magnetic field analysis that uses thermal equivalent circuits. Detailed evaluations
that obtain temperature distribution are possible for coupled analysis. While the evaluation of temperature distribution is not possible whenusing thermal equivalent circuits, magnetic field analysis that accounts for temperature increases can be performed at high speeds with ease.
In this example, magnetic field analysis is performed using a thermal equivalent circuit, and changes in average torque due to magnet
demagnetization are confirmed.

Coil

Thermal Equivalent Circuit

Part Temperature, Copper Loss
The thermal equivalent circuit during thermal equilibrium and the heat flow vectors ofeach component are shown in Fig. 2.
The coil temperature, magnet temperature, external case temperature, and copper loss
during initial temperature and during thermal equilibrium are shown in Table 1.
From Fig. 2 it is understood that heat is transferred from the magnet to the external case
via the rotor core and the shaft, then dissipated to the ambient air. The magnet and heatdissipation path temperatures are as per Table 1; the ambient air is 20 deg C, the externalcase temperature is 134.5 deg C, and the magnet temperature is 136.1 deg C. Because thedifference in temperature between the external case and the magnet is small, it cantherefore be determined that the cause of the rise in temperature for the magnet ismainly the small heat dissipation that occurs from the external case to the ambient air.
It is understood from Table 1 that copper loss also increases in accordance with the rise incoil temperature. In this analysis there are no changes in current due to temperature
because drive occurs with a specified current. Specifically, the temperature increases thecoil electric resistance and copper loss.

Average Torque
The average torque of the initial temperatures and during thermal equilibrium is shown in
Fig. 3.
It is understood that average torque decreases during thermal equilibrium. As mentioned
previously, there are no changes in current due to temperature. It can therefore be
determined that the decrease in average torque is due to the thermal demagnetization of
the magnet.

116

JSOL Corporation

Coil component

Ambient

External caseCopper loss

Heat source component(copper loss)
Shaft

Coil temperature

MagnetHeat source component(magnet eddy current
loss)

Inside rotor core→：Heat ﬂow

Fig. 2. Thermal Equivalent Circuit During
Thermal Equilibrium

Fig. 1. Thermal Equivalent Circuit

2.5

Item

External casetemperature,deg C

Magnettemperature,
deg C

Coil
temperature,deg C

Copper loss,
W

During initial
temperature

20

20

20

24.7

During thermal
equilibrium

134.5

136.1

144.2

36.4

2
Average torque, Nm

The thermal equivalent circuit is displayed in Fig. 1.
Loss is expressed by heat source components. Heat sources are copper loss and the eddy
current loss of magnets.
Coil temperature calculated with the thermal equivalent circuit is fed back into coilcomponents. Coil components retain temperature dependency in resistance values, andcopper loss values are updated. The feedback of coil temperature and updates to copper
loss are repeated, and part temperatures during thermal equilibrium are obtained.
Magnets demagnetize due to heat generation from eddy current loss, in addition to heattransmitted from the coils through parts and air gaps.

1.5

1

0.5

0
During initial temperature

Table 1. Temperature History

Fig. 3. Average TorqueDuring thermal equilibriumJMAG Application Catalog

Case

206

Temperature Analysis of an Oil-Immersed Transformer
Using a Coupled Magnetic Field and Thermal Fluid Analysis
module: FQ, LS

Overview
Large power transformers are used to transform high voltage power generated at a power station into power that can be transmitted. For the
structure of the transformer, oil-immersed transformers, which fill the inside of the tank that contains the transformer body with insulating oil
for cooling and insulation, are often used.. The insulating oil for the oil-immersed transformer is sometimes forced to circulate using a pump,and sometimes made to flow naturally.
In order to estimate the temperature of the winding and core of the oil-immersed transformer, heat transfer from the insulating oil must be
estimated. Various structures exist inside the tank besides the core of the transformer and winding, such as clamps for supporting cores andshields for preventing magnetic field leakage. Whether circulation is forced or natural, the flow channel is complex, and estimating heattransfer is not an easy task. Especially in areas where structures block flow channels and other areas with large amounts of localized heatgeneration, hotspots with high temperatures may be formed. In order to understand and accommodate for these phenomena, a loss
distribution analysis, as well as a thermal fluid analysis accounting for the flow path of the insulating oil are necessary.
This document introduces, examples and steps will be shown to obtain heat distribution in a magnetic field analysis for a forced circulation
type oil-immersed transformer, map the thermal distribution to thermal fluid analysis model, and then obtain the temperature by handlingthe insulating oil as a fluid in a thermal fluid analysis.

Connect

Insulating oil entrance
Tank
ShieldU PhaseV PhaseW PhaseInsulation oil exit
Secondary winding
Primary winding

Loss

Temperature

Iron loss density distribution occurring in the core of the transformer obtained in themagnetic field analysis is shown in fig. 1, and distribution of copper loss and stray load
loss (joule loss density distribution) occurring outside the core are shown in fig. 2. The
main components for the loss amounts are iron loss in the core and copper loss in thewinding, but when looking at the loss density, it shows that losses are also high in theconnector.

Temperature distribution that can be obtained in a thermal fluid analysis is shown in fig.
4. As estimated from the flow rate in fig. 3, looking at each phase shows the U-phase withthe lowest temperature, and W-phase has a higher temperature than U-phase. Since theflow rate for V-phase is small and there is no thermal transfer to the tank side because it islocated in the center, the temperature is extremely high, and the core temperatures rise
locally by around 110 degC. It shows that there is a large amount of heat generation, theinner-most core generates highest temperature, and the temperature of the connectors
between the core and the winding is extremely high.

Flow Rate Distribution (Streamline)
Flow rate (streamline, contour) of the insulating oil obtained in the thermal fluid analysis
is shown in fig. 3. When looking at the streamline, the insulating oil flowing in from theentrance keeps moving downward, and flow rate in the U-phase winding is larger. On theother hand, we can see that flow rate is smaller in the W-phase winding side comparedwith the U-phase winding side. Furthermore, because the flow passage for the V-phase in
the center is narrow, insulating oil cannot easily enter, and the flow rate is also very small.

(Unit: W/m^3)

Fig. 1 Iron loss density distribution(Unit: W/m^3)

Fig. 2 Joule loss density distribution(a) Streamline display

Fig. 3 Flow rate distribution(b) Streamline display

Fig. 4 Temperature distribution

https://www.jmag-international.com/

117

Case

207

Transfer Torque and Eﬃciency Analysis of Magnetic Gear

Inner yoke
Outer yoke
Outer magnet

Inner magnet

module: DP

Overview
Gears are used to transfer power. Velocity and torque are transformed depending on the gear ratio. Originally, mechanical gears using
conventional wheels had issues with noise due to the meshing of wheels, and its maintenance requiring lubricating oil. As a gear to solvethese issues, there is the magnetic gear. Magnetic gears are coupled with the magnetism of things like permanent magnets, and can transfer
power without contact, as well as reducing noise and maintenance costs. When prioritizing the size of the transfer torque, rare earth sinteredmagnets with high energy product are selected for permanent magnets used in magnetic gear. However, since rare earth sintered magnets
have high electric conductivity (not as strong as metals such as copper though), eddy current will occur in the permanent magnet andbecome the cause for decrease in transfer efficiency of magnetic gear. Eddy current of permanent magnets will occur due to variations inmagnetic flux density of the gap; however, due to the principle of magnetic gears using pole pieces, the magnetic flux density distribution ofgaps have various harmonics components, and for this reason, magnetic field analysis is effective in accurately estimating eddy current.
In this example, the transfer torque and transfer efficiency are obtained with magnetic field analysis for the flux modulated type magnetic
gear using permanent magnets for the inner and outer rotor and positioning a pole piece between the magnets. In addition, how transfer
torque and transfer coefficient are affected due to revolution speed is researched.

Pole piece

Torque, Transfer Eﬃciency
0

2.4
Ideal torque ratio

Inner rotor

-20

2.3
Torque ratio

Outer rotor
Torque (N・m)

As an example of a torque waveform that works on the inner and outer rotor, the results
for the inner rotor with revolution speed of 2100 (rpm) will be shown in Fig. 1. The torqueis negative for both the inner and outer rotor, but as the inner rotor is forcibly rotated in
the counter-clockwise direction, which is the positive direction in terms of rotation
direction, it shows that external force equivalent to torque is necessary for rotation. On
the contrary, the outer rotor is rotated in the clockwise direction, which is the negativerotation direction. As torque is negative, it shows that the force rotating the inner rotor istransferred as the force rotating the outer rotor.
Table 1. shows the transfer efficiency from the inner rotor to the outer rotor. As it is withtorque ratio, it can be seen that efficiency decreases with increase in revolution speed.

-40
-60

2.2
2.1

-80

2

-100

1.9

-120
0

0.002

0.004

0.006

1.8
0

1000

2000

3000

4000

5000

Revolution speed of inner rotor (rpm)Time (sec)Fig. 1 Torque waveform (Inner rotor revolution
speed 2100 rpm)

Fig. 2 Torque ratio ‒ revolution speed

Joule Loss
As an example of joule loss (eddy current) density distribution occurring in the magnet,
Fig 3 shows the results when the revolution speed of the inner rotor is 2100 (rpm).
However, the contour plot shows the average rotation between 48 (deg) to 90 (deg) of
the inner rotor. The figure shows that joule loss is high on the surface for the innermagnet and the side for the outer magnet.

Magnetic Flux Density Distribution, Magnetic Flux LineAs an example of magnetic flux density distribution and flux line, the results (final step)for inner rotor revolution speed of 2100 (rpm) is shown in Fig. 4. However, it is shown
expanded to 1/1 (360 deg). It can be seen that the magnetic saturation in the pole piece isdrastic.

Revolution speed of inner rotor (rpm)Transfer Eﬃciency (%)

700

97.2

2100

91.8

3500

87.1

4900

82.7
Inner rotor revolution speed 2100 rpm

Table 1 Revolution speed

Fig. 3 Joule loss density(Unit :T)
Inner rotor revolution speed 2100 rpm

Fig. 4 Magnetic Flux Density Distribution, Magnetic Flux Line

118

JSOL Corporation(Unit: W/m^3)

JMAG Application Catalog

Case

208

High Frequency Induction Hardening Analysis Accounting
for Deformation (Abaqus Link)
module: FQ, ABAQUS2

Gear

Core

Overview

Coil

Since gears are products that demand mechanical intensity, high frequency induction hardening is run for surface heat treatment. Since
induction hardening can rapidly heat only the surface of the gear in a short amount of time, the hardness of the surface is continually
increased while retaining the toughness inside. On the other hand, the surface of the gear is exposed to high temperatures in the heatingprocess prior to induction hardening; therefore, there is a possibility of thermal expansion.
This document introduces evaluates how heating ranges change depending on the deformation amount at high temperatures anddeformation during heating, with heat generation analysis in JMAG and linking of heat transfer / deformation analysis in Abaqus.

Deformation AmountTime History of Temperature

The contour plot of the Z component of displacement is displayed in Fig.1, and thecontour plot of the absolute value of displacement is displayed in Fig.2.
With the rise in temperature, deformation where the tip of the helical gear is open up anddown the Z-axis direction has occurred, and deformation expanding in thecircumferential direction can be confirmed.

The process in which temperature is rising in the center part of the top edge of the gear
tip is displayed in Fig.4. After temperature has exceeded the Curie point, the process inwhich the speed of the temperature rise is decreasing can be confirmed.

Temperature Distribution
The temperature distribution at the end of the heating process is displayed in Fig.3. Eddy
current is flowing at the gear tip part close to the heating coil, and the process oftemperature rise can be confirmed.

Fig.1 Displacement contour plotof the Z component

Fig.2 Contour plot of the absolutevalue of displacement

Fig.3 Temperature distribution

Stress Distribution
Mises stress distribution at the end of the heating process is displayed in Fig.5. The
process in which stress is concentrated at the gear bottom due to temperature increase /thermal expansion at the gear tip of the helical gear can be confirmed.

Fig.4 Temperature versus time

Fig.5 Mises stress distribution

https://www.jmag-international.com/

119Magnetic Measurement Analysis Accounting
for Eddy Currents

Case

209

Magnetic
Excitation coilmodule: TR

Overview
Magnetic properties affect the performance of products composed of magnetic circuits using magnetic material. This requires that themagnetic properties are assessed accurately. The H-coil method and excitation current method can be used to measure magnetic propertiesof magnetic material. The excitation current method will be used in this Application Note.
An excitation coil is wrapped around a ring test piece and a current is run through it. Doing this generates a magnetic field inside theexcitation coil, and magnetic flux flows though the test piece placed inside the coil. Output voltage will be measured in the B coil wrappedaround the ring test piece. Magnetic flux is obtained from the output voltage waveform, and then magnetic flux density is calculated from across section of the target. When the excitation current frequency used in the measurement is high, eddy current is generated in themagnetic material, so effects also appear in the waveform of the measured magnetic properties.
In this example, an analysis model is created based on the excitation current method, and the differences in magnetic properties obtained
when the frequency is changed are shown.

Relationship between Frequency and Magnetic Properties (B-H
Curve)The relationship between the average magnetic flux density and magnetic field of themagnetic materials obtained when changing the frequency of the alternating current
passed through the excitation coil to 1Hz and 10Hz is shown in Fig. 1 (B-H curve),
respectively. When eddy current is generated, responsiveness is lowered in the measured
average magnetic flux density due to eddy current, and a hysteresis loop is drawn.

B CoilMagnetic Flux Density Distribution Contour Plot
Magnetic flux density contour plot is shown in Fig. 4. Generated magnetic flux is nearly
constant.

Eddy Current Density Contour Plot
Current density contour plot is shown in Fig. 5. Eddy currents concentrate on the outside
perimeter of the magnetic material due to the skin effect.

Current Waveform Flowing Through Excitation CoilThe current waveform passing through the excitation coil is shown in Fig. 2. In order toclearly show the areas with volatile changes in magnetic flux in the graph, time intervals
where the current alternates between positive and negative are shortened.

Magnetic Waveform Produced by B CoilThe history graph of the B coil magnetic flux when the excitation current frequency is 1Hz
is shown in Fig. 3.

1.0

0.0
-120000 -60000
0-1.0

60000 120000

-2.0

100.0

3.0
2.0

50.0

1.0

0.0
-120000 -60000
0-1.0

60000 120000-2.0

Current,AMagnetic Flux Density, T

2.0

Magnetic Flux Density, T

150.0
3.0

0.0
-50.0

-3.0
Magnetic Field, A/m-3.0
Magnetic Field, A/m

1 Hz

10 Hz

-100.0-150.00

Fig. 1. Relationship Between the Magnetic Flux Density
and the Magnetic

Magnetic Flux of FEM Coil, Wb

0.001
0.000750.00050.000250-0.00025-0.0005-0.00075-0.001
0

0.2

0.4

0.6
0.8
Time, sec

1

1.2

Fig. 3. History Graph of B Coil Magnetic Flux

120

JSOL Corporation

1.4

0.2

0.4

0.6
0.8
Time,sec1

1.2(Unit: T)

1.4

Fig. 2. Current Waveform Flowing Through Excitation CoilFig. 4. Magnetic Flux Density Contour Plot

(Unit: A/m^2)

Fig. 5. Current Density Contour PlotJMAG Application Catalog

Case

210

Outer Yoke

Torque Decoupling of In-out Magnet Coupling

MagnetJig

Jig

module: DP

Overview
Magnet coupling has two rotors, the driving side and the passive side. Unlike magnetic gears, the driving side and the passive side rotate
synchronously. If torque exceeds the allowable torque, connection from the driving side to the passive side will slip, causing torque
Inner Yokedecoupling. The allowable torque is evaluated with the maximum value of the holding torque that keeps the relative positional relationshipbetween the driving side and the passive side.
In this example, an analysis for finding holding torque and an analysis for transient response during torque decoupling.

Holding Torque
In general, the holding torque has a periodicity of two poles (90 deg), and maximum
20

torque is reached when the relative position is shifted by 1/4 pole (22.5 deg). When the15

passive side is fixed and the driving side is rotated, relative position is changed and theFig. 1 shows the holding torque for 90 deg when the inner part is rotated. Peak torque at

10
Torque, Nmholding torque waveform for 90 deg is obtained.
22.5 deg is about 16.2 Nm.

5
0
-5
-10

Katt

-15

Tload ：Load torque, Nm-20：Relaxation factor, Nm / (deg / sec)

RPM ：Rotation speed, r / min
0

15

30

Starting Response and Delay Angle

45

60

75

90

Mechanical Angle, deg

Load torque given to the passive side is based on the supposition of load occurring fromcircumstances such as fluid in the pump. Also, load torque dependency on rotation speed

Fig. 1. Holding TorqueFig. 2. Conversion of Load Torque to Relaxation Factor

is considered, and the equation shown in Fig. 2 is used to assign the equation of motion as
the relaxation factor.

800

The rotation speed of the passive side is displayed in Fig. 3 for when the load torque600

Notice that the passive side is able to follow the rotation of the driving side until a load
torque of 16.5 Nm is reached. Values of 17 Nm and 18 Nm do not follow the rotation, andthe sides become decoupled.
Fig. 4 shows the delay angle of the passive side at the time the driving side makes one

120

400

Delay Angle, degrotation speed of the driving side is also displayed in the figure as the “Reference” line.

140

Rotation Speed, r/min

assigned to the passive side is 10 Nm, 14 Nm, 16 Nm, 16.5 Nm, 17 Nm, and 18 Nm. The

200
Reference
10 Nm
14 Nm
16 Nm
16.5 Nm
17 Nm
18 Nm

0
-200
-400

100
80
60
40
20
0

-600
0

0.02

0.04

0.06

0.08

0.1

10

11

12

Time, sec

13
14
15
Torque Load, Nm

16

17

18

rotation. The delay angle when the load torque is 16.5 Nm is about 22 deg. For load
torque of 17 Nm and 18 Nm, it can be seen that slippage is about 90 deg.

Fig. 3. Passive Side Rotation Speed When Load Changes

Fig. 4. Delay Angle on the Passive Side

Case

211

MagnetStator Core

IPM Motor Wire Joule Loss Analysis

Coilmodule: DP

Overview
The square wires and flat wires of segment coils and edgewise coils, etc., are being used for the purpose of improving lamination factors, thereduction of end parts, minimization, improvements in heat dissipation, and improvements in productivity in IPM motors. On the other
hand, from the viewpoint of higher efficiency and increasing output, etc., motor coil resistance to loss is becoming a necessity as so to

Rotor Coreobtain a reduction of loss in electrical appliances in general.
When increasing the number of motor rotations, concentration of electric current occurs within the coil interior due to the eddy and current
joule loss value increases. Especially at high rotation such as 10,000 r/min, joule loss may increase from 3 times to 4 times compared withlow speed. In running accurate evaluations of joule loss and handling concentration of electric current within the wire correctly, it is also
important to properly manage the eddy current generated by the distribution of magnetic flux density that links the wire and temporalchanges to magnetic flux density. For these reasons, simulation via magnetic field analysis is a necessity.
In this example, evaluation of coil joule loss that includes eddy currents during motor rotation is introduced here.

Comparison of Winding Joule loss
250

r/min is displayed in fig. 1. It is understood that joule loss increases over 4 times more

200

when the eddy current is accounted for, compared to when the eddy current is notaccounted for.

Joule Loss Density Distribution When Accounting for Eddy Current

Winding Joule loss, W

Coil joule loss when using fundamental frequency and PWM waveforms as input at 10,000

150

100

50

0

Eddy current notaccounted for

Wire joule loss density distribution (at a time of maximum U phase current) when the(Unit: W/m^3)

Eddy current
accounted foreddy current is accounted for is displayed in fig. 2. It can be seen that gap side wire loss
density is high and offset is large. From this it is understood that there is considerable

Fig. 1 Winding Joule loss

Fig. 2 Joule Loss Density Distribution

eddy current generation at this position.

Flux Linkage per Component
Magnetic flux lines when eddy current is not taken into account are investigated and therelationship between magnetic flux linking the strand and its Joule loss is confirmed. Fig. 3
shows the flux lines.
In fig. 3, the rotor and the wires which are near the airgap are linked magnetically by theflux line. Observing PWM waveform on the other hand shows linkage with the entire wire.
From this results, joule loss concentration within wires near the gap is thought to be

Fundamental frequency (1 kHz)PWM waveform (20 kHz)

caused by the link with fundamental frequency flux components.
Fig. 3 Flux Linehttps://www.jmag-international.com/

121

Case

Winding

AC Loss Analysis of a Claw Pole Alternator

212

Stator coremodule: TR

Rotor coilElectric Machines are affected primarily by 3 types of losses. Iron losses in steel lamination sheets, joule losses in the conducting components

Claw poleand joule losses in coils.

Shaft

Overview

When we primarily focus on AC losses in coils, one way to reduce them is by using a “Bar winding”. Bar winding is used large conductor bars,
so this improves fill factor and reduces DC resistance which in turn reduce DC losses. Large cross section of these windings cause make themvulnerable to AC losses like skin effect and proximity losses. The AC losses are frequency dependent so if the machine operates on higher
frequencies and higher speeds.
In order to examine the AC losses we use a bar wound claw pole alternator which is a three dimensional machine. This allows us to examine
the AC losses in greater depth and get their full impact. These losses cannot be calculated via the equivalent circuit method. Hence finite
element method has to be used.
In this example, we introduce the case study that evaluates AC losses in coils of claw pole alternator.

Loss Analysis
Fig. 1 shows AC losses vs DC losses. Fig. 2 shows loss distribution of coils. Fig. 3 shows loss distribution and flux line of end windings.
From fig. 1, we can observe that the effect of AC losses is significant on the overall efficiency, thus they cannot be ignored.
In fig. 2, most of the losses are concentrated on the edges of the coil geometry than the insides. This is one of the characteristics of AC Losses.
From fig. 3, we can confirm the relationship of loss distribution and magnetic flux in the end windings.

90.00080.00080.7

70.000Losses, W

60.00050.00047.7

40.00030.00020.00010.000(Unit: W/m^3)

0.000(Unit: W/m^3)
DC Losses

AC Losses

Fig. 1 AC Losses VS DC Losses

Case

213

Fig. 2 Loss contour plot

Fig. 3 End Winding Losses and Flux Line

Circuit/Control Simulation of a Wound-Field
Synchronous Motor

Shaft
Rotor Core

module: DP, RT

Armature coil

Overview
Traditionally, motor control design and motor design are often independent processes, with cooperative design being difficult. Meanwhile,for advanced motor control design, a plant model that is more detailed and behaves like the actual machine is required for control

Stator Coresimulation.

Field coil

In JMAG, it is possible to create a plant model (JMAG-RT motor model) suitable for the actual machine, taking into account magnetic
saturation characteristics and space harmonics of the motor. By importing this plant model into a control/circuit simulator, it is possible tocarry out coupled simulation taking into account both detailed motor characteristics such as magnetic saturation and space harmonics, andmotor driver control characteristics.
In this example, JMAG-RT is used to obtain the torque of a wound-field synchronous motor (below WFSM) and the coil inductance current
dependence, and the spatial harmonic components contained in them. Furthermore, we will simulate the WFSM current control byimporting a JMAG-RT motor model created as a plant model into the control/circuit simulator.

Current Waveform, Torque Waveform, Inductance Waveform

3.0
70

waveforms in Fig. 4 and Fig. 5. From each waveform, it is understood that a large current
flows momentarily immediately after turning on the power, and the torque also increases.
Moreover, it can be confirmed that a steady-state condition is gradually converged to. On

Torque, Nm

armature current waveforms in Fig. 2, the torque waveform in Fig. 3, and the inductance

Luu

2.5

60

2.0

50

1.5
Inductance, mH

The field current waveform when vector control is carried out is shown in Fig. 1, the

40
30
20

1.0
0.5
0.0
-0.5

-1.5
0

0
0

0.03

motor, the current and torque waveforms which include space harmonic components

0.06

0.09

0.12

0.15

Time, sec

0.03

0.06

Fig. 3. Torque waveform

Fig. 4. Inductance waveform

60
Iv

Iw

450

40

Inductance, mH

Current, A

Current, A

6

0
-20

4

-40

2

Lﬀ

350

20

8

Mfu

400

10

300
250
200
150
100
50
0

-60

0
0

0.03

0.06

0.09

0.12

0.15

0

0.03

0.06

0.09

0.12

Time, sec
Time, sec

Fig. 1. Field current waveform

122

JSOL Corporation(Iu: U-phase current, Iv: V-phase current, Iw: W-phase current)

Fig. 2. Armature current waveform

0.15

-50
0

0.03

0.06

0.09

0.12

0.15

Time, sec
(Mfu: ﬁeld coil U-phase mutual inductance, Lﬀ: ﬁeld coil self-inductance)

Fig. 5. Inductance waveform

0.09

0.12

0.15

Time, sec
(Luu: U-phase coil self-inductance, Muv: U-V phase mutual inductance,
Muw: U-W phase mutual inductance)

caused from the slot shape can be confirmed in steady state.

Iu

Muw

-1.0

10

the other hand, since the JMAG-RT motor model takes into account the shape of the

12

Muv

JMAG Application Catalog

Monitoring the Radial Force Acting on the Teethof IPM Motors Using Circuit Control Simulation

Case

214

MagnetCoil
Rotor Core

module: DP, RT

Overview
Motors used in electric vehicles are generally quieter than engines, but vibration and noise may be a problem because the drive range is
wide. This vibration and noise also occur due to electromagnetic excitation forces and eigenmode resonances of the motor.
Among the electromagnetic excitation forces, for the radial force acting on the teeth the second harmonic becomes dominant as theStator Core

magnet passes by two poles in one electrical angle. On the other hand, it is known that the sixth harmonic can easily excite the zero-orderring mode which is the most likely mode to radiate sound because the excitation force of each tooth has the same phase. Because of this,for countermeasures against vibration and noise, it is useful to grasp the excitation force for each frequency component.
In JMAG, it is possible to create motor models that can model actual machines in detail taking into account magnetic saturation
characteristics and spatial harmonics in the motor. By capturing this motor model, the JMAG-RT model, in a control/circuit simulator, it ispossible to perform coupled simulation taking into account both the magnetic saturation characteristics and spatial harmonics of the motor
and the control characteristics of the motor driver. In addition, radial forces acting on the teeth when driving the motor can be monitored
providing feedback for motor design and control design.
In this example, an IPM motor as a JMAG-RT model is captured in a control/circuit simulator, and radial forces acting on the teeth during
motor driving is monitored while changing the operating point (current phase).

Control Circuit

Radial Force Waveform Acting on the Teeth

The control circuit uses the current amplitude and current phase command values, and a

Fig. 6 shows the radial force waveform acting on the teeth for each current phase whenvoltage command value is connected via an inverter to the motor.

the IPM motor is driven by vector control, and Fig. 7 shows the frequency components. In

The control method specification realizes vector control from PI control.

Fig. 6, when a values is negative the radial force is acting in the direction toward theFig. 1 shows the control section of the control circuit, and Fig. 2 shows the circuit section

center axis of the motor.

and the motor. The command values are current amplitude 84.8 A, and current phases ofAs the current phase changes, the positional relation between the stator magnetic poles

0 deg, 45 deg, and 75 deg. The voltage command values are input to the motor via theand the rotor magnetic poles changes and the current waveform changes. This shows that

inverter.

the radial force acting on the teeth also changes. From Fig. 7, it is possible to check theradial force acting on the teeth for each frequency component.

Current Waveform
Figures 3 to 5 show the current waveform for each current phase when the IPM motor isdriven by vector control. By advancing the current phase, the magnetic field is weakened
and the magnetic saturation is relaxed. As the magnetic saturation is relaxed and theinductance increases, the harmonic content contained in the current waveform decreases.

InverterCurrent amplitude command value

Place of measurement
U PhaseCircuit section and motor

V PhaseW Phase100
Motor0(deg)

80

20
0
-20
-40
-60
-80
0.192

0.194

0.196

0.198

0.2

-800

-1200
0.19

Time, sec

V PhaseU PhaseW Phase60
Current, A

40
20
0
-20

1

80

0.9

60

0.8

40

0.7

20
0
-20
-40

-60

-60

-80

-80

-100
0.19

-100
0.19

0.196

0.198

0.2

Time, sec

Fig. 3. Current Waveform (Current Phase: 0 deg)0(deg)

W Phase

100

-40

0.194

V PhaseFig. 5. Current Waveform (Current Phase: 75 deg)

Amplitude, N

U Phase

0.192

0.194

0.196

0.198

0.2

Time, sec

Fig. 2. Control Circuit (Circuit Section and Motor)

80

Current, A-600

Output port for speciﬁed physical quantities

100

0.192

-400

-1000

-100
0.19

Fig. 1. Control Circuit (Control Section)

75(deg)-200
Radial force, N

Current, A

40

Current phase command value

45(deg)06045(deg)Fig. 6. Radial Force Waveform Acting on the Teeth

75(deg)

0.6
0.5
0.4
0.3
0.2
0.1

0.192

0.194

0.196

0.198

0.2

Time, sec

Fig. 4. Current Waveform (Current Phase: 45 deg)

0
1

2

3

4

5

6

7

8

9

Frequency component, orderFig. 7. Frequency Components of the Radial Force Acting on the Teeth

https://www.jmag-international.com/

123

Case

215

Simulation of an IPM Motor with a Delta Connection
Using a Control Simulator and JMAG-RT

Shaft

Coil

Rotor coremodule: DP, RT

Overview
Traditionally, motor control design and design of the motor itself have mostly been performed independently, and cooperative designs havebeen difficult to carry out. Also, for advanced motor control design, a motor model showing more details and conformity to the behavior of
the actual machine is required by control simulation.
In JMAG, it is possible to create a detailed model that conforms to the actual machine as well as account for spatial harmonics and magnetic
saturation characteristics that are included in the motor. Importing this JMAG-RT motor model to a control/circuit simulator makes it possible
to perform a linked simulation that accounts for a motor's magnetic saturation and spatial harmonics as well as a motor drive's control
characteristics. Additionally, if the motor has a delta connection, cyclic currents will flow. This leads to an increase in copper loss and torqueripple, so it is beneficial to monitor cyclic currents while the motor is activated and feed back that information to control design and motor
design.
In this example, the control and circuit simulator have been incorporated as a JMAG-RT model, and the cyclic current the IPM motor.is
monitored.

Magnets

Stator core

Control Circuit

Current Waveform

The command value is set at 1,800 r/min and the d-axis current is set at 0 A, and thevoltage command value is transmitted to the motor via the inverter.
Fig. 1 shows the control circuit, a control unit, and Fig. 2 shows the circuit and motor.

Fig. 5 shows the line current waveform at steady-state, Fig. 6 shows the phase current
waveform, and Fig. 7 shows the circulating current waveform.
In a motor with a delta connection, the imbalance of the back electromotive forces causes
circulating current to flow, which can cause copper loss and torque ripple.
In this example, a circulating current of about half of the line current or phase current
flows, and it can be seen that its contribution to loss is large.

