Dynamic Model of Induction Machine with Faulty

Cage in Rotor Reference Frame

Vanja Ambrozic, Klemen Drobnic, Rastko Fiser, and Mitja Nemec

University of Ljubljana, Faculty of Electrical Engineering, Trzaska 25, 1000 Ljubljana, SLOVENIA

E-mail: vanjaa@fe.uni-lj.si

Abstract- This paper presents a simplified model of an induction machine with broken rotor bars. The model is founded on the fact that broken bars cause asymmetries in the normally homogeneous cage’s resistance and inductance. Thus, the rotor becomes pseudo-salient. This effect could be best comprised through the rotor time constants that differ in the orthogonal axes of a rotor reference frame. A method for their evaluation is also presented. Consequently, a choice of rotor co-ordinates for the model of a faulty machine to be developed in, is obvious. Experiments show good agreement with simulation results obtained from the developed model.

I. INTRODUCTION

Operation of a healthy induction motor (IM) can be

adequately described by a set of ordinary differential

equations. In this way good results in both steady-state as

well as transient modes can be achieved therefore avoiding

the use of partial differential equations which form a

foundation for finite elements analysis (FEA) [1-3]. FEA

is a very time consuming process, especially when

transients and asymmetrical geometry have to be taken

into consideration [4].

As induction motor squirrel cage is essentially a system

of intertwined windings located in space, one can model

an IM by describing particular loop with an electrical

equation. Knowing actual resistance, inductance and

capacitance of electrical circuit, a very accurate insight

into behavior of the machine can be obtained. This

approach is known as winding function. Unfortunately,

this type of modeling inevitably produces complex

mathematical problem and demands adequate

computational power to solve it [5]. Moreover, a common

drawback of both FEA and winding function approach is

the necessity of number of design data that could be

unavailable. When IM model forms a part of larger

simulation model (e.g. controlled electrical drive) the non-

complex IM scheme is advantageous for the analysis of the

control system as a whole. Logical action towards less

complex model is reducing the number of differential

equations by various types of transformations which are

based on specific (symmetrical) properties of IM [4, 6].

Ultimate step of this simplification is a well known two-

axis model of IM [7].

There are two fundamental effects that develop in the

machine due to rotor broken bars. In current spectrum two

distinctive components at (1-2s)f0 and (1+2s)f0 arise

which are acknowledged as basic signs of rotor asymmetry

(s, f0 denote per-unit slip and electrical supply frequency,

respectively) [8]. In addition, alternating component with

frequency of 2sf0 adds to the electromagnetic torque of

machine. In order to model these two characteristic effects

it is sufficient to use models that are taking into account

only fundamental component of stator magnetomotive

force, i.e. two-axis representation of motor windings.

Fault condition of IM degrades its symmetrical

characteristics and puts all simplified models under

question. For example, breakage of rotor bars is a typical

fault that distorts existing magnetic field by changing

electrical parameters of a machine [9]. Recent papers from

the authors [10-12] tackle this issue by devising new ways

to measure/evaluate changing parameters and by studying

different reference frames in the light of their suitability to

describe broken rotor bars scenario.

II. PSEUDO-SALIENCY OF A FAULTY ROTOR

The basic premise adopted in this paper is the awareness

that broken rotor bars cause asymmetry in the rotor cage

[13]. Consequently, rotor resistance and inductance differ

depending on the number of broken bars. Usually, when

the fault occurs it starts to propagate by damaging bars in

immediate proximity of the original fault location. In this

paper, measurements on a machine with a healthy and

faulty rotor (7 broken out of 44 bars) are presented (Fig.

1,2). The severity of a fault in question could be

considered exaggerated; nevertheless, it was the only one

available as the same rotor was used for previous

measurements and analysis. The sinusoidal dependence of

both parameters on rotor angle is clearly visible, thus

justifying the future modeling.

0 20 40 60 80 100 120 140 160 1800,30

0,40

0,50

0,60

0,70

0,80

0,90

healthy rotor

7 broken bars

R (

Ω)

ε (°)

Fig. 1: Measured resistance versus rotor position for healthy rotor and

rotor with 7 broken rotor bars.

IEEE PEDS 2011, Singapore, 5 - 8 December 2011

978-1-4577-0001-9/11/$26.00 ©2011 IEEE 77

0 20 40 60 80 100 120 140 160 1801,20

1,30

1,40

1,50

1,60

1,70

1,80

healthy rotor

7 broken bars

L (

mH

)

ε (°)Fig. 2: Measured inductance versus rotor position for healthy rotor and

rotor with 7 broken rotor bars.

III. MODEL OF A FAULTY INDUCTION MACHINE IN ROTOR

REFERENCE FRAME

As demonstrated, the almost sinusoidal distribution of

rotor resistance allows for establishment of a rotor

reference frame whose alignment is determined by the

number and arrangement of broken rotor bars in normally

symmetric rotor (Fig. 3). Therefore, the choice of so

defined rotor reference frame – RRF as a basis for a model

development, seems to be the most obvious one.

b

a

dq

Fig. 3: Definition of rotor reference frame.

