Loss Minimization Design
Using Magnetic Equivalent Circuit
for a Permanent Magnet Synchronous Motor
Daisuke Sato

Jun-ichi Itoh

Department of Electrical Engineering
Nagaoka University of Technology
Nagaoka, Niigata, Japandsato@stn.nagaokaut.ac.jp

Department of Electrical Engineering
Nagaoka University of Technology
Nagaoka, Niigata, Japan
itoh@vos.nagaokaut.ac.jp

Abstract— This paper proposes a simple magnetic
equivalent circuit using permeance method in order todesign rough configurations of the lowest loss motor. First,
the iron loss is calculated by the proposed method andcompared with the analysis result by finite element method(FEM). As a result, the error of the calculation results
between the proposed method and the FEM is 2.9%. Next,
the motor losses are calculated by the proposed method
when the several specifications are changed. As a result,these results agree in principle with that of the FEM.
Therefore, the validity of the proposed method is confirmed.
In addition, the method of the motor design is considered interms of the loss.
Keywords— Permanent magnet synchronous motor, Lossminimization design, Permeance method, Finite element
methodI.

INTRODUCTION

Recently, the permanent magnet synchronous motor(PMSM) is actively researched in order to achieve thehigh efficiency in the motor drive system. In addition, the
PMSM is not only high efficiency but also smaller than
the induction motor. Therefore, the PMSM is applied tothe electric vehicle and the home electric appliance [1-4].
The analysis of the loss, the torque and the backelectromotive force are important in terms of further
improvement of the motor. Generally, the characteristicsof the motor are analyzed by the finite element method(FEM) [5-7]. However, the FEM spends much analysis
time. Moreover, the development of the motor model isrequired as often as modifying the configuration of the
motor. Hence, the FEM is not suitable for the calculationof the optimized solution such as the maximum torquedensity or the lowest loss point because much time oftrial and error is needed to find the optimization point.
On the other hand, the permeance method is used inorder to design the motor simply in the past. In thepermeance method, a magnetic circuit is replaced with anelectric circuit. In this method, the magnetic flux and themagnetomotive force can be expressed as the current andthe voltage. Therefore, this method is suitable for the
optimized solution because of shorter calculation time,
although the calculation accuracy of the permeancemethod is less than that of the FEM. This is because themagnetic circuit of the motor is primarily determined as
the distributed constant circuit.
The permeance network method has been proposed as
one of the motor characteristic analysis using thepermeance method [8-11]. This method expresses thestator, the rotor and the air gap on a magnetic circuit. The
torque and the back electromotive force can be calculated
by calculating the magnetic flux in the air gap from themagnetic circuit. However, the magnetic resistance of the
air gap depends on a position of the magnetic pole.
Therefore, the construction of the magnetic circuit iscomplex. On the other hand, the method, whichcalculates eddy current loss in the permanent magnet, hasbeen proposed [12]. In this method, the eddy current loss
can be accurately calculated in comparison with the FEM.
However, the calculation process is complicated. In
addition, the equation is changed when the shape of the
motor is changed.
Generally, first, rough configurations such as size,pole number, teeth width are decided in the design of
IPMSM. After that, the details such as flux barrier are
decided. Therefore, it is considered that the simple loss
calculation is enough in the phase of rough design.
In this paper, the simple magnetic equivalent circuit
using permeance method is proposed in order to design
the rough configurations of the lowest loss motor. The
characteristic of the equivalent circuit is separation
between rotor and stator. Therefore, the complex
calculations are not needed because the magnetic
resistance of air gap can become constant. In addition, if
the shape of motor is changed, the magnetic resistanceand the magnetomotive force are changed only. Thus, thecircuit is not changed.
First, this paper presents the calculation method offlux density in stator core. Next, the loss of IPMSM is
calculated and the result is compared with the analysis
result of FEM and the measurement result. Finally, thelowest loss motor is designed by the loss calculation in
the changing parameter of IPMSM

II.