Rotation Speed Waveform, d-Axis Current Waveform
Fig. 3 shows the rotation speed waveform, and Fig. 4 shows the d-axis current waveform.
It can be seen that both tend toward their command values.

Id command value

a

Ia

b

Ib

c

Ic

a
U

W

Circuit unit and motor

Ib

V
Iu

Ic

6

6

4

4
Current, A

Current, A

Ia

2
0
-2

-6
0.98

Fig. 1 Control Circuit (Control Unit)

Fig. 2. Circuit and Motor

0.985

0.99
Time,sec

c

500

6
4

-0.5
-1

0

0

0.2

0.4

0.6
Time, sec

Fig. 3. Rotation Speed Waveform

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0.8

1

2
0
-2

-1.5
-500

V

0.5

Current, A

Current, A

Rotation speed, r/min

U

W
b

0

-4

-2
0

0.2

0.4

0.6
Time, sec

Fig. 4. d-Axis Current Waveform

0.8

1

-6
0.98

V

2
0

0.985

0.99
Time,sec

0.985

0.99
Time,sec

0.995

Fig. 6. Phase Current Waveform

1

1000

c
Iw

-2

-6
0.98

1

a

1.5
1500

0.995

Fig. 5. Line Current Waveform

2

2000

Iv

Iw Iu
Iv

-4

-4
Rotation speed command value

U

W
b

0.995

Fig. 7. Circulating Current Waveform

1

1

JMAG Application Catalog

Case

216

Simulation of a Claw-Pole Type Alternator
Using a Control Simulator and JMAG-RT

Shaft
Rotor Core

module: RT, TR

Overview
The demand for low fuel consumption of automobiles has been increasing every year, and types of auxiliary systems and devices such as
power steering and cooling pumps are being electrified to support this. For this reason, the amount of electricity used has increased every
year for general gasoline-powered vehicles, and the demand for higher output power for alternators which supply electricity is increasing.
However, since it is not allowable to increase the vehicle size to increase the generating capacity, it is necessary to increase the power density.
Furthermore, it is necessary to achieve high efficiency.
The rotation speed of the alternator changes as the vehicle speed changes while driving. But at the same time, the reference voltage of
automotive electrical components is set at 12 V. Because of this, it is necessary to monitor the output voltage with a voltage regulator andcontrol the field current considering the engine speed and electric load.
In this example, the output voltage and field current of a claw-pole type alternator is checked when the rotor speed is changed.

The control circuit is shown in Fig. 1. The command value is set to a voltage of 12 V, andthe current controller is connected to the field winding of the alternator. The initial valueof the alternator is set at 2400 r/min, and at 0.1 sec later changed to 1800 r/min.

Output voltage

Field current

Rotation speed

20

Voltage, V

16

3000

8

2500

7

2000
12
1500
8
1000
4

500

0

0
0

Current controller

Fig. 1 Control Circuit (Control Section)

0.05

0.1
Time, sFig. 2 shows the rotation speed waveform and the output voltage waveform, and Fig. 3
shows the rotation speed waveform and the field current waveform.
From Fig. 2, it can be seen that the output voltage is kept at 12 V even if the rotation
speed changes. This is because, as shown in Fig. 3, when the rotation speed decreases,
increasing the field current increases the flux linkage of the coil and increases the output.

0.15

0.2

Fig. 2 Rotation speed waveform and Output voltage
waveform

Field Current, A

Alternator

Rotation speed, r/min

Output voltage command value

Rotation Speed Waveform, Output Voltage Waveform,and Field Current Waveform

Rotation speed
3000
2500

6
2000

5

1500

4
3

1000

2

Rotation speed, r/min

Control Circuit

500

1
0

0
0

0.05

0.1
Time, s0.15

0.2

Fig. 3 Rotation speed waveform and Field current waveform

Case

217

Rotor Coil
Stator Core

Cage (bar)Creating Eﬃciency Maps for 3-Phase Induction Motors

Rotor coremodule: FQ, LS, EFFICIENCYMAP

Overview
Due to advances in the variable speed operation technology of induction motors, applications to areas requiring a wide range of operation areprogressing. As an effective way to evaluate the characteristics of induction motors in such areas, efficiency maps can be used in theoperating region.
To create efficiency maps using actual measurements or computations, it is necessary to search for the highest efficiency point by using
voltage and slip as parameters for each rotation speed and load point. Doing this makes the analysis execution enormous, taking time to
output the results as well. With JMAG, efficiency maps can easily be obtained by creating a JMAG-RT model which is a plant model andusing the efficiency map creation function in JMAG-RT Viewer.
In this example, JMAG-RT Viewer is used to create efficiency maps of a 3-phase induction motor when the driving temperature has beenchanged.

Coil
Stator coreEﬃciency Map
Fig. 1 and Fig. 2 show the efficiency maps when the drive temperature is changed to 20
deg C and to 120 deg C respectively. From the figures, the efficiency for each drive
condition can be understood. Also, as the temperature rises, it can be seen that theefficiency becomes worse.

Loss Map
The stator copper loss map for when the drive temperature is changed to 20 deg C and to
120 deg C is shown in Fig. 3 and Fig. 4 respectively, and the rotor copper loss map isshown in Fig. 5 and in Fig. 6.
When the temperature is high, it can be seen that the copper loss on the primary side andthe secondary side is large. This is because the electrical characteristics of the primary coils
and the secondary conductors deteriorate and the electrical resistances increase, whichleads to a decrease in efficiency.

Fig. 1. Efficiency Map (20 deg C)

Fig. 2. Efficiency Map (120 deg C)

Fig. 3. Stator Copper Loss Map (20 deg C)

Fig. 4. Stator Copper Loss Map (120 deg C)

Fig. 5. Rotor Copper Loss Map (20 deg C)

Fig. 6. Rotor Copper Loss Map (120 deg C)

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Case

218

High-Frequency Induction Hardening Analysis
Accounting for Deformation

Coil
Gear

module: DS, FQ, HT

Overview
Gear surfaces are generally hardened to ensure abrasion resistance of the tooth surfaces, but it is necessary to harden the surfaces while
maintaining the overall toughness of the gear. For high-frequency induction heating, which is one of the surface hardening methods, it ispossible to harden only the tooth surfaces by rapidly heating only these surfaces using a high-frequency power supply.
On the other hand, in order to uniformly heat the surfaces of the gear, it is necessary to consider several factors such as how to adjust thegeometry, arrangement, frequency, and size of the heating coil.
Eddy currents generated by high-frequency varying magnetic fields are generated near the surface of the teeth. As temperature rises material
characteristics change drastically, and the object heated is thermally deformed by thermal stress. It is necessary to compute heat generation
amounts and deformation amounts in numerical analysis using finite element methods (FEMs).
In this example, using the JMAG magnetic field - thermal stress coupled analysis function, analysis is performed evaluating gear deformationfrom induction heat hardening.

Coil lead part

Eddy Current Loss
A high-frequency (30 kHz) current flows in the coil causing a magnetic field fluctuation
which induces induced current in the gear. Due to the skin effect, current concentrates on
the tooth tips of the gear.
Skin thickness can be calculated by the equation shown in Fig. 1.
From magnetic field analysis, induced current in the model can be checked. The loss dueto the induced current is used as a heat source for thermal stress analysis. The
temperature distribution in and on the gear and deformation of the gear obtained fromthermal stress analysis are input as conditions in magnetic field analysis.
Fig. 2 shows the eddy current loss density distribution on a cut surface of a gear tooth tip.

Temperature Distribution

δ：skin depth, m：frequency, Hz
μ：magnetic permeability, H/m
σ：electrical conductivity, S/m(Unit: W/m^3)

Fig. 2. Eddy Current Loss Density Distribution over 0 to 0.2 sec

Fig. 1. Equation of the Skin Depth

Loss due to induced current acts as a heat source increasing temperature. In thermal
stress analysis, the loss obtained from magnetic field analysis is set as an input condition.
Since losses concentrate primarily on the tooth tips which are outside the model, thisregion naturally becomes the region of maximum temperature change. Material
magnetization characteristics and temperature characteristics of electrical characteristics
are taken into account using an obtained temperature distribution as an input condition
for magnetic field analysis.
Fig. 3 shows the temperature distribution on the gear, and Fig. 4 shows the temperaturevs. time of a tooth tip. Fig. 4 shows the change in temperature at the tooth width centershown in Fig. 3.

800

Temperature, deg C

700
600
500
400
300
200
100
(Unit: deg C)
Measuring point

0

0.0

0.5

1.0

1.5

2.0
Time, sec

2.5

3.0

3.5

Thermal Stress (Deformation)
The deformation of and temperature distribution on the gear at a 4 sec are shown in Fig.
5, and the deformation of the gear is shown in Fig. 6. The thermal expansion deformationof the model is confirmed at 100 times magnification.
As the gear temperature changes, deformation occurs due to thermal expansion andthermal contraction. During this time, thermal stress is produced due to for example
constraints and an uneven temperature distribution. From the figure, it can be seen thatthe areas of highest temperature expand the most. After heating for 4 sec, the tips are
deformed to a maximum of 0.18 mm. Furthermore, the gear is drawn toward the groundby gravity. The influence of gravity on the deformation can also be confirmed.

Fig. 3. Gear Temperature Distribution

Fig. 4. Tooth Tip Temperature vs. TimeOriginal gear geometry(Unit: mm)(Unit: deg C)

Fig. 5. Gear Temperature Distribution

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Fig. 6. Gear Deformation

4.0

JMAG Application Catalog

Magnet yokeCase

219 Axial Gap Type Motor Cogging Torque Analysis

Magnetmodule: TR

Overview

Stator yokeAn axial gap motor has a motor structure in which the rotor and stator are arranged in a disk shape facing each other and rotate using
magnetic flux in the axial direction. The advantages of this motor are that this motor can be made thinner than a radial gap motor, and thathigher torque can be obtained for a given magnetic flux density if properly designed. This motor has been developed for a variety of uses
including automobiles, and can respond to market demands with its unique design.
In an axial gap motor, 3D electromagnetic field simulation is necessary because the magnetic flux passing through the facing rotor andstator is a 3D magnetic circuit.
In this example, presents an example of cogging torque analysis of an axial gap type motor.

Coil

Top: entire motor
Bottom: stator

Cogging Torque Waveform
0.05
0.04
0.03
0.02
Torque (Nm)A cogging torque waveform is shown in Fig. 1.
This result shows that the amplitude is approximately 0.044 Nm. The cogging torqueperiod is determined by the number of slots and the number of poles. Since this analysis
target has 8 poles and 6 slots, the cogging torque period is 15 deg obtained by dividing
360 by 24, the least common multiple of 6 and 8. The torque waveform result is periodic
and has a period of 15 deg as well, and since the coils are not excited the torquewaveform has half-wave symmetry.

0.01
0.00
-0.01
-0.02
-0.03
-0.04

Magnetic Flux Density Distribution-0.05

Fig. 2 shows the magnetic flux density distribution for a rotation angle of 0 deg and for3.75 deg.
The symmetry of the magnetic flux density distribution seen at the rotation angle of 0 deg
is not seen at the rotation angle of 3.75 deg. Cogging torque is generated by thisphenomenon.

Case

220

0

1

2

3

4

5

6

7

8

9

Rotation angle of 0 deg

10 11 12 13 14 15

Rotation angle of 3.75 deg(Unit: T)

Mechanical angle (deg)Fig. 1. Cogging Torque Waveform

Fig. 2. Magnetic Flux Density Distribution

Analysis of the High-Frequency Induction Hardening
and Cooling of a Gear

Coil

Workpiece (gear)

module: FQ, HT

Overview
In order to ensure the abrasion resistance of teeth surfaces where teeth are in contact with each other, a gear must maintain the toughness ofeach entire tooth while increasing the hardness of the surface, necessitating making the teeth with tenacity. Inductive hardening, which is oneof a number of surface hardening methods, can locally and rapidly heat and harden just the teeth surfaces.
In the induction hardening heating process, it is required to uniformly heat the surface of the gear. Also, to suppress oxidation,
decarbonization, and deformation in the cooling process, it is necessary to complete the cooling in a short time. Therefore, the cooling rate isan important factor affecting the mechanical properties of the gear heated.
In order to accurately obtain the temperature change in the gear from induction heating analysis, it is necessary to not only accurately express
the heat generated in the gear by electromagnetic induction, but also to obtain temperature distributions taking into account the change intemperature and physical characteristics due to heat generation. Because of this, electromagnetic field analysis - thermal analysis two-way
coupled analysis is essential.
In this example, deals with induction heating and cooling analysis. An example is presented where the cooling rate for a gear differs
depending on the cooling conditions.

Coil lead part

Temperature Distribution of the Gear and the Change
in Temperature vs. Time

Change Among the Gear Temperature Distributionsin the Cooling Process

The temperature distribution of the gear at the end of the heating process is shown in Fig.
1, and the change in temperature vs. time at the tooth tip is shown in Fig. 2. From thetemperature distribution, it can be seen that the tooth tip is generating heat due to eddy
currents. Fig 2 shows the change in temperature at the tooth tip width center (point of
measurement). It can be seen that water cooling is faster than air cooling so the effectfrom water cooling is higher.

Fig. 3 shows the temperature distribution at various times in the cooling process. It can beseen that in the vicinity of the teeth tips significant cooling can be achieved with watercooling in a short time.
In addition, when comparing the temperature distribution between water cooling and air
cooling, it can be seen that with water cooling the cooling of the teeth tips proceeds more
quickly than the more interior portion of the gear. This is thought to be because the heatdissipation from the teeth tips in contact with the cooling water proceeds faster than theheat conduction interior to the gear.

Water coolant
Water coolant

Air coolant

800

Temperature, deg C

700
600
500
Air coolant

400
300
200
100
0

Point of measurement(Units: deg C)

0

1

2

3

4

5

6

7

8

9

Fig. 1. Temperature Distribution After 4.0 Sec(Unit: deg C)

10 11 12 13 14 15

Time, sec

Fig. 2. Temperature vs. Time at Tooth Tip

5.0 (sec)

7.5 (sec)

10.0 (sec)Fig. 3. Changes Among Temperature Distributions vs. Timehttps://www.jmag-international.com/

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Case

221

Conﬁrmation of the Inﬂuence from the Reduction ofHigh-Frequency Resistance of a Magnetically Plated Wire Choke Coilmodule: TS, FQ
CoreOverview
A choke coil is an electronic component aimed at filtering high frequency current. The current generated inside a choke coil has a
distribution in a single wire, and among the wires, due to current bias from the skin effect, proximity effect, and leakage flux near the gap.
By using a magnetically plated copper wire with a ferromagnetic thin film layer for the coil wire, high frequency loss can be reduced
compared to conventional polyurethane copper wire. This is due to the proximity effect generated in the coil being reduced by themagnetic plating layer, and due to the effective resistance of the conductor being lowered.
In this example, high frequency resistance values are obtained from choke coil copper loss analysis results using JMAG's layer coating meshfunction, and the effectiveness of reducing the resistance value of the coil using magnetically plated wire is confirmed.

Speciﬁcation of Copper WireC

Coil
BobbinB
C

Polyurethane copper wire

Fig. 1 shows the specification of copper wire used in this case study.

Magnetically plated copper wire

Comparisons of AC Resistance Value Frequency Characteristics

Copper partA

Polyurethane
copper wire

Magnetically plated
copper wire

φ0.33(mm)φ0.33(mm)Ferromagnetic thin ﬁlm layerB-

0.6(μm)Insulating coating layerC

10(μm)

9.4(μm)OD

D

φ0.35(mm)φ0.35(mm)Fig. 1 Specification of Copper Wire

Fig. 3 show the Joule loss density distribution and isolines for polyurethane copper wire
and magnetically plated wire at a frequency of 75 kHz.
From fig. 3, it can be seen that joule loss when using magnetically plated wire is lower
than when using polyurethane copper wire.
This shows that there is an influence from the proximity effect of coil wire being reduced
when magnetically plated wire is used.

Fig. 4 shows the magnetic flux density distributions in coil wires near the gap when using
polyurethane copper wire and magnetically plated wire.
The figure shows that the magnetic flux density in the copper wire when using
magnetically plated wire is smaller than when using polyurethane copper wire.
As shown in the figure, when magnetically plated wire is used, the number of isolines in
the copper wire is large indicating that the magnetic flux concentrates on the copper wire
surface, and the difference in the magnetic flux density between the copper wire interiorand surface is higher than that for polyurethane copper wire.
Since a magnetic circuit is formed around the copper wire in the magnetic layer when amagnetically plated wire is used, and since the magnetic flux generated by the proximity
effect in the wire interior is lower, it is seen that this is effective in reducing current bias.

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6
4
2
0
0

25

50

75

100

Frequency f, kHz

Fig. 2 Comparisons of AC Resistance Value Frequency
Characteristics

Enlarged
view

Gap

Gap

Comparison of Magnetic Flux Density Distributions

8

Left : Polyurethane copper wire 75 kHz
Center : Magnetically plated copper wire 75 kHz
Upper right : Whole model(Unit: W/m^3)

Fig. 3 Comparison of Copper Loss Density Distributions(Copper wire expansion near the gap)

Gap

Comparison of Copper Loss Density Distributions

Left : Cross section of the polyurethane copper wire
Right : Cross section of the magnetically plated copper wire

10

Gap

In fig. 2, the frequency resistance values from the Joule loss values of the coil when
polyurethane copper wire and when magnetically plated wire are used are calculated, andthe frequency resistance values at each frequency are compared.
From fig. 2, it can be seen that when using magnetically plated wire, the AC resistance islower than when polyurethane copper wire is used.

AC Resistance Value R, ohm

D
A

12

Left : Polyurethane copper wire 75 kHz
Right : Magnetically plated copper wire 75 kHz(Unit : T)

Fig. 4 Comparison of Magnetic Flux Density Distributions(Copper wire expansion near the gap)JMAG Application Catalog

Case

222

Irreversible Thermal Demagnetization Analysis
of Incompletely Magnetized Magnets

MagnetRotor Core
Yoke

module: DP, ST

Overview
It is important to solve the problem of temperature rise in order to achieve high motor efficiency and high output. Magnets are used in
motors and display large variation in their properties verse temperature. For rare earth magnets at temperatures over hundred degrees
demagnetization can occur. Since the presence or absence and degree of demagnetization are determined by a combination of thedemagnetizing field and the temperature applied, it is necessary to pay attention to magnetic circuit design in addition to the heat resistanceof magnets themselves. In particular, since it is easier for demagnetization to occur at locations where the coercive force is not sufficient due
to incomplete magnetization, it is desirable to at the same time evaluate the coercive force distribution of magnets.
Using magnetic field analysis simulation that uses finite element method analysis, it is possible to evaluate motor characteristics at hightemperatures and high loads by taking into account the thermal demagnetization characteristics verse coercive force distribution values
obtained from magnetization analysis.
In this example, analysis is performed while varying the temperature of magnetized magnets, and then the influence on the torque waveform
and magnetic flux density distribution is evaluated.

MagnetStator Core

Coil

Rotor Core
motor

Position of rotor and magnetsagM

0.8e+06

t
ne
g
izin

1.2e+06

t
ne

ag
M

ic

ld,

1.6e+06

,A
ld
Fie

etic

A/

m

(Unit: A/m)

Fie

gn

Ma

3.2e+06

Input current of 25 kA Input current of 35 kA Input current of 45 kA/m

Since demagnetization characteristics after magnetization also depend on temperature, acharacteristic table was prepared verse several different temperatures where thetemperature is for after magnetization from an magnetizing magnetic field. This table wascreated for magnetizing magnetic field, temperature, and demagnetization characteristics(Fig. 1).
When used as magnet material after magnetization, a table of temperature demagnetization characteristics verse magnetizing magnetic field at the time ofmagnetization is referenced. Since the magnetizing magnetic field is not uniform inside
the magnet, the demagnetization characteristics vary in the magnet.

magnetization yoke and rotor－ 20 deg C－ 60 deg C－ 100 deg C－ 140 deg C

Flux Density, T

Magnetizable Material with Thermal Demagnetization
Information

Magnetization coilFig. 1. Magnetizing magnetic field - temperature demagnetization characteristics

Fig. 2. Magnetization distribution after magnetization

Magnetic Field Distribution After Magnetization

Induced Voltage Waveform
Fig. 3 shows the U-phase induced voltage waveform when the magnets are magnetized inplace with an input current of 25 kA, 35 kA, and 45 kA. The rms value of the induced
voltage over this time is shown in Table 1. From an evaluation of only the induced
voltages it can be confirmed that there is a difference of about 5 % between the 25 kA
case and the other cases.

25 kA

35 kA

45 kA

160
U-phase induced voltage,vFig. 2 shows the magnetic field distribution of two magnets magnetized in place (in therotor core) at input currents of 25 kA, 35 kA, and 45 kA. As the input current becomeslarger the portion of the magnets near the bridge, which is least magnetized, becomes
more magnetized. Where magnetization is low the magnetization is insufficient.

140
120
Item

100
80
60

Speciﬁcation

Induced voltage rms value, V

25

35

45

Magnetization current, kA

108.3

113.2

114.2

40
20
0

0

10

20
30
Mechanical angle,deg

4050Fig. 3. Induced voltage

Table 1. Applied current during and induced voltage rms
values over the magnetization period

Change in Torque Due to Thermal Demagnetization
25 kA
45 kA
35 kA before demagnetization
25 kA after demagnetization
45 kA after demagnetization

35 kA
25 kA before demagnetization
45 kA before demagnetization
35 kA after demagnetization

95

76

85
80

57

Position of rotor and magnets

75
Torquue, Nm

Using magnetized magnets in place for input currents of 25 kA, 35 kA, and 45 kA thetorque waveforms are shown in Fig. 4 for when the magnet temperature changes from 60
to 125 deg C at the first 1/2 electrical angle period (mechanical angle of 45 deg) and from
125 to 60 deg C at the second 1/2 electrical angle period.
From Fig. 4, it can be seen that the average torque decreases due to thermal
demagnetization when the magnets are heated to 125 deg C. In addition, even if returnedto 60 deg C, since the average torque decreases by 3 to 11 % compared to with before thetemperature increase, it can be seen that irreversible demagnetization occurred during
the temperature rise. Furthermore, it can be confirmed that the degree of deterioration of
torque characteristics due to demagnetization is dependent on the input current during
magnetization.

3%

70

3
8

11 %

65

19

60
55

0

50
45

60 deg C0

125 deg C
0.0025

Fig. 4. Torque waveform

0.005
Time,sec(Unit: %)60 deg C
0.0075

Input current of 25 kA

Input current of 35 kA

Input current of 45 kA

Fig. 5. Demagnetization ratio distribution

Demagnetization Ratio Distribution
Fig. 5 shows the demagnetization ratio distribution when the temperature is returned to
60 deg C referenced to the magnetization state before the temperature rise. The
demagnetization shown here is irreversible demagnetization caused by exceeding theknee point, which is an indicator of the deterioration of the magnet characteristics.
From the figure, even if returned to 60 deg C, significant demagnetization appears over a
wider region than before the temperature rise. Moreover, when the input current during
magnetization is small, it can be confirmed that large demagnetization occurs near thebridge.

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Case

223

Multi-Objective Optimization of an IPM Motor Accounting
for Stress at High-Speed Rotation

Stator Core
Magnetmodule: DP, DS

CoilOverview
As so to prevent mechanical issues from occurring during high-speed rotation for IPM motors, it is necessary to realize highly accurate designs
accounting for the trade-off between torque and the stress that generates from centrifugal force at high-speed rotations. The phase at which
maximum torque can be obtained will also change alongside changes to geometry, and it is necessary that optimization calculations accountfor this. FEA is essential for accuracy in these circumstances, and the use of genetic algorithms is recommended as a tool for handling theissues in this trade-off relationship.
In this example, the optimum current phase that achieves high torque per each geometry is obtained, and the torque characteristics (1,000
r/min) and stress due to centrifugal force at high-speed rotation (10,000 r/min) obtained with that optimum phase are both taken as objectivefunctions for a multi-objective genetic algorithm used to carry out the optimization of rotor geometry.

Rotor Core

Optimization Conditions
Design variables in Fig. 1 are displayed in Table 1. 6 rotor shape design variables are
selected. Magnet positions, magnet geometry, and flux barrier geometry that areconsidered to have a large influence on von Mises stress and torque are parameterized.
This time obtains geometry with high torque at 1,000 r/min and low von Mises stresswithin the rotor during rotations at 10,000 r/min. In accordance with this, 2 evaluation
items shown in Table 1 are used and the performance of rotor geometry is evaluated.

Design variable range(4) Slit depth(2) Magnet width(3) Magnet 1/2 length(1) Magnet distance from center(5) Slit angle

Item

Minimum value(1) Magnet distance from center, mm26(2) Magnet width, mm

2

5.49

(3) Magnet length, mm

3

9.15

(4) Slit depth, mm

Maximum value
36.6

0.1

1.83

(5) Slit angle, deg

45

140

(6) Slit clearance, mm

0.1

1.6

Evaluation items
Evaluation items(objective functions)

Optimization Results

Selecting Rotor Phase
Fig. 3 shows a graph of the phase of each geography and torque. From the maximum
phase geometry obtained from the torque formula, it is understood that the phaseshowing maximum torque differs. The phase showing the maximum torque value isshown in red.

(6) Slit clearance

Fig. 1 Optimization Design Parameters

Von Mises stress maximum value, MPa

Fig. 2 shows the result of multi-objective optimization using genetic algorithms with the
maximum value of von Mises stress and average torque objective functions. Performance
improvements can be confirmed from a comparison of all individuals from the initial and10th generations. It can also be observed that von Mises stress and torque are in a
trade-off relationship in the 10th generation.
The case with the lowest von Mises stress (1) and the case with the highest average torque(2) in the initial generation are emphasized. Cases A and B are 10th generation cases withthe maximum average torque for approximately the same maximum von Mises stress ascases (1) and (2) respectively, and are shown in Table 2. While each have approximately
the same maximum von Mises stress, it can be confirmed that average torque improves
between (1) and A by 93 %, and by 16 % between (2) and B.

Torque average value

Torque characteristics are evaluated using average
torque values.

Von Mises stressmaximum value

It is considered that the lower the von Mises stressmaximum value is in the model, the less likely
breakage will occur. The von Mises stress maximum
value in the model is used for evaluations.

Table 1. Design Variable Ranges, Evaluation Items

600
Item

Speed 10,000 r/min
Von Mises stress maximum
value, MPa

400

Initial individuals (1)(low stress)

110.5

22.3

300

Initial individuals (2)(high torque)

479.0

50.5

10th generation A

105.0

43.0

10th generation B

470.6

58.5

(2)

500

B

200
(1)

0

0

10

20

30
Torque, Nm

40

50

60

Fig. 2. Initial Generation and 10th Generation
Population PerformanceA

Low speed 1,000 r/min
Average torque, N・mA

100

Rotor Interior Magnetic Flux Flow

B

(1)

Table 2. Comparison of Performance of Individuals

(2)

60
50

Torque, Nm

The magnetic flux density distribution and magnetic flux lines of each geometry are
shown in Fig. 4. For thick-bridge geometry (A), it is thought that the magnetic flux flows
through the bridge to the adjacent magnetic pole, causing a decrease in torque.
Conversely, for thin-slit geometry (B), a high torque value is obtained because theshort-circuited rotor interior magnetic flux is suppressed and flows to the stator.

Description

40
30
20
10

Rotor Interior Magnetic Flux Flow
The magnetic flux density distribution and magnetic flux lines of each geometry are
shown in Fig. 4. For thick-bridge geometry (A), it is thought that the magnetic flux flows
through the bridge to the adjacent magnetic pole, causing a decrease in torque.
Conversely, for thin-slit geometry (B), a high torque value is obtained because theshort-circuited rotor interior magnetic flux is suppressed and flows to the stator.

0
0

10

20

30

40

50

60

Phase, deg

AFig. 3 Mises Stress Distributions for 10th Generation
Individuals

Von Mises Stress Distribution
The von Mises stress distribution of geometry A and B shown in Fig. 5. It can be confirmedthat for thin-bridge geometry (B), there is a tendency for stress to be higher at thoselocations, while von Mises stress remains low for thick-bridge geometry (A). From this it
can be observed that maintaining bridge thickness is effective in controlling von Mises
stress.
A

B(Unit:MPa)Fig. 5. The Von Mises Stress Distribution of 10th
Generation Individuals

130

JSOL Corporation

B(Unit:T)

Fig. 4. 10th Generation Rotor Geometry and Magnetic
Flux Density Distribution

JMAG Application Catalog

Case
Chassis224 Wireless Power Transfer Device Stray Loss Analysis

Ferrite core(power receiving side)

Coil(powerreceivingside)

module: FQ

Overview
In recent years, using electromagnetic induction type power supply systems for wireless power transfer systems for electric vehicles(hereinafter referred to as EVs) and plug-in hybrid cars (hereinafter referred to as PHEVs), for which practical applications are sought, havebeen investigated. Wireless power transfer systems for EVs and PHEVs are required to maintain power transmission with high efficiency. Many
studies on magnetic resonance coupling (magnetic resonance) schemes with resonance capacitors have been performed.
Considering corrosion resistance and strength, steel plates such as SUS steel plates are used for the casings of wireless power transfer
equipment. Because of this, if the leakage magnetic flux component generated from the power supply device interlinks with the metal casing,stray loss occurs resulting in a decrease in efficiency. Therefore, when the loss component is high, it is necessary to suppress the loss byproviding for example an aluminum shield.
Since in order to accurately calculate stray loss it is necessary to accurately calculate the flow of magnetic flux interlinking with steel plates oraluminum shields and the eddy current distribution caused thereby, electromagnetic field analysis using finite element methods is necessary.
In this example, an example is presented investigating by using electromagnetic field analysis whether stray loss can be suppressed using an
aluminum shield.

Coil(power supply side)

AluminumshieldFerrite core(power supply side)

Stray Loss Distribution
Fig. 1 shows the Joule loss distributions of the chassis and aluminum shield. Regardless ofthe presence or absence of the aluminum shield, it can be seen that Joule loss due toleakage flux is generated around the ferrite cores. After installing an aluminum shield on
the surface of the chassis, it is possible to confirm that almost no loss is generated in the
chassis due to this shielding with loss occurring only on the aluminum shield surface. In
addition, it can be confirmed that there are places with higher Joule loss density with thealuminum shield installed as compared to before the aluminum shield was installed.

Stray loss, W

Chassis surface

Difference in Stray Loss With and Without an Aluminum Shield
Table 1 shows Joule loss values due to eddy currents. It can be confirmed that the strayloss could be reduced when an aluminum shield was installed even though the Joule loss
density on the surface was higher. This is because the skin thickness decreased and thevolume where the loss occurred decreased since aluminum was used which has higher
conductivity than the SUS material used for the chassis.

