Improvement of induction motor analysis accuracy
in reluctance network analysis
Tomoyuki Umesaka*, Katsubumi Tajima*, and Yukihiro Yoshida***Cooperative Major in Life Cycle Design Engineering, Graduate School of Engineering and Resource Science, Akita Univ.,
1-1Tegatagakuen-machi, Akita 010–8502, Japan**Department

of Electrical and Electronic Engineering, Graduate School of Engineering and Resource Science, Akita Univ.,

1-1Tegatagakuen-machi, Akita 010–8502, Japan

In previous studies, we applied reluctance network analysis (RNA) to the dynamic analysis of the induction motor.
However, the accuracy of our analysis of the motor characteristics using the RNA model was not high because thedetailed magnetic flux distribution in the rotor was not taken into account. In this paper, we propose a 3D RNA modelthat considers the detailed magnetic flux distribution in the rotor and skew of the rotor bar. Using the proposed RNAmodel, the winding currents of the induction motor under no-load and locked rotor conditions can be calculated withhigher accuracy than with the previous RNA model.
Keywords: Induction motor, Reluctance network analysis, Analysis of dynamic characteristics, Squirrel cage rotor, Skew,
High-speed calculation

1.

Introduction

2. Capacitor motor (single-phase induction motor)

In Japan, the proportion of thermal power to the total〈2･1〉 Structure and principles of the capacitor motor

amount of power used has become large since most of the

The basic circuit of the capacitor motor is shown innation’s nuclear power plants have ceased operation. The

Fig. 1, where Cr is a running capacitor, Cs is a starting

proportion of fossil energy to total energy in Japan was

capacitor, v is the applied voltage, and im and ia are theabout 90% as of in 2011, and most of the fossil fuels weremain

imported. It has been pointed out that the stable supply

respectively.

windingandauxiliary

winding

currents,

of future power in Japan will not be easy (1) because the

By connecting Cr, the phase difference between the

power consumption of the world is increasing due toauxiliary and main winding currents is almost 90

population growth and economic growth in emerging

degrees. As a result, the rotating magnetic field iscountries.

generated around the squirrel cage rotor, and the motor

Therefore, improving the efficiency of motors used ingenerates a driving torque.

various applications is strongly required. Of particularinterest are induction motors, which are used inextensive applications ranging from large industrial
equipment to compact household appliances.
To

improvehigh-precision

theefficiency

calculation

of

such

methodof〈2･2〉Specifications of analysis object
The test motor is a capacitor start-type capacitor run
motor (SKD-DBKK8) made by Toshiba. Its specifications

motors,theaand dimensions are listed in Table 1. The main winding

motor

and auxiliary winding are both distributed windings.

characteristics of all potential materials and dimensions

Figure 2 and Table 2 show the arrangement and number

is required. The finite element method (FEM) hasof turns, respectively (2).

cv

Therefore, in a previous work the authors appliedcgreat deal of computing time and memory are required.

w

past, but for dynamic analysis of the induction motor, asgenerally been used for electromagnetic devices in the

reluctance network analysis (RNA) to the dynamic
analysis of the induction motor (2).
In the current work, we present an analysis of thew

accuracy improvement of the dynamic characteristics of
the induction motor based on RNA considering thedetailed magnetic flux distribution and skew of the rotor.

Fig. 1

Circuit configuration of a capacitor motor.

3.
34 35 36 1 2 3
4
33

32
31
30
29
28
27
26
25
24
23
22

〈3･1〉 Overview of Reluctance Network Analysis

5
6

21 20 19 18 17 16

2-dimensional RNA model of capacitor motor

7
8
9
10
11
12
13
14
15

The

authors

previously

proposedamethodofreluctance network analysis (RNA) that is suitable fordynamic simulations of an orthogonal-core type variableinductor, the SR motor and IPM motor, because of its
simple modeling, high calculation accuracy, and ease of
coupled analysis. Moreover, in calculation based on RNA,
it is possible to use a general-purpose circuit simulation
program such as SPICE.

(a) Main windings

We explain the flow of construction of the RNA modelof the capacitor motor below.

