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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 22, NO. 2, JUNE 2007 233

Simulation Studies of the Transients ofSquirrel-Cage Induction Motors

Masahiro Ikeda and Takashi Hiyama, Senior Member, IEEE

AbstractA new simulation approach is proposed in consid-eration of a saturation and a deep bar effect for the study oftransients of three-phase squirrel-cage type induction motors. Themathematical model of an induction motor is expressed by the sixdifferential equations of three-phase instantaneous voltage andcurrent. The torque of an electric equation is related to the motionequations of motor and driven machine in the mathematicalmodel. The values of reactance of stator and rotor are changedby the saturation of core caused by starting current. Also boththe values of reactance and resistance of rotor bar are variedby the deep bar effect in the rotor core during starting. Thecalculation method of circuit constant that adds the influenceof saturation and deep bar effect is proposed in this paper. Thecircuit constant of simulation model in consideration of saturationand the deep bar effect are decided by these computation methodsin accordance with the conditions of rotation speed and current.If the large current flows, the leakage reactance of the stator andthe rotor decreases by saturation. Moreover, the resistance of therotor gradually decreases when the rotational speed rises fromstop to synchronous speed, and the leakage reactance increasesgradually. The calculated values were compared with the observedvalues of the examination machine of 1100 kW4P and an excellentagreement was obtained demonstrating the accuracy of theproposed simulation. Consequently, it is shown that the saturationand the deep bar effect are the essential factors to perform theaccurate simulations of the induction motor. After checking thevalidity of the proposed approach, the simulation of the groundingfaults was performed. In this study, all the simulation programshave been developed in the Matlab/Simulink environment.

IndexTermsDeep bar effect, grounding fault, induction motor,saturation, transient phenomenon.

I. INTRODUCTION

THREE-PHASE squirrel-cage type induction motors are

commonly utilized in the industries from the capacity of

several kilowatt to thousands of kilowatt as the driving units for

fans, pumps, and compressors.

Usually, the motors are maintained periodically. However,

when the ground fault occurs at the motor terminal, a seri-

ous damage may be brought to the motor. In the worst case,

the motor is unable to start after the restoration of the power

supply. Therefore, it is significant to understand the transient

phenomena under abnormal conditions for the optimal design

of induction motors.

There are a good number of papers already published on

the transient phenomena of the three-phase squirrel-cage type

Manuscript received December 1, 2005; revised December 1, 2005. Paperno. TEC-00017-2003.

M. Ikeda is with the Large Rotating Machinery Department, ToshibaMitsubishi-Electric Industrial Systems Corporation, Nagasaki 852-8004, Japan(e-mail: [email protected]).

T. Hiyama is with Kumamoto University, Kumamoto, 860-8555 Japan(e-mail: [email protected]).

Digital Object Identifier 10.1109/TEC.2006.874203

induction motors. In these former studies, the d q axis based

models have been utilized in the consideration of the deep bar

effect and/or the saturation. However, there are some limita-

tions for the two-axis model such as the fixed motor resistance

and also the fixed motor inductance together with the balanced

voltage. To overcome this situation, new types of two-axis mod-

els have been proposed in [1] and [2]. However, it is easy to

calculate the current with the unbalanced voltage by six differ-

ential equations of three phases (we call direct method in this

paper) [3] in Section II.

This paper presents a new approach for the simulation of

the induction motor transients using direct method consider-

ing both the saturation and the deep bar effect. The leakage

reactance of the stator slot is variable in accordance with the

saturation of magnetic-wedge caused by the large current such

as the starting current. The leakage reactance of the rotor also

varies with the saturation of the slot tip part. Therefore, in order

to have the accurate simulation results, the leakage reactance

of the slots should be changed through the modification of the

permeance [4] of the slots. Both the resistance of the rotor bar

and the leakage reactance of rotor slot should also be changed

by the deep bar effect [5] caused by rotor current. By using

the proposed calculation method in Sections III and IV, all the

constants of the electric circuit of the induction motor have beenmodified for the accurate simulations.