Case

225

Aluminum shieldsurface

Chassis surface(a) Aluminum shield
not installed

Chassis

Aluminum shield

Total

Aluminum shield
not installed

77.7

-

77.7

Aluminum shield
installed

0.5

15.5

16.0

(Unit: W/m^3)(b) Aluminum shield installed

Fig. 1. Joule Loss Density Distribution

Table 1. Comparison of Joule Loss Values

Induced Voltage Analysis on Memory Motors Using
Variable Magnets

Stator core
Fixed magnetic
force magnet

module: ST, DP

VariablemagnetOverview
There are many kinds of variable flux motors, but the kind that makes a permanent magnet magnetization variable is called a memory motor.
The magnetic field for magnetizing and demagnetizing memory motor magnets is generated by the same three-phase coil current as in thedrive system, so the magnetic force can be changed without adding any special devices.
In this example, the change in motor characteristics are checked as the memory motor variable magnet magnetization is changed using thethree-phase coil current. One example of modeling using a variable magnet user subroutine is presented as well.

Coil

Rotor core

Alteration of Variable Magnetized Magnet Induced Voltage
Vv

Vu

Vw

80
60
Induced voltage, V

Fig. 1 shows the induced voltage when the variable magnets are fully magnetized, thendemagnetized, and then magnetized again.
In Table 1, the induced voltage state for each interval in Fig. 1 is shown as well.
In interval (3) in Fig. 1, from demagnetizing the variable magnets, only the induced
voltage due to the fixed flux magnets can be seen. The maximum change of induced
voltage from this model is approximately 50 %.

40
20
0

Variable Magnet Operating Points
The variable magnet operating points are shown in Fig. 3. While changing the input
magnetization characteristics, it can be seen that the residual magnetic flux density Br
changes when a magnetization/demagnetization current is applied.

Current state0No-loadMagnetization

(2)

-10

Demagnetizing current applied

Demagnetization

Variable magnet magnetization state(3)0No-loadDemagnetization

(4)

+200

Magnetizing current applied

Magnetization(5)0No-load

-60

(1)

(2)

(3)

(4)

(5)

0.01

0.02

0.03

0.04

0.05

Magnetization

-80
0

0.06

Time, sFig. 1. Change in Variable Magnet Magnetization
State and Induced Voltage

Table 1. Induced Voltage State

1.5

Magnetic ﬂux density, T

Fig. 2 shows the flux lines at no-load when the variable magnets are magnetized anddemagnetized.
The left is when magnetized and the right is when demagnetized. When the variablemagnets are demagnetized, it can be seen that the magnetic flux from the magnets
flowing in the stator is decreased.
Also, from the enlarged views of two variable magnets it can be seen that in the left viewfor the magnetization state several flux lines penetrate the variable magnets passing fromone magnet core to the other and returning back again.
On the other hand in the right view for the demagnetization state, the flux lines do not
penetrate the variable magnets, and it can be seen that the flux lines generated from thefixed magnetic force magnets flow around the variable magnets. This means that astronger magnetic flux is generated from the fixed magnetic force magnets than from thevariable magnets.

Id (A)

(1)

-20
-40

Flux Line Changes Under No-load

Interval

①

1

④
0.5

⑤

0
②

③

-0.5
-1
-1.5
-200

1st step(completely magnetized, no load)

Fig. 2. Flux Lines at No-load

97th step(completely demagnetized, no load)

-150

-100

-50

0

50

100

150

200

Coercive ｆorce, kA/m

Fig. 3. Variable Magnet Operating Points

https://www.jmag-international.com/

131

Case

Case

226 Vibration Characteristics Analysis of SPM Motors

Bearing
Rotor coremodule: DP, DS

MagnetOverview

Stator core

Electromagnetic force acting on a motor causes vibration and noise in the form of electromagnetic excitation force. Vibration and noise also
are produced when this electromagnetic excitation force resonates with an eigenmode of the motor. In order to accurately evaluate thisphenomenon, it is necessary to ascertain each frequency and spatial mode in detail about the electromagnetic force and eigenmodes of the
motor.
In this example, the acceleration is evaluated by obtaining the electromagnetic force generated in the stator core of the SPM motor at multiple
rotation speeds and coupling with the eigenmodes of the motor. In addition, the frequency components of the electromagnetic force andeigenmodes and the spatial modes are analyzed, and the resonance modes are confirmed.

Flange bracket
Shaft

Eigenmodes
Notice the spatial 4th-order mode is focused on. From the eigenmode analysis results, theeigenfrequency for each vibration mode is found.
The spatial 4th-order mode results are shown in Fig. 1 and Table 1.
Acceleration, m/sec^2

330
315

0

15

1.471
45

0

spatial 4th-order42, 4310.8 kHz

195

180

165

1.471
0.721

0.173
0.085

12000

15000

Fig. 3. Amplitude of Acceleration of Vibration Due to
Excitation from Electromagnetic Force

0.042

26th-order22th-order

4000

0.020
0.010

0
4500 6000 7500 9000 10500 12000 13500
Rotation speed, r/min

Acceleration(Unit: m/sec^2)Upper: 22nd-order 8.8 kHz
middle: 26th-order 10.4 kHz
lower: 34th-order 13.6 kHz

Fig. 4. Eigenmodes Dependent on Electromagnetic Force
Order and Number of Poles

0

345
3

15

0

285

-3

270

-6

255

105
120
135
195

180 165

150

Fig. 5. Resonance Vibration Modes at 12,000 r/min

22th-order26th-order34th-order1
24
23 0.06
30
0.04
22
45
0.02
0
21
60
-0.02
-0.04
75 20
-0.06
-0.08
90 19
-0.1

Case surface acceleration (m/sec^2)
at phase 0 deg

JSOL Corporation

9000

34th-order

12000
spatial4th-order mode
8000

210

132

6000

Rotation speed, r/min

16000

225

Table 1. Eigenfrequencies

3000

Acceleration(Unit: m/sec^2)

20000

3.000

0.353

0
0

150

Fig. 3. Amplitude of Acceleration of Vibration Due
to Excitation from Electromagnetic Force

240

Fig. 1. Spatial 4th-order Mode Eigenmodes

0.010

135
210

Measurement points

20000
10000

0.020

120

225

30000

0.042

105

240

300
Eigenfrequency

0.085

90

255

330
315

Orders

0.173

75

-2

22th-order26th-order34th-order

Eigenmode

0.353

60

270

40000

0.721

-1

285

50000

3.000

30

300

Frequency, Hz

Coupled analysis of magnetic field transient response analysis and structural frequency
response analysis is performed to evaluate the acceleration of the vibration excited by theelectromagnetic force generated in the stator.
Fig. 2 shows a radar graph of the cylindrical coordinate system R component of theacceleration at a measurement position with a phase of 0 deg. As a measurement
example, the graph shows the acceleration computed from the electromagnetic force
2nd-order component. Also, the amplitude of the acceleration is plotted on graphs withaxes of rotation speed and frequency shown in Fig. 3 and Fig. 4.
Among the plotted accelerations, some particularly large values can be seen. The
electromagnetic forces which are the excitation forces of these vibrations are 22nd-, 26th-,and 34th-order, and the acceleration increases because the rotation speed rises and theelectromagnetic force 22nd-, 26th-, and 34th-order frequencies become close to the
4th-order mode eigenfrequency. This is due to specific frequency components ofelectromagnetic force strongly resonating with certain eigenmodes.
The 22nd-, 26th-, and 34th-order vibration modes at 12 kHz are shown in Fig. 5. All the
spatial 4th-order vibration modes are shown.
For the 22nd, 26th, and 34th-order components of the electromagnetic force, when thecylindrical coordinate R component at phase 0 deg is shown in a radar graph similar to theacceleration radar graph above, it can be seen that there is a strong correlation between
the acceleration and the electromagnetic force (Fig. 6). On the other hand, there are
frequency components close to the spatial 4th-order mode eignenfrequency, but it canbe seen that for small order vibrations there are different spatial modes as shown in Fig. 7.

345 1

Frequency, Hz

Rotation Speed, Frequency, and Acceleration Graphs

2

23
22

3
5
6

1

28th-order230th-order

3

5

-0.01

20

-0.02

19

-0.03

32th-order

4

0

21

4

24
0.01

6
7

7
18

18

8

8

17

9
16

10
15

14

13

12

11

Electromagnetic force R component (N)at phase 0 deg for each toothFig. 6. Spatial 4th-Order Mode Vibration andElectromagnetic Force

17

9
16

10
15

11

14
12
13
(Unit: N)
The 28th- and 32nd-order electromagnetic forces are in spatial 8th-order mode,and the 30th-order electromagnetic force is in spatial 12th-order mode.

Fig. 7. Spatial Non-4th-Order Electromagnetic ForceJMAG Application Catalog

Case

227

Circuit Control Simulation for Three-Phase
Induction Motor

Coil
Rotor coremodule: FQ, RT

Overview
Since control design and motor design are designed by different designers, coordination of the designs is difficult. However, in order to realize
the sophisticated demands of recent years, it is important to identify problems during the coordination between motor design andcontrol/circuit design in the early stages of development. In cooperative design, it is possible to utilize circuit simulation using a highly
accurate plant model.
Characteristics of the induction motor depend on motor shapes, materials, and drive states. For example, it is necessary to calculate theresistance of the secondary side by taking into account the current distribution in the secondary conductor, but the phenomenon iscomplicated and it is difficult to calculate manually. JMAG-RT can create a highly accurate plant model (JMAG-RT model) by calculating the
FEA while taking the motor shape, material, and driving condition into account, then extracting the characteristics of an induction motor.
In this example, we conduct a circuit simulation to control current and speed by incorporating the JMAG-RT model of an induction motor into
the control/circuit simulator.

Cage (bars)
Stator core

Control Circuit

Rotation Speed Waveform and d-Axis Current Waveform

The control circuit is shown in Fig. 1. The command values are 2,000 r/min for rotation
speed and 30 A for the d-axis current, and the voltage command value is connected to themotor via the inverter.

Fig. 2 shows the rotation speed waveform and Fig. 3 shows the d-axis current waveform. It
can be seen that both become steady at their command values.

Command value
Id command value

Command value
Circuit unit and motor

Rotation speed

150

Current, A

2000

1500

1000

0
0.0

Fig. 1 Control Circuit (Control Unit)

100

50

0

500

Rotation speed command value

Id

200

2500

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Fig. 2. Rotation Speed Waveform

-50
0.0

0.2

0.4

0.6

0.8

1.0
1.2
Time,sec

1.4

1.6

1.8

2.0

Fig. 3. d-Axis Current Waveform

Case

228

Stator Core

Induction Motor Thermal Analysis

Coilmodule: HT, LS, TR
Cage

Overview
Induction motors can be found in a wide number of uses, from industrial machines to home appliances, as they are simple construction in
addition to being small, light, affordable, and maintenance-free. An induction motor is a motor in which a rotating magnetic field in thestator winding causes an induced current to flow in an auxiliary conductor. This current and magnetic field exert force on the auxiliary
conductor in the rotation direction and cause the motor's rotor to rotate. When this occurs, a large amount of joule heat is generated in the
motor due to the influence of induced currents.
In order to correctly evaluate increases in temperature induction motor joule loss and iron loss, the use of coupled analyses is advantageous
as so to obtain temperature distribution from thermal analyses using iron loss density distribution obtained from magnetic field analysis.
This makes it possible to confirm overall temperature distribution and locations where temperatures grow particularly high.
In this example, evaluating induction motor temperature distribution.

Shaft
Rotor Core

Loss Density Distribution
The joule loss density of the cage and windings is displayed in Fig. 1 and Fig. 2
respectively. A large portion of joule loss is distributed in the end ring.
Iron loss analysis results for the stator and rotor are displayed in Fig. 3 and Fig. 4.

Temperature Distribution
Stator temperature distribution is displayed in Fig. 5, and rotor temperature distribution isdisplayed in Fig. 6.
For the stator, each part is a heat source, and it is understood that temperatures rise until
approximately 63 deg C. On the other hand, the main source of rotor heat is the cage, andit can be confirmed that heat is transferred to the rotor core. Because it is additionallydifficult to emit the heat trapped within the stator, this becomes a high temperature of
156 deg C or above.

(Unit: W/m^3)(Unit: W/m^3)

Fig. 4. Rotor Iron Loss

Fig. 3. Stator Iron Loss

(Unit: deg C)

Fig. 1. Cage Joule Loss

(Unit: deg C)(Unit: W/m^3)(Unit: W/m^3)

Fig. 2. Winding Joule Loss

Fig. 5. Stator Temperature

Fig. 6. Rotor Temperature

https://www.jmag-international.com/

133

Case

229

Analysis of the Eﬀect of PWM on the Iron Loss of an
Induction Motor

Rotor coreCage

module: DP, LS

Overview
An induction motor is a motor in which the rotating magnetic field of the stator coils causes induced current to flow in an auxiliary
conductor, which produces force in the rotational direction. Induction motors are widely used in everything from industrial machines tohome appliances because they have a simple construction without parts that experience wear from abrasion, and can be used simply byconnecting them to a power source.
Usual simulation use sinusoidal inputs for their studies of induction motors but in reality electrical machines are generally fed using Inverter
drives that generates PWM signals. For this reason looking at PWM input in FEA simulation would allow to give us a better idea of the losses
in an induction motor.
In this example, we introduce a case study to obtain the iron losses of the induction motor that accounts for the PWM input.

Stator core

PWM
The PWM of the induction motor will be generated using a modified macro of thestandard PWM Inverter of JMAG. The PWM Inverter uses the angular positon of the rotor
to create the PWM but this cannot be used in a Induction machine setting as the machine
does not turn at synchronous speed. We need therefore to Set the PWM to besynchronized with a frequency, in our case 120 Hz and give us a voltage of 40.8 V. The
switching frequency was set at 10 kHz.
Fig.1 illustrates the circuit used to create the PWM and fig 2 shows the waveform.

50
40
30

Voltage, V

20
10
0
-10
-20
-30
-40
-50
0

Losses in Induction Motor
Fig. 1 PWM Circuit

0.015

0.02

0.025

Fig. 2 PWM Inverter waveform

Measuring points

With PWM

Without PWM

1.E+00
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05(Unit: T)

1

10

100

1,000

10,000100,000

Frequency, Hz

Iron Loss Study
An iron loss study will then be applied on the results of the magnetic study. Here, compare
the iron losses in the induction machine with and without the use of PWM.
Table 1 shows the different losses in the rotor and stator iron. We can observe that the
total losses in the rotor iron increased by 28 % and 17 % in the stator Iron.

0.01

Time, sec

Magnetic ﬁeld ﬂux, T

The study was set so that induction machine rotating at a speed of 3,300 r/min (a slip of
0.0833). To obtain the currents and magnetic flux density in the induction machine we will
use a magnetic transient study. The time step was set so that each period of PWM has atleast 30 points.
Fig. 3 shows the magnetic flux density of the induction machine. This highlights the areas
where the magnetic filed is the most concentrated. The higher magnetic field flux density
will also induce more Iron losses.
Fig. 4 and fig. 5 shows the FFT of magnetic flux density at the rotor teeth and stator teeth
with PWM and without. These show that the losses considering PWM will probably
increase due to the harmonics present in the Iron.
Fig. 6 shows the iron loss density distribution.

0.005

Fig. 3 Magnetic Flux Density

Fig. 4 Magnetic Flux in Rotor Teeth

With PWM

Without PWM

Magnetic ﬁeld ﬂux, T

1.E+00

1.E-01

1.E-02

1.E-03
10

100

1000

10000

(Unit: W/m^3)

Frequency, Hz

Fig. 6 Iron Loss Density

Fig. 5 Magnetic field flux In Stator Teeth

Part

Iron Loss (Total)
With Without
PWM
PWM

Diﬀ

Joule Loss
With Without
PWM
PWM

JSOL Corporation

With Without
PWM
PWM

Diﬀ

Rotor
Stator

3.23 W 2.76 W +17 % 2.23 W 1.81 W +23 %Table 1 Iron Loss Comparison

134

Hysteresis LossDiﬀ

0.69 W 0.54 W +28 % 0.60 W 0.47 W +28 % 0.09 W 0.07 W +28 %1W

0.95 W +5 %

JMAG Application Catalog

Case

Rotor Core230 6-Phase SPM Motor Inverter Fault Simulation

U2

U1

V2

Magnet

V1
W2

W1

W1

W2
V1

module: DP, RT

U2
U1

V2
U1

Overview

V2

U2

In recent years, multiphase design has been adopted for automotive motors due to the merits of cost reduction and improvement infunctions. For example, for power supplies there are 6-phase motors with 3-phase double windings that use two inverters, having a 30 deg
phase shift in electrical angle. From this, by using multiple inverters, enhanced redundancy of the in-vehicle motor system is acheived. The
motor system can be stopped safely without falling to be in an uncontrollable state even if a failure occurs in a part of an inverter. However,circuit simulation is necessary to accurately evaluate safety. When an inverter malfunctions, from overcurrent flowing and from the range ofthe magnetic field in and around the steel plate where the magnetization properties are nonlinear, inductance exhibits nonlinear behavior as
well. Because of this, a highly accurate plant model cannot be created using only linear methods. In JMAG, it is possible to create a plant modelthat models actual motors in detail taking into account magnetic saturation characteristics and spatial harmonics exhibited in motors.
In this example, a JMAG-RT model of a 6-phase SPM is loaded into a control/circuit simulator, and circuit simulation for an inverter failure is
carried out.

V1

W1

W2
W2

W1
V1

V2

U1

U2

Coil

Stator Core

Control Circuit

Current Waveform, Torque Waveform

The control circuit is shown in Fig. 1. A command value of current amplitude of 8 Apeak
and current phase of 0 deg is used, and the voltage command value is sent to the motor
via the inverter. Inverter 1 is connected to the U1, V1, and W1 phase terminals, andinverter 2 is connected to the U2, V2, and W2 phase terminals. In addition, a fault alarm is
set to go off at 0.02 sec after the start of simulation execution.

The currents for the U1, V1, and W1 phases, for the U2, V2, and W2 phases, and the torquewaveform are shown in fig. 2, fig. 3, and fig. 4 respectively.
From fig. 2, it can be seen that the U1-phase current decreases somewhat after theoccurrence of the fault, and overcurrent flows through the V1-phase and W1-phase coils.
Meanwhile, from fig. 3 it can be seen that the U2-phase, V2-phase, and W2-phase currents
are stable.
As shown in fig. 4, although the torque exhibits a large periodic ripple after the fault, its
size remains steady. From this, it can be judged that the motor system can safely be
brought to a stopped state without falling into an uncontrollable state.

Time of fault

Inverter fault command

U1 phase

V1 phase

Time of fault

W1 phase

15

15

10

10

5

5

U2 phase

V2 phase

Time of fault

W2 phase

6

Current command

0

-5

-10

-10

0.01

0.02

0.03

0.04

0.05 0.06
time, s

0.07

0.08

0.09

Fig. 2. Current Waveform (U1, V1, and W1 Phases)

0.1

4
3
2
1

-15
0

Fig. 1. Control Circuit

0

-5

-15
Inverter 2

Torque, Nm

Current, A

Motor

Current, A

5
Inverter 1

0

0.01

0.02

0.03

0.04

0.05 0.06
time, s

0.07

0.08

0.09

Fig. 3. Current Waveform (U2, V2, and W2 Phases)

0.1

0
0

0.01

0.02

0.03

0.04

0.05
time,s

0.06

0.07

0.08

0.09

0.1

Fig. 4. Torque Waveform

https://www.jmag-international.com/

135

Case

231

Vector Control Simulation of a Synchronous
Reluctance Motor

Rotor Core

module: DP, RT

Overview
Since control design and motor design are designed by different designers, cooperative design is difficult. However, in order to meet thesophisticated demands in recent years, it is important to identify problems while coordinating motor design and control/circuit design at theearly stages of development. In cooperative design, it is possible to utilize circuit simulation using a highly accurate plant model.
When operating a synchronous reluctance motor (hereinafter referred to as a SynRM), since the nonlinear range of the magnetization
properties of an electromagnetic steel plate is used, inductance also exhibits nonlinear behavior. Because of this, a highly accurate plantmodel cannot be created using only linear methods. In JMAG, it is possible to create a plant model that models actual motors in detail taking
into account magnetic saturation characteristics and spatial harmonics exhibited in motors.
In this example, circuit simulation is carried out simulating the control of current and speed by loading a JMAG-RT model for a SynRM into acontrol/circuit simulator. In addition, comparison with an ideal motor model is performed.

Stator Core

Control Circuit

Rotation Speed Waveform, d-Axis Current Waveform

The control circuit is shown in fig. 1. The rotation speed command value is set to 600 r/min
and the d-axis current command value is set to -5 A, and the voltage command value issent to the motor via the inverter.

Fig. 2 shows the rotation speed waveform for a JMAG-RT model and an ideal motor
model, and fig. 3 shows the d-axis current waveform.
It can be seen that both converge to the command value. Moreover, it can be seen that
there is a difference in behavior between the two as regards the overshoot.

Id command value

Command value

Circuit section and motor

JMAG-RT model

Ideal motor model

900

Rotation speed, r/min

700
600
500
400
300
200
100
0
-100
0.00

0.05

0.10

0.15

Fig. 1. Control Circuit (Control Section)

U-phaseIdeal motor model

-4
-6
-8

0.15

0.20
Time,sec

0.25

Fig. 3. d-Axis Current Waveform

136

0.35

0.40

JSOL Corporation

0.30

0.35

0.40

V-phaseW-phaseU-phaseV-phaseW-phaseJMAG-RT model

25

25

6

20

20

5

15

15

10

10

5
0

5
0

-5

-5

-10

-10

-15

-15

-20
0.00

0.05

0.10

0.15

0.20
Time,sec

0.25

0.30

0.35

Fig. 4. Coil Current Waveform (JMAG-RT Model)

0.40

-20
0.00

Ideal motor model

4
Torque, Nm

Current, A

Current, A

-2

0.10

0.30

Current, A

JMAG-RT model

0

0.05

0.25

Fig. 2. Rotation Speed Waveform

2

-10
0.00

0.20
Time, sec

Rotation speed command value

Command value

Coil Current Waveform, Torque Waveform
The coil current waveforms when using the ideal motor model and the JMAG-RT model
are shown in fig. 4 and fig. 5, and the torque waveform is shown in fig. 6.
Regarding the rise, the JMAG-RT model has more current flowing, but the torque being
generated is lower. This is due to the magnetic saturation of the core being precisely
taken into account. In addition, looking at the steady-state torque waveform, it can beseen that there is torque ripple due to spatial harmonics.

800

3
2
1
0
-1

0.05

0.10

0.15

0.20
0.25
Time,sec

0.30

0.35

Fig. 5. Coil Current Waveform (Ideal Motor Model)

0.40

-2
0.00

0.05

0.10

0.15

Fig. 6. Torque Waveform

0.20
Time,sec

0.25

0.30

0.35

0.40

JMAG Application Catalog

Case

232 Reactor Joule Loss AnalysisCore

module: FQ

CoilOverview
When designing a high-frequency reactor which is a key component in a converter, inductance and loss, which are parameters required forvoltage conversion, are important factors.
By setting gaps in the core of the reactor, it becomes difficult for the core to become magnetically saturated making a stable inductanceobtainable. However, the position of the air gaps may affect Joule loss and eddy currents in the coil.
Therefore the position of the air gaps is an important issue in the design of a reactor. It takes however an inordinate amount of time andmuch cost to perform studies in trial production. By using FEA, it is easy to visualize leakage magnetic flux and compare Joule losses against
air gap positions easily.
In this example, investigating Joule loss with respect to the positions of air gaps in a reactor is presented.

SpacerFlux Leakage
A sine wave current of 0.5 A at 10 kHz is supplied to the reactor. If there are spacers (airgaps), there is leakage magnetic flux. In addition, this leakage magnetic flux induces
current in the coil, generating Joule loss.
The leakage magnetic flux in a certain area is shown in Fig. 1, and the induced current
generated in the coil is shown in Fig. 2.

Coil inner sideenlarged

Joule Losses Comparison(Unit: A/m^2)

Fig. 1. Leakage Magnetic Flux

Fig. 2. Induced Current Generated in a Coil

0.18
0.16
0.14
Joule Losses, W

Next, losses are compared when changing the spacer positions over 19 intervals.
Fig. 3 shows a Joule loss comparison graph and Fig. 4 shows leakage magnetic flux
comparisons.
From Fig. 3 it can be seen that the position of the spacers greatly affects the Joule loss inthe coil.
When the two spacers are placed side by side, some of the magnetic flux leaks from thecore center.
As shown in Fig. 4a, when the two spacers are placed close to each other, it is seen thatthe magnetic flux leaks to the sides, is short-circuited, and flows near the core center. In
addition, as shown in Fig. 4b, if spacers are placed at the mid-interval position from thecenter, though leakage magnetic flux is generated, it can be seen that it is over a smaller
region. However in Fig. 4c it can be seen that the leakage magnetic flux extends to the
nearby portion of the core as with the magnetic flux short-circuit portion of the core in
Fig. 4a.

0.12
0.1

a：1 mm0.08
Enlarged

0.06

b：10 mmc：19 mmSpacer position
Enlarged
Enlarged

0.04
0.02
0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19(Unit: T)

Spacer position, mm

Fig. 3. Joule Losses ComparisonsFig. 4. Leakage Magnetic Flux Comparisons

https://www.jmag-international.com/

137

Case

233 Induction Heating Coil Optimization

Coil

Gear

module: FQ, HT

Overview
As one of the surface hardening techniques for steel, high-frequency induction hardening has been applied to many kinds of mechanicalparts. In general, prototyping and testing high-frequency induction hardening equipment requires a great deal of time and money, and issometimes accompanied by danger. Moreover, in addition to the complex phenomenon called induction heating, material properties alsochange greatly due to heating exceeding the Curie point, making it difficult to estimate the heating state. For this reason, simulation basedon finite element analysis which can model detailed phenomena is useful.
When hardening is performed, uniform heating is desirable, but there are significant factors to consider such as the geometry andarrangement of heating coils and the adjustment of current frequency and amount. When there are many design variables or items toevaluate, applying automatic computation using an optimization function for simulation can greatly reduce the workload.
In this example, an example of using an optimization function to design a coil to use for induction hardening a gear is presented.

Coil Design Variables
The coil has two turns, the distance from the center of the workpiece, CPosition, and thecoil width CWidth are design variables. In addition, input current lamp is also defined as adesign variable.
Fig. 1 shows the coil dimension design variables, and Table 1 shows all the design
variables and range settings.
Fig. 2 shows the temperature evaluation points, and Table 2 shows the objective functions
for coil optimization.The objective functions have the following three objectives.
- Minimizing input current- Minimizing the standard deviation Tdev from the target temperature T̲R using theevaluation point temperatures T̲i (Fig. 2)- Minimizing the bias Tbias from the target temperature T̲R using the evaluation pointtemperatures T̲i (Fig. 2)

Variable nameMinimum value

Iamp

3

8

CPosition

1

20

CWidth1025

CWidthCPosition

Table 1. Design Variables

Fig. 1. Design Variables (Coil Dimensions)

Initial Design

T1

Fig. 3 shows the temperature distribution with the initial design.
As shown in Fig. 3, the workpiece temperature is high at the temperature evaluation
points T2 and T4 at the center of the workpiece, and low at the temperature evaluation
points T1 and T3 near the end faces of the workpiece.
In the initial design, since the difference in temperature among the temperatureevaluation points is large, improvements need to be made.

T3

T2

Item

Expression

Objective

Input current, kA
Input

3≦Iamp≦8

Minimum value

Sensitivity Evaluation Using a Correlation Matrix
Fig. 6 shows a correlation matrix, Fig. 7 shows a CPosition-Tdev scatter plot, and Fig. 8
shows a comparison of two temperature distributions from two optimization cases withthe same current where in one case Tdev is larger.
As seen in Fig. 6, CPosition is the most sensitive among the design variables, and the coilwidth CWidth has low sensitivity. In addition, as in Fig. 7, as CPosition increases, Tdev also
increases.
From Fig. 8, it can be seen that as the coil positions move to the ends of the workpiece,
the temperature also becomes higher near the ends of the workpiece and thetemperature variation becomes larger.
From these results, it can be seen in this Fig. 8 example that CPosition has high correlation
with Tdev, heating occurs more uniformly for cases when CPosition is small, and CWidth
has low correlation with Tdev.

Bias
Tbias

∑

Minimum value

1
𝑇𝑖−𝑇𝑅
4 𝑖=1

∑

𝑇𝑏𝑖𝑎𝑠 =

Minimum value

𝑇𝑖 : temperature at a measurement point
𝑇𝑅 : target temperature = 850(deg C)
𝑇𝑑𝑒𝑣 : standard deviation
𝑇𝑏𝑖𝑎𝑠 : bias

Table 2. Objective Functions

Fig. 2. Temperature Measurement Points

T1

Case
ParetoInitial design
Optimized value
Case with a lower current value than the optimized value case

T3

500
T2

450
400

Tdev

350
T4

300
250
200
150
100
50

(Unit: deg C)

0Fig. 3. Initial Design Temperature Distribution
Contour Plot

2.5

3

3.5

4

4.5
5
Input, kA

5.5

6

6.5

Fig. 4. Input Current Input vs. TemperatureStandard Deviation Tdev

Iamp CPosition CWidth T1

T2

T3

T4

Input Tbias

Tdev

Iamp
CPosition
CWidth
T1
T2
T3
T4
Input
Tbias
Input current: 4.7 A

Input current: 4.2 A(Unit: deg C)

Fig. 5. Temperature Distribution Comparison for
an Optimized Geometries vs. Input Current

Fig. 7. Scatter Plot

JSOL Corporation

1(𝑇𝑖−𝑇𝑅 )4(𝑖=1)

𝑇𝑑𝑒𝑣 =

4

Tdev

Fig. 6. Correlation Matrix

CPosition：3

138

4Standard deviation
Tdev

T4

Pareto Curve Produced From a Multi-objective Optimization
Optimization using a multi-objective genetic algorithm with a population size of 15 and
10 generations is performed. Fig. 4 shows a graph of input current Input and the standard
deviation of temperature, Tdev, obtained by the multi-objective optimization.
The standard deviation of temperature, Tdev, one of the design variables with theobjective of being minimized, is shown. The optimized value case, the case with theminimum Tdev value, may lower the input current value, and also reduces the standard
deviation compared with the initial design.
Whether the current is lower after the optimization is checked. As shown in Fig. 4, if thecurrent value is lowered, the temperature does not rise sufficiently at any of the four
evalution points, and the standard deviation Tdev is larger.
Fig. 5 shows a comparison of the temperature distribution for an optimized geometry vs.
input current.