MAGNETIC EQUIVALENT CIRCUIT OF PMSM

Generally, in the permeance method, the magnetic
circuit is replaced with an equivalent electric direct
current circuit. However, it takes the time variation of themagnetic flux density to calculate the iron loss.
Therefore, the conventional permeance method is notapplied directly. In addition, it is difficult to express as
one circuit because the rotor is rolling and the stator is
not rolling. Therefore, the magnetic flux in the air gap iscalculated by the equivalent circuit of the rotor. Next, thetime variation of the magnetic flux density in the teeth
and yoke are calculated by the equivalent circuit of thestator. It is noted that the input of the stator circuit is themagnetic flux in the air gap.
Fig. 1 shows the model of the IPMSM and Table I
shows the parameters of the IPMSM. The motor is theconcentrated winding IPMSM. In addition, the permanent
magnets are magnetized in parallel and mounted in the
motor. Furthermore, the motor has periodicity every 60
degrees.
Fig. 2 shows the equivalent magnetic circuit of therotor. It is noted that the equivalent circuit comprises of
the half of one pole region. This is because the region of
30 degrees is able to be regarded as symmetry. The
magnetic resistance R is given by (1).

R=

l
μS

(1)

It is noted that μ is the magnetic permeability, S is the
cross section of the material, and l is the length of thematerial. The magnetic resistance of the rotor core is nottaken into consideration because it is sufficiently smaller
than the magnetic resistance of the magnet. In addition,
the magnetomotive force of the magnet Fm is expressedasFm = H e l m

TABLE IPARAMETERS OF IPMSM.

Core
MagnetNumber of poles
Number of slots
Coil turns per teeth
Outer diameter of stator
Inner diameter of stator
Outer diameter of rotor
Inner diameter of rotor
Air gap length
Iron stack length
Magnet dimensions

35H300 / Nippon SteelNMX-41SH / Hitachi Steel
1218
11
100 mm
66 mm
64 mm
25 mm
1 mm
50 mm
12 .5 mm × 25 mm × 4 mm

Fig. 2 Magnetic equivalent circuit of rotor. The circuit consists of
the half of one pole region. In this circuit, the flux in air gap iscalculated.

(2)

where He is the coercive force of the magnet and lm is thelength of the magnet. Further, the magnetic saturation isexpressed by the magnetic resistance Rsat which varies by
the magnetomotive force because the magnetic saturation
occurs in the area between the magnet and the outer ofthe rotor. Thus, the magnetic flux in the air gap iscalculated.
Fig. 3 shows the equivalent magnetic circuit of thestator. This circuit comprises the magnetic resistance of
the air gap, the teeth, the yoke and some magnetomotive
forces. The magnetomotive force by the flowing current
to the armature winding Fi is expressed as

Fi = NI m sin(ωt + θ )Fig. 1 Concentrated winding IPMSM. The magnets are magnetizedin parallel. The motor has periodicity every 60 degrees.

(3)

where N is the number of winding turns per the teeth,
Imsin( ωt+θ) is the instantaneous current. In addition, the

Ry11

Ry12

Ry21

Ry22

Ry31

Rt1

Rt2

Rt3

Fi1

Fi2

Fi3

Ry32

Rg11

Rg12

Rg21

Rg22

Rg31

Rg32

Fg11

Fg12

Fg21

Fg22

Fg31

Fg32

Fig. 3 Magnetic equivalent circuit of stator. The AC magnetomotive
force Fg and Fi are set. The magnetic resistance is considered themagnetic saturation. In this circuit, the flux densities in teeth andyoke are calculated.

AC magnetomotive force Fg is set as the elementgenerated the magnetic flux in the air gap. As a result, theeffect of the rolling rotor can be considered.
Fig. 4 shows the waveform of magnetomotive force Fg.
it is assumed that Fg is trapezoidal waveform owing to
the layout of the magnet. The waveform of Fg isdetermined from the proportion of the magnet width andthe teeth width. The phase of the magnetomotive force φgis determined from the position of the teeth and expressed
as

φg =

2π

θp

θt

(4)

where θt is the position of the teeth and θp is the angle ofone period.
Further, it is necessary for the magnetic resistance ofthe teeth and the yoke to consider the magnetic saturation.
Therefore, the magnetic resistance depends on themagnetomotive force in the magnetic saturation region.
Accordingly, the time variation of the magnetic flux
density in the teeth and the yoke are calculated. Moreover,the iron loss in the stator core can be calculated by themagnetic flux density of the teeth and the yoke.