From the comparisons between the simulation results and the

experimental ones, the accuracy of the proposed scheme has

been clarified in this study. Namely, the simulation results have

been compared with the experimental results tested by JEC-

37, the standard of the Japanese Electro-Technical Committee

[6]. In the design process of large induction motors, highly

accurate investigation is inevitable to investigate the transient

performance. Therefore, the proposed scheme is quite helpful

for the optimization of the design process.

Already there are a good number of papers published on the

investigation of transients of the three-phase induction motors.However, only few papers have investigated the transients of

the grounding faults. Following the rapid progress in computer

technologies, the direct simulations are now available based on

the three-phase instantaneous voltages and currents without the

above limitations. Therefore, it is possible to simulate the un-

balanced voltage problems. In addition, the example simulation

results are also shown in Section VI about the grounding faults.

II. MATHEMATICALMODEL OFINDUCTIONMOTOR

The mathematical description of the squirrel-cage type in-

duction motors is given by the following matrix differential

equation shown in (1). In this equation, L means the self and

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234 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 22, NO. 2, JUNE 2007

mutual inductances of stator and rotor windings. The diagonal

elements of R show resistances of stator and rotor, and thenondiagonal elements show the induced voltages by the rotor

rotation.

E= d(LI)

dt + RI=L

dI

dt +

R +

dL

dt

I

=LdI

dt + RI (1)

where1

The components of the matrix elements are shown as follows:

M= 2

3LM=

2

3

Xm2f

Ms= Mr =M2

Ls= Xs2f

+ M Lr= Xr2f

+ M

G1=

3

2

P

2M G2=

13

P

2(LrMr)

where

Ea, Eb, Ec three-phase instantaneous stator voltages;ea, eb, ec three-phase instantaneous rotor voltages;Ia, Ib, Ic three-phase stator currents;ia, ib, ic three-phase rotor currents;Ls stator leakage inductance;Lr rotor leakage inductance;Rs resistance of stator winding;Rr resistance of rotor conductor; angle velocity of three-phase induction motor;Xm magnetizing reactance;Xs stator leakage reactance;

Xr rotor leakage reactance;P number of poles;f frequency of power source.

Mechanical equations are also given by (2)

JMd2M

dt2 + KM(M L) =TM

JLd2Ldt

2 + TLKM(M L) = 0 (2)

1

where

TM motor torque;TL torque of the driven unit;JM inertia of the motor rotor;JL inertia of the driven unit;KM spring constant between the motor rotor and the driven

unit;M torsional angle of the motor rotor;L torsional angle of the driven unit.The torques ofTMand TLare given by (3) and (4) as follows:

TM= G1(Iaib+ Iaic+ Ibia Ibic Icia+ Icib) (3)TL= T0(1 S2). (4)

The term T0is the rated torque of the driven machine and thetermSdenotes the slip.

S=0

0, 0=

4f

P

where

0 synchronous angle velocity of the rotor; instantaneous angle velocity of the rotor.

III. MAGNETICSATURATION

A. Leakage Reactance of Stator Slot

When the current flowing into the stator windings increases

than the certain value, the wedge part in the stator slots is mag-

netically saturated. Here,KS1 denotes the permeance [4]of thestator slot. Dimensions of the stator slot are shown in Fig. 1.

For the motor supplied to the examination, the relation between

current ratioI/I0 and specific permeability s is assumed as

shown in Fig. 2. Here, I0 is the rated current. When the mag-netic wedges are used, the coefficient KS1 is calculated by usingthe value of specific permeability sshowninFig.2.Atthesametime, Carters coefficient of statorKg1is calculated by (A1) inAppendix and Xzigis calculated by (A3) using Kg1. Therefore,there exists a relation between the stator leakage reactance Xsand the specific permeabilitysin (5). Then, when the specificpermeability decreases by larger current, the leakage reactance

E=

EaEbEceaebec

L=

Ls Ms Ms M M/2 M/2Ms Ls Ms M/2 M M/2Ms Ms Ls M/2 M/2 MM M/2 M/2 Lr Mr Mr

M/2 M M/2 Mr Lr MrM/2 M/2 M Mr Mr Lr

I=

IaIbIciaibic

R=

RsRs

RsG1 G1 Rr G2 G2

G1 G1 G2 Rr G2G1

G1 G2

G2 Rr

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IKEDA AND HIYAMA: SIMULATION STUDIES OF SQUIRREL-CAGE INDUCTION MOTORS 235

Fig. 1. Configuration of stator slot.