Maximum value

CPosition：16(Unit: deg C)

Fig. 8. Temperature Distribution Comparison vs. CPositionJMAG Application Catalog

Case

Case

234 Vibration Characteristics Analysis of an Induction Motor

BearingBar

Rotor coremodule: DP, DS

Stator core

Overview
Because the construction of induction motors is simple, easy, durable, and highly effective, they have been widely used throughout industry ingeneral for an extensive period of time. In recent years however, there has been an increase in words raised concerning issues of pollution andimprovements to work environments, and the noise that emanates from induction motors is also becoming an issue. Electromagnetic force
acting on the induction motor is the source that generates torque, but at the same time the excitation force of the motor itself generates
electromagnetic noise.
To favourably evaluate both noise caused by electromagnetic force and vibrations, it is necessary to accurately grasp the electromagnetic
force distribution acting on the stator core which is the source of radiated sound, and to obtain the eigenmode of the entire motor that
includes the connected case. Because electromagnetic force distribution and eigenmode are dependent on the geometry of the stator core, itis necessary to obtain this via analyses such as the finite element method.
In this example, electromagnetic force generated in the induction motor stator core is obtained, and an example of evaluating sound pressure
by linking it with the eigenmode of the motor is shown.

Shaft

Flange bracket

Eigenmode
Attention is paid to the mode of vibration depending on the number of poles. Because a4-pole model is being used, it is understood that this mode exists near 3,700 Hz byconfirming the rectangular vibration mode. The confidition of deformities is shown in Fig. 1.

Acceleration, m/s^2

2.5

Acceleration at Each Frequency Generated by Electromagnetic
Force
The frequency dependence of acceleration generated by electromagnetic force during
rotation is displayed in Fig. 2. One side is chosen for evaluation points in the diagram forthe case shown, maximum value is extracted from the effective value of each point and istaken as the evaluation value of that frequency. It is confirmed that acceleration is at
maximum when near eigenmode at 3,700 Hz as shown in Fig. 1, and resonance isconsidered to occur also.

2
1.5
1
0.5
0
Frequency, Hz

Fig. 1. Square Form EigenmodeFig. 2. Acceleration Frequency Dependence

Acceleration and Sound Pressure Distribution in Square Form
Eigenmode
Acceleration at 3,700 Hz and sound pressure levels are each displayed in Fig. 3 and Fig. 4.
It can be seen that both acceleration and sound pressure levels are high due to resonance.
It can be additionally confirmed that eigenmode spatial characteristics are reflected in the
distribution of both.

(Unit: dB)(Unit: m/s^2)

Fig. 3. Acceleration Distribution at 3,700 Hz

Fig. 4. Sound Pressure Level Distribution at 3,700 Hz

Case

235

Core

Reactor Core Stray Loss Analysis

BobbinCoilmodule: LS, TR

Overview

SpacerWhen designing a high-frequency reactor which is a key component in a converter, inductance and loss, which are parameters required forvoltage conversion, are important factors.
Copper loss is dependent on coil resistance and current, and iron loss is classified as eddy current loss or hysteresis loss. Other losses arecalled stray loss.
Stray loss is a loss caused by eddy currents in the core portions near coil ends and conductive parts near the coils which in turn are caused
by mainly leakage flux from the coil ends.
Although the stray loss is not always a large part of the entire loss, heat generation may occur locally sometimes causing operational
problems. Although it is difficult to capture stray loss by actual measurement and calculation by hand, it is possible to visualize core surface
eddy currents which result in stray loss or leakage flux from the coils to the core by JMAG transient analysis.
In this example, reactor core stray loss is described.

Leakage Flux and Eddy Currents
Hysteresis loss
Lamination
direction

Eddy current loss

Stray loss

30

Spacer (gap)25
Lamination
direction

20
Losses,W

Fig. 1 shows the flux lines near a gap and a core eddy current distribution.
From the enlarged view of the flux lines, it can be confirmed that fringing occurs in which
the magnetic flux bulges due to the magnetic flux cross sectional area becoming larger
than the core cross sectional area at the core gap. In addition, in-plane eddy currents are
generated on the surface right under the right coil near the gap, and it can be seen thatthe magnetic flux is generated perpendicularly to the laminating direction.

15
10

Reactor Loss
Fig. 2 shows the loss in the reactor.
By performing magnetic field analysis and iron loss computations together, analysis
results can be obtained by separating out hysteresis loss, eddy current loss, stray loss, and
loss.

5
0
(Unit: A/m^2)

Fig. 1. Leakage Flux and Eddy Currents

Fig. 2. Reactor loss

https://www.jmag-international.com/

139

Case

236

Magnetic Circuit Optimization for Plastic Magnets
in an Axial Magnetic Field

Yoke
S

MagnetN

Cavity spacemodule: ST

Overview
Plastic magnets are characterized by light weight with high strength and have a high degree of freedom in shaping, and very suitable for mass
production.
Anisotropic plastic magnets with good magnetic properties need magnetic field orientation. For this reason, a magnetic circuit is created togenerate the magnetic flux necessary for magnetic field orientation in the product cavity in the mold used.
Design requirements for the magnetic circuit are that the magnetic flux generated in the cavity has the magnitude necessary and sufficient forthe magnetic field orientation, that flux parallel to the analysis target main axis be produced, and that the magnetic circuit be formed as small
as possible.
In the case of simple magnetic circuits, these requirements can be met by hand calculation, but FEA is effective for obtaining a generated
spatial magnetic flux density distribution, angles, etc. in detail in the cavity. In addition, since trial and error cannot be avoided in an optimum
design that minimizes geometry size while satisfying the above requirements, and time is consumed, automation of analysis combined with
optimization tools is widely used.
In this example, a use case is presented in which a magnetic circuit of a product-forming mold, in a magnetic field aligned with the mold's
main axis, is optimized, with the mold's dimensions used as design variables.

Optimization Results
The response graphs for the results after optimization is shown.
Fig. 2 and Fig. 3 show graphs of Bmin and degy̲theta from the objective functions,
respectively, vs. the radius of the magnetic circuit. The yellow plot is the optimum value.
The optimum value is the minimum value of the radius of the magnetic circuit when theobjective function constraints are satisfied.The optimum value plot shown in Fig. 2 and in
Fig. 3 are for the same case.
Fig. 4 shows the results magnetic flux density contour plots and flux lines obtained with
the initial geometry and with the optimum geometry. The initial values of the variables a,b, c and the values obtained after optimization. It can be seen that in the region of thecavity surrounded by the red rectangle, for the initial geometry some of the flux enters at
an angle, however, for the optimized geometry obtained the flux has parallel flux lines andis nearly uniform.

aUnit: mmb c

Design variable

Initial value

Minimum value

Maximum value

5

a
b
c

3.0
1.0
1.0

3.0
1.0
1.0

10
10
10

4

Bx
By

Objectivefunctionsdegy̲theta
B, T

Y
Z X
Expression

Description

a＋b＋cMagnetic circuit radius -> minimum value

By ≧ 0.3 T

Minimum value of the magnetic flux density absolutevalue not less than 0.3 T

degy̲theta ≦ 5 deg

Angle of magnetic flux generated in the cavity spacedegy̲theta =JSOL Corporation

2

0

0

5

10

15
size, mm

20

25

30

Fig. 2. Magnetic Circuit Radius and Cavity Magnetic Flux
Angle (When Satisfying the Objective Function Constraints)

0.7

0.6

0.5

0.4

(Unit: T)
Initial geometry

0.3

0

5

10

15
size, mm

20

25

30

Fig. 3. Magnetic Circuit Radius and Cavity Magnetic Flux
Density (When Satisfying the Objective Function
Constraints)

140

3

1

Fig. 1. Optimization Conditions

Bmin, T

Fig. 1 shows design variables and objective functions as optimization conditions.
The design variables are the magnet radius a, the non-magnetic material width b, and theyoke width c. Settings are performed for an optimization function that gives the highest
priority to minimizing the magnetic circuit radius a+b+c.
In addition, the angle formed by the magnetic flux density B generated in the cavity withrespect to the Y-axis and the threshold for the magnitude of B are specified.
In this example, the angle is within an error of 5 deg and the value of B is 0.3 T or more.
Also, these variables are used in objective functions.

Nonmagnetic materialNdeg̲theta, deg

Optimization Conditions

S

Optimized geometry

Design variableab

Unit: mmc

Initial value
Optimized value

3.0
8.4

1.0
1.0

1.0
1.0

Fig. 4. Optimized Results (Magnetic Flux Density Contour
Plots, Magnetic Flux Lines)JMAG Application Catalog

Case

Winding

237 AC Loss Analysis of an IPM Motormodule: TR
MagnetOverview

Rotor Core

Primary power loss drivers in PM machines are iron, magnets and the winding. The main focus of this analysis is the winding losses,
particularly those which occur in the end winding. The flux patterns occurring in the end winding region are different from those within thecore winding length. These losses can be defined by their two components namely DC and AC losses , DC winding loss is easily analyzed bycircuit analysis of the windings and are within the realm of arithmetic calculation. The AC loss components are caused due to various effectsof the magnetic fields generated by the conductors in proximity of each other. This used to be estimated by creating prototypes and doing
tedious measurements on the coil parts. This used to make the design workflow expensive and time consuming.
However by using a 3-D Finite Element Analysis on these coils the losses can be analyzed relatively quickly and economically.
In this example, we introduce the case study that evaluates AC losses in coils of an IPM motor.

Stator core

Loss Analysis
The difference between DC and AC losses can be gleaned from Fig. 1.
After considering AC losses and subsequently adding DC and AC losses we get a total
increase of 45 % in losses. This is quite a significant amount of increase.
The AC losses in the end winding as seen in Fig. 2 depend on the shape of the winding aswell as the flux leakage through the end windings. This happens due to mutual flux
linkages between individual conductors which create eddy currents flowing throughsystem.
In Fig. 3, the flux linkages between the stator end (fringing effect) and the coil and mutual
flux linkages in the conductors can be observed. These are the main drivers behind AC
losses in the end winding.

AC Loss

DC Loss

120
100

38.207

Loss, W

80
60

69.24840
20(Unit: W/m^3)(Unit: W/m^3)0Fig. 1 AC Losses VS DC Losses

Fig. 2 Joule Loss Density Distribution

Fig. 3 End Winding Loss and Flux Line

Case

238

Core

Hysteresis Loss Analysis in a Reactor with DC Bias

BobbinCoilmodule: DP, LS

Overview

SpacerWhen designing a high-frequency reactor which is a key component in a converter, inductance and loss, which are parameters required forvoltage conversion, are important factors.
A reactor are usually used in power electronics and experience high frequency signals. Therefore they will be subject to hysteresis losses that
we can calculate with JMAG.
In this example, we will observe the difference between the hysteresis losses calculation method loop count and the play model for the modelthat is subjected to a DC bias filed on top of a AC field.

Hysteresis Loss Comparison
Table 1 shows a difference between the two method of calculations. This is due to the way
the method calculate the hysteresis losses in the Iron core.
The loop count method uses a Steinmetz method which takes in account only AC
variation of the field. Whereas the play model takes in account the AC variation on top of
the any DC component that may be present. This will be highlighted in the Fig. 1.

Play Model

Loop Count

3.08 (W)

0.55 (W)

Magnetic Field Flux, T

0.8Play Model

0.6
0.5
Measuring point

0.4
0.3
0.2
0.1
0
0

0.00010.00020.00030.00040.0005

Time, sec

Table 1 Hysteresis Loss Comparison

Fig. 1 Magnetic Field Flux in the Reactor

6.25 µm56.25 µm
106.25 µm
156.25 µm
206.25 µm-0.555-0.56

18.75 µm68.75 µm118.75 µm
168.75 µm
218.75 µm

31.25 µm81.25 µm131.25 µm
181.25 µm
231.25 µm

43.75 µm93.75 µm143.75 µm
193.75 µm
243.75 µm-0.565-0.57

Magnetic Field Flux, T

Magnetic Field Flux, T

The Reactor is supplied by a 10 kHz, 2 A sinusoidal current as well as 20 A DC current. This
provides us with a Magnetic field flux containing an AC and DC component, as illustrated
in Fig.1.
The core being laminated each of them will be subjected to a 1D lamination FEA study tofind the eddy currents induced in the plates at different depths. From there we can extract
the average Magnetic Field Flux depending on the depth as shown in Fig.2. We observethat the field is the highest around the skin of the laminated sheets.
The play model will take in account even the DC component of the field as Illustrated in
Fig.3 where we can observe the hysteresis loops of the Magnetic field flux at different
depths in the steel sheets. This is the main difference between the loop count and playmodel. Since loop count only considers AC fields we have a difference in the resulting
hysteresis loops.
Therefore for an accurate estimation of those looses in a system that includes an AC andDC component of magnetic field flux, it is recommended to use a play model hysteresis
loss calculation especially since in this case the hysteresis losses represent more than halfof the AC losses in the core. 2.75 W of eddy current loss compared to the 3.08 W of
hysteresis loss.

0.7-0.575-0.58-0.585
-0.59

0

0.05

0.1

0.15

Depth, mm

Fig. 2 Hysteresis loop (Average Field Flux)

0.2

0.25

0.8
0.3
-0.2
-0.7
-1.2
-150

-100

-50

0

50

100

150

Magnetic Field, A/m

Fig. 3 Hysteresis loop

https://www.jmag-international.com/

141

Case

239

Stator coreFault Analysis in an IPM Motor

Coilmodule: DP, RT

Overview
Electrical faults are a commonly occurring conditions during operation of machines. These can cause an adverse effect on the powerelectronics or the power system connected to these machines. It is necessary to run many test cases with changing fault conditions for acomprehensive analysis of motor behavior during a fault.
JMAG-RT Model can simulate the effect of harmonics on the system and magnetic saturation effect that are calculated from FEA.
In this example, we introduce the case study that simulate the fault of an IPM motor using JMAG-RT. The fault simulated in this case study is an
open circuit fault which is caused due to failure of an IGBT, this is one of the most commonly occurring faults in PMSM operation.

MagnetRotor coreFault Description
The Motor is a 4 pole machine rated at 1 kW. It is operating at 600 r/min and 10 A.
The fault occurs due to an IGBT failure at 0.3 sec into operation start.
This causes an open circuit in the six step inverter. This causes the current through thatparticular IGBT to fall to zero.
We are particularly looking for the effect of this kind of fault on the torque and current
characteristics of this motor.

Inverter fault command

JMAG-RT Model

LdLq (Constant) Model

32

Torque, Nm

2
1
1
0
-1

Fault Analysis
Fig. 1 Motor Control Circuit

0.2

0.3

0.4

0.5

0.6

Time, sec

Fig. 2 Torque Characteristic comparison (10 A)

JMAG-RT Model

LdLq (Constant) Model
JMAG-RT Model

20

LdLq (Constant) Model

10

15
8
Torque, Nm

10
5
0

-15

Fault analysis at higher loading

6
4
2

-5

0

-10
0.1

0.2

0.3

Time of Fault

0.4

0.5

0.6

Time, sec

Ld

-2

0.1

0.2
Time of Fault

Fig. 3 Current characteristic comparison

0.3

0.4

0.5

0.6

Time, sec

Fig. 4 Current characteristic comparison

Lq

8.00E-03
7.00E-03
Inductance, H

There is a difference between the average values of torque when using LdLq (Constant)Model and JMAG-RT Model in Fig. 4. This difference is however not observed in Fig. 2. The
main difference here being the supply currents. Fig. 2 has 10 A while Fig. 4 has 28.3 A.
The reason behind this difference is the saturation effect, this can only be observed in the
JMAG-RT Model and not the LdLq (Constant) Model.
From Fig. 5, the values of Ld and Lq decrease as the input current increases. This happens
due to the cores being saturated . Due to saturation the Iron in the cores starts acting likeair. This causes a decrease in the average torque observed in Fig. 5. The saturation can beobserved in FEA from Fig. 6.
The behavior prediction in the fault thus also depends on characteristics not observed bya simple LdLq (Constant) Model thus it is useful to use a High Fidelity one.

0.1

Time of Fault

Current, A

There is a fundamental difference between the high fidelity model (JMAG-RT Model) and
low fidelity model (LdLq (Constant) Model). The JMAG-RT model take into account theangular position of the current vectors. This requires FEA. Thus with JMAG-RT model it ispossible to capture torque ripple before and during fault.
This is beneficial in prediction of motor behavior due to a IGBT fault while doing a systemanalysis. The torque behavior is demonstrated in Fig. 3. The point at which the fault occurs
is also visible. By running higher time resolution analysis on such a model it is also possible
to predict the phenomenon while reclosing or after fault is resolved.
This difference can also be observed in the current waveform after the fault occurs. From
Fig. 3, the difference in currents is minimal this causes the difference in the average valuesof torque for two models to be minimal.

-1

6.00E-03
5.00E-03
4.00E-03
3.00E-03
2.00E-03
5

10

15

20

25

30

28.3(A)

10(A)Input Current, A

Fig. 5 Inductance Analysis

142

JSOL Corporation

Fig. 6 Magnetic Flux Density Distribution(Unit: T)

JMAG Application Catalog

Case

240

NT Characteristics Analysis of an IPM Motor
Using PAM Drive

MagnetStator coremodule: DP

Overview
A PMSM usually requires a drive system to operate. There are various types of drive systems such as PAM drive and PWM drive systems.
The concept of PAM drive is simple: a carrier signal modulates a supply signal to generate pulses and controls the amplitude of these pulses bycontrolling the amplitude of the supply signal. By applying appropriate switching control, this concept can be used in a PMSM drive device.
Furthermore, due to the simplicity of the switching control, PAM drive tends to be cheaper and more robust than PWM drive.
In addition, with IPM the back electromotive voltage increases during high-speed rotation, which reduces the torque and the current flowing
to the coils. When designing a motor, it is necessary to understand the characteristics for a wide range of operating points. The evaluation of
NTI characteristics with FEA at the design study stage contributes greatly to development speed.
In this example, an example of evaluating the performance of an IPM with PAM operation is presented.

Coil
Rotor core

PAM Drive
Fig. 1 shows the magnetic flux density distribution at a rotation speed of 1,800 r/min.
It can be seen that the teeth near the coils being excited have a magnetic flux density
close to saturation.
Fig. 2 and Fig. 3 show the torque and current waveforms respectively at an operating
point of 1,800 r/min when using PAM drive.
Fig. 4 shows the NTI curves, which predicts motor performance at various operating
points. The torque of the IPM motor decreases as the rotation speed increases andincreases as the current increases.

15

5
4
3

Rotation speed, r/min

Current, A

10
5
0
-5
-10

2

-15
1
0
0.01

(Unit: T)

Fig. 1. Magnetic Flux Density Distribution (1,800 r/min)

Case

241

-20
0.015

0.02

0.025

0.015

0.02

0.025

0.03

50

2000

40

1500

30

1000

20

0

Time, sec

Time, sec

Fig. 2. Torque Waveform (1,800 r/min)

60

2500

500

-25
0.01

0.03

Current

3000

Current, A

20

6Torque, Nm

Rotation speed

25

7

10

0

2

4

6

8
10
Torque, Nm

12

14

0
16

Fig. 4. N-T-I Curve

Fig. 3. Current Waveform (1,800 r/min)Single-Phase Synchronous Motor N-T-I Characteristics
Analysis

MagnetStator Core

module: DP

Overview
The widely used three-phase synchronous motor can be driven efficiently and stably but costs tends to increase due to using an inverter.
Whereas since single-phase synchronous motors do not require inverters, they can reduce costs. Furthermore, because they are brushless,maintenance costs can also be lowered.
In a single-phase synchronous motor, the back emf and the current flowing through the coils change in accordance with the rotation speedof the rotor. At low rotation speed, since the back emf is small, a large current flows through the coils, and magnetic saturation occurs in themagnetic circuit.
On the other hand, the back emf increases at high speed, so the current flowing through the coils and torque decrease. Because of thiswhen designing a motor it is important to understand characteristics for a wide range of operating points. Obtaining N-T-I characteristics
from performing analysis in JMAG at the design study stage contributes greatly to the speed of motor development.
In this example, a use case in which the N-T-I characteristics for a single-phase synchronous motor are obtained is presented.

Coil
Rotor Core

N-T-I Characteristics

Fig. 2 shows the magnetic flux density distribution at 2,000 r/min and at 12,500 r/min.
Since a large current flows at a low driving speed, magnetic saturation can be seen in thestator core as well. On the other hand, the current flowing through the coils decreases athigh driving speeds, so it can be seen that the magnetic saturation in the stator core isrelaxed.

Rotation speed, r/min

Magnetic Flux Density Distribution

Rotation speed

Current

15000

100

13500

90

12000

80

10500

70

9000

60

7500

50

6000

40

4500

30

3000

20

1500

Current, A

Fig. 1 shows an N-T-I characteristics graph.
From Fig. 1, torque for each rotation speed, maximum driving rotation speed, and so on
can be seen. The current and torque are roughly proportional.

10

0

0
0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

2,000 (r/min)

12,500 (r/min)(Unit: T)

Torque, Nm

Fig. 1. N-T-I Characteristics (Rotation Speed - Torque- Current Characteristics)

Fig. 2. Magnetic Flux Density Distribution

https://www.jmag-international.com/

143

Case

242

Solenoid Valve Attractive Force Analysis Taking
into Account Movable Part Motion

Fixed Core

Coil

module: ST

Overview

Yoke

Solenoid valves are used to adjust the inflow or outflow of liquids or gases by translational movement of the iron core. If an electric current ispassed through the coil, an electromagnet is created and an electromagnetic attractive force is generated between the movable part and thefixed parts. Since high responsiveness is required for valve opening and closing, an important item to evaluate is whether or not the powersource used for driving and the solenoid valve meets the required responsiveness and attractive force.
Solenoid valves are often used for hydraulic valves and injectors, and the movable part itself is placed in high pressure liquid and encounters
flow resistance. Since the movable part uses a guide to move, there is error with the possibility to cause an inclination. In the case of
inclination, the electromagnetic attraction force is unbalanced, so it is necessary to confirm beforehand whether or the force acts in the proper
direction.
It is useful to examine by FEA beforehand whether there is a change in the attractive force when there is an inclination and whether an
unbalanced force causes the movable part to become unaligned.
In this example, an example is presented where the influence on the electromagnetic force by the presence or absence of inclination is
confirmed at each position of the movable part.

Movable Iron Core

Position and Inclination of The Movable Part
Fig. 1 shows the position and inclination of the movable part. The position of the movablepart is considered for three gap lengths - 3.6 mm, 1.9 mm, and 0.2 mm. At each position,
there is an inclination of the movable part. Comparisons to the situation of having no
inclination is carried out. The center axis of the inclination is not dependant on theposition of the movable part and has fixed coordinates. The inclination is 2 deg.

Fig. 4 to Fig. 6 show the electromagnetic force distribution around the movable part ateach position.
As shown in Fig. 4, at the gap length of 3.6 mm almost no change in electromagnetic force
due to inclination is observed. On the other hand as shown in Fig. 5, it can be seen thatthe electromagnetic force increases at the position where the fixed iron core and movableiron core are closest with an inclination at the gap length of 1.9 mm. As a result, theattractive force has increased. With the gap length of 0.2 mm shown in Fig. 6, theelectromagnetic force acts not only in the motion direction but in the horizontal direction
due to the inclination of the movable part. It can be seen that this is the reason for thedecrease in the attractive force.

Attractive force, N
Center axis of inclination

Gap length of 3.6 mm

Gap length of 0.2 mm

3.6

Gap length, mm

Fig. 2. Attractive Force

0.10
0.08
Positive directionof the torque

0.06
0.04
0.02
0.00
-0.02

3.6

1.9

0.2

Gap length, mm

Fig. 3. TorqueWithout Inclination

With Inclination(Unit: N/m^2)

Fig. 4. Electromagnetic Force Distribution(Gap length 3.6 mm)

With Inclination

Fig. 5. Electromagnetic Force Distribution(Gap length 1.9 mm)

JSOL Corporation

0.2

10

0

Fig. 1. Position and Inclination of the Movable Iron Core

Without Inclination

144

1.9

15

5
Center axis of inclination

Torque, Nm

Electromagnetic Force Distribution

With Inclination

20
Gap
lengthInfluence on Electromagnetic Force
Fig. 2 shows the attractive force from the inclination at each movable part position, andFig. 3 shows the torque around the center axis of the inclination.
As can be seen in Fig. 2, the inclination has almost no affect on the attractive force at the
gap length of 3.6 mm, but the attractive force increases by 20% without inclinationcompared with inclination at 1.9 mm, and decreases by 3% at 0.2 mm.
As shown in Fig. 3, with inclination it can be seen that the torque becomes larger as the
gap length becomes shorter. In this example, since the positive direction of the torqueand the direction of the center axis of inclination are made to coincide with each other, it
is seen that the unbalanced force generated in the movable part acts in a direction toincrease the inclination.

Without Inclination
25(Unit: N/m^2)

Without Inclination

With Inclination

Fig. 6. Electromagnetic Force Distribution(Gap length 0.2 mm)(Unit: N/m^2)

JMAG Application Catalog

Case

243

Voice Coil Motor Control Simulation
Using a Control Simulator and JMAG-RT

Yoke
MagnetDirection oftranslational
movement

module: DP, RT

CoilOverview
Linear actuators are used in machine tools and other applications because they can achieve high speed and high acceleration/deceleration,and because they have high positioning accuracy. In particular, coreless type linear actuators have a small driving force compared to linear
actuators having a core, but because they do not generate cogging their driving force fluctuations are small. Utilizing this characteristic, they
are used in environments requiring high-precision positioning such as head drives for electronic component mounting machines and for veryfine movements when precision is needed.
The change in driving force from the change in actuator positions greatly affects the positioning accuracy. Furthermore, the driving force iscontrolled by the current. Moreover, because of the nonlinearity of the material and due to the magnetic path not being fully contained in themagnetic material, it is difficult to calculate by hand the driving force for each current value and position of the voice coil motor. Therefore,taking into account dynamically changing characteristics, in order to control the voice coil operating at high speed, a plant model containing
characteristics dependent on magnetic saturation and the position of the mover is required.
JMAG-RT can create a highly accurate plant model in a JMAG-RT model by performing FEA taking into account the geometry, materials, anddriving conditions, and extracting the characteristics of the voice coil motor.
In this example, circuit simulation is performed to control the position of a mover by loading a JMAG-RT model for a voice coil motor into acontrol/circuit simulator.

Cut open view

Control Circuit

Coil Position vs. Time

The control specifications are shown in Table 1. The control circuit control section isshown in Fig. 1.

Fig. 2 shows the coil position vs. time, and Fig. 3 shows the excitation current vs. time.
As shown in Fig. 2, the movement from the 0-mm position to the 0.8-mm position and the
movement from the 0.8-mm position to the 1.6-mm position are both a movement of 0.8mm, but the electromagnetic force is different, and the trajectory of the movement isdifferent because it is dependent on inductance current and on position and is influenced
by the spring.
From Fig. 3, it can be seen that a surge current is generated the moment the coil starts tomove. Since surge current may damage peripheral electrical equipment, examination ofthis equipment is necessary when it is excessive.

Voice coil motor

Control method

Position control realized by chopping control

1.0

Position

0.9

1.8

0.8

1.6

0.7

1.4

Current, A

Details

Position, mm

Item

Command Value
2.0

1.2
1.0
0.8

0.6
0.5
0.4
0.3

0.6

0.2

0.4

0.1

0.2

0.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time, sec

Time, sec

Command value

Table 1. Control Specifications

Fig. 1. Control Circuit

Fig. 2. Change in Coil Position vs. Time

Fig. 3. Change in Excitation Current vs. Time

TMR element

Case

244 Magnetic Field Strength Analysis at a Sensor Position

Pin layer
Insulation
layerFree layer

Sensor position

Side view
Free
layer

module: DP

Pin
layer

Overview

Top view

To control the position or speed of a motor or actuator, it is necessary to accurately detect the position of the mover and calculate command
values based on that position. To detect the position, a method to detect leakage flux produced from an analysis target is widely used.
There are various detection methods, from using electromagnetic induction from a pickup coil to using the Hall effect or the magneto resistive
effect. The latter methods have begun to be used in recent years because they can obtain a signal corresponding to the magnitude of themagnetic flux density regardless of the frequency.
To predict the signal produced in a sensor by the magnetic flux density generated at the sensor position and apply it to the displacement andspeed control of the mover, electromagnetic field analysis that can accurately calculate the magnetic flux density and magnetic field
distribution generated from the analysis target is useful.
In this example, with the analysis target being a motor and the detection sensor being a TMR sensor, an example where a detection signal
waveform is obtained that is based on magnetic field values computed from electromagnetic field analysis is presented.

Magnetic Field Radial Component at the Sensor Position

Pin layer direction
2A

Fig. 1 shows the magnetic field strength radial component at the sensor position.
Together with this value increasing as the motor current is increased, it can be confirmedthat the rotor position achieving the maximum value shifts to a lower angle.

Fig. 2 shows the percentage change in resistance when the sensor pin layer is fixed in theradial direction, and the free layer magnetization direction is the same as the external
magnetic field direction. The TMR ratio maximum value is 100 %, and the percentagechange in resistance is for when the angle between the pin layer magnetization direction
and the free layer magnetization direction is 90 deg.
When the current changes, it can be confirmed that the percentage change in resistanceshifts to a lower rotation angle, and it can be seen that the relationship between the fieldstrength and the position of the motor changes with the current amplitude. It can be seenthat it is necessary to ascertain the relationship between the sensor output and the rotor
position when the current changes in order to detect the rotor position accurately.

4A

8A

200

2A

150

60

100

40

50

4A

8A

20
ΔR, %

Hr, A/mResistance Change Rate of Each Sensor

45 deg

0
-50

0
-20

-100

-40

-150

-60

-200

0

30

60

90
120
Rotation angle,deg

Fig. 1. Magnetic Field Radial Component
at the Sensor Position

150

180

0

30

60

90
120
Rotation Angle, deg

150

180

Fig. 2. TMR Ratio vs. Rotor Rotation Angle

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145

Case

Stator Core

245 Injector Control and Eddy Current Eﬀects

Coilmodule: DP

Overview
Solenoid injectors used in engines open a valve by injecting fuel as a plunger moves from the magnetic force of an electromagnet. For
injectors in engines, high responsiveness to applied voltages is required for suppressing flow rate fluctuations and improving fuel economy.
MATLAB/Simulink and JMAG can be linked directly when performing injector circuit control simulations. Taking into account eddy current
effects as well, this allows injector characteristics to be reflected in more detail and allows evaluation of control systems that accurately
model actual control systems.
In this example, an example of evaluating the difference in responsiveness of an injector when eddy currents are taken into account or not ispresented.

Yoke
Mover

Control Circuit
The 18-V DC power supply of the circuit energizes the coil via a switch controlled by
MATLAB/Simulink.
The switch is controlled by hysteresis band control and maintains the current in a range
around a reference current.
When current flows through the injector, the reference current is maintained until theplunger reaches its end position. After that the reference current is lowered to maintainthe plunger at a specified position and finally the current value is returned to 0 after acertain time as shown in Fig. 2.
Fig. 1 shows the control circuit specifications, and Fig. 2 shows the control model.