TABLE II CALCULATION CONDITIONS.

Armature current
Motor speed
Electrical frequency
Teeth

18 Arms
3600 r/min
360 Hz7.0 × 104 A/Wb

Yoke

5.9 ×104 A/Wb
3.4 × 106 A/Wb

Magnetic resistanceGap

CALCULATION OF MAGNETIC FLUX DENSITY AND
IRON LOSS

The magnetic flux density and the iron loss arecalculated by the permeance method. Furthermore, thecalculation results by applying the permeance method arecompared with that of the FEM. Table II lists thecalculation conditions of the permeance method. The
armature current is sinusoidal waveform.
Fig. 5 shows the FEM model of the concentrated
winding IPMSM. The magnetic flux density and the ironloss are analyzed by two-dimensional FEM using JMAG
Designer (JSOL). It is noted that the number of elements
is 6,525 and the time interval per 1 step is 2.71 μs (1024steps).
Fig. 6 shows the waveform of the magnetic flux
density calculated by the permeance method and the
FEM. Accordingly, the waveform of the teeth by thepermeance method is saturated at 1.75 T. Moreover, it issimilar to the waveform of the teeth by the FEM.
However, the peak of the magnetic flux density in theyoke by the permeance method is drastically changed.
Therefore, the waveform of the yoke by the permeancemethod is different from the waveform by the FEM.
Fig. 7 shows the harmonic analysis result of themagnetic flux density. From Fig. 7, the fundamental
component of the permeance method almost agrees withthat of the FEM. However, the error occurs at the oddorder harmonic component by the magnetomotive force.
Especially, the 3rd order and 9th order components by thepermeance method do not occur because the equivalentcircuit of the stator is the balanced three-phase circuit.
Next, the iron loss in the stator is calculated from themagnetic flux density and the iron loss curve of theFig. 5 FEM model of the concentrated winding IPMSM. The
number of elements is 6,525. Time interval per 1 step is 2.71 μs(1024 steps).

2
Magnetic flux density [T]III.

Fig. 4 Waveform of magnetomotive force. The waveform isdetermined from the proportion of the magnet width and the teeth
width.

Saturation magnetic flux
density: 1.75 T

Teeth (Per.)

Yoke (FEM)

Teeth (FEM)
1.5

1

0.5

0

Yoke (Per.)

0.001

Time [s]0.002

Fig. 6 Waveform of magnetic flux density by permeance method
and FEM. The waveform of the teeth by the permeance method issimilar to that by the FEM

material.
Fig. 8 shows the iron loss curve of the material

(35H300 / Nippon Steel). As shown in Fig. 8, the ironlosses at the required frequency are not shown. For thisreason, the iron losses at the required frequency arecalculated by the linear interpolation. As the result, it isconfirmed that the iron loss in the stator is 30.4 W andthe calculated iron loss by the FEM is 31.3 W. The error
between the permeance method and the FEM is 2.9%.
COMPARISON WITH IRON LOSS MEASUREMENTRESULT

The motor loss is measured and compared withcalculation result in order to confirm the validity of theloss calculation by the proposed method. The IPMSM istypically driven by the inverter. If the carrier frequency islow, the motor loss by harmonic component of the output
voltage should be considered. However, the loss can be
reduced by the high carrier frequency. In addition, the 3level inverter can reduce not only the harmoniccomponent of the output voltage but also the switchingloss compared with the 2-level inverter. Therefore, in thispaper, the motor loss by the carrier harmonics is ignoredon the assumption that the IPMSM is driven by the 3level inverter.
Fig. 9 shows the construction of IPMSM loss
measurement system. The parameters of the test motor
are shown Fig.1 and Table 1. The 3-level T-type neutral
point clamped inverter is applied to IPMSM loss
measurement system. In addition, the input power of themotor
ismeasured
by
the
power
meter
(WT1800/YOKOGAWA) and the shaft power iscalculated based on the measurement result of the output
torque by the torque meter (TH-2105/Ono sokki).
Fig. 10 shows the measurement result of the motor
loss at 3600 r/min. The motor loss is reduced due to theincreasing carrier frequency. The reason is that theharmonic component in the output voltage of the inverter
is reduced. However, the motor loss is almost unchanged
over 7520 Hz. Therefore, the output voltage is equivalent
to sinusoidal voltage over 7520 Hz. In this section, themotor loss at 7520 Hz is compared with the calculation
results of the proposed method and the FEM because thecarrier frequency is ignored.
Next, the iron loss PFe is separated from the motor loss
Ploss. The copper loss PCu and the mechanical loss Pm areexpressed by (5) and (6), respectively. The mechanicalloss is calculated by the empiric formula of windage loss
because it is assumed that the friction loss of bearing is
negligible small. Therefore, the iron loss is expressed by
(7).