Fig. 2. Relation betweens andI/I0.

also decreases.

XS= Xslot1+ Xend1+ 0.5 XzigXslot1= A KS1 Xzig=f1(Kg1)

KS1= d

3b+

3Z

8b +

W1

b +

W2

b (5)

whereXslot1 stator slot leakage reactance;Xend1 stator coil end leakage reactance;Xzig zig-zag leakage reactance;A coefficient determined by the motor electric design;Kg1 Carters coefficient of stator slot (see Appendix);d depth of the conductor (see Fig. 1);Z thickness of the insulation (see Fig. 1);s specific permeability of the magnetic wedge (see

Fig. 2).

B. Leakage Reactance of Rotor Slot

The leakage reactance of the rotorXr can be calculated by(6). If the current in the rotor bar increases beyond the certain

limitation, the leakage reactance of the tip part of rotor slot

teeth becomes small by themagneticsaturation because thelarge

current magnetically saturates the tip part of the slot tooth. Here,

KS2denotes the permeance [4]of the rotor slot. Dimensions ofthe rotor slot are shown in Fig. 3. For the motor supplied to

the examination, the slot coefficient KS2 can be calculated bychanging the dimension of width of slot opening from b1 to b2in accordance with the current ratio I/I0shown in Fig. 4. At thesame time, Carters coefficient of rotorKg2is calculated by the(A2) in Appendix. And Xzig is calculated by (A3) using Kg2.

Therefore, when the current in the rotor bar increases beyond

Fig. 3. Configuration of rotor slot.

Fig. 4. Relation betweenbandI/I0 .

the certain limitation, the leakage reactance decreases by the

influence of the saturation of the tip part of rotor slot.

Xr= Xslot2+ Xend2+ 0.5 Xzig

Xslot2= BKS2 Xzig = f2(Kg2)KS2= Kx

d

3b2+

2a2b1+ b2

+a1b1

(6)

where

Xslo2 rotor slot leakage reactance;Xend2 rotor coil end leakage reactance;Xzig zig-zag leakage reactance;B coefficient determined by the motor electric design;Kg2 Carters coefficient of rotor slot (see Appendix);a1 depth of slot opening (see Fig. 3);a2 depth of slot slanting part (see Fig. 3);

b1 width of slot opening (see Fig. 3);b2 width of slot bottom (see Fig. 3);d depth of slot rectangular part (see Fig. 3).

IV. DEEPBAREFFECT

A. Coefficient of Deep Bar Effect

When starting the squirrel-cage type induction motor, the

deep bar effect is significantly important. The rotor resistance

Rrand the rotor reactanceXr shall change by rotational speedunder the influence of the deep bar effect. This effect [5]can be

taken into account to determine Rr and Xr by using (7). The

coefficientKr for increasing the resistance and the coefficient

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Fig. 5. Relationship betweenKrandKh.

Kxfor decreasing the reactance are expressed as (7)

Kr= sin h2+ sin 2

cosh2cos 2

Kx = 3

2

sin h2sin 2cosh2

cos 2

= d, = 2

(sf/)1011 (7)

where

d height of the rectangular bar (see Fig. 3); specific resistance of conductor.In case of rectangular-type rotor bar, the coefficient Khfor the

deep bar effect can be defined in the relation with the rotational

speed as (8).

Kh= = Khs

S (8)

Here,Khs is the coefficient of deep bar effect at the slip of

1.0.

B. CoefficientKrfor Increasing Resistance

The relationship between the coefficient Kr and the coeffi-cientKh is assumed as shown in Fig. 5 in case of rectangularbar. If the rotational speed is decided, Khcan be calculated by(8).Kris calculated ifKhis decided becauseKhand Krhavethe relation as shown in Fig. 5.