1

Period, sec

0.1

Pulse width (percentage, % of period)

0.1

0.4

Reference current maintained

0.3
0.2
0.1
Time, sec

Fig. 2. Control Model

Fig. 1. Circuit Diagram, Control Specifications

Eddy currents taken into accountEddy currents taken into accountEddy currents not taken into account"

Eddy currents not taken into account"

110
100
90
80
70
60
50
40
30
20
10
0

14
12
10

6

2
0
0

0.002

0.004

0.006

0.008

0.01

Time, sec

0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.002

0.004

0.006

Time, sec

Fig. 5. Current

0.002

0.004

0.006

Fig. 4. Force Acting on the Plunger

Eddy currents taken into account00Time, sec

Fig. 3. Plunger Position

Current , A

8

4

Eddy currents not taken into account"

JSOL Corporation

Final position

0.5

0

0.5

146

Reference current, A

Amplitude
Pulse generator

Eﬀects on Coil Current Due to Eddy Currents
Fig. 5 shows the current.
The model taking into account eddy currents has 30 % or more current flowing in theinjector. That is, the power consumption increases.

Speciﬁcation

Force, N

Fig. 3 shows the plunger position and Fig. 4 shows the force acting on the plunger.
Looking at the position of the plunger in Fig. 3, it is seen that when taking into accounteddy currents, it takes longer to reach the final position and return to the initial position.
This is because as seen from the force acting on the plunger in Fig. 4, when eddy currents
are taken into account, the force generated on the plunger becomes smaller. The rate ofchange in force is also smaller. This phenomenon is caused by the generation of eddy
currents which impede fluctuations of the magnetic field.

Item

Position, µm

The Position of the Plunger and the Force Acting on the
Plunger

0.6

0.008

0.01

0.008

0.01

JMAG Application Catalog

Case

246

Analysis of Electromagnetic Force Acting Between
a Coil and a Magnet

Coil
Current direction

module: ST

Overview
With the miniaturization of electrical equipment, even effects from very small interactions are a concern. Therefore, it is important to ascertain
the magnetic and electromagnetic forces acting on parts. Moreover, in an analysis model where the magnetic circuit extends into the air
region, it is necessary to use a sufficiently fine mesh over a large region since obtaining an accurate magnetic field distribution in the air region
is directly related to electromagnetic force computation accuracy.
In circumstances such as these, the adaptive mesh function can be used to again subdivide mesh from air region error distribution, and meshthat obtains a high analysis accuracy can be generated. By additionally using electromagnetic force calculation correction methods, it ispossible to obtain the electromagnetic force acting between parts that would otherwise be difficult to calculate.
This case study obtains electromagnetic force acting between a coil and magnet using the electromagnetic force calculation correction
method and the adaptive mesh function.

Forces Acting on the Magnet
Upon confirming the electromagnetic force distribution generated on the magnet shownin Fig. 1, both magnetic forces, one acting on the lower face and the other acting on theupper face of the magnet, differs by 4 order of magnitude from the net force on the entire
magnet. The forces acting on the two magnet faces are obtained from the nodal forces.
Since the net force on the entire magnet is obtained from the cancellation of theelectromagnetic forces distributed in this way, the electromagnetic force must be
computed with very high accuracy. The forces acting on the magnet are shown in Fig. 2.

Magnetization direction

N
S

Magnetization direction

Force acting on the coilElectricmagnetic force on the upper and on thelower magnet face, N

The current flows in a direction perpendicular to the attractive force acting between thecoil and the magnet. During such a time, the forces acting between the coil and themagnet are balanced. Fig. 1 shows the attractive and repulsive forces acting between thecoil and the magnet.

Force acting on the magnet
0.6
Electromagnetic force,mN

Attractive Force Acting Between Coil and Magnet

Magnet

0.4
0.2
0
-0.2
-0.4
-0.6

Coil

MagnetFig. 1. Attractive and Repulsive Forces Acting Between
Coil and Magnet

3

2

1

0

-1

-2

-3
Top

BottomFig. 2. Forces Acting on the Magnet

Magnetic Field Lines Extending in Air
Since magnetic flux lines giving rise to electromagnetic force extend far in the air, accuratecomputation is required over a wide region. Fig. 3 shows the lines of magnetic force
extending in the air.

Fig. 3. Magnetic Field Lines Extending in Air

https://www.jmag-international.com/

147

Case

247

Magnetic Circuit Optimization
for Plastic Magnets in a Radial Magnetic Field

Outer
diameter yoke
Innerdiameter yoke

Magnetmodule: ST

Cavity space

Overview
Plastic magnets are magnets formed from pellets made by mixing magnetic powder and resin binder. Depending on the manufacturingmethod, compression molding, extrusion molding, or injection molding are used. Plastic magnets are light weight with high strength andhave a high degree of freedom in shaping, and very suitable for mass production.
Anisotropic plastic magnets with good magnetic properties need magnetic field orientation. For this reason, a magnetic circuit is created togenerate the magnetic flux necessary for magnetic field orientation in the product cavity in the mold used.
The design requirements of the magnetic circuit are that the magnetic flux generated in the cavity obtain a necessary and sufficientmagnitude for magnetic field orientation, that uniform and parallel magnetic flux can be obtained, and that the magnetic circuit be formed as
small as possible. Furthermore, the price of the magnetic circuit magnets are high compared to the other magnetic circuit components, so
reducing the size of the magnets is effective for lowering the cost.
In the case of simple magnetic circuits, these requirements can be met by hand calculation, but FEA is effective for obtaining a generated
spatial magnetic flux density distribution, angles, etc. in detail in the cavity. In addition, since trial and error cannot be avoided in an optimum
design that minimizes geometry size while satisfying the above requirements, and time is consumed, automation of analysis combined with
optimization tools is widely used.
In this example, a use case of magnetic circuit optimization in a metal mold with radial magnetic field orientation used for ring-shaped magnet
production is presented.

Optimization Conditions

Units: mma
Design variables

Initial value

Minimum value

Maximum value

a
b
c

10
5
20

4
1.2
3

10
5
20

f
Bx
By
B(T)
Y
Z

degx̲thetaeExpression

Objectivefunctionπfa^2

1Magnet volume => minimum value0

Bmin ≧ 0.4(T)Minimum value of magnetic flux
density Bx is 0.4 T or moredegx̲theta≦3(deg)Angle, deg of magnetic flux generated in thecavity space degx̲theta =Constraints

0

1000

2000

3000

4000

5000

6000

7000

size (mm^3)

Fig. 2. Magnet Volume and Angle Graph(Cases Satisfying Constraints)

1
0.9

Bmin (T)

0.8
0.7
0.6
0.5
0.4
0

1000

2000

3000

4000

5000

6000

7000

size (mm^3)

Initial geometry

Fig. 3. Magnet Volume and Magnetic Flux Density Graph(Cases Satisfying Constraints)

Units: mm
Design variablesaeInitial value
Optimized values

f

10.0

5.0

20.0

7.7

2.6

3.0

Table 1. Initial Values and Optimized Values

JSOL Corporation

2
1.5

0.5

Description

Fig. 1. Optimization Conditions

148

3
2.5

X
Item

Optimization Results
Fig. 2 and Fig. 3 show the graphs of the response values degx̲theta and Bmin vs. size
obtained by optimization. The cases where the Bmin and degx̲theta constraints are notsatisfied are not included in the graphs.
The initial value is shown in red and the optimum value is shown in yellow. For the initialvalue shown in red, size indicating the volume of the magnet is approximately 6,300mm^3. Whereas for the optimum value shown in yellow the volume of the magnet is 555
mm^3, which is about 11 times smaller than the initial value. The optimum value plotshown in Fig. 2 and in Fig. 3 are for the same case.
Fig. 4 shows the magnetic flux density contour plot and flux lines for the initial geometry
and for the optimized geometry. It can be confirmed that the magnetic flux density of thecavity satisfies both the Bmin and degx̲theta constraints, and that a parallel and uniform
magnetic flux density can be obtained.
Table 1 shows the initial values of the variables a, e, and f and the same values obtained
from optimization.

N
Magnetization
direction

degx̲theta (deg)Fig. 1 shows the design variables, objective function, and constraints used for
optimization conditions.
The design variables are a radius a and a height f of one magnet and e is the height of theyoke. The objective function is defined so that the volume πfa^2 of the magnet isminimized.
The constraints specify the maximum limit of the angle formed by the magnetic flux
density B generated in the cavity with respect to the X-axis, and the minimum limit of themagnitude of B. In this use case, two constraints are defined. One is that the angle formed
with the X-axis be within ±3 deg and the other is that Bx be greater than or equal to 0.4 T.

Cavity member(nonmagnetic)

S

Optimized geometry(Unit: T)

Fig. 4. Optimized Results (Magnetic Flux Density
Contour Plots, Magnetic Flux Lines)JMAG Application Catalog

Case

249

Dielectric Heating Analysis of a High-Frequency
Preheater with Parallel Plate Electrodes
module: EL, HT

Overview

Electrode

When creating a resin mold with thermosetting resin, there is a method to beforehand raise the resin temperature to just before the curing
temperature. This allows molding using a small molding pressure.
To raise the resin temperature, a high-frequency preheater using dielectric heating is used.
The electrodes of the high-frequency preheater consist of parallel plate electrodes and roller electrodes. The parallel plate electrodes are
suitable when there are many heating materials, and by using auxiliary electrodes it is possible to control the range where temperature is
raised. Since the temperature distribution is dependent on for example the shape of the auxiliary electrodes, it is necessary to check whether
the target temperature is reached in a range containing the high temperatures of interest. With FEA, it is possible to visualize the temperature
distribution inside a material which is difficult to actually measure, making evaluations easy.
In this example, a use case evaluating the influence on the temperature distribution of materials (tablets) with and without auxiliary electrodes
is presented.

Auxiliary
Electrode

TabletPositional Relationship Between a Tablet and Its AssociatedElectrodes
Fig. 1 shows the positional relationship between a tablet and its associated electrodes.
The generated electric field is affected greatly by the distance between the electrodes.
The distance between an electrode and the tablet is fixed whether or not an auxiliaryelectrode is used.

5(mm)

5(mm)
Without auxiliary electrode

Without auxiliary electrode

With auxiliary electrode

Electric Field Loss Density Distribution
The dielectric loss density distribution is shown in Fig. 2, and the electric field strength
distribution is shown in Fig. 3.
From Fig. 2, it can be seen that when there is no auxiliary electrode the loss on the sidesurface (perpendicular to the radius) is high, and when there is an auxiliary electrode theloss toward the center radially is high.
From Fig. 3, it can be seen that the electric field concentrates toward the center radiallywith an auxiliary electrode. The dielectric loss is affected by the physical characteristics,
frequency, and electric field. In this use case since the physical properties and frequencies
are the same with and without an auxiliary electrode, the difference in the dielectric loss
distribution with and without an auxiliary electrode can be said to be due to thedifference in the electric field strength distribution with and without auxiliary anelectrode.

(Unit: W/m^3)
With auxiliary electrodeFig. 1. Positional Relationship Between Electrodeand a Tablet

Fig. 2. Dielectric Loss Density Distribution

Without auxiliary electrode

Without auxiliary electrode((Unit: V/m)

Temperature Distribution
The temperature distribution after heating for 90 sec is shown in Fig. 4.
Differences also appear in the temperature distribution with and without an auxiliaryelectrode. From this it can be seen that with an auxiliary electrode it is possible toconcentrate greater heat near the center of an electrode extending the greater heat area
to the center of the tablet.

(Unit: deg C)
With auxiliary electrodeFig. 3. Electric Field Intensity Distribution

With auxiliary electrodeFig. 4. Temperature Distribution (After 90 sec)

https://www.jmag-international.com/

149

Dielectric Heating Analysis
of a High-Frequency Preheater with Roller Electrodes

Case

250

TabletElectrodes

module: EL, HT

Overview
When creating a resin mold with thermosetting resin, there is a method to beforehand raise the resin temperature to just before the curing
temperature. This allows molding using a small molding pressure.
To raise the resin temperature, a high-frequency preheater using dielectric heating is used.
The electrodes of the high-frequency preheater consist of parallel plate electrodes and roller electrodes. The roller electrodes rotate thetablets and heat them, so they are suitable for uniform preheating. Furthermore, by changing the pitch of the roller electrodes, thetemperature distribution can be controlled. With FEA, it is possible to visualize the temperature distribution inside a material which isdifficult to actually measure, making evaluations easy.
In this example, a use case is presented where the influence of the pitch of roller electrodes on the temperature distribution of a material(tablets) is evaluated.

Roller Electrodes
Rotation Direction

Roller Electrode - Tablet Positional Relationship

Temperature Distribution

Fig. 1 shows the positional relationship between the roller electrodes and the tablets.
The pitch of the roller electrodes is set to 90 deg and 40 deg. Also, the distance between
the roller electrodes and the tablets is fixed.

The temperature distribution during heating is shown in Fig. 4 to Fig. 6, and thetemperature change between the radial centers and circumferential surfaces of the tablets
is shown in Fig. 7 and Fig. 8.
Due to the differences in pitch of the roller electrodes, a difference also is exhibitedamong the temperature distributions. It can be seen that when the pitch is 90 deg,
heating is more prominent towards the circumferential surface of the tablets, but when
the pitch is 40 deg the temperature difference between the radial centers and at thecircumferential surfaces of the tablets is small during heating.

Dielectric Loss Density
The dielectric loss density distribution is shown in Fig. 2, and the electric field strength
distribution is shown in Fig. 3.
From Fig. 2, it can be seen that when the pitch of the roller electrodes is 90 deg the loss at
the circumferential surface of the tablets is high, and when the pitch is 40 deg the loss
interior to the circumferential surface of the tablets is high.
From Fig. 3, it can be seen that by reducing the pitch of the roller electrodes, the electricfield inside the tablets concentrates interior to their circumferential surfaces. The dielectricloss is affected by the physical characteristics, frequency, and electric field. In this use case,since the physical characteristics and frequency are common for both analysis targets, thedifference in the dielectric loss distribution from the difference in the pitch of the rollers
can be said to be due to the difference in the electric field strength distribution.

45(deg)70(deg)90(deg)40(deg)Pitch: 90 deg

Pitch: 40 deg

Pitch: 90 deg

Fig. 1. Positional Relationship Between Roller Electrodes
and Tablets

Pitch: 40 deg(Unit: W/m^3)

Fig. 2. Dielectric Loss Density Distribution

Pitch: 90 deg

Pitch: 40 deg(Unit: deg C)

Pitch: 90 deg

Fig. 5. Temperature Distribution (After 40 sec)

Pitch: 40 deg(Unit: deg C)

Fig. 6. Temperature Distribution (After 60 sec)

Surface
Surface
CenterCenterPitch: 40 deg(Unit: V/m)
Pitch: 90 deg

Pitch: 40 deg(Unit: deg C)

0

10

20

30

40

Time, sec

Fig. 3. Electric Field Intensity Distribution

150

JSOL Corporation

Fig. 4. Temperature Distribution (After 10 sec)

Center

SurfaceTemperature, deg C

Pitch: 90 deg

Temperature, deg C

Center
100
90
80
70
60
50
40
30
20
10
0

Fig. 7. Temperature Change (Pitch: 90 deg)

50

60

Surface

100
90
80
70
60
50
40
30
20
10
0
0

10

20

30

40

Time, sec

Fig. 8. Temperature Change (Pitch: 40 deg)

50

60

JMAG Application Catalog

Case

Rear bracket

251 Alternator Vibration Analysis

Statormodule: DS, TR

Overview
Electromagnetic force in the form of an electromagnetic excitation force acting on a motor or an alternator causes vibration and noise.
Vibration and noise are produced when this electromagnetic excitation force resonates with an eigenmodes of the motor. In order toaccurately evaluate this phenomenon, it is necessary to ascertain each frequency and spatial mode in detail about the electromagnetic force
and eigenmodes of the motor. FEA is useful in advance at the design stage for ascertaining the situations where large vibrations and noiseoccur, and for analyzing the factors involved and taking countermeasures.
In this example, the electromagnetic force generated in the stator core of a claw pole alternator is obtained, and acceleration is evaluated with
an electromagnetic force - structural coupled analysis on the alternator. In addition, the frequency components of the electromagnetic force
and the eigenmodes and spatial modes are analyzed to obtain the sources of resonance.

Rotor
Front bracket

Electromagnetic ForceSound Pressure Level Evaluation and Vibration Analysis

The electromagnetic force distribution for one electrical angle period at steady-state is
obtained. When rotating a 16-pole rotor at 1,500 r/min, 200 Hz is the fundamental
frequency. Fig. 1 shows the electromagnetic force frequency components.
In general, the electromagnetic force acting on a stator tooth has a frequency component(secondary component) which is twice the fundamental frequency, but if the rotor is aclaw-pole rotor, since the N-pole and S-pole widths are different, the fundamental
component is included as well. In particular, the magnitude of the fundamental
component included differs between the axial center and either end of a claw pole. Fig. 2
shows the electromagnetic force time history.

Vibration analysis is performed for each electromagnetic force frequency component
obtained from magnetic field analysis in order to determine sound pressure levels. Fig. 5
shows the sound pressure levels.
The ring modes can be checked by computing the vibration spatial modes fromaccelerations of the stator outer circumference at 3.6 kHz and 7.2 kHz. Fig. 6 shows thespatial mode order analysis.
It can be seen that the effects from the 3rd- and 4th-order ring modes are large. Fig. 7
shows the sound pressure level distribution.

Eigenmode
The stator is fixed between the front and rear brackets. Since each part is connected using
bolt holes, rigid body conditions are used to model these connections. In addition, thebolt holes for mounting the rear bracket are completely constrained simulating amounted bracket.
Fig. 3 shows the constraint conditions. Fig. 4 shows the 2nd- to 4th-order ring modes andtheir eigenfrequencies of the stator.

Opposing rotor face

Surface force density measurement point

0.025
2

3

0.015
2.5
0.02

0.01

1
2

0.005
0.015
0

0
0

0.01

200

400

600

1.5

800
1

Surface force density, Mpa
0.005

Total electromagnetic force on one tooth, N

0.5

0

0
0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

Measurement point surface
force density, MPa

3.5

0.02

Total electromagnetic force for one tooth, N

Measurement point surface force density, MPa

0.03

0.025

0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0.055

9,000

0.02

(1)

30

End

3

(1) 3.6 kHz

35

Sound pressure level, dB

0.03

40

Center

1st harmonic

Center0

20

1

Measurement point

0.057
0.058
Time, sec

0.059

3rd-order ring mode

4th-order ring mode

5.0 kHz6.9 kHz

Fig. 4. Eigenmodes

7

8

9

0
2

3

4

5

6

7

8

9

Ring mode orderFig. 5. Sound Pressure Levels(1) 2.4 kHz

Fig. 3. Constraint Conditions

6

0.01

1

3.2 kHz

5

0.005

Frequency, Hz

2nd-order ring mode

4

0.015

0.06

Fig. 2. Electromagnetic Force Time History3(2) 7.2 kHz
0.02

10

0
0.056

2

Ring mode order
15

5

End

0.01
0.005

25

Frequency, Hz

Fig. 1. Surface Force Density and Electromagnetic Force
Frequency Components

0.015

(2)

Fig. 6. Spatial Mode Order Analysis(2) 7.2 kHz(Unit:dB)

Fig. 7. Sound Pressure Level Distribution

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151

Case

Stator Core

252 Geometry Optimization of a Solenoid Valvemodule: DP

Yoke

Overview
For the good functioning of a solenoid valve, we want that the solenoid valve is as responsive as possible to the opening and closing
commands. For this, we want to optimize the rise time of the solenoid valve's mover.
But at the same time we want to make sure that the volume of the solenoid valve doesn't increase much.
In this example, we introduce the case study that optimize the shape of a solenoid valve using a multi-objective optimization algorithm.

CoilMovable Core

Optimization Conditions

Results

The rise time of the mover is measured form the time the coil is feed to the time themover reaches its final position.
We will therefore look at the impact of shape of the mover has on the rise time of mover.
Fig. 1 shows geometrical design parameters that will be changed in the solenoid valve.
The dimensions presented are all defined by two parameter: Radius andShoulder̲coefficient. Equations in shows the relation between all the geometricparameters.
For this case study, we will set the optimization to minimize the rise time of the mover aswell as the volume of the mover.
The constraint and set in this optimization study are all described in Table 1.

Looking simply at the rise time depending on the volume of the valve in Fig. 2. We can seethat increasing the Radius / Volume decreases the time needed for the mover to reach its
final position. The minimal rise time being set 12.8 msec with a Radius of 7.9 mm and a
Volume of 15.6 cm^3.
This represent an decrees of 50 % in Rise time but an increase of 56 % in Volume from thebase model.
To find the best compromise between rise time and volume we introduce a volumetricrise time. It will be the product of the rise time of a geometry by its volume.
This is illustrated in Fig. 3 where we found that the best compromise is a rise time of 14.6
msec and a Radius of 6.1 mm. It increases the volume of the valve by 20 %. but on theother hand, it decreases the rise time by 40 %. Giving a better rise time for the space used.

Min. value

Max. value

Radius

5 mm

8 mm

0

0.8

Shoulder Stator
Shoulder̲coeﬃcient

0.6

15

0.5

14
13
12

Shoulder Radius
Radius16

Volimetric Rise Time, s.cm^3

Design parameters
Stator Radius

Volume, cm^3

Stator Radius ＝ Radius
Shoulder Radius ＝ Radius × Shoulder̲coefficient
Shoulder Stator ＝ 1.125 × Radius × Shoulder̲coefficient

Constraint
Radius × Shoulder̲coeﬃcient ≧ 2

11
10
0

0.01

0.02

0.03
Rise time, sec

Fig. 1 Geometrical design parameters

152

JSOL Corporation

Table 1 CAD constraint

Fig. 2 Rise time - Volume

0.04

0.05

0.06

0.4
0.3
0.2
0.1
0

5

5.5

6

6.5
Radius, mm

Fig. 3 Radius - Volumetric Rise Time

7

7.5

8

JMAG Application Catalog

Case

254

Insulating oil
Tank

Vibration Analysis of a Large Oil-Filled Transformer

Clamp

Slats

module: DS, FQ

Overview
Fig. 1 shows the electromagnetic force distribution of the core, Fig. 2 shows the magnetostrictive force distribution of the core, and Fig. 3
shows the electromagnetic force distribution for the tank. Also, the leakage flux and eddy current distribution for the tank are shown in Fig. 4.
The core and tank excitation sources are focused on. The excitation sources of the core are the electromagnetic and magnetostrictive forces
generated in and on the core. Furthermore, the vibration source of the tank is the electromagnetic force affecting eddy currents generated inand on the tank which are generated from leakage flux from the core reaching the tank.
From Fig. 1 to Fig. 3, it is clear that the magnetostrictive force is the dominant force in the excitation force. From Fig. 4, it can be seen that theleakage flux interlinks with the tank generating eddy currents.

Core
Shield plate

Secondary winding
Primary winding

Excitation Force Distribution
Core electromagnetic force distribution is displayed in Fig. 1, core magnetostrictive force
distribution is displayed in Fig. 2, and tank electromagnetic force distribution is displayedin Fig. 3. Tank magnetic flux leakage and eddy current is additionally displayed in Fig. 4.
It is understood that magnetostrictive force is dominant as the magnitude of theexcitation force, as indicated Fig. 1 to 3.
From Fig. 4, it can be seen that a generation of eddy currents follows the linkage of tankmagnetic flux leakage. The electromagnetic force acting on these eddy currents is thetank's main source of vibration.

Acceleration Distribution

Nodal force(Unit: N)

Magnetostrictive
force(Unit: N)

Magnetic
ﬂux density(Unit: T)

Fig. 1. Core Electromagnetic Force Distribution

Magnetic
ﬂux density(Unit: T)

Fig. 2. Core Magnetostrictive Force Distribution

Fig. 5 shows the acceleration distribution of the core, and Fig. 6 shows the acceleration
distribution of the tank.
The excitation force was dominated by the magnetostrictive force of the core, but thevibration is seen to be larger in the tank than in the core. It is thought that this is due tothe low rigidity of the tank. In addition, since the vibration on the side where the shielding
plates are not installed is large, it is thought that by adding shielding plates to this side as
well excitation force and vibration can be reduced.
Nodal force(Unit: N)

Magnetic
ﬂux density(Unit: T)(Unit: A/m^2)

Fig. 3. Tank Electromagnetic Force Distribution

Fig. 4. Tank Leakage Flux and Eddy Currents(Unit: m/s^2)(Unit: m/s^2)

Fig. 6. Tank Acceleration Distribution

Fig. 5. Core Acceleration Distribution

Case

255

Evaluating the N-T Curve of an IPM motor
Using 3D Correction

Rotor Core
Magnetmodule: TR

Shaft

Overview
Since control design and motor design are designed by different designers, cooperative design is difficult. However, in order to meet thesophisticated demands in recent years, it is important to identify problems while coordinating motor design and control/circuit design atthe early stages of development. In cooperative design, it is possible to utilize circuit simulation using a highly accurate plant model.
In JMAG, it is possible to create motor models that can model actual machines in detail taking into account magnetic saturation
characteristics and spatial harmonics in the motor. By incorporating the JMAG-RT motor model into a control/circuit simulator, it is possible
to perform coupled simulation taking into account both the magnetic saturation characteristics and spatial harmonics of the motor and thecontrol characteristics of the motor driver.
In recent years, due to motors becoming thinner and power output increasing, 2D model approximations during plant model creation are
no longer able to be used. However, by performing 3D analysis for some operating points and correcting the characteristics of the plantmodel used, it is possible to improve the model accuracy while suppressing the computation time.
In this example, obtains N-T curves using the correction function.

N-T curves

3D

2D

Stator Core

2D correction

250

200
Torque, Nm

An N-T curve for a 3D model, a 2D model, and a 2D model with correction are shown in
Fig. 1.
It can be seen that there is a large difference in the maximum torque at low speed
between the 3D model and the 2D model. This is because for the 2D model the torque isoverestimated because the magnetic saturation at the motor end is underestimated. N-T
curve becomes the same as 3D by using correction.

Coil

150

100

50

0
0

2000

4000

6000

8000

10000

Speed,r/min

Fig. 1. N-T curves

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153

Case

Stator core

257

Magnet

IPM Motor Topology Optimization
Coilmodule: DP, DS

Overview
Motor design requirements are becoming more sophisticated. While satisfying torque ripple and back electromotive voltage, line voltage
limits, magnet demagnetization resistance, the stress on cores, and displacement constraints due to centrifrugal force, it is required that
torque is maximized at both low and high speeds.
Optimization that uses FEA is effective in satisfying these strict requirements. Parameter optimization that can change dimensions toparametrics depends on the topology of the initial geometry that is the optimum geometry. Topology optimization capable of making
significant changes to part layout can make changes to topology and perform searches in wider design spaces.
In this example, rotor core maximum stress and radial direction maximum displacement in the radial direction are set to certain values or
less, and rotor geometry is designed that minimizes the torque ripple and maximizes the average torque value. Sensitivity analysis is thenperformed afterwards.

Rotor core

Optimization Conditions

Sensitivity Analysis Results

Fig. 1 shows the design region, and Table 1 shows the evaluation items and objectivefunction.
Fig. 1 shows that the design region is situated near the magnet and the gap which areconsidered to have a large effect on von Mises stress and torque.

Torque waveforms and magnetic flux lines are shown in Fig. 4 and Fig. 5 respectively forwhen grooves are present and absent in the rotor core surface for the geometry obtainedwith optimization. Unlike case optimization, no smoothing is performed for the boundary
faces between materials, meaning that results with grooves differ from optimization
results.
Although optimization uses the grooves because the torque ripple rate improves when
the grooves are present, the effect is rather small, and no significant changes to magnetic
flux flowing from the rotor to the stator teeth can be observed. As such, it is thought thateven an absence of grooves is acceptable for the design.

Optimization Results
The transitions in the torque average values and torque ripple rates for the best cases ofeach generation are shown in Fig. 2. The von Mises stress distribution for the best cases
throughout all generations are shown in Fig. 3, with various quantities shown in Table 2.
A state of convergence where torque average values grow larger with each generationand torque ripple rates grows smaller can be confirmed
In is understood that in the best cases, results that satisfy von Mises stress and radial
direction displacement constraints are obtained.

With grooves

Without grooves

25
Evaluation item

Objective function

Torque average value

Maximize the torque average value.

Torque ripple

Torque ripple rate is minimized.

Item

Value

Average torque, Nm

20.7

Von Mises stressmaximum value

Rotor core internal von Mises stress is a
maximum value of 200 MPa or less.

Displacement
maximum value

Rotor core radial direction displacement is
a maximum value of 200 MPa or less.

Torque ripple, %2.47

Stress maximum value, MPa

86.4

Displacement maximum value, mm

0.00915

Table 1. Evaluation Items and Objective Functions

0

20

15

15

10

10

5

5

Torque ripple rate, %Torque average value, Nm

25

20

0

Grooves

0
200

300

400

500

(Unit: MPa)

Generation

Fig. 2. Transitions in Torque Average Values andTorque Ripple

154

Table 2. Various Evaluation Value Quantities

Torque ripple rate

25

100

JSOL Corporation

1000.00010.00020.0003

Time, sFig. 1. Design Region

0

15

5

■ : Design region

Torque average value

Torque, Nm

20

With grooves

Fig. 3. Von Mises Stress Distribution

Fig. 5. Magnetic Flux Lines

Without grooves

Fig. 4. Torque Waveform

0.00040.0005JMAG Application Catalog

Case

Coil

258 Magnetic Shield Geometry Optimization

Shield setting
regionmodule: ST

Overview
In electrical device design, due to increasing demand, optimal design using trial and error is being studied. In recent years, topology
optimization that can significantly change the geometry itself has been attempted, and novel designs heretofore not considered are being
created.
By using FEA, it is possible to optimize a sensitivity analysis (density method) topology using the material density in the design region as adesign variable. For relatively simple problems, from evaluating the influence (sensitivity) of design variables on objective functions such as
for maximizing torque or minimizing loss it is possible to speed up convergence and achieve a topology optimization with a small numberof cases.
In this example, a case study in which the geometry of a magnetic shield is optimized using topology optimization that uses a sensitivity
analysis (density method) is presented. Due to the material density distribution, topology changes, magnetic energy changes, and the
optimal case magnetic flux lines can be checked.

Evaluation
space

Changes in Topology

Optimal Case Magnetic Flux Lines

Fig. 1 shows the material density distribution (during changes in topology) during
topology optimization calculation process.
As the optimization progresses, the geometry gradually gets determined, resulting in a
multi-layer shield geometry from minimizing the magnetic energy in the evaluation
region.

Fig. 3 shows the magnetic flux lines flowing in the shield design region for the optimalcase.
It can be seen that the most of the magnetic flux flows along the shield. Also, it can beseen that the leakage magnetic flux has not entered the evaluation region due to theinnermost layers.

Changes in Magnetic Energy During Optimization Processing
Fig. 2 shows the history of magnetic energy changes during the topology optimization.
From the air layer and the magnetic layer alternating going from the outermost layer tothe innermost layer, it can be seen that the magnetic resistance increases and themagnetic energy decreases in the direction of the magnetic flux.

Case 1

Magnetic energy, J

1E-08

Case 50

8E-09
6E-09
4E-09
2E-09

Case 200

Case 1000(Unit: none)(In this document, a material density of 0 is set for the air
and a material density of 1 is set for the magnetic material.)