(5)

PCu = 3Ra I 2
Pm = 8 D (l + 0 .15 )v a2

(6)

PFe = Ploss − PCu − Pm(7)

where, Ra is wire resistance per one-phase, I is root mean

Fig. 7 Harmonic component of magnetic flux density. The
fundamental component by applying the permeance method almost
agrees with that of the FEM.

2Iron loss[W/m3]

IV.

(×106)
2 kHz

1 kHz

1.5
5 kHz
10 kHz1
0.5

400 Hz100 Hz

0

0.5
1
1.5
2
Magnetic flux density [T]Fig. 8 Iron loss curve of 35H300 (Nippon Steel). The iron losses at
the required frequency are calculated by the linear interpolation.

Fig. 9 IPMSM loss measurement system. The test motor is driven
by the 3-level T-type neutral point clamped inverter. The input
power of the motor is measured by the power meter. The shaft
power is calculated based on the measurement result of the output
torque.

square value of line current, D is outer diameter of rotor, lis length of rotor and va is peripheral speed of rotor.
Fig. 11 shows the measurement and calculation resultsof the iron loss at 3600 r/min and 2.3 Nm. The error
between the calculation result by proposed method andthe analysis result by FEM is caused by the loss in therotor and the magnets. In addition, the error between the

V.

MOTOR DESIGN OF LOWEST LOSS

The motor loss is calculated by the proposed method
when the mechanical parameter is changed. Moreover,the lowest loss motor is designed based on the calculation
results.
Fig. 12 shows the flowchart of the motor design of thelowest loss. 1) Topology of magnetic equivalent circuit isdecided based on the configuration of IPMSM. 2)
Specifications of the motor are decided. 3) Magnetic
resistance and Magnetomotive force in the rotor circuit
are calculated. 4) Flux in the air gap is calculated by theequivalent circuit of the rotor. 5) Magnetomotive force is
set to the equivalent circuit of the stator from the flux inthe air gap. 6) The flux density in stator core is calculated.
7) Iron loss of the stator core is calculated by the flux
density and iron loss curve of the core. In addition,copper loss is calculated by the wire resistance and theline current. 8) After, the losses are calculated in several
specifications, the lowest loss motor is selected. In thissection, the cases of changing pole/slot number, lengthsof axial/radial direction and length of teeth are considered.

A. Changing Pole and Slot Number
If the proposed magnetic equivalent circuit is applied
to the model changing pole number, the frequency of themagnetomotive force has to be changed only. Therefore,
the reconstitution of the circuit is not required. In thissection, it is assumed that the ratio between pole number
and slot number is constant.
Fig. 13 shows the analysis models of 8 poles 12 slots
motor and 16 poles and 24 slots motor. The angles of one
period are 90 and 45 degrees, respectively. In addition,
the 12 poles 16 slots motor is also calculated and shownin Fig.1. The outer diameter of the rotor and the stator,
the lengths of the air gap and the axial direction, the turnnumber of the coil and the volume of the magnets areconstant. The area of slots is changed by the changing
slot number. Therefore, the space factor of the coil isconstant (50%) and the area of wire (i.e. the resistance ofwire) is changed. Moreover, the speed is 3600 r/min. In
terms of the output torque, the amplitude of current isadjusted so as to become 3.9 Nm in the FEM. In addition,
the amplitude of current in the proposed method isdecided by the flux in the air gap.
Fig. 14 shows the motor loss by the changing pole
number and slot number based on the constant volume.
The motor loss is standardized by the volume. Therefore,
if it is assumed that the current density and the magnetic
flux density are constant, the calculation result can also
be applied to the motor of the different volume. The
errors between the proposed method and the FEM areless than 10% in the all models. In addition, the trend ofchanging loss in the proposed method agrees with that in

140
120
Motor loss [W]analysis result and the measurement result is less than 1%.
Therefore, it is assumed that the error is the stray load
loss. Thus, the validity of the proposed method is
confirmed.