IfKr is determined, the resistance of rotorRrcan be calcu-lated by (9)as follows:

Rr= 6.4 N2e1106 (KrRbi+ Rbo+ Rring) (9)

whereNe1 number of effective series conductors;Rbi resistance of rotor bar in the core;Rbo resistance of rotor bar out of the core;Rring resistance of rotor end ring.

C. CoefficientKxfor Decreasing Reactance

The relationship between the coefficient Kx for decreasingreactance and the coefficientKhis assumed as shown in Fig. 6in case of rectangular bar. If rotation speed is decided, Kh canbe calculated by (8). The relation betweenKh andKx shownin Fig. 6determinesKx. Therefore, we can calculate the rotor

reactanceXrby (6).

Fig. 6. Relationship betweenKxandKh.

Fig. 7. Block diagram of mathematical model.

V. SIMULATIONS

A. Configuration of Mathematical Model for SimulationsA new mathematical model has been developed with consid-

ering the saturation and the deep bar effect for the simulations

of the three-phase induction motor in the Matlab/Simulink en-

vironment.

When supplying the voltage to the motor, current is calcu-

lated by (1). At this time, if a large current flows, reactance

is changed because of saturation. Therefore, when the current

exceeds 400% of full load current, Xs andXr are changed de-pending on (5)and (6). When coefficient of deep bar effect Khexceeds 1.0 while motor is starting, Rr and Xr are changed,respectively, byKrand Kx.

The two stages of simulation are proposed in this study. Thefirst stage is a period of starting process of a motor. The motor

accelerates from the stop to the rated speed and then rotates at

rate speed stably. The second stage is the period of the grounding

faults. The block diagram of the mathematical model at the first

stage and the second stage is shown in Fig. 7. The specifications

of 1100 kW motor used for simulations are shown in Table I.

B. Accuracy of Simulations

Two different types simulations CAL1 and CAL 2 have been

performed. In CAL1, the saturation and the deep bar effect have

not been considered. However, in CAL2, both the saturation and

the deep bar effect have been taken into account. The simulated

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TABLE ISPECIFICATIONS OF1100KW MOTOR

Fig.8. Simulated current during the startingprocess considering the saturationand the deep bar effect.

Fig. 9. Measured current during the starting process.

current of CAL2 is shown in Fig. 8 during the starting pro-

cess and the measured current is shown in Fig. 9. The current

waveforms in Figs. 8 and 9 are quite similar. This fact clearly

indicates the accuracy of the proposed scheme.

As shown in Table II, the simulated current of CAL2 con-

sidering the saturation and the deep bar effect is in same level

compared with measured one.

TABLE IICOMPARISON OFCURRENTS

TABLE IIICOMPARISON OFSIMULATIONS ANDEXPERIMENTS

The comparisons between the examination results of large

motors and CAL1 and CAL2 are indicated in Table III. In this

table the examination results Test (JEC-37) have been per-

formed by JEC-37 [6]. Comparisons of the starting torque and

the maximum torque are also performed in addition to that ofthe starting current.

Table II and III clearly indicate that the accuracy of the pro-

posed calculation method is excellent.

VI. GROUNDFAULT

Because it is possible to simulate the unbalanced voltage

problem by using this direct model, we show the result of

simulation.

A. Single-Phase Ground Fault

Regarding the single-phase to ground fault at the stator termi-

nal of phase A,the simulationresult isshownin Fig. 10. The fault

occurs 2 s after the starting of the motor. The waveforms of the

stator current of phase A, the stator current of phase B, the stator

current of phase C, the generating torque of the motor, and the

rotational speed are shown in order in Fig. 10. The currents of the

normal phases of B and C are smaller than that of the grounding

fault phase of A. The current levels in the phases of B and C are

from 3.7 to 4.5 p.u. The level of the torque exceeds the 5.0 p.u.

immediately after the ground fault. Although average torque

balances with the demand torque of load, it includes vibration

of2fingredient exceeding the single-sided amplitude 1.2 p.u.

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Fig. 10. Single-phase ground fault.

Fig. 11. Two-phase ground fault.

Fig. 12. Three-phase ground fault.