Fig. 1. Material Density Distributions During
Optimization Processing

0

0

200

400
600
Number ofcases

800

1000

Fig. 2. The Progressing of Optimization Processing
and Changes in the Magnetic Energy(Unit: none)

Fig. 3. Optimal Case Magnetic Flux Lines

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155

Case

259

Automatically Searching for Current Phase that Obtains
Maximum Torque and Evaluating Torque Characteristics

Stator core

Rotor coremodule: DP

MagnetOverview
In the low speed side of an N-T curve, there is a region where the maximum torque is constant regardless of the rotation speed because ofcurrent limitations. In this region, the current phase that maximizes torque is selected. When designing motor components, many designs
are compared using dimensions as design variables. In such situations, the current phase with maximum torque will differ depending on
motor geometry. In order to obtain this, what is required is an evaluation of the phase with maximum torque per geometry in advance using
FEA.
When comparing a large number of designs using simulations, torque characteristics can be evaluated by automatically searching for thecurrent phase that provides the maximum torque.
In this example, a search is performed for the current phase that maximizes torque for each of 3 rotor geometries. Motor average currents
and torque ripples are evaluated using the obtained current phases.

Evaluation Details

3 conditions for rotor geometry
Case 1Case 1

Fig. 4 shows the graph comparing the torque value obtained with the torque formula ateach current phase for Case 1 geometry, and the average torque value obtained with FEA.
Because the torque value obtained with the torque formula is of a fixed inductance, thevalue is high when compared to the FEA results that account for the influence of magnetic
saturation. In this model, torque shows the maximum at a current phase of 30 deg forboth FEA and the value obtained from the torque formula, and it can be confirmed thatcurrent phase values match.

156

JSOL Corporation

Search for the current phase that provides the maximum torquewith the torque formula

Torque, Nm

50

Ld, Lq, Iq, Id, and φ calculation

40
30
20

T = (Npoles/2)・(φ・Iq+(Ld-Lq)・Id・Iq)
10

T: torque, Nm Npoles: number of poles, φ: magnetic ﬂux, Wb
Id: d-axis current, A Iq: q-axis current, A
Ld：d-axis inductance, H Lq: q-axis inductance, H

0

0

10

20

30

40

50

60

50

60

Current phase, deg

Motor characteristics analysis (average torque / torque ripple)

Fig. 1. Evaluation flow

Fig. 2. Current Phase - Torque Graph

Average torque

Ripple ratio

FEA
45.0

70

Torque formula

70

40.0

60

60

35.0
50

30.0

40

25.0

30

20.0
15.0

20

50
Torque, Nm

Average torque, Nm

Comparison Between Torque Values Obtained With the Torque
Formula and FEA

Case 2

Inductance calculation

Average Torque and Ripple Ratio
Fig. 3 shows the average torque and ripple ratio at the current phase for maximum torque.
These values are calculated from the current phase showing maximum torque obtained
from the inductance shown in Fig. 2.

Case 360

Performed automatically

Current Phase - Torque Graph
Fig. 2 shows the current phase - torque graph. From this, the current phase where thetorque is maximum is found for each rotor geometry. The current phases for the
maximum torque for cases 1 to 3 are shown in red.

Case 270
Case 3

Ripple ratio, %Inductance values are calculated at 10 deg increments in the current phase range of 0 to
60 deg calculated from 3 rotor geometry cases, obtaining values Ld, Lq, Id, Iq, and φ.
From inductance evaluation results, torque is obtained with torque formulas for each rotor
geometry, and a search is performed for the current phase that obtains maximum torque.
Motor torque analysis is performed with the obtained current phase, from which theaverage torque and ripple ratio are obtained. Fig. 1 shows the evaluation flow.

coil

40
30
20

10.0
10
0
Current phase

Case 130 degCase 240 degCase 340 degFig. 3. Average Torque and Ripple Ratio at theCurrent Phase Showing Maximum Torque

5.0

10

0.0

0
0

10

20

30
Phase, deg

40

Fig. 4. Comparison Between Phase and Torque Curvefrom FEA and the Torque Formula (Case 1 geometry)JMAG Application Catalog

Stator yokeCase

260 Creating Axial Gap-Type Motor Eﬃciency Maps

Coilmodule: LS, TR

Overview

Frame

Axial gap-type motors can consist of a thinner construction than that of radial gap-type motors, and are used for in-vehicle motors whereavailable space is otherwise compromised. Because in-vehicle motors are additionally driven at a wide range of operating points, motor
performance is evaluated using efficiency maps.
Efficiency maps require a significant amount of calculations, and organizing results can also take a considerable amount of time. Although it
is possible to estimate efficiency by calculating the torque, voltage, and current from motor voltage equations and torque equations, there
exists several issues in the difficulty involved in using this method to estimate loss and account for the effects of the nonlinear
magnetization characteristics of the motor iron core, as well as accurately calculating efficiency, which can also prove to be similarly
challenging.
Through simulations using the finite element method, however, efficiency maps that account for the effects iron loss and nonlinear
magnetization characteristics can be written with ease. Comparing motor performance in varying design proposals is also a simple process.
In this example, the motor performance of axial gap-type motors with changes made to the flatness ratios.

MagnetAbove: Whole view
Below: Rotor

Comparison Conditions
Flatness ratios are defined by axial length/diameter. Design proposal A that has a flatness
ratio of 0.24 is compared with design proposal B that has a flatness ratio of 0.42.
Motor volume, gap length, magnet quantity, and windings remain the same for bothdesign proposals.

Eﬃciency Maps
The efficiency maps for design proposals A and B are shown in Fig. 2, while the copper loss
maps are shown in Fig. 4.
From Fig. 2, it is understood that maximum torque is large in low-speed regions for design
proposal A, and design proposal B reaches high efficiency in high-speed regions. A low
flatness ratio means a larger gap surface area where torque can be generated, therefore
increasing maximum torque. A high flatness ratio decreases the field magnetic flux that
occurs due to permanent magnets, and the load d-axis current required for weakening
magnetic flux decreases. This leads to a reduction in copper loss, thus leading to highefficiency (Fig. 3, Fig. 4).
It is thereby understood that motor performance can be adjusted by making changes tothe flatness ratio of axial gap-type motors.

Design proposal A
Flatness ratio 0.24

Design proposal B
Flatness ratio 0.42

Fig. 1. Design Proposals and Flatness Ratios

Design proposal A
Flatness ratio 0.24

Design proposal B
Flatness ratio 0.42

Fig. 3. Id Maps (High-Speed Region Magnification)

Design proposal A
Flatness ratio 0.24

Design proposal B
Flatness ratio 0.42

Fig. 2. Efficiency Maps

Design proposal A
Flatness ratio 0.24

Design proposal B
Flatness ratio 0.42

Fig. 4. Copper Loss Maps (High-Speed Region Magnification)

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157

Case

261

SPM Motor Magnet Coercive Force Optimization For
Obtaining Sinusoidal Magnetization

Stator core
Shaft

module: DP

Overview
In general, when the magnetic flux density of a gap approaches a sine wave, harmonic components are suppressed and efficiency improves.
A method exists to locally change the coercive force of magnets used in SPM to bring magnetic flux density closer to a sine wave. However,the proper coercive force of magnets is difficult to find through a process of trial and error. In situations such as these, by using optimization,
coercive force distribution can be obtained effectively.
In this example, a magnet is segmented into blocks and coercive force is defined. The coercive force of each block is calculated so that theinduced voltage waveform approaches a sine wave through optimization. The voltage waveforms are compared between the case where
the coercive force distribution obtained by the optimization is used and the case where the same coercive force distribution is used for all
magnet blocks.

Magnet block
Coil

Rotor Magnetization
The number of poles of the target motor is 4. Therefore, the angle between two poles is 90
degrees.
1 pole uses a magnet block divided into 9 every 10 deg. These are magnetized in theradial direction.
Fig. 1 displays the rotor magnet orientation direction.

Spatial magnetic ﬂux density
evaluation area

Spatial Magnetic Flux Density Evaluation

Optimization Conditions
Fig. 3 shows magnet blocks as design variables, design variable ranges, and objectivefunctions.
Design variables are the coercive forces of each magnet block, and the minimum andmaximum values indicate the search range.
The primary goal is to minimize THD, but in order to prevent sufficient torque not being
obtained due to low spatial magnetic flux density, a constraint is imposed to maximize thefundamental.
By satisfying these, the generated magnetic flux density is at a maximum and the
waveform is close to a sine wave.
Objective functions are set to maximize the fundamental and minimize THD using theabove gap magnetic flux density evaluation formula.

: Magnetic ﬂux density spatial distribution
nth-order component

N

S: Magnetic ﬂux density spatial distribution
fundamental wave (primary) component

Fig. 1. Rotor Magnet Orientation Direction

9 8<Magnet blocks (1 to 9)>

7

6

5

Fig. 2. Spatial Magnetic Flux Density Evaluation

0.6
4
3
2
1

0.5
0.4

<Design variable ranges>
Magnet block positions

Variable

1, 9Minimum Value

Maximum Value

Description

Coe1

5305001061000

2, 8

Coe2

5305001061000

3, 7
4, 6

Coe3
Coe4

5305005305001061000
1061000

Coercive force ofeach magnet
block, A/m

9

Coe5

5305001061000

THD

Fig. 2 shows the spatial magnetic flux density evaluation.
The spatial magnetic flux density is found when the rotor is fixed at no load.
To determine whether close to a sine wave, the total harmonic distortion (THD) that
represents the degree of signal distortion is used. Since THD is the ratio of the square rootof the sum of the squared harmonic components to the fundamental component, thesmaller the THD the closer the induced voltage waveform is to a sine wave.
A different coercive force for each magnet block is set and the THD value is evaluated.

0.3
0.2

<Objective functions>
Expression

Type

THD

MaximumValue

fundamental

Maximum
Value

0.1

Description
The value from the following formula.
Bn is the magnetic ﬂux density spatial distribution nth-order component.
fundamental is the fundamental wave (primary) component.

=

2

2

+

3

2

+

4

2

+

5

2

⋯+

0

2

0.6

0.7

0.8

0.9

1

1.1

1.2

Fundamental, T

Fig. 3. Optimization Conditions

Fig. 4. Pareto Solution

1.E+06

Fig. 4 shows the Pareto curve of the optimization results.
The point shown in green is the initial design, and the points shown in blue are the results
obtained from optimization. Here, the design proposal with the smallest THD among thedesign proposals on the Pareto curve is adopted. The design proposal obtained by
optimization is smaller in both fundamental and THD compared to the initial case.

Magnet Block Coercive Force

9 8

7

6

5

Coercive force, A/m

1.E+06

Pareto Curve43

8.E+05
6.E+05
4.E+052
1

0.E+00

Position of magnet block

1

2

3

4

5

6

7

8

Fig. 5. Magnet Block Coercive ForceTable 1. THD

InitialOptimized

1.5

15

1

10

0.5

5

0
-0.5

30

60

90

120

150

180

Angle, deg

Fig. 6. Spatial Magnetic Flux Density Waveform

InitialOptimized

14

Amplitude, V

10
8
6
4
2

1

2

3

4

5

6

7

-15

0

0.01

0.02

0.03
Time, sec

12

8

9

10

11

Order

Fig. 8. Induced Voltage Frequency Components

JSOL Corporation

0

-10

0

Spatial Magnetic Flux Density Waveform

0

Optimized

-5

-1
-1.5

158

0.17

9

Voltage, V

Spatial magnetic ﬂux density, T

InitialBoth initial case and optimized case THD is shown is Table 1.
Compared to the initial case, optimized case THD becomes a value of 1/2 or less. From
this, it can be described that the optimized case has a lower ratio of spatial harmonics tothe fundamental wave than the initial case, and is a waveform close to a sine wave.

Fig. 7 shows the induced voltage waveform, and Fig. 8 shows the induced voltage
frequency components.
Similar to the prior THD and spatial magnetic flux density waveform evaluation results, it
can be confirmed that this nears a sine wave through optimization.

Optimized case

0.37

Magnet Block

THD of the Initial Case and Optimized Case

Induced Voltage Waveform

Initial case

2.E+05

Fig. 5 shows the magnet block coercive force at the blue point obtained with
optimization.
It is understood from the graph that coercive force near the pole center increases.

Radial direction spatial magnetic flux density waveforms are displayed in Fig. 6.
Peaks and valleys are thought to be produced by the positions and angles of the magnets.
However, by optimizing the magnet block coercive forces, it can be confirmed that the
spatial magnetic flux density has become a waveform close to a sine wave. This isconsistent with the results in Table 1 above.

THD

Fig. 7. Induced Voltage Waveform

0.04

0.05

JMAG Application Catalog

IPM Motor Thermal Demagnetization Analysis Accounting
for Dy-Diﬀusion Magnet Incomplete Magnetization

Case

262

Rotor core
Magnetmodule: DP, ST

Yoke

Overview

Magnetization coil
Magnetization yoke

Dy-diffusion magnets can increase demagnetization resistance by distributing a high coercive force on their magnet surfaces. On the other
hand, if the magnetization is incomplete, the coercive force is reduced creating areas where demagnetization is likely. Therefore, it isimportant to consider both the effect of Dy diffusion and the effect of incomplete magnetization.
Using finite element methods, it is possible to define the coercive force distribution and thermal demagnetization characteristics withincomplete magnetization. Motor performance at high temperatures can be evaluated using defined magnet characteristics.
In this example, the thermal demagnetization resistance of a motor is analyzed using Dy-diffusion magnets with a coercive force distribution
under incomplete magnetization.

Stator core
MagnetCoil

Rotor core
Motor

Magnetization Material Determined from Demagnetization
Data
Since the demagnetization characteristics after magnetization also depend on thetemperature, prepare a characteristic table based on the applied magnetic field during
magnetization and the temperature to be used after magnetization. Since the appliedmagnetic field is not uniform in the magnet, each element has different demagnetization
characteristics.
Fig. 1 shows the table of applied magnetic field-temperature-demagnetization
characteristics.

The Dy-diffusion magnet maximum coercive force is distributed inside the magnet. Fig. 2
shows the coercive force distribution (complete magnetization) of a Dy-diffusion magnet
used in this case study.
During incomplete magnetization, the coercive force distribution changes vs. the appliedmagnetic field.

Fig. 4 shows the average torque when the magnet temperature is varied.
When the magnetization current is 25 kA, it can be seen that even at 100 deg C the torqueis lower than with complete magnetization. From this it is predicted that compared withcomplete magnetization a decrease in the overall torque over the entire temperature
range results due to the entire magnet not being sufficiently magnetized due to the effectof incomplete magnetization at the magnetization current of 25 kA.
Also, the magnetization characteristics change as the temperature rises. Even withcomplete magnetization, when the temperature is 130 deg C or more, the torquedecreases greatly and thermal demagnetization occurs. At the magnetization current of
45 kA, the torque up to 130 deg C is almost the same as that at complete magnetization,
but when 130 deg C is exceeded, the torque decreases more than at complete
magnetization. This is because with incomplete magnetization there is an area wheredemagnetization is large, thus at high temperature there is more demagnetization thanwith complete magnetization. From this the average torque is reduced.

25 kA

Coercive force(normalized)

0.8e+06Appli
ed

1.2e+06mag
ne

1.6e+06

Ma

e
gn

tic

ld,

tic
ﬁe

3.2e+06

Complete magnetization

55

Rotor and magnet positions

ﬁe

45 kA

56

(Unit: A/m)m
A/

Average torque, Nm

Magnetic ﬂux density, T

Fig. 3 shows the magnet magnetization distributions obtained by in-place magnetization
at input currents of 25 kA and 45 kA. The smaller the input current, the smaller the magnet
magnetization near the bridge. Incomplete magnetization results where themagnetization is small.
In addition to the effect of the magnitude of the magnetization, the Dy-diffused magnets
are also affected by the coercive force distribution. Demagnetization is likely to occur
where there is low coercive force and incomplete magnetization.

TorqueDy-diﬀusion Magnet Coercive Force Distribution

─ 20 deg C
─ 60 deg C
─ 100 deg C
─ 140 deg C

Magnetization Distributions

54
53
52
51
50
49
48
47

ld

(Unit: None)

46
100

,A

Input current of 25 kA Input current of 45 kA Complete magnetization

105

110

115

120

125

130

135

140

Magnet temperature, deg C/mFig. 1. Applied Magnetic Field - Temperature Demagnetization Characteristics Table

Fig. 2. Dy-diffusion Magnet Coercive Force Distribution(Complete Magnetization)

Fig. 3. Magnetization Distributions (Under No-Load)

Fig. 4. Magnet Temperature - Average Torque Characteristics

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Case

263

IPM Motor Stress Analysis Taking Into Account Friction
between Magnets and the Rotor Core

Stator core

Rotor coremodule: DS
MagnetOverview
In an IPM motor, magnets experience centrifugal force during high-speed rotation, generate friction while sliding, and are pressed against
the rotor core, generating large stress. If the stress acting on a part exceeds its yield stress, plastic deformation occurs, which may cause achange in its properties or cause it to break. Therefore, it is necessary to create a design where yield stress is not exceeded even withcentrifugal force.
By using simulation, the effects of centrifugal force can be checked and reflected in design before actually making the motor. During
simulation, it is important to accurately simulate plastic deformation and part contacts.
In this example, in order to evaluate how the contacts between the magnets and the rotor core in an IPM motor affect the stress
distribution, the von Mises stress is checked when there are rigid contacts and when the magnets are allowed to slide while the friction
taken into account.

Core Material Characteristics

Coil

400

Fig. 1 shows the core material characteristics. The yield stress of the material used is 330
MPa.

300
Stress, MPa

Von Mises Stress Distribution
Fig. 2 shows a comparison between von Mises stress distributions. Since for a rigid
connection there is no slipping, a large stress is generated at the contacts. When taking
into account friction, the magnet slides tangentially along contact area with the rotor
core, generating large stress at the bridge. The distribution differs between the rigid
connection case and the case when friction is taken into account. In actual phenomena,
since it is thought that there is sliding and not a contact with a continuous rigid
connection, it is understood that it is necessary to correctly model the contact state for
accurate evaluation.
The von Mises stress maximum value exceeds the yield stress of 330 MPa. If left as is,plastic deformation occurs and the geometry changes, which may result in changes incharacteristics or breakage, indicating that measures such as changing the material arenecessary.

Yield stress

350

250
200
150
100
(Unit: MPa)

50
0

0

0.001

0.002

0.003

0.004

0.005

Rigid body coupling
Maximum stress: 330.35 MPa

StrainFig. 1. Core Material Characteristics

Fig. 2. Von Mises Stress Distribution

Case

Magnet264 Creating 6-Phase IPM Motor Efficiency Maps

Friction taken into accountMaximum stress: 330.41 MPa

Rotor coremodule: DP, LS

Overview
For traction motor, multi-phase design is sometimes employed with the objective of improving redundancy and reducing torque ripple
during drive. multi-phase design includes the likes of 6-phase drive using two inverters. Efficiency maps are created for both motor design andcontrol design in order to fully comprehend the characteristics of motors such as Traction motors that are capable of drive within such a wide
range.
Torque, voltage, and current can be calculated from motor voltage equations and torque equations for estimating efficiency, but there exists
the issue of difficulty in estimating iron loss, as well as accounting for the nonlinear magnetization characteristics of the iron core. Throughsimulations using FEA, however, efficiency maps that account for the effects of iron loss and nonlinear magnetization characteristics can beobtained with ease.
In this example, the efficiency map of a 6-phase IPM motor is created with loss ratios at each operating point also evaluated.

Coil

Stator core

Efficiency Map and Power Factor Map

Loss Ratios at Each Operating Point

The efficiency map is displayed in Fig. 1, and the power factor map is displayed in Fig. 2.
Motor mode and generator mode efficiency can be understood from Fig. 1. It can be seenthat the high-efficiency region for this motor on the motor mode extends to thehigh-speed area on the generator mode.
For this motor, it can be understood from Fig. 2 that the ratio of reactive power becomeslarger at a low rotation speed in high-torque regions, and the ratio of reactive power
becomes smaller in low-torque regions.

Fig. 3 displays pie charts showing loss ratios at low speed and high load, low speed andmedium load, medium speed and medium load, and lastly, high speed and low load.
Copper loss decreases in accordance with load. It can be seen that the loss of both thestator core and the rotor core increases in accordance with rotation speed, with eddy
current loss increasing in particular.

N=500, T=120

N=2000, T=60

N=5000, T=50

N=12000, T=10

500 r/min

2,000 r/min

5,000 r/min

12,000 r/min

N Rotation speed
T TorqueCopper loss
Stator core eddy current loss
Stator core hysteresis loss
Rotor core eddy current loss
Rotor core and hysteresis loss
Mechanical loss

Fig. 1. Efficiency Map

160

JSOL Corporation

Fig. 2. Power Factor Map

Fig. 3. Loss Ratios at Each Operating PointJMAG Application Catalog

Case

Rotor core

265 Creating Synchronous Reluctance Motor Eﬃciency Maps
module: DP

Overview
With the increase in price for rare earth magnets, synchronous reluctance motors that do not utilize permanent magnets have been
attracting attention for the likes of industrial purposes. Because they do not use permanent magnets or secondary conductors, synchronous
reluctance motors can be anticipated to be variable speed motors that are both inexpensive and highly efficient. Efficiency maps are createdfor both motor design and control design in order to fully comprehend the characteristics of these motors that are capable of drive withinsuch a wide range.
Torque, voltage, and current can be calculated from motor voltage equations and torque equations for estimating efficiency, but there
exists the issue of difficulty in estimating iron loss, as well as accounting for the nonlinear magnetization characteristics of the iron core.
Through simulations using FEA, however, efficiency maps that account for the effects of iron loss and nonlinear magnetization
characteristics can be obtained with ease.
In this example, the efficiency map of a synchronous reluctance motor is created with loss ratios at each operating point also evaluated.

Stator coreEﬃciency Map and Power Factor Map

Loss Ratios at Each Operating Point

The efficiency map is displayed in Fig. 1, and the power factor map is displayed in Fig. 2.
Motor mode and generator mode efficiency can be understood from Fig. 1.
It can be seen from Fig. 2 that this motor has a power factor of 0.7 or less, which isparticularly low in both the high-torque region and the low-torque region.

Fig. 3 displays pie charts showing loss ratios at low speed and high load, low speed andmedium load, medium speed and medium load, and lastly, high speed and low load.
Copper loss decreases in accordance with load. It can be seen that the loss of both thestator core and the rotor core increases in accordance with rotation speed, with eddy
current loss increasing in particular.

N=2000, T=2

N=4000, T=10

N=8000, T=5

N=14000, T=2

2,000 r/min

4,000 r/min

8,000 r/min

14,000 r/min

N Rotation speed
T TorqueCopper loss
Stator core eddy current loss
Stator core hysteresis loss
Rotor core eddy current loss
Rotor core and hysteresis loss

Fig. 1. Efficiency Map

Fig. 2. Power Factor Map

Fig. 3. Loss Ratios at Each Operating Point

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Case

267

IPM Motor Topology Optimization Using the Density
MethodStator Core

MagnetCoilmodule: DP, DS

Overview
Motor design requirements are becoming more sophisticated. While satisfying the torque ripple rate and back electromotive voltage, line
voltage limits, magnet demagnetization resistance, the stress on cores, and displacement constraints due to centrifugal force, it is requiredthat torque is maximized at both low and high speeds.
Optimization that uses FEA is effective in satisfying these strict requirements. Design proposals can be obtained with a high degree of
freedom because topology optimization in particular can perform searches that are not dependent on initial shapes.
Magnetic field and structural coupled analysis is run in this example. While accounting for rotor core strength, topology optimization thatuses the density method is used to perform searches for a rotor shape that maximizes average torque.

Rotor Core

Optimization Conditions
Fig. 1 shows the design region, Table 1 shows the evaluation item and objective functions.
The red frame shown in Fig. 1 is the design region. Topology accounts for the symmetry
per each 1/2 pole while accounting for the rotor strength.

Optimization Results

Fig. 1. Design Region

Torque ripple rate

Torque ripple rate is minimized.

Mean compliance

Mean complianceis is minimized.

Table 1. Evaluation Items and Objective Functions

Accounting for rotor strength
Not accounting for rotor strength

0.3

Item

Rotor strength
accounted for

Rotor strength
not accounted for

Average torque, Nm

23.14125.799Torque ripple rate, %

0.064

0.187

0.25
0.2
0.15
0.1
0.05
0

10

15

20
Torque, Nm

25

30

Fig. 2. Average Torque-Torque Ripple Rate for All Cases

Rotor strength accounted forTable 2. Optimization Case Average Torque and Torque
Ripple Rate

Flux barrier

Rotor strength not accounted forFig. 3. Optimization Case Material Density Distribution
Contours and Flux Lines

JSOL Corporation

Torque average is maximize.

0.35

Bridge162

Objective functions

Average torque■ : Design region

Torque ripple, %The graph for the average torque-torque ripple rate for all cases when both accounting forand not accounting for rotor strength is shown in Fig. 2, and the optimum case average
torque and torque ripple rate are shown in Table 2.
Average torque increases when rotor strength is not accounted for.
The results showing the optimum case material density distribution contours andmagnetic flux lines when both accounting for and not accounting for rotor strength are
shown in Fig. 3.
Flux barriers are generated in either circumstance to suppress the magnetic flux thatshort-circuits in the rotor, which results in larger average torque. When accounting for
rotor strength, however, the formation of a bridge makes it easier for the magnetic flux toshort-circuit. It is therefore thought that average torque is larger when not accounting for
rotor strength.
The results showing R direction displacement contours for the optimum case when bothaccounting for and not accounting for rotor strength are shown in Fig. 4.
Not accounting for rotor strength means that the bridge is not generated, and there exists
the structural issue of the increasingly large rotor displacement. Accounting for rotor
strength means that the bridge is generated, and the displacement is suppressed.
From this we understand that optimization accounting for rotor strength can be useful.

Evaluation item

Rotor strength accounted for(Maximum displacement:
0.008 mm)

Rotor strength not accounted for(Maximum displacement:
24414 mm)

(Unit: mm)

Fig. 4. Optimization Case R Direction Displacement
Contours

JMAG Application Catalog

Case

Shaft

268

SPM Motor Teeth Geometry Topology Optimization
module: DP

MagnetOverview
Basic policies for motor design are the maximization of torque and low oscillation. The smaller the torque ripple, the lower the vibrations of
the motor. Torque ripple is generally affected by stator teeth geometry. Due to the complexity of the magnetic flux distribution in gap
vicinities, expressing detailed teeth geometry with magnetic equivalent circuits can prove to be of considerable difficulty.
Performing optimization that uses FEA capable of analyses based on actual geometry can be effective in satisfying these requirements.
Optimization results can be obtained with a high degree of freedom because topology optimization additionally performs searches that are
not dependent on initial geometry.
In this document, the on / off method is used to search for stator teeth geometry that maximizes average torque and minimizes torque ripple
with the entire areas of stator teeth tips as the design region.

Stator core

Optimization Conditions

Sensitivity Analysis

Fig. 1 shows the design region, and Table 1 shows the objective functions.
As shown in Fig. 1, the design region is the stator teeth parts, which are thought to have alarge influence on torque and torque ripple.

As so to confirm the influence of teeth geometry on the torque characteristics, sensitivity
analysis is performed for each best case.
Consider best case A, which has maximum average torque. Analysis patterns are shown in
Table 2, and analysis results are shown from Fig. 6 to Fig. 8. From a comparison between A
pattern 1 and A pattern 3, it is understood that the presence or absence of slot opening
parts has a small effect on torque characteristics. In A pattern 2, the end parts of the teeth
tips are thick, and it can be considered that torque ripple increases because of the steep
permeance with the slot opening parts. In A pattern 4, it can be considered that the
average torque decreases because the magnet magnetic flux short-circuits at the teeth tip
parts.
Consider best case B, which has the minimum torque ripple rate. Compare best case B andA pattern 3 which share similar geometry. Table 3 shows the geometry of A pattern 3 andbest case B, and the analysis results are shown from Fig. 9 to Fig. 11. Because magnetic
flux flows easier in the iron region for A pattern 3, the maximum torque increases. It canbe considered that the reason the torque ripple rate increases is because the magnetic
flux where force acts in the opposite direction of rotation flows easily.

Optimization Results
Fig. 2 shows a correlative diagram between the torque average value and torque ripple
rate, and Fig. 3 shows a larger view of the range for the maximum torque and minimum
torque ripple from Fig. 2. In Fig. 3, the maximum torque average value is shown at point A
as the best case, and the minimum torque ripple rate is shown at point B.
Topology optimization geometries for best cases A and B are shown in Fig. 4 and Fig. 5
respectively.
For best case A which has the maximum torque, material inside the teeth consists entirelyof electromagnetic steel sheets. The presence of slot opening parts can also be observed
here, whereas in best case B where torque ripple is at a minimum, teeth are connectedaround the entire arc, and there are no slot opening parts.

Name
Description

A pattern 1
Geometry with maximum
average torque

A pattern 2
Electromagnetic steel sheets
other than the slot opening part

Average torque

Torque ripple rate

0.3

10%
9%

0.295
Objective

Average torqueMaximize

Torque ripple

MinimizeName

A pattern 3

Description

No slot opening partA pattern 4
Teeth parts are entirely
electromagnetic steel sheets: Design region

8%
7%

0.29

6%
5%

0.285

4%
0.28

3%
2%

0.275

1%

Diagram
0.27

Fig. 1. Design Region

Table 1. Objective Functions

Table 2. Best Case a Sensitivity Analysis Patterns

A pattern 1
30%

200%

A pattern 3

A pattern 2

A pattern 3

A pattern 40%Fig. 6. Torque Characteristics (Best Case a Sensitivity Analysis)

A pattern 4

0.315
25%

120%
100%
80%
60%
40%

0.31
0.305

20%
15%
10%

0.22

0.24

0.26

0.28

0.3

0.32

Torque average, Nm

Fig. 2. Correlation Curve Between
Torque Average Value and Torque Ripple

0%
0.295

A pattern 1

A pattern 2

A pattern 3

A pattern 4

0.29
0.285

5%

20%

0.3
0.295

0.28

A

0.275

B
0.296

0.297

0.298

0.299

0.27

0.3

0

5

Torque average, Nm

Fig. 3. Correlation Curve Between
Torque Average Value and Torque Ripple

10

15

20

25

30

Rotation angle, deg

Fig. 7. Torque Waveform(Best Case a Sensitivity Analysis)Fig. 8. Air Gap Vicinity Magnetic Flux Lines(Best Case a Sensitivity Analysis)

Average torque
A pattern 3

Description

Slot opening part

Torque ripple rate

0.3

6%

0.295

5%

0.29

4%

0.285

3%

0.28

2%

B
Geometry with minimum
torque ripple rate

No slot opening part

Torque average, Nm

Name

Air region
Diagram

1%

0.275
0.27

Fig. 5. Topology Optimization Geometry (Best Case B)

Table 3. A Pattern 3 and Best Case B

A pattern 3

A pattern 30%BFig. 9. Torque Characteristics (A Pattern 3 and Best Case B)

B

0.31
0.305
0.3
Torque, Nm

Fig. 4. Topology Optimization Geometry (Best Case A)

Torque ripple rate

140%

Torque, Nm

Torque ripple ratio

160%
Torque ripple ratio

A pattern 2

A pattern 1

0.32

180%

0%
0.2

Torque ripple rate

Evaluation item

Torque average, Nm

Diagram

0.295
0.29
0.285
0.28
0.275
A pattern 3

B

0.27
0

5

10

15

20

25

30

Rotation angle, deg

Fig. 10. Torque Waveform (A Pattern 3 and Best Case B)

Fig. 11. Air Gap Vicinity Magnetic Flux Lines(A Pattern 3 and Best Case B)

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Case

269

Creating Three-Phase Induction MotorEﬃciency Maps Accounting for Harmonics

Stator coreCoilmodule: DP, LS

Overview
With advances in variable speed operation technology for induction motors, progress is being made in applications to fields where wide
operating regions are required. An effective method of evaluating the characteristics of induction motors in fields like these is to create
efficiency maps for the operating regions.
Simple calculations that do not account for harmonics that enable the faster creation of efficiency maps are effective for concept designs thatare narrowed down from a larger number. When concept designs have then been narrowed down to a certain extent, efficiency maps canthen be created which do account for harmonics, which then allows for more accurate evaluations to be run.
In this example, efficiency maps are created for a three-phase induction motor accounting for PWM carrier and slot harmonics. These arecompared with efficiency maps when harmonics are not accounted for.