0.7 Nm
1.5 Nm
2.3 Nm

100
80
60
40
5000

7500
10000
12500
Carrier frequency [Hz]Fig. 10 Measurement result of IPMSM loss at 3600 r/min. The loss
is almost unchanged over 7520 Hz because the harmonic component
in the output voltage of the inverter is low.

Fig. 11 Measurement result and calculation result by permeancemethod and FEM of iron loss (3600 r/min, 2.3 Nm). The error
between the calculation result by proposed method and the analysis
result by FEM is caused by the loss in the rotor and the magnets.

the FEM. Furthermore, in this case, the loss of 12 poles
16 slots motor is the lowest.
The loss of the 12 poles motor is smaller than that of
the 8 poles motor. The reason is that the current can be
become small under the same output torque by increasingpole number. However, the resistance of wire increases
because the area of slot becomes small. Therefore, theloss of the 16 poles motor is larger than that of the 12
poles motor. In addition, the width of the teeth canbecome large by decreasing pole number and slot number.
Therefore, the iron loss decreases by the decreasing flux
density. On the other hand, if the pole number and slot
number are increased, the iron loss increases because thestator core is saturated magnetically.

B. Changing Lengths of Axial and Radial Direction
Fig. 15 shows the schematic of the changing radial
direction length and axial direction length. The
calculation conditions are as follows; the motor shown in
Fig. 2 is used. The volume of the motor is constant andthe axial direction length D and the radius r are changed.
The air gap length is set to 1 mm by adjusting the size ofthe rotor. The amplitude of the armature current isFig. 13 IPMSM models of changing pole number. The angles of one
period are 90 and 45 degrees, respectively. The outer diameter of therotor and the stator, the lengths of the air gap and the axial direction,
the turn number of the coil and the volume of the magnets areconstant.

Fig. 14 Motor loss at 3600 r/min and 3.9 Nm by the changing pole
number and slot number based on the constant volume. The loss isstandardized by the volume. The errors between the proposed method
and the FEM are less than 10% in the all models. The loss of 12 poles
16 slots motor is the lowest in this case.

Fig. 12 Flowchart of the motor design of the lowest loss. The
lowest loss motor can be designed by the calculating the losses inseveral specifications.

calibrated because the output torque is set to constant.
The motor speed is 3600 r/min. The copper loss and theiron loss in the stator are calculated.
Fig. 16 shows the motor loss when the axial direction
length and the radius are changed. The radius r is
standardized by the axial direction length D and the loss
is standardized by the volume of the motor. From thecalculation result by the permeance method in Fig. 15,
the motor loss decreases when the size is enlarged to theradial direction. In addition, the analysis results by the
FEM agree in principle with that of the permeancemethod. Therefore, the validity of the loss calculation byusing the permeance method is confirmed.
If the size is enlarged to the radial direction, themagnetic flux in the air gap decreases because themagnetic resistance of the air gap and the magnet
increases. Therefore, the large current is required in orderto keep the magnetic flux. However, the wiring resistancecan be decreased because the cross section of the coilwinding can be enlarged and the length of the coilwinding can become short. Thus, the copper loss can be
decreased. On the other hand, if the size is enlarged to theaxial direction, the magnetic flux increases in order todecrease the magnetic resistance. However, the magnetic
flux density decreases because the cross section of thestator core is enlarged at a great rate. Therefore, the ironloss can be decreased. For these reason, the copper loss
and the iron loss are in the relationship of trade-off. As aresult, the motor loss is decreased by enlarging to theradial direction because the variation of the copper loss islarger than that of the iron loss. If the current density
becomes low, the variation of the iron loss has asignificant impact on the motor loss and the lowest loss
point exists.