34 35 36 1 2 3
33
4

32
31
30
29
28
27
26
25
24
23
22

〈3･2〉RNA model of stator (3)5621 20 19 18 17 16

7
8
9
10
11
12
13
14
15

The cross section of the capacitor motor is roughly
composed of a stator, a rotor, and the air gap, as shownin Fig. 3.
Because the stator has 36 slots in a circumferential
direction,

wedivide

it

into

36

parts

in

the

circumferential direction, as shown in Fig. 4. The
division of each part includes the surrounding space so
as to consider the linkage fluxes into multiple elements,

Fig. 2(b) Auxiliary windings

as shown in Fig. 5. In the figure, each tooth is divided

Arrangement of main windings.

into

two

equal

parts

becausetheconcentrated

magnetomotive forces due to winding current are
Table 1

Specifications of specimen motor.

Parameter
Frequency
Voltage
Current
OutputNumber of poles
Rated speed
Running capacitor
Starting capacitor
Number of stator slots
Internal diameter of stator
Outer diameter of stator
Gap width
Number of rotor slots
Outer diameter of rotor
Iron core length

Table 2

Value
50 Hz
100 V
12.6 A750 W
41440 rpm
40 mF
350 mF36
90.0 mm146.0 mm
0.3 mm44
89.4 mm93.0 mmarranged as shown in Fig.5. The divided elements can be
expressed by a unit two-dimensional magnetic circuit, asshown in Fig. 6.
StatorAir gaps

Rotor

Fig. 3

Cross section of a capacitor motor.

Number of turns of windings.

Main windings
Slot number
35-2,11-8,17-20,29-26
34-3,12-7,18-21,30-25
33-4,13-6,15-22,31-24
32-5,14-5,14-23,32-23

Turns
7
11
14
7

Auxiliary windings
Slot number
Turns
1-9,18-10,19-27,36-28
36
2-8,17-11,20-26,35-29
18
3-7,16-12,21-25,34-30
5Fig. 4

Division in the radial direction of stator.

The rotor has a chain structure consisting of end ringsAirand conductor bars. The magnetic flux  of the stator

Yokeflows to the rotor and the electromotive force e isgenerated by Faraday's law of electromagnetic inductionin the rotor to flow the eddy current i. In addition,Slotaccording to the law of Lenz, eddy current i acts to

Slotprevent the change of the magnetic flux flow, andgenerates a magnetomotive force in the opposite
direction of the magnetic flux. We obtained a chain
electrical circuit composed of 44 circuits (Fig. 9), where

Magnetomotive Forcefthe size of the conductor bar is Rbar = 1.14×10-4  thesize of the end ring is Rend = 4.54 × 10-6  and theresistivity is 75℃ of aluminum.

Teeth
Fig. 540
39
38

RNA model of the stator (1/36).

41

42

43 44 1

2 3

4

5

6
7
8

37
35

9
10

34

11

36

S
l

Rm

33

12

32

13

31

14

30

15
16
17
18

29
28
27

Fig. 7
Fig. 6

Magnetic circuit of element.

26

25 24

23 22 21

20

19

Division in the radial direction of the rotor.

Rotor bar

Reluctance Rm in this unit magnetic circuit can be

Magnetomotive force

represented by equation (1) with magnetic permeability

m, magnetic path length l [m], and sectional area S [m2]:

Rm 

l
mS

（1）

Rotor core

Magnetomotive force is proportional to the number of

Shaft

turns N and current i of the coil. The magnetomotive
force source is placed in consideration of the distributed

Fig. 8RNA model of the rotor (1/44).

winding of the main and auxiliary winding between
stator slots (2).

Rbar

〈3･3〉RNA model of the rotor (3)The rotor of the capacitor motor used in this paper isthe squirrel cage rotor. It has 44 slots in the

Rend

Rend


i

circumferential direction, so we divide the rotor into 44
parts in the circumferential direction, as shown in Fig. 7,

Rbar

where each part is represented as a magnetic circuit
network, as shown in Fig. 8. Therefore, by linking 44
circuits, we can derive the RNA model of the rotor.

Fig. 9

Electric circuit model of the rotor.