Rotor core

Cage (bar)Circuit Diagram

High-Speed Region Loss Components

The circuit diagram of the induction motor is shown in Fig. 1. Control is accounted for by
the circuit.

The breakdown of loss components when harmonics are or are not accounted for in thehigh-speed region (1600 rpm, 180 Nm) is shown in Fig. 7. When harmonics due to PWM
carriers and slots are accounted for, both iron and secondary copper losses are larger thanwhen harmonics are not accounted for because the corresponding iron and secondary
copper losses are included.

Eﬃciency Map, Ratio Map
An efficiency map not accounting for harmonics is shown in Fig. 2 and an efficiency map
accounting for harmonics is shown in Fig. 3. Ratio maps for the total loss, iron loss, andcopper loss are shown in Fig. 4, Fig. 5, and Fig. 6 respectively when harmonics are or are
not accounted for.
Efficiency at operating point 1600 rpm, 180 Nm is 96.5% when harmonics are notaccounted for and 04.7% when harmonics are accounted for, showing a 1.8 point
difference. A look at the high-speed region shows a large difference in loss for both ironand copper.

Control input

1600 rpm, 180 Nm

Fig. 1 Induction motor circuit

Fig. 2. Efficiency map not accounting for harmonics

Fig. 5. Iron loss ratio map

Primary copper loss

Secondary copper loss

Fig. 6. Copper loss ratio map

Iron loss

Mechanical loss

1800
1600
1400

1600 rpm, 180 Nm

Loss, W

1200
1000
800
600
400
200
0
Not accounting for harmonics

Fig. 3. Efficiency map accounting for harmonics

Fig. 4. Total loss ratio map

Accounting for harmonics

Fig. 7. Breakdown of both loss components in the 1600 rpm, 180 Nm

Stator yokeCase

271 Axial Gap Type Motor AC Copper Loss Analysis

Coilmodule: TR

Overview

Frame

The primary loss of axial gap type motors occurs in the core, magnets, and coils. This document focuses on the loss that occurs within coils.
Coil loss can be defined in accordance with both DC copper loss and AC copper loss. It is entirely possible to calculate DC copper loss withrelative ease from the coil geometry, electrical resistivity, and the current. Conversely, AC copper loss is influenced by the likes of magnetic
fields generated by conductors that are positioned close together. There are instances of using square wire to increase the coil fill factor, but
in this circumstance, because the coil wire cross section is large, the current is easily distributed, and making estimates from calculations
done manually can prove difficult.
By using finite element analysis, coil loss can be visualized, and analysis is made entirely possible without the need for prototyping any realmachines.
In this example, the AC copper loss of an axial gap type motor using square wire is evaluated.

MagnetAbove: Whole view
Below: Rotor

AC Copper Loss
Coil AC copper loss is shown in Table 1, and coil cross section loss density distribution andmagnetic flux density distribution are both shown in Fig. 1.
It can be understood from Fig. 1 that loss becomes larger in the vicinity of the air gap. It
can be considered that the distribution of loss occurs due to flux leakage in the vicinity of
the air gap linking with the coils.

Coil
Loss density distribution

Air gap(Unit: W/m^3)
AC copper loss (W)

1,344
Coil

Magnetic ﬂux density distribution
Air gap(Unit: T)

Table 1. AC Copper Loss

164

JSOL Corporation

Fig. 1. Coil Cross Section Loss Density Distribution
and Magnetic Flux Density Distribution

JMAG Application Catalog

Case

272 Thrust Analysis of Linear Actuator with a Curved Section
module: TR

Overview
Linear motors are used for machine tools and similar devices because of their ability to exhibit high speeds, high acceleration anddeceleration, in addition to high-accuracy positioning. Among them, linear actuators with curved sections also possess the ability to realize
curved motions in addition to straight motions. High thrust that is also stable is required as a matter of improving performance, but because
high thrust is affected by curved motion sections, prior investigations are necessary.
Differences in gap magnetic flux density between linear motion sections and curved motion sections can be difficult to comprehend in virtual
calculations. High-accuracy estimations are possible through the use of FEM, which can take into account the effects of material non-linearityand magnetic flux leakage in space.
In this example, the thrust characteristics of a linear actuator with a curved section are obtained via analysis, and thrust at the transition from a
linear motion to a curved motion is evaluated.

MagnetMover yoke

Direction
Coil
Stator yoke

Thrust Characteristics
Thrust characteristics are shown in Fig. 1.
It is understood that thrust in the curved section is lower compared to the straight section.
This is because, as shown in Fig. 2 and Fig. 3, the space between magnets becomesnarrower on the inner diameter side of the curved section and widens on the outer
diameter side, so the area where the mover and the stator are situated opposite eachotheris reduced, and the effective magnetic flux is also reduced.
It is understood that changes in the amount of thrust are large in the curved section.
Similarly, as changes occur where the mover and the stator are situated opposite
eachother, the magnetic field of magnets becomes non-sinusoidal, but because coilmagnetomotive force is a sine wave, this results in the generation of unintended thrust.

ThrustMoving average

160
140

Thrust, N

120
100
80
60
Straight
motion40

Curved motion

Straight
motion

20
0
0

5

10

15

20

25

30

35

Time, sFig. 1. Thrust Characteristics

Fig. 2. Mover and Stator Situated Opposite
Eachother in a Straight Section

Fig. 3. Mover and Stator Situated Opposite Eachother in a Curved Section

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Case

273

Creating IPM Motor Eﬃciency Maps Accounting
for AC Loss

Stator core
Magnetmodule: DP, LS

Overview
IPM motors use rare-earth sintered permanent magnets with strong magnetic forces, and in addition to the magnet torque that occurs as aresult of the magnetic fields of magnets as well as rotating magnetic fields, IPM motors are also capable of using reluctance torquegenerated from the difference between d-axis inductance and q-axis inductance. IPM motors are therefore in possession of wide operating
regions, and are highly efficient. It is for these reasons that IPM motors are used for the likes of traction motors for electric vehicles.
Creating efficiency maps for both motor design and control design can be useful due to the fact that motor efficiency varies in accordancewith rotation speed and load.
An example of creating efficiency maps is shown in JAC165. PWM iron loss is accounted for in JAC165 in pre-processing, but input issinusoidal, and coil AC loss is ignored. Because power electronics are used for traction motors, ignoring the effects from PWM can lead tothe possibility of overestimating motor performance.
In this example, an IPM motor efficiency map that accounts for AC loss from a PWM is created, and a comparison with an efficiency map that
does not account for AC loss is performed.

Eﬃciency Map Not Accounting for AC Loss
The efficiency map not accounting for AC loss is shown in Fig. 1, and the operating points
extracted to create an efficiency map that does account for AC loss are shown in Fig. 2.
In order to create the efficiency map accounting for AC loss from the PWM, an efficiency
map not accounting for AC loss, similar to that of Fig. 1, is created first. Refer to JAC165 formore information on efficiency maps not accounting for AC loss.
It is from this efficiency map that operating points for the purpose of creating an efficiency
map accounting for AC loss are extracted.
This document categorizes three regions consisting of a low speed region, a medium
speed region, and a high speed region. Operating points are selected to include maximum
torque points.

Coil
Rotor coreLow speed region

Fig. 1. Efficiency Map Not Accounting for AC Loss

Medium speed region

High speed region

Fig. 2. Operating Points to Extract

Modeling to Account for AC Loss
Control circuit

The control circuit is shown in Fig. 3.
The control circuit is created to account for the influence of the PWM. As so to additionallycalculate AC loss, the coils model wire geometry similar to the schematic diagram.
A JMAG-RT model is used in the control circuit. Calculations of a transient state at the startof analysis are performed with the JMAG-RT model, and by performing FEA only with anelectric angle period after reaching a steady state, analysis time can be reduced.

Eﬃciency Map Accounting for AC Loss

Confirm the loss that occurs during low-speed, low-load and high-speed, low-load shownin Fig. 6.
A breakdown of loss components during low-speed, low-load is shown in Fig. 7, and thestator core eddy current loss frequency components when accounting for AC loss areshown in Fig. 8.
PWM carrier frequency is 6,000 Hz. From Fig. 7 and Fig. 8, it is understood that the PWM
harmonic components have a large effect on eddy current loss at low-speed, low-load. For
analysis not accounting for AC loss, PWM iron loss is obtained in post-processing. This
reduces the difference from when accounting for AC loss.
A breakdown of loss components during high-speed, low-load is shown in Fig. 9, and flux
lines and current density distribution frequency components when accounting for AC loss
are shown in Fig. 10.
From Fig. 9, it is understood that the influence of eddy current loss from the PWM at
high-speed, low-load is small, but also that whether AC loss is accounted for or not results
in a large difference in copper loss. From Fig. 10, it can be considered that the difference inthis copper loss is due to the eddy currents that occur as a result of the leakage flux within
the slots.

Fig. 3. Control Circuit

Fig. 4. Efficiency Map Accounting for AC Loss

Low speed low load

Fig. 5. Efficiency Difference Map

Hysteresis loss

Eddy current loss

High speed low load

Fig. 6. Copper Loss Comparison Map

PWM eddy current loss

Copper loss

6

300
5
250
4
Loss, W

Loss Component Breakdown

JMAG-RT model

200
Loss, W

The efficiency map accounting for AC loss from the PWM is shown in Fig. 4. Additionally,
as a comparison between accounting for AC loss and not accounting for AC loss, anefficiency difference map is shown in Fig. 5, and a copper loss comparison map is shownin Fig. 6.
From Fig. 5, it is understood that the difference between accounting for AC loss and notaccounting for AC loss is approximately one point in low speed and medium speed
regions. There is also a difference of two to five points or more in the high speed region.
When there is a desire to reach high efficiency of 90% or higher like that of a traction
motor, there exists the possiblity that this difference could have an effect on accuracy.
From Fig. 6, it is understood that whether AC loss is accounted for or not results in a large
difference in copper loss in the high speed region. It is thought that the differences in
efficiency shown in Fig. 5 occur as a result of this.

Coils

Inverter

3

150
2
100
1
50
0
67

0
Not accounting for AC loss

Accounting for AC loss

Fig. 7. Breakdown of Loss Components at Low-Speed,
Low-Load

Hysteresis loss

Eddy current loss

PWM eddy current loss

6000

11933
Frequency, Hz

17867

23800

Fig. 8. Stator Core Eddy Current Loss Frequency
Components at Low-Speed, Low-Load

Copper loss

2500

2000

Loss, W

1500

1000

500

0

(Unit: A/m^2)
Not accounting for AC loss

Accounting for AC loss

Fig. 9. Breakdown of Loss Components at High-Speed,
Low-Load

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JSOL Corporation

Fundamental wave components PWM harmonic components

Fig. 10. Frequency Components for High-Speed,
Low-Load Flux Lines and Current Density Distribution

JMAG Application Catalog

Case

274

SPM Motor Thermal Demagnetization Analysis Using
Bonded NdFeB Magnets

Rotor magnet

Stator core

Rotor core

Stator coilmodule: DP

Overview
Because SPM motors are, due to their structure, succeptible to demagnetizing fields from rotor magnets, it is necessary to consider the
influence of magnet demagnetization on motor characteristics.
In order to properly grasp this demagnetization, the demagnetizing characteristics that magnets possess should be correctly accounted for.
When there exists magnetization characteristics that make it difficult to observe knickpoints on demagnetization curves like those ofbonded NdFeB bonded magnets, it becomes difficult to recreate demagnetization curves with linear approximation. By correctly expressing
demagnetization curves using point arrays, the influence of demagnetization on various characteristics can be accounted for at a high levelof accuracy.
In this example, an SPM motor thermal demagnetization analysis is performed using bonded NdFeB magnets. Both the influence on torquewaveforms when changing rotor magnet temperatures and the distribution of the demagnetization ratio within the magnets are evaluated.

Demagnetization Curves

0.7
Point Array @60degC
Bilinear Approximation @60degC
Point Array @140degC
Bilinear Approximation @140degC

0.6
0.5
Magnetic ﬂux density, T

Fig. 1 shows the demagnetization curves of the rotor magnets used in this document. The
blue lines show characteristics at 60 deg C, and the red lines show characteristics at 140
deg C. The dashed lines are the demagnetization curves obtained with bilinear
approximation, and the solid lines are the actual demagnetization curves plotted withpoint arrays.
It can be observed that differences exists in the demagnetization curves obtained withbilinear approximation compared to the actual demagnetization curves. From this, it canbe considered that a difference in results occurs when accounting for thermal
demagnetization using the two types of demagnetization curves shown.

0.4
0.3
0.2
0.1
0
-0.1
-0.2

-500

-400

-300

-200

-100

0

Fig. 1 Magnetization Curves(Unit: %)Fig. 2 Demagnetization Ratio Using Demagnetization
Curves Obtained with Bilinear Approximation

Eﬀect on Motor Characteristics

Point Array

Bilinear Approximation

0.08
0.07
0.06
Torque, Nm

The influence on torque characteristics was evaluated when magnet temperatures werechanged from 60 deg C to 140 deg C and back to 60 deg C again when running at an
excitation condition of a phase current amplitude of 5 A at an angle 80 deg.
Fig. 2 and Fig. 3 shows the demagnetization ratio of the rotor magnets after irreversible
demagnetization with contours. The demagnetization ratio is defined as the amount ofvariation for magnet residual magnetic flux density prior to additional load and heating. It
is understood that a difference occurs between the bilinear approximation of thedemagnetization curves and using the actual demagnetization curves.
Fig. 4 shows torque characteristics, where it understood that differences in torquecharacteristics occur. Compared to the average torque after irreversible demagnetization,
it is understood that there is an error of 8 % that occurs when using the bilinear
approximation demagnetization curves.
This shows that demagnetization can be correctly accounted for by specifying
demagnetization curves with arbitrary point arrays when using magnets whereknickpoints on demagnetization curves are difficult to observe.

100

Magnetic ﬁeld strength, kA/m

0.05
0.04
0.03
0.02

0
(Unit: %)140 (degC)

60 (degC)

0.01
0

180

60 (degC)

360

540

Rotation angle, deg

Fig. 3 Demagnetization Ratio Using Actual
Demagnetization Curves

Fig. 4 Torque Characteristics

https://www.jmag-international.com/

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Case

275

IPM Motor Iron Loss Analysis Accounting for the Eﬀect
of Residual Strain

Stator core

Rotor magnet

Stator coil

Rotor coremodule: DP

Overview
As electromagnetic appliances become more and more efficient, evaluations of iron loss at a high accuracy also become increasingly sought
after. Actual manufacturing sees processes of electromagnetic steel sheets being punched out with a press, laminated, fixed into place withcaulking, welding, with bolts tightened, etc., with the laminated iron core either press-fitted into frame or shrink-fitted. Stress and residualstrain can occur as a result of these processes, which compared to more ideal conditions, causes increases in iron loss.
In simulations using the finite element method, iron loss can be calculated taking into account the effect of this stress and distortion.
In this example, an analysis of an IPM motor accounting for the residual strain that occurs as a result of punching processes, and evaluates
the effect on torque, magnetic flux density, and iron loss.

Expressing Deterioration as a Result of Residual Strain

Eﬀect on Iron Loss

The deterioration that occurs as a result of residual strain is expressed via the deterioration
coefficient in the cut plane, and the deterioration function from the surface.
Fig. 1 shows the permeability deterioration coefficient of the magnetic field dependency,and Fig. 2 shows the hysteresis loss correction coefficient of the magnetic flux density
dependency.

Fig. 5 to Fig. 8 show the differences in the breakdown of iron loss, iron loss density, andhysteresis loss density, all from the presence or absence of residual strain.
It is understood from Fig. 5 that iron loss increases by approximately 13 % by accounting
for residual strain.
From Fig. 6 to Fig. 8, it is understood that differences occur not only in hysteresis loss, but
Joule loss also.
The increase in Joule loss occurs as a result of the deterioration of magnetization
characteristics at the edges due to residual strain, as well as the increase in magnetic flux
density at the center of the teeth.

Eﬀect on Magnetic Flux Density and TorqueFig. 3 shows differences in magnetic flux density due to the presence or absence ofresidual strain, while Fig. 4 shows the comparison of average torque.
It can be understood from Fig. 3 that by accounting for residual strain, magnetic flux
density grows smaller in the vicinity of the core cut plane. This is because permeability
deteriorates due to manufacturing.
From Fig. 4, it can be understood that the effect on average torque from residual strain isapproximately 1 %.

hys. Loss, stator

1.4

1
0.8
0.6
0.4
0.2

hys. Loss, rotor

joule loss, stator

joule loss, rotor

100

1.2

90
1

80
70

0.8
Loss, W

Hysteresis loss correction coeﬃcient

Permeability correction coeﬃcient

1.2

0.6

60
50
40

0.4

30

0.2

20
0

0
0

200

400

600

800

1000

1200

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Magnetic ﬂux density, T

Magnetic ﬁeld, A/m

Fig. 1. Permeability Correction CoefficientFig. 2. Hysteresis Loss Correction Coefficient

1.6

10

Without residual strain

With residual strain(Unit: W/m^3)0
Without residual strain

With residual strain

Fig. 5. Iron Loss Breakdown

Fig. 6. Iron Loss Density Distribution

10
9.5

Average torque, Nm

9
8.5
8
7.5
7
6.5
6
5.5
Without residual strain

With residual strain(Unit: T)

Without residual strain

5
Without residual strain

Fig. 3. Magnetic Flux Density Distribution

168

JSOL Corporation

Fig. 4. Average TorqueWith residual strain(Unit: W/m^3)

Without residual strain

With residual strain

With residual strain

Fig. 7. Hysteresis Loss Density Distribution

Fig. 8. Joule Loss Density Distribution(Unit: W/m^3)

JMAG Application Catalog

MagnetCase

Rotor core

276 Six-Phase IPM Motor Harmonic Iron Loss Analysis

module: DP, LS

Overview
For traction motors, polyphase design is sometimes used with the objective of improving output density, increasing redundancy during
drive, and reducing torque ripple. In polyphase design, six-phase drive with two inverters are used. The typical control method is current
vector control, and PWM motors are widely used to create command currents.

Coil

Stator core

It is necessary to be aware of iron loss to increase the efficiency of motor drive systems. When power is converted using PWM, carrier
harmonics produced by PWM are superimposed on the current and IPM motor core magnetic flux density waveforms, causing an increase iniron loss. Iron loss accounting for PWM carrier harmonics can be evaluated by modeling control circuits that include PWM inverters andcoupling them with magnetic field analysis.
In this example, confirming the effects of carrier harmonics on the iron loss of a six-phase IPM motor are explained here.

Operating Point to Evaluate

Loss Breakdown

The efficiency map and the operating point for evaluating this motor are shown in Fig. 1.

Hysteresis loss and Joule loss that occurs during sinusoidal current drive and PWM inverter

This evaluates harmonic iron loss at an operating point where rotation speed is 2,800

drive are shown in Fig. 3.

r/min and torque is 75 Nm. This operating point also has a current amplitude of 76.6 A and

Loss increases during PWM inverter drive for both the rotor core and the stator core. In

a current phase of 36 deg.

particular, there is a large increase in Joule loss approximately 30% more increase for both
the rotor core and stator core.

Modeling to Account for Harmonic Iron Loss
The control circuit is shown in Fig. 2.

Joule Loss/Loss Density Distributions

The control circuit is created to account for the influence of the PWM. A JMAG-RT model is

Joule loss frequency components are shown in Fig. 4 and Fig. 5 and loss density

used in the control circuit. Calculation is performed with the JMAG-RT model when

distributions are shown in Fig. 6 and Fig. 7. Note that, the vertical axis for Fig. 4 and Fig. 5

analysis begins in a transient state. Analysis time is then reduced by switching to FEA after

are deliberately narrowed to clarify the difference.

reaching a steady state.

Iron loss mainly increases in the secondary component of the PWM carrier frequency (12kHz) (the red frame in the center of the diagram). It is recognized that the increase in Jouleloss during PWM inverter drive is caused by the carrier frequency. Therefore, it is necessary
to account for the effects of carrier harmonics when evaluating detailed motor
characteristics.

Controller

Inverter

Coil (FEA)

Operating point to evaluate

Joule loss (Stator core)

Hysteresis loss (Stator core)

Joule loss (Rotor core)

Hysteresis loss (Rotor core)Sinusoidal

PWM Inverter

1,200

Loss, W

1,000
800
600
400
JMAG-RT, FEA
Switch2000JMAG-RT modelFig. 2. Control Circuit

10

50

8

40

Loss, W

Loss, W

Fig. 1. Efficiency Map and Operating Point for Evaluation

6
4

2

4

5

7

9
10 12 14Frequency, kHz
Sinusoidal current drive

15

17

19

20

0

2

4

5

50

7

9
10 12 14Frequency, kHz
Sinusoidal current drive

15

7

15

17

19

20
Joule loss density
Hysteresis loss density
Iron loss density(Sinusoidal current drive) (Sinusoidal current drive) (Sinusoidal current drive)

Joule loss density
Hysteresis loss density
Iron loss density(Sinusoidal current drive) (Sinusoidal current drive) (Sinusoidal current drive)

40
Loss, W

8
Loss, W

20
0

0

10
6
4
2
0

30
10

2
0

Fig. 3. Loss Breakdown

30
20

2

4

5

7

9
10 12 14Frequency, kHz
PWM inverter drive

15

17

19

20

Fig. 4. Rotor Core Joule Loss Frequency Component(Unit: W/m^3)

10
0

0

0

2

4

5

9
10 12 14Frequency, kHz

17

19

20

PWM inverter drive

Fig. 5. Stator Core Joule Loss Frequency Component

Joule loss density(PWM inverter drive)

Hysteresis loss density(PWM inverter drive)

Iron loss density(PWM inverter drive)

Fig. 6. Rotor Core Loss Density Distributions(Unit: W/m^3)

Joule loss density(PWM inverter drive)

Hysteresis loss density(PWM inverter drive)

Iron loss density(PWM inverter drive)

Fig. 7. Stator Core Loss Density Distributions

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Case

Case

Shaft

277 Vibration Analysis of Axial Gap-Type Motor

Stator Core

Coil

module: DS, TR
Rotor

Overview
Axial gap type- motors can be can have a thinner construction than radial gap-type motors, and are used for in-vehicle motors where spaceis limited. In general, the motor is quiet compared to the engine, however, the wide rotation speed range may at times cause the vibrationnoise to be a problem.
In order to evaluate noise/vibration caused by electromagnetic force, it is necessary to accurately obtain the source of sound radiation, thetotal eigen mode of the motor, including the cases it is linked to must be perceived and taken into account at the same time. It is effective
to use the finite element method in simulations before creating a prototype to assess the level of vibration caused by air distribution fromelectromagnetic force and eigen mode.
In this example, phenomena confirmed in axial gap-type motors that occur at maximum sound pressure level, from an electromagnetic
excitation force and eigen mode perspective.

Frame

MagnetAbove：Whole view
Below：Rotor

Frequency Characteristics of Maximum Sound Pressure Level

Spatial Mode of Electromagnetic Force Acting on Teeth

Sound pressure level target plane for stress evaluation is shown in Fig. 1 and frequency
characteristics of maximum sound pressure level is shown in Fig. 2.
As shown in Fig. 1, sound pressure level target plane for stress evaluation is on a 200 mm
diameter sphere.
As shown in Fig. 2, sound pressure level reaches maximum at 1,333 Hz.
The phenomena that occur when reaching maximum sound pressure level is confirmed
from an electromagnetic excitation force and eigen mode perspective.

The spatial mode of electromagnetic mode with large frequency component of 1,333 Hz isshown in Fig. 5. The circumferential figures represent the teeth numbers and the radial
direction figures represent the size of the electromagnetic force. Note that, the size ofmagnetic force is normalized at maximum value.
Since there are 6 peaks in one period, you can see that the sixth spatial component isdominant in the magnetic force at 1,333 Hz. As shown in Fig. 6, this is thought to bebecause the winding arrangement is made so there are six phases in the circumferential
direction.

Electromagnetic Force History Acting on Teeth / Frequency
Component
Electromagnetic force history acting on stator teeth wrapped with U phase coil is shownin Fig. 3 and frequency components are shown in Fig. 4.
In Fig. 3, you can see that the historical waveforms of electromagnetic force are not thesame even for teeth wrapped in the same phase coil. Note that, the waveform for teeth
wrapped in V phase or W phase coils will be out-of-phase.
In Fig. 4, you can see that the electric angle secondary component 1,333 Hz, is dominant
in both teeth. This is because two rotor magnetic poles pass through in one electricalangle period.

Acceleration Distribution
The acceleration distribution by frequency at maximum noise pressure level is shown in
Fig. 7.
You can see that acceleration of the case is larger in the area connecting to the stator.
Note that, spatial distribution of acceleration is in spatial 2nd order and is different fromspatial distribution of electromagnetic force.

Eigen Mode
Eigen modes that are similar to acceleration distribution and change in spatial 6th order
are shown in Fig. 8.
Electromagnetic force distribution is in spatial 6th order, but the natural frequency of
eigen mode in spatial 6th order is extremely high at 18 kHz. Therefore, the natural
frequency is thought to be greatly affected by the eigenmode of spatial 2nd order (9 kHz)
which is closer to the electromagnetic excitation force frequency.

Maximum sound pressure level, dB

70

18

2
3

V-phase

40

0.2

15

V-phaseW-phaseU-phase5

W-phaseU-phase030

V-phaseU-phase40.4

W-phaseW-phase14

U-phase6

20

W-phase
13

10

7
12

0
667

1,333 2,000 2,667 3,333 4,000 4,667 5,333 6,000

W-phaseV-phase8
11

U-phaseV-phaseU-phaseV-phase910Frequency, Hz

Fig. 1. Sound Pressure Level Measured Part

1

0.6

16

50

1
0.8

17

60

Fig. 2. Frequency Characteristics of Sound Pressure Level

Fig. 5. Spatial Mode of Electromagnetic Force

Fig. 6. Winding Wire Arrangement

T3
T1

T2
T1

T2

T3
Acceleration(axis direction)

16

Eigen Vector(axis direction)

T1

T2

Force (Axis direction), N

14
T3

Force (Axis direction), N

35
30
25
20
15
10

10
8

9,066 Hz

6
4
2

5
0

0
0

15

30
Rotor Angle, deg

45

Fig. 3. Electromagnetic Force History Acting on Teeth

170

12

JSOL Corporation

667

1333

2000

2667

3333

4000

4667

5333

6000

6667

(Unit: None)(Unit: m/s^2)
19,956 Hz

Frequency, Hz

Fig. 4. Frequency Component of Electromagnetic ForceActing on Teeth

Fig. 7. Acceleration Distribution (1,333 Hz)

Fig. 8. Eigen ModeJMAG Application Catalog

Case

Rotor core
Cage (bar)

278 Creating Six-Phase Induction Motor Eﬃciency Maps

module: FQ, LS

Overview
With advances in variable speed operation technology for induction motors, progress is being made in applications to fields where wide
operating regions are required. With this, polyphase design may be used for the purpose of enhancing redundancy. Polyphase design
includes the likes of six-phase drive using two inverters. Efficiency maps are created for both motor design and control design in order tofully comprehend the characteristics of these motors that are capable of drive within such a wide range.
It is necessary that operating region rotation speed and each loss point results are obtained as so to draw efficiency maps with actual
measurements and calculators. This means that a large amount of calculations and time to organize the results are required. In concept
design, evaluating efficiency maps within short amounts of time is also required. In JMAG-Designer, however, efficiency maps that accountfor the effect of iron loss and nonlinear magnetization characteristics can be created with ease.
In this example, a six-phase induction motor efficiency map is created in the concept design stage, and each operating point slip and ironloss are evaluated.

Shaft
Coil

Stator coreEﬃciency Map, Slip Map

Loss Ratios at Each Operating Point

The efficiency map is displayed in Fig. 2, and the slip map is displayed in Fig. 3.
Efficiency can be understood from Fig. 2.
Slip can be understood from Fig. 3. Slip with maximum efficiency at each operating point
can also be understood.

Fig. 4 displays pie charts showing loss ratios at low speed and low load, low speed andhigh load, medium speed and low load, medium speed and high load, and lastly, highspeed and low load.
It is understood that iron loss dominates more than copper loss during high rotation
speed.

W2

W1
V2
V1
U2
2
U1
U
N=750, T=10

N=750, T=120

N=5000, T=10

N=5000, T=80

N=12000, T=20

750 r/min

750 r/min

5,000 r/min

5,000 r/min

12,000 r/minNRotation speed

T

TorquePrimary copper loss
Secondary copper loss
Stator core iron loss
Mechanical loss

Fig.1 Distributed Winding

Fig. 2. Efficiency Map

Fig. 3. Slip Map

Fig. 4. Loss Ratios at Each Operating Point

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279

Thermal Analysis Accounting for Coolingof the IPM Motor

Rotor core

MagnetStator coremodule: ExpressFR

Overview
EV/HEV traction motors operate at ranges such as high torque and high power ranges. When running at such operating points, thetemperature of parts rises as the loss becomes the heat source. Cooling system design using water jackets and ATF is put into practice asnormal cooling by air allows the parts to exceed their heat resistant temperature.
It is possible to run an analysis that takes into account the cooling and the heat resistance between the parts by using the thermal
equivalent circuit.
In this example, thermal equivalent circuit is used to run thermal analysis of a 3-phase IPM motor that accounts for cooling, and check thetemperature change in the coil.