C. Changing Length of Teeth Width
Fig. 17 shows the schematic of the changing teeth
width. The losses are calculated when the width of theteeth is changed based on the constant width of the yoke.
In addition, the area of the coil is changed by thechanging teeth width. Therefore, the space factor of thecoil is constant (50%) and the resistance of wire ischanged. Moreover, the speed is 3600 r/min and theoutput torque is 3.9 Nm.

Fig. 15 Schematic of the changing radial direction length D and axial
direction length L The air gap length is set to 1 mm by adjusting thesize of the rotor. The amplitude of the armature current is calibrated
because the output torque is set to constant. The motor speed is 3600
r/min. The copper loss and the iron loss in the stator are calculated..

1000
800

350

FEM (Total)
Permeance (Total)
Permeance (Copper loss)
Permeance (Iron loss)

600
400
200
0

2
4
6
8
10 12 14 16
Radial direction length / Axial direction length [p.u.]
Fig. 16 Motor loss at 3600 r/min and 3.9 Nm when the radial
direction length and the axial direction length are changed based on
the constant volume. The radius r is standardized by the axial
direction length D and the loss is standardized by the volume of the
motor. The motor loss decreases when the size is enlarged to theradial direction.

Fig. 18 shows the motor loss when the teeth width ischanged based on the constant volume. The teeth width isstandardized by the yoke width and the loss isstandardized by the volume of the motor. The trend ofchanging loss in the proposed method agrees with that in
the FEM. In addition, when the teeth width is 1.8 p.u., theloss is the lowest in the calculation results by the
proposed method. On the other hand, in the results by the
FEM, when the width is 2.0 p.u., the loss is the lowest.
The difference between the loss at 1.8 p.u. and 2.0 p.u. isvery few. Thus, it is considered that the teeth width ofminimum loss can be designed by the proposed method.
If the teeth width is become large, the magnetic
resistance decreases and the flux in the core increases.
Therefore, in less than 1.6 p.u., the iron loss increases dueto the increasing flux density of yoke. However, in over
1.6 p.u., the iron loss decreases because the flux densityof the teeth decreases. In addition, the flux in the air gap
increases by the increasing flux in the core. If the output
torque is same, the current can be decreased. Thus, thecopper loss can be decreased. However, in over 1.8 p.u.,
the copper loss increases because the area of coil become
small and the resistance of wire become large.
From these results, when the motor parameters are

FEM
Permeance (Total)
Permeance (Copper loss)
Permeance (Iron loss)

300
Motor loss [W/dm3]

Motor loss [W/dm3]

1200

Fig. 17 Schematic of the changing teeth width. The yoke width isconstant. The area of the coil is changed by the changing teeth width.
The space factor of the coil is constant (50%) and the resistance ofwire is changed

250
200
150
100
50
0

1

1.5
2
Teeth width/Yoke width [p.u.]
Fig. 18 Motor loss at 3600 r/min and 3.9 Nm when the teeth width ischanged based on the constant volume. The loss at 1.8 p.u. is thelowest in the calculation results by the proposed method.

changed, it is confirmed, the validity of loss calculationby the proposed method. Therefore, the minimum loss
motor can be designed. For example, if the mechanicalstrength is considered and the diameter is less than 2 p.u.
as a constrained condition, the parameter of the minimum
loss motor is as follows: the pole number is 12, thediameter is 2 p.u. and the teeth width is 1.8 p.u..
VI.

CONCLUSIONSThis paper proposed the simple magnetic equivalentcircuit in order to design the rough configurations of thelowest loss motor. As a result, the error of the iron loss
between the proposed method and the FEM is 2.9%. In
addition, the motor losses were calculated by thepermeance method when several parameters werechanged. As a result, these results are almost agreed inprinciple with the results of the FEM. Therefore thevalidity of the proposed method was confirmed.
The future works are as follow; first, the eddy current
loss in the permanent magnet will be calculated by thepermeance method. Second, the motor loss considered
the carrier harmonics will be calculated.
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