4. 3-dimensional RNA model(5)〈3･4〉Driving expression of the rotor (4)In an earlier work (5), we proposed an RNA model of the rotor
when the rotor is driving. This model consists of an electrical〈4･1〉New RNA model of the rotor

circuit to evaluate the induced currents on the rotor bars andAs described in <3.3>, the previous RNA model of theend rings and a magnetic circuit to express the MMFs

rotor in Fig. 8 is a reverse T-shaped simple magnetic

corresponding the induced currents on the rotor, as shown incircuitFig. 10. In the figure, there are eight stator slots and eight rotor

distribution of the core and skew of the rotor bars. To

slots .

improve the analysis accuracy of the motor, we have

The flux flowing into the first slot of the rotor r1 shown in Fig.
10 (a) is obtained by

r1 

that

did

not

considerthemagnetic

flux

developed a new RNA model that represents the flux
distribution in the rotor and skew. We divide a part of 
4      s1    s 2   0     ,  4 
4 

the rotor shown in Fig. 8 into 19 elements corresponding(3)

to the shape of the conductor bar and the iron core of therotor to make the new RNA model.

where  is the rotating angle of the rotor. The magnetomotive
force generated by the induced current in the rotor electric circuit MMF1〈4･2〉 Driving expression of new RNA model
Here, we consider the magnetic flux r passing throughshown in Fig. 10 (b) is expressed by

MMF1  
4      ir1    ir 8   0     ,  44

the rotor core. In the previous model (Fig. 8), all r(4)passed through the rotor core. In contrast, in the model
proposed here (Fig. 11), r flows in both the rotor core andwhere ir1 and ir8 are the electric currents calculated by the electric circuit

the conductor bar. In other words, r’ passes through themodel.

rotor core between the conductor bars.
We calculate r’ on the rotating state as follows.
① Flux r can be given by using Fig. 10 (a): it flows from thestator to 1/44 of the rotor model (Fig. 11).
②

However, the magnetic flux r’ flowing between the
conductor bars is different from r. For this reason, we use
another RNA model to calculate r’ from r (Fig. 12).

③ To obtain a reaction field magnetomotive force ir in theelectric circuit model of the rotor using the r, we give theoriginal magnetic circuit MMF1–MMF44 with reference to Fig.
10. (b).

r(a) Fluxes flowing into the rotor.

ir 8

θ

MMF1

ir1

Rotor
Rotor bar
bar

ir 2

r’

Magnetomotive force
Rotor coreCore(b) MMFs acting on the stator
Fig. 10
stator.

Shaft
Shaft

Expression of coupling between the rotor andFig. 11

New RNA model of the rotor(1/44).

r1

rotor1

rotor2

r2

r3

rotor3

r2’

r1’

r3’

R’

l
S

・・・

Fig. 15

r44’

r43’

r44

rotor44

r43

5.

r42’
rotor42capacitor motor by using the OrCAD PSpice 16.6

Calculation circuit of r1’–r44’ fromr1–r44.

Fig. 12Results and discussion
We examined the validity of the new RNA model of ther42

rotor433-dimensional magnetic circuit of element.

(Cadence Design Systems, Japan) as a solver for the
analysis〈4･3〉Consideration of skew

andaregular

PC(Intel®Core(TM)

i7-3770KCPU @ 3.5 GHz 3.5 GHz 16.0 GB RAM).

The conductor bars of the squirrel-cage rotor of the

Because the same flux distribution appears at every 180

induction motor are skewed as shown in Fig. 13. Because

degrees in the four-pole structure, we use a half model.

the 2-dimensional RNA model cannot express the skew, aWith the stator and rotor core material, we assume a3-dimensional model is needed to consider the axial

non-oriented silicon steel sheet 50H600 (NIPPON

direction structure. In this paper, we constructed a

STEEL CORPORATION)

3-dimensionalrotor

permeability of 4000. The rotor conductor bars were thestructure of the axial direction to consider the skew ofsame relative permeability as air because their material

two slots. The rotor bar shape was approximated using a

is non-magnetic.

three-layer

RNA model

structureasrepresenting

shown

in

the

Fig.