Cooling Speciﬁcations
Cooling specifications are shown in Fig. 1.
The water jacket is installed on the upper 90 deg sector of the stator.
Assume that oil in the ATF pool accumulates inside the lower 90 deg sector of stator.
Housing, coil end and shafts are modeled in the thermal equivalent circuit.

Coil

Air gap

Housing
Water jacket

Water jacket
Connection between
stator core surface
and ATF Pool
Stator core side
(Water jacket)

Stator core side

Stator core side
(ATF pool)

Housing

Thermal Equivalent Circuit
The thermal equivalent circuit is shown in Fig. 2.
The heat transfer boundary component labeled as FEM is connected to the finite element
model.
The sides of stator core are divided into three parts: the water jacket part, the ATFpool
part, and the rest of the remaining parts, which are connected to the housing and outside
air through thermal resistance component.
Core surfaces other than the above and coil ends are thermally connected to the shaft andthe housing through the air inside the housing.

Connection between housing and outside air

Coil end
Housing

Shaft
ATF pool

Coil end

ATF pool

Shaft

Coil end
Circumferential direction

Axial direction

Fig. 1. Cooling Specifications

Fig. 2. Thermal Equivalent Circuit

Center of water jacket
Tip of water jacket

Center of water jacket

Center of ATF

Tip of water jacket

Tip of ATF

130

Temperature,deg C

120

Temperature Change in CoilThe temperature evaluation point of the coil is shown in Fig. 3, the temperature change ofthe coil is shown in Fig. 4, and the temperature 15 minutes after the motor starts is shownin Table 1.
You can determine whether the temperature is within the allowable temperature range ofthe heat resistance class by checking the coil temperature.
In this case study, the ATF pool part is cooler than the water jacket part when checking thetemperature 15 minutes after the motor starts. You can see that the ATF has a higher
cooling effect.
In the ATF pool part, the end portion is cooler than the central portion. You can see thatcooling through the coil end is more effective than cooling through the core.

JSOL Corporation

100
90
80

Center of
ATF

70

Tip of
ATF

0

Fig. 3. Temperature Evaluation Point of Coil

100

200

300

400
500
Time, sec

Fig. 4. Temperature Change of Coil

Item

Center ofwater jacket

Center of ATF

Tip of water
jacketTip of ATF

Temperature(deg C)

123.5

115.4

123.4

110.5

Table 1. Coil Temperature 15 Minutes After the Motor Starts

172

110

600

700

800

900

JMAG Application Catalog

Case

280 Topology Optimization of Induction Heating Coilmodule: FQ, HT

Overview
When designing an induction heating coil, it is necessary to investigate both the heating time and the coil shape in order to uniformly heat
the workpiece. Such investigations are run through a process of trial and error based on both the knowledge and experience of skilled
engineers, as well as experiments with actual machines.
Optimization using FEA is effective in reducing prototyping costs and reducing the construction period. In parameter optimization using
design variables and dimension ranges, although knowledge and experience is required to determine parameters, because searches can be
performed for topology optimization without depending on the initial shape, design proposals with a high degree of freedom can beobtained.
In this example, magnetic field and thermal coupled analysis is run, and using topology optimization, searches are performed for a heating
coil that maximizes the power factor and minimizes the apparent power.

CoilWorkpiece

Optimization conditions

Sensitivity Analysis

The design region is shown in Fig. 1, the constraint conditions are shown in Table 1, andthe objective functions are shown in Table 2.
Define the following four points in the objective functions.
Power factor maximization
Apparent power minimization
Minimizing the standard deviation Tdev from the target temperature T̲i using theevaluation point temperature T̲i (Fig. 2)
Minimizing the bias Tbias from the target temperature T̲i using the evaluation pointtemperature T̲i (Fig. 2)
The temperature evaluation points are shown in Fig. 2.

In order to confirm the effect that the wavy coil shape on the surface facing the workpiecehas on the temperature distribution of the workpiece, sensitivity analysis is run for thebest case.
Fig. 5 shows Optimum solution 1 which is the case where the temperature standard
deviation Tdev is the minimum value, as well as the temperature distribution of the casewhere the coil surface facing the workpiece is changed to a shape that is near to a straightline.
In the case where the coil surface is changed to a shape that is near to a straight line, theupper part of the workpiece is heated first. The workpiece surface temperature is not
uniform even if heat transfer occurs within the workpiece. Conversely, in Optimum
solution 1, multiple points where the coil is asymptotic to the workpiece are heated first.
The workpiece surface temperature then becomes uniform due to the heat transfer. It is
understood that the wavy shape of the coil surface is important for uniform temperature.

Optimization Results
The correlation diagram between the coil power factor and the temperature standard
deviation Tdev is show in Fig. 3.
The case with the minimum temperature standard deviation Tdev is Optimum solution 1,
the case with the maximum power factor is Optimum solution 3, and the one casebetween Optimum solution 1 and Optimum solution 3 is Optimum solution 2.
The topology optimization shapes for optimum solutions through 1 to 3, and thetemperature distribution contours during heating at 7.0 sec are both shown in Fig. 4.
In each optimum solution, the coil surface facing the workpiece has a wavy shape,whereas the lower part of the coil has a shape that follows along the workpiece. It is
understood that the workpiece surface is heated evenly in each case.

Item

Objective

Workpiece surface minimum temperatureHT̲min900 deg C or more

Workpiece surface maximum temperatureHT̲maxItem

Expression

Objective

Power factor
PF

ActiveP1/ActiveP3

Maximize

Apparent power
ActiveP3

ActiveP3

Minimize

Standard deviation
Tdev

Minimize

Bias
Tbias

Minimize●:
Evaluation
points36 divisions

1300 deg C or less
ActiveP1: Active power
ActiveP3: Apparent powerm : Number of evaluation points: evaluation point temperature: target temperature = 1000(deg C): standard deviation: bias■：Design region

Fig. 1. Design region

Enlarged diagram
6 divisions
6 divisions

Table 1. Objective functions

Table 1. Constraint conditions

Fig. 2. Temperature evaluation points

Optimum solution 1 (Standard deviation minimization)
Optimum solution 2
Optimum solution 3 (Power factor maximization)
0.12
0.1

Tdev

0.08
0.06
0.04

3.5 sec

00.35

7.0 sec

3.5 sec

7.0 sec(Unit: deg C)(Unit: deg C)

0.02

7.0 sec
0.37

0.39

0.41

0.43

Power Factor

Fig. 3. Power factor vs temperature standard
deviation Tdev

0.45

7.0 sec

7.0 sec● : Optimum solution 1 ● : Optimum solution 2 ● Optimum solution 3(Standard deviation(Power factor
minimization)
maximization)

Fig. 4. Optimum solution temperature distribution

Optimum solution 1(Standard deviation minimization)

The coil surface facing
the workpiece is changed
to a shape nearing a straight line

Fig. 5. Sensitivity analysis temperature distribution

https://www.jmag-international.com/

173

Case

284

Analysis of Minute Electromagnetic Forces Acting
on a Magnet

Fixed

module: TR

Overview
With the miniaturization of electrical equipment, the amount of equipment that uses minute electromagnetic forces is increasing. Therefore,
it is important to understand the minute electromagnetic forces that act between parts, such as between multiple magnets.
With magnetic field analysis, you can estimate minute electromagnetic forces at a high accuracy.
In this example, we obtain the minute electromagnetic forces acting on a magnet that is moving translationally.

Electromagnetic Forces Acting on the Magnet

Translation
MovableElectromagnetic force ,μN

150

The electromagnetic force acting on the movable magnet in the direction that it moves in
is shown in Fig. 1.
The movable magnet is moving translationally away from the fixed magnets. As a result,as the distance moved increases, the electromagnetic forces acting on the movablemagnet decrease. You can check these trends even with minute electromagnetic forces.

120

90

60

30

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Displacement, mm

Fig. 1. Electromagnetic forces acting on the magnet

Case

285

Integrated Magnetization Analysis of an SPM Motorby Magnetizing Multiple Times

Rotor core
Stator core
Magnetmodule: DP

Overview
Permanent Magnet Synchronous motors (PMSM) sometimes use a method known as integrated magnetization. This is where magnets areassembled without magnetization then magnetized after the assembly. Depending on the combination of the number of poles and slots, themagnets may not be sufficiently magnetized by just one magnetization. The magnets are then magnetized more than once by rotating therotor or changing the current values. In situations like these, the design must prevent the irreversible demagnetization of the magnets thathave been previously magnetized.
By using a magnetic field analysis simulation which uses the finite element method, you can check the magnetization distribution that occurs
in the magnets as the result of the magnetization conditions. By accounting for magnetic hysteresis during this process, the magnetization
distribution in the magnets can be evaluated at a higher accuracy.
In this example, the magnets are magnetized more than once with the rotor at different positions to evaluate the magnetization distribution
in the magnets while also accounting for hysteresis.

Coil

Shaft

Magnetization Field Distribution

Magnetization Distribution of the Magnet Positioned in the Air

The magnetization field distribution and the magnetic flux line diagram when themagnetization current is 1 kA are shown in Fig. 1.
The magnets are evaluated for the 1st and 2nd magnetizations. In the analysis, themagnets are evaluated as being arranged in the same positions. In the diagram, the statorside is shown as having been rotated. Note that the magnets are arranged in a radiallyoutward orientation direction.
In the 1st magnetization, the magnetic fields are weak in the center of the magnet
because the center of the magnetic pole faces the slot opening part. To ensure that themagnets are fully magnetized, for the 2nd magnetization, the magnetic pole instead faces
the center of the tooth. Looking at the magnetization distribution of the 2nd
magnetization, we see this occurring in the opposite direction of the orientation direction
at the ends of the magnet.

The magnetization distribution (radial component) of the magnet positioned in the airwhen the 1st and 2nd magnetizations end respectively is shown in Fig. 2.
Looking at the magnetization distribution when the 1st magnetization ends, weunderstand that the magnetization is weaker at the center of the magnetic pole. Looking
at the magnetization distribution when the 2nd magnetization ends, the magnetization
has increased at the center of the magnetic pole, but compared to the 1st magnetization,
magnetization has reduced at the ends of the magnet. From this we understand that
irreversible demagnetization has occurred. As shown by the magnetization field
distribution in Fig. 1, this is because magnetic fields are generated in the opposite
direction in the 2nd magnetization.

1st magnetization

1st magnetization(Unit: A/m)
(Unit: A/m)
2nd magnetization

2nd magnetization

Fig. 1. Magnetization field distribution andmagnetic flux diagram

174

JSOL Corporation

Fig. 2. Magnetization field distribution andmagnetic flux diagram

JMAG Application Catalog

IPM Motor Iron Loss Analysis Accounting for PWM: Evaluation with Diﬀerent Modulation Methods

Case

286

Stator core
MagnetRotor coremodule: DP, LS

Overview
PWM control is widely used in the control of IPM motors. Because currents include carrier harmonics, this results in an increase in iron loss.
Many different types of control methods are available. Modulation methods are selected according to the objective and application of thecontrol. The carrier harmonic components of a current will vary depending on the modulation method. Assessing iron loss is essential in
improving the efficiency of IPM motors, and the modulation method must also be accounted for in order to evaluate the iron loss accurately.
By using a magnetic field analysis simulation that accounts for control, the iron loss accounting for the PWM control carrier harmonics canbe assessed.
In this example, the effects on iron loss when the modulation method for PWM control is changed are evaluated.

Coil

Current Waveforms

Iron Loss

The current waveforms for the PWM 3-phase modulation and 2-phase modulation are
shown in Fig. 1, and the frequency components of the current are shown in Fig. 2.
In this example, we compare the components at approximately 6,000 Hz, which is thecarrier frequency of the PWM inverter. The carrier amplitude of the 2-phase modulation islarger than that of the 3-phase modulation.

The iron losses for the PWM 3-phase modulation and 2-phase modulation are shown in
Fig. 3.
The iron loss of the 2-phase modulation increases compared to that of the 3-phasemodulation. This may be due to the carrier amplitude of the 2-phase modulation current,
which is larger than that of the 3-phase modulation.
In general, switching loss is smaller with the 2-phase modulation. However, looking at theanalysis results, we see that the iron loss of the 2-phase modulation increases. Whenconsidering overall system efficiency, the effect that different modulation methods haveon iron loss must be obtained correctly.

3-phase modulation

2-phase modulation

3-phase modulation30

2-phase modulation

2

200

1.8

175

1.6

0

150

1.2
-30

Iron loss, W

Current, A

1.4

1
0.8

25

0.2
0.03

0.031
Time, sec

Fig. 1. Current waveforms

287

75
50

0.4

Case

100

0.6

-60

-90
0.029

125

0.032

0.033

0
5,200

5,600

6,000

6,400

6,800

0

7,200

3-phase modulation

2-phase modulation

Frequency, Hz

Fig. 2. Frequency components

Fig. 3. Iron loss

Analysis of Current Conditions for the Maximum
Eﬃciency of a WFSM

Shaft
Armature coil
Rotor
core

module: DP, LS

Overview
Wound-field synchronous motors (hereinafter referred to as WFSM) possess the capability of varying field magnetomotive force and feature
characteristics where high efficiency can easily be achieved in the mid-to-high speed and low torque range. In recent years, one area ofapplying WFSM is in traction motors for electric vehicles, where there is demand for improved motor efficiency.
In designing high-efficiency motors, it is important to understand the combination of field current and armature current that can achieve
maximum efficiency. However, understanding these by using actual measurements requires great effort. On the contrary, by using
simulations, these can be understood with ease in a short amount of time.
In this example, we determine the motor efficiency in relation to the field current for a WFSM when the motor output is kept constant.

Stator core

Field coil

Relationship Between Field Current, Motor Eﬃciency,
and Loss Breakdown
Copper loss in the armature

95

Copper loss in the ﬁeld

Iron loss

350
300
90
Motor Losses,W

Motor Eﬃciency, %The motor efficiency in relation to the field current when the rotation speed is 600 r/min
and the torque is 28 Nm is shown in Fig. 1, and the loss breakdown for when the fieldcurrent is 3, 6, and 10 A are shown in Fig. 2.
From Fig. 1, motor efficiency is reaching maximum when the field current is 6 A. From Fig.
2, it can be seen that the copper loss is taking a large proportion in the motor loss. When
reducing the field current to less than 6 A, copper loss of the armature increases andefficiency goes down since it is necessary to increase the current amplitude of thearmature in order to maintain the same output. On the other hand, when increasing thefield current to over 6 A, the efficiency goes down likewise since the copper loss of the
field increases.
From these results, it can be concluded that determining the optimal combination of thearmature and field currents is important in order to achieve maximum efficiency in the
WFSM. Moreover, by using simulations, it is possible to comprehend the details of the loss
which is a determinant of motor efficiency.

85

250
200
150
100
50

80
0

5

10
Field Current, A

Fig. 1. Motor efficiency in relation to field current

15

0
3

6
Field Current, A

10

Fig. 2. Loss breakdown in relation to field current

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175

Case

Synchronous Reluctance Motor Topology Optimization

288

Coil
Rotor coremodule: DP

Overview
The synchronous reluctance motor is a low-cost, highly efficient variable speed motor because it does not use permanent magnets or
secondary conductors. Since itsʼ torque characteristics are particularly affected by the flux barrier shape, optimization with the flux barrier
shape as a design variable is most effective in improving torque characteristics.
Topology optimization allows for bold changes in material arrangement when exploring the design region, and is an effective method forgenerating concepts at the early stages of design when there are few geometry constraints.
In this example, topology optimization is used to minimize the torque ripple and maximize the average torque when exploring the rotor
geometry.

Stator core

Optimization Condition

q-axis InductanceFig. 1 shows the design region and Table 1 shows the evaluation items and objectivefunctions. Fig. 1 shows that the rotor perimeter and area near the shaft are excluded fromthe design region with consideration for the actual manufacturing of the motor.

Fig. 2 shows the transitions in the torque average values and torque ripple rates for thebest case of each generation. Fig. 3 shows the rotor core geometry in conventional design
and the best case throughout all generations, with various quantities shown in Table 2.
As the generation progresses, it can be seen that the torque average value grows larger
and the torque ripple rate becomes smaller as they converge.
It is understood that in the best case, a comparison to the conventional design shows thatthe average torque is raised by 4 % and the torque ripple rate is lowered by 70 %.

Fig. 5 shows the waveform of the q-axis inductance. Fig. 6 shows the magnetic flux line
near the q-axis for each timing with Position A (0 deg) as the timing when magnetic flux
flows most easily in the q-axis direction and with Position B (20 deg) as the timing whenmagnetic flux is least likely to flow in the q-axis direction in a conventional design.
In Fig. 5, it can be seen that in the best case the change in q-axis inductance is small
compared to conventional designs.
In Fig. 6, it can be seen that magnetic flux flows easily when the core on the q-axis isfacing the teeth and does not flow well when it is not facing the teeth. In the best case,compared to the conventional design, the flux barrier generated on the q-axis makes it
difficult for the magnetic flux to flow easily when the q-axis is facing the teeth and makes
it flow easily when it is not facing the teeth. It can be seen that the change is kept small
and this reduces the torque ripple.

Reluctance Torqued-axis InductanceFig. 4 shows the torque formula for reluctance torque.
In a reluctance torque, the average value is larger when the difference between the d-axis
inductance and q-axis inductance is large, and the ripple decreases as the change in
difference is small.

Fig. 7 shows the waveform for the d-axis inductance. Fig. 8 shows the magnetic flux lines
near the d-axis with Position C (22.5 deg) as the timing when magnetic flux flows mosteasily in the d-axis direction in a conventional design.
In Fig. 7, it can be seen that the d-axis inductance is smaller in the best case compared tothe conventional design.
In Fig. 8, it can be seen that the magnetic flux flowing in the d-axis direction is suppressedby the flux barrier which extends in the q-axis direction in the best case. This increases thedifference between the d-axis and q-axis inductance, resulting in a higher average torque.

Optimization Results

Conventional design : Lq

Best case : Lq

1.20E-01

Objective function
Maximize the torque average value

Torque ripple

Minimize the torque ripple rate

1.19E-01
Lq, H

Evaluation item
Torque average value

Conventional design (Left: Position A, Right: Position B)
1.18E-01
A
1.17E-01

0

B

5

10

： Design Region

20

25

30

Rotor position, deg

Fig. 1. Design region

Table 1. Evaluation items and objective functions

Torque average value

15

Fig. 5. q-axis inductance

Torque ripple rate

Fig. 6. Magnetic flux lines near the q-axis

Conventional design : Ld

1.2

30

1

25

0.8

20

0.6

15

0.4

10

0.2

5

q-axisBest case (Left: Position A, Right: Position B)

d-axisBest case : Ld

5.20E-02

0

0

200

400
600
Generation

0
1000

800

Fig. 2. Transitions in torque average value andtorque ripple rate

Ld, H

Torque ripple rate, %Torque average value, Nm

q-axis
5.00E-02

4.60E-02

Conventional design

Best case

Conventional design

Best case

Average torque, Nm

0.93

0.97

Torque ripple rate, %

21.2

6.4

Table 2. Various evaluation value quantities

176

JSOL Corporation

4.40E-02

C

0

5

10

15

20

Rotor position, deg

Fig. 3. Geometry of rotor core

Tr = Pn (Ld –Lq) Id Iq
Item

4.80E-02

Tr ： reluctance torque, Nm
Pn ： pole logarithm
Ld ： d-axis inductance, H
Lq ： q-axis inductance, H
Id ： d-axis current, A
Iq ： q-axis current, A

Fig. 4. Torque FormulaFig. 7. d-axis inductance25

30
Conventional design (Position C)

Best Case (Position C)

Fig. 8. Magnetic flux lines near the d-axis

JMAG Application Catalog

Case

289

Eﬃciency Map Creation for Wound-Field
Synchronous Motors

Field coil

Rotor coremodule: DP, LS

Overview
Wound-field synchronous motors are gaining attention as traction motors for electric vehicles that require a wide drive range because thefield can be strengthened or weakened simply by adjusting the current. In motor design and control design, efficiency maps are created tounderstand the characteristics of motors that are driven over a wide range.
A wound-field synchronous motor has many different combinations of currents to obtain the same torque, and from within thosecombinations, the most efficient (lowest loss) combinations must be found.
When using FEA, you can account for detailed geometry and also run calculations for multiple operating points at the same time. You canobtain efficiency maps easily by also automating the results processing.
In this example, efficiency maps for a wound-field synchronous motor are created to check the combinations of currents that result in
maximum efficiency. In addition, the loss breakdowns are also evaluated for the representative operating points.

Shaft
Armature coil

Stator coreEﬃciency Map

Loss Breakdowns at Each Operating Point

Fig. 1 shows the efficiency map.
It can be seen that for this motor, the maximum torque is 280 Nm, the maximum rotation
speed is 12,000 rpm, and the maximum efficiency is 95 %.

Fig. 8 displays pie charts showing the loss breakdowns at low speed and high load (2,000
r/min, 150 Nm), low speed and medium load (2,000 r/min, 30 Nm), medium speed andmedium load (8,000 r/min, 30 Nm), and lastly, high speed and medium load (12,000 r/min,
30 Nm).
The DC copper loss breakdown for the stator core is large at low speed and high load. The
AC copper loss breakdown for the stator core is large at medium speed and medium load,
as well as high speed and medium load. Among the iron losses for the stator core, theeddy current loss in particular increases with rotation speed, but at 12,000 r/min, the ACcopper loss is more dominant. On the contrary, it can be seen that the breakdown for theentirety of the rotor core loss is small.

Loss and Current Maps
Fig. 2 shows the iron loss map, Fig. 3 shows the field coil copper loss map, Fig. 4 shows thestator copper loss map, Fig. 5 shows the field current map, Field. 6 shows the current
amplitude map, and Fig. 7 shows the current phase map.

Fig. 1. Efficiency map

Fig. 2. Iron loss map

Fig. 5. Field current map

Fig. 6. Current amplitude map

N
T

Fig. 3. Field coil copper loss map

Fig. 4. Stator copper loss map

Fig. 7. Current phase map

Rotation speed
TorqueDC copper loss

Mechanical loss

AC copper loss

Field coil copper loss

Stator core PWM eddy current loss

Rotor core PWM eddy current loss

Stator core eddy current loss

Rotor core eddy current loss

Stator core hysteresis loss

Rotor core hysteresis loss

Fig. 8. Loss breakdowns at each operating pointhttps://www.jmag-international.com/

177

Design Exploration of IPM Motors,
Including Evaluating Part Temperature and Stress

Case

290

Stator core
Magnetmodule: ExpressFR

Overview
In the initial stages of motor design, if only the magnetic design is completed and the design proceeds to the later stages, rework may occur
when it is discovered that the part temperature and stress do not meet the requirements. Such rework can be reduced by running
multifaceted evaluations from the initial design stages, including not only magnetic design but also thermal and stress.
By using the templates included in JMAG-Express, you can create motor models and evaluate design plans both quickly and easily. In
addition, by combining the prepared scenarios, you can run magnetic design, thermal design, and structural design all at the same time.
In this example, the part temperature and stress are evaluated at the same time for the motor design plan, and a design plan that meets therequirements is explored.

Coil
Rotor core

Design Requirements, Design Variables

Design Plan Adjustments

Table 1 shows the design requirements, Fig. 1 shows the rotor core and geometry that are
the design variables, and Table 2 shows the cooling specifications.
The rotor core slit depth and the cooling jacket flow velocity are also design variables.

The design variables are adjusted based on the results of the initial design plan.
In this example, the evaluation is run again with the rotor core slit depth changed to 1.5mm and the cooling jacket flow velocity changed to 2 msec.
Fig. 5, Fig. 6, and Fig. 7 show the obtained efficiency map, average temperature of eachpart, and maximum stress position, respectively. Table 4 shows the maximum stress value.
From Fig. 5, it can be seen that the maximum torque exceeds the requirement of 280 Nm
and the efficiency at the rated point exceeds the requirement of 95 %.
Fig. 6 shows that the coil temperature is 130 deg C or less, and Fig. 7 and Table 4 showthat the maximum stress occurs at the bridge, where it is 200 MPa or less. From this, it canbe seen that the design plan meets the requirements after the adjustments have beenmade.

Initial Design Plan Evaluation
Fig. 2, Fig. 3, and Fig. 4 show the efficiency map, average temperature of each part, andmaximum stress position, respectively, for a rotor core slit depth of 0.5 mm and a coolingjacket flow velocity of 0.5 msec. Table 3 shows the maximum stress value.
From Fig. 2, it can be seen that the maximum torque exceeds the requirement of 280 Nm
and the efficiency at the rated point exceeds the requirement of 95%.
On the contrary, Fig. 3 shows that the coil temperature is 130 deg C, and Fig. 4 and Table 3
show that the maximum stress occurs at the bridge where it exceeds 200 MPa. This
indicates that adjustments are necessary in terms of the temperature and stress.

MagnetCoil

Rated point

Objective

Maximum torque

280 Nm or more

Eﬃciency at rated point

95 % or more

Coil temperature (*1)

130 deg C or lessMagnet temperature (*1)

100 deg C or lessvon Mises stress

200 MPa or lessRotor coreslit depth

Average temperature, deg C

150
Evaluation item

140
130
120
110
100
90
80
70

*1 1,800 sec during continuous operation at rated point

Table 1. Design requirements

60

Fig. 1. Design variables

Fig. 5. Efficiency map (Adjusted design plan)

0

200

400

600

800 1000 1200 1400 1600 1800
Time,sec

Fig. 6. Average temperature of each part
(Adjusted design plan)Max
Rated point

Item

Value

Housing cooling method

Cooling Jacket

Coolant Type

Water

Table 2. Cooling specifications

MagnetFig. 2. Efficiency map (Initial design plan)

Coil

Average temperature, deg C

150
140
130
120
110
100
90
80
70
60

0

200

400

600

Fig. 3. Average temperature of each part
(Initial design plan)

Item

Value

Maximum stress value

220 MPa

Table 3. Maximum stress value (Initial design plan)

178

Max

800 1000 1200 1400 1600 1800
Time,sec

JSOL Corporation

Fig. 4. Maximum stress position(Initial design plan)

Fig. 7. Maximum stress position (Adjusted design plan)

Item

Value

Maximum stress value

175 MPa

Table 4. Maximum stress value (Adjusted design plan)JMAG Application Catalog

Case

291

Control Simulation of Switching Number of Poles
in a 6-Phase Induction Machine

Shaft

module: DP, RT

Overview

Multiphase design may be used in drive motors to increase high power density, enhance redundancy during operation and reduce torqueripple. In a cage inductance motor, the number of poles is determined by the rotating magnetic ﬁeld generated by the stator side. A wide
range of motors with good characteristics can be built by switching the number of poles at both the low and high-speed sides. For practicaluse, we want to minimize the change in torque when switching the number of poles.
To accurately evaluate the behavior of switching the number of poles during a simulation, the motor model to be controlled must also behighly accurate. The JMAG-RT model is a high-accuracy behavior model created by FEA-based calculations that can express motor
characteristics that account for magnetic saturation and slip dependency.
In this document, the behavior of a 6-phase induction motor during the switching of the number of poles is evaluated by simulation.

Control Circuit

The control circuit is shown in Fig. 1.
A controller and inverter are arranged for both the 4-pole and 8-pole drive. Use only the
8-pole drive for the time between 0 to 4 sec. For each current command value between 4
and 6 sec, decrease the values for the 8-pole drive and increase the values for the 4-pole
drive. After 6 sec and beyond, use only the 4-pole drive.
The switching pattern is shown in Fig. 2. The linear and exponential switching patterns areevaluated here.

Coil Current Waveform (Steady State)

The coil waveforms in the steady state for both the 8-pole drive and 4-pole drive areshown in Fig. 3 and Fig. 4.
In an 8-pole drive, the phase diﬀerence is oﬀ by 120 deg, similar to a 3-phase sine wave,
and in a 4-pole drive, the phase diﬀerence is oﬀ by 60 deg, similar to a 6-phase sine wave.

Command Value
Coeﬃcient

Motor

1.2
1
0.8
0.6
0.4
0.2
0

4
5
6
Time, sec
Switching Pattern (Linear)

7

8

5
6
7
Time, sec
Switching Pattern (Exponential)

8

D-PhaseE-PhaseF-PhaseA-Phase
120
100

80

80

60

60

40

40

20
0
-20

3

4

B-PhaseC-Phase

D-PhaseE-Phase

-60

-80

-80

1.8205 1.821

1.8215 1.822

1.8225 1.823

1.8235 1.824

Time, sec

Fig. 3. Coil current waveform (8-pole drive)

-120
6.787

6

8

2

4

6

Im command value
It command value

Im
It

8

Time, sec

180
160
140
120
100
80
60
40
20
0
-20

2

4

6
Time, sec

Im command value
It command value

Im
It

8

180
160
140
120
100
80
60
40
20
0
-20

2

4

6

8

Time, sec

Fig. 6. Current command value and current waveform(Switching pattern: exponential)

Switching pattern (Exponential)

100

80

60

40

20

-100

-100

4

Switching pattern (Linear)

F-Phase0-60

2

Im command value
It command value

Im
It
180
160
140
120
100
80
60
40
20
0
-20

Fig. 5. Current command value and current waveform(Switching pattern: linear)

20

-40

Im command value
It command value

Time, sec
2

-20

-40

-120
1.82

8-pole

180
160
140
120
100
80
60
40
20
0
-20

Current, A

3

Torque, Nm

C-PhaseCurrent, A

Current, A

B-Phase100

The torque waveform is shown in Fig. 7.
There is a large ﬂuctuation when the number of poles are switched and you can see thatthe ﬂuctuation is smaller when the switching pattern is linear. However, for practical use, itis necessary to design a system that can further reduce this ﬂuctuation.

8-poleFig. 2. Switching pattern

120

Torque Waveform

Current, A

2

Inverter (8-pole)Fig. 1. Control circuit

The current command value and current waveform for a linear switching pattern is shownin Fig. 5 and the current command value and current waveform for an exponential
switching pattern is shown in Fig. 6.
It can be seen that, the current waveform follows the current command value in eachswitching pattern.

ImIt

4-poleController (8-pole)

Current Waveform on the Controller Side

Current, A

Command Value
Coeﬃcient

Inverter (4-pole)

1.2
1
0.8
0.6
0.4
0.2
0

Cage (Bar)
Stator Core

Current, A

4-poleController (4-pole)

A-PhaseCage (End Ring)

Rotor Core

6.789

6.791

6.793

Time, sec

Fig. 4. Coil current waveform (4-pole drive)

6.795

0

2

4

6

8
Time, sec

10

12

14

Fig. 7. Torque waveform

https://www.jmag-international.com/

179For more information about JMAG, visit : https://www.jmag-international.com*The names of products and services described herein are the trademarks or registered trademarks of the respective owners.

JMAG Business Company
JSOL Corporation
KUDAN-KAIKAN TERRACE 11th Floor
1-6-5,Kudanminami,Chiyoda-ku,Tokyo 102-0074, Japan
TEL +81(0)3-6261-7361
E-mail : info@jmag-international.com