B. Two-Phase Ground Fault

The simulation result of the two-phase ground fault at the sta-

tor terminals of phase A and B is shown in Fig. 11. Fault currents

flow into both phase A and phase B. When the grounding fault

happens, the level of the current in the healthy phase C reaches

8 p.u. After the fault, the transient torque increases immediately

to the level of 8 p.u., however, the torque decreases rapidly asshown in Fig. 11.

C. Three-Phase Ground Fault

The simulation result of the three-phase ground fault at the

stator terminals is shown in Fig. 12. The amplitude of the tran-

sient current of each phase is from 8 to 10 p.u. immediately after

the fault. These currents rapidly decrease within 0.2 s after the

fault. The amplitude of the transient torque is also near 7 p.u.

immediately after the fault, however, it rapidly decreases within

0.2 s from the fault.

VII. CONCLUSION

Thenew simulation approach for thetransient of squirrel-cage

induction motor has been proposed in this paper. In this method,

it is possible to simulate the unbalanced voltage problem in the

situation of varying the constants of resistance and reactance.

The calculations of transient phenomena were carried out in

consideration of the saturation of core and the deep bar effect

of rotor. It is clear from the comparison between the simulation

result and the test result that this method is effective to calculate

the transient phenomena.

The various types of ground faults had been simulated by this

method. The results of simulations were the same contents that

we had thought experimentally. Therefore, the proposal method

is effective for calculating the ground faults.

APPENDIX

Definition of Carters coefficientKg1and Kg2:Carters coefficient of statorKg1is defined to be the function

of specific permeabilitysas

Kg1(s) =Kg1n

1 0.0611 + e2(s2)

(A1)

whereKg1nis the Carters coefficient at s= 1.0.

Carters coefficient of statorKg2is the function of the dimen-sion of width of slot openingb as

Kg2(b) = C1+ C2 b

C1+ C2 b b2 (A2)

whereC1 andC2 are the coefficients determined by the motorelectric design.

Then zig-zag leakage reactanceXzig is calculated by

Xzig= D1

{Kg1(s)}2 +

D2

{Kg2(b)}2 (A3)

whereD1 and D2 are the coefficients determined by the motor

electric design.

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REFERENCES

[1] C. F. Landy, W. Levy, M. McCulloch, and A. S. Meyer, The effect ofdeep-bar properties when assessing reswitching transients in squirrel cageinduction motor, in IEEE Ind. Appl. Soc. Annu. Meeting, 1991, vol. 1,pp. 3539.

[2] A. C. Smith, R. C. Healey, and S. Wiliamson, A transient inductionmotor model including saturation and deep bar effect, IEEE Trans.

Energy Convers., vol. 11, no. 1, pp. 815, Mar. 1996.[3] B. T. Ooi and T. H. Barton, Starting transients in induction motors with

inertia loads, IEEE Trans. Power App. Syst., vol. PAS-91, pp. 18701874,Sep.Oct. 1972.

[4] R. Richter,Elektrische Maschinen. Berlin, Germany: Springer-Verlag,1924, pp. 268271.

[5] P. L. Alger,Induction Machines. New York: Gordon and Breach, 1970,p. 265.

[6] JEC-37, Standard of the Japanese Electro-Technical Committee, pp. 6162, 1979.

Masahiro Ikeda receivedthe B.E. andPh.D.degrees

in electrical engineering from Kumamoto University,Kumamoto, Japan in 1973 and 2005, respectively.

In 1973, he joined Mitsubishi Electric Corpora-tion, Japan and in 1999, he joined TMA Electric Cor-poration, Japan. Since 2003, he has been workingwith Toshiba Mitsubishi-Electric Industrial SystemsCorporation, Japan. His current research interests in-clude design, control, and power saving of electricmachines.

He is a member of IEE of Japan.

Takashi Hiyama(M86SM93) received the B.E.,M.S., and Ph.D. degrees from Kyoto University,Kyoto, Japan in 1969, 1971, and 1980, respectively,all in electrical engineering.

Since 1989, he has been a Professor at the Depart-ment of Electrical and Computer Engineering, Ku-mamoto University, Kumamoto, Japan. His currentresearch interests include intelligent system applica-tions to power system operation, control, and man-agement.

He is a member of IEE of Japan, SICE of Japan,and Japan Solar Energy Society.

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