14.

and a relative magnetic

The

In the case of applying the sinusoidal voltage of 50 Hz,

3-dimensional unit magnetic circuit used is shown in

the waveforms of the currents in the previous andFig. 15. For the laminated structure of the rotor core, theproposed model with locked rotor and under no-load are

reluctance R’ in the laminated direction was calculated

shown in Figs. 16 and 17. As we can see, the calculation

as a relative permeability of 25 using the methods ofaccuracy is improved. Moreover, from Fig. 17 shows thatreference 6.

the current pulsation occurring the previous model isreduced. This pulsation is thought to stem from errors in
the magnetic flux and the magnetomotive forces of the
Stator core

Stator slot

linear interpolation for use in driving expression, as
Air gap

described in <3.4>. However, by considering the detailed

Rotor barmagnetic flux distribution and the skew, we are able toreduce the magnitude of the pulsation.

Shaft

Fig. 13

The effective values Im, Ia obtained from these current

Rotor core

waveforms are shown in Tables 3 and 4. Calculation

Stator and rotor of capacitor motor.

errors of each winding current are compared with thecalculation results by the previous model. Results showthat they are greatly reduced. In particular, the2 slots
Layer 1

Layer 2

Rotor barLayer 3zr

Fig. 14

Rotor barConsideration of skew for the RNA model.

calculation error of the main winding current of the
motor with locked rotor is reduced.
Figure 18 shows the change of Im, and Ia to slip s. As
we can see in figure, there is good agreement around thesynchronous speed. However, the error of the main
winding current is increased with the increase of slip s.
Twenty cycles of computation took about 30 minuteswith the locked rotor, and about one hour in therotational state.

30

0
0.08
－10
-10

0.085

0.09

0.095

Current im ,ia (A)Current im ,ia (A)

ia

10

20

im (meas.)im (calc.)
ia (meas.)ia (calc.)

im

20

0.1

－20
-20
－30
-30

Time t (s)(a)

Previous model

Current im ,ia (A)

ia

0
0.38
－10
-10

0.385

0.39

Im(prev.)Im(conv.)Im(prop.)Im(prop.)10

Ia(meas.)Ia(meas.)5

Ia(prev.)Ia(conv.)ia0

im (meas.)im (calc.)
ia (meas.)ia (calc.)im

10

Im(meas.)Im(meas.)im

Ia(prop.)Ia(prop.)

0

30
20

15

0.395

0.02

0.04

0.06

Slip sFig. 18

Load characteristics of motor.

0.4

－20
-20

6.

－30
-30

Time t (s)(b)
Fig. 16

Proposed model

Current waveforms of the motor with locked

rotor.

Conclusions
We proposed a 3D RNA model that considers thedetailed magnetic flux distribution in the rotor and skew
of the rotor bar. Experimental results showed that theCurrent im ,ia (A)

15
10

ia

5

0
0.08
－10
-5

calculation accuracy of the winding currents underim (meas.)im (calc.)
ia (meas.)ia (calc.)

im

0.085

0.09

0.095

no-load and locked rotor conditions are improved
compared with the previous RNA model. Also, thepulsation of winding current waveforms is reduced while

0.1

maintaining high-speed calculation. This is much-needed

－20
-10

progress in the high-precision characteristic calculation

－30
-15

Time t (s)(a)
Current im ,ia (A)

15

ia

0
0.38
-5
－10

0.385

However, the calculation error of the main windingim (meas.)im (calc.)
ia (meas.)ia (calc.)

im

10
5

of the induction motor.

Previous model

0.39

0.395

current of the motor with load became large in the low
speed area. Therefore, in our future work we intend to
improve the accuracy of the load characteristics of the

0.4

induction motor to apply the proposed method to

－20
-10

induction motor design.

－30
-15

Time t (s)(b)
Fig. 17

Proposed model

Current waveforms of the motor under no-load.

References(1)

Table 3 Im and Ia of the motor with locked rotor.
（Input voltage 25.6[Vrms]）
Im [Arms]
previous
Measurement

Ia [Arms]

proposedprevious

12.6

proposed
3.45

(2)
(3)
(4)

Calculation

18.1

14.4

2.68

4.11

Relative error

43.7%

14.3%

22.3%

19.1%

(5)
(6)

Table 4 Im and Ia of the motor under no-load.
（Input voltage 100[Vrms]）

Measurement
Calculation
Relative error

Im [Arms]
previous
proposed
7.35
6.45
6.99
12.2%
4.9%

Ia [Arms]
previous
proposed
2.57
2.66
2.56
3.5%
0.4%

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Received Nov. 16, 2015; Revised Feb. 29, 2016; Accepted Mar.
21, 2016