Since most control schemes adopt rotor field reference

frame, the model could be also developed in field co-

ordinates. In that case, more complicated terms involving

slip angle appear due to different angular frequencies

between RRF (in which the anomalies of the rotor are

defined) and field reference frame. The latter model will

not be discussed in this paper.

In RRF, the stator and rotor equation of cage IM would

normally be [14]

( ) ( )( )

( ) ( )RRF

SRRF RRF RRF

S S S S

d tt R t jp t t

dtω= + +

ψu i ψ (1)

( )( )

0

RRF

RRRF

R R

d tR t

dt= +

ψi (2)

However, in a case of a faulty rotor, the rotor resistances

differ in both rotor axes (indexes d and q) [12].

Consequently, uniform RR in (2) should be replaced with

appropriate resistances RRd and RRq after defining the

vectors in RRF.

( )( )

0Rd Rq

Rd Rd Rq Sq

d jR i jR i

dt

ψ ψ+= + + (3)

From (3), derivatives of rotor fluxes in both co-ordinates

can be separately expressed as functions of their respective

currents. Rotor currents (in d, q co-ordinates) can be

obtained from (4). From both equations, we can also

obtain the term for rotor flux without using rotor currents

(L0 denotes mutual inductance).

0S S S RL L= +ψ i i (4)

0R R R SL L= +ψ i i (5)

After this manipulation, rotor flux derivative is

calculated and, together with calculated rotor currents,

inserted into (3). Consequently, a full set of equations for a

voltage model of an IM with a faulty rotor is obtained.

Sd

Sd S Sd Sq

Sq

Sq S Sq Sd

du R i p

dt

du R i p

dt

ψωψ

ψωψ

= + −

= + +

(6)

Sd Sd

S Rd S Sd Rd Sd

Sq Sq

S Rq S Sq Rq Sq

di dL L i

dt dt

di dL L i

dt dt

ψσ τ τ ψ

ψσ τ τ ψ

+ = +

+ = +

(7)

In the presented equations common symbols for motor

parameters are used. The most important achievement is

the elimination of separate resistances and inductances in

both rotor axes. Instead, they are represented through

different time constants (τRd and τRq). In such a way, there

is no need for separate measurements of resistance and

inductance.

Another important issue is the assumption of fault-

independent total leakage factor σ. Although this

parameter depends on the inductance, due to its definition,

which involves ratio of inductances, it is assumed

constant. This simplification, in our opinion, does not

influence heavily the model accuracy [12].

Being the cross-product of flux and current vectors, the

torque can be expressed in arbitrary reference frame as

( )1.5el Sd Sq Sq Sd

T p i iψ ψ= − (8)

Note that the main goal of this whole manipulation is to

obtain the dependence of the stator current components in

rotor reference frame (as outputs) on the stator voltages (as

inputs). If necessary, current components can be further

easily transformed into arbitrary reference frame (e.g.

stator co-ordinates). Fig. 4 shows the block diagram of IM

in RRF.

IV. MEASUREMENT OF ROTOR TIME CONSTANTS

As already mentioned, different rotor time constants in

both rotor axes are the basis for modeling the faulty rotor.

Therefore, a simple method for their determination has

been developed. It is based on producing only pulsating

field without torque generation (Fig. 4) [15]. In order to

get the dependence of resistance and inductance (thus also

a time constant) on the rotor angle ε (Fig. 1,2) the

78

measurement procedure described below has to be repeated at different angles.

Fig. 4: Block diagram in RRF of an IM with a faulty rotor.

Fig. 5: Single phase measurement of machine parameters at standstill

(assuming 2-pole machine).

In order to simplify the calculation, equation for the

squared impedance has been appropriately modified (9).

Impedance is determined at steady state by impressing the

stator voltage at different frequencies ω1 and then dividing

by the measured stator current.

( ) ( )2 2 2 2

12 2 2

1 2 2

1

2 1 1

1

S R S RS

S S S

S R

R LVZ R L

I

σ τ σ τ ωω

τ ω

− + + = = +

+ (9)

Stator parameters, as well as the total leakage factor

(taking its value for the healthy rotor, for simplification

[15]), are already known, so the rotor time constant(s) can

be calculated by fitting (9) to the measured impedance

frequency characteristics, shown in Fig. 6. In this case,

only the impedance for the healthy rotor (ZS2) and

impedances at the extremities of the co-ordinate system

(ZSd2 and ZSq

2), are shown. Note that the measurement is

performed only up to the pull-out slip frequency (say 10

Hz). At higher frequencies, skin effect causes modification

of (9) [15].

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0 10 20 30 40 50 60

Fig. 6: Squared impedance vs. frequency for healthy and faulty (in d- and

q-axis) rotor.

Measurement and fitting described above gave the

results summarized in Table I. Although both the

inductance and resistance in a faulty rotor increase

throughout the whole rotor angle range, it is obvious that

the increase of the resistance is much higher (as could be

expected). Consequently, rotor time constants of a faulty

rotor are considerably lower in both axes than the one of a

healthy rotor. The latter does not depend on the rotor

angle, except for very small change due to the rotor

asymmetry caused by the manufacturing process.

TABLE I

DETERMINED ROTOR TIME CONSTANTS

Healthy rotor Faulty rotor

TR (s) TRd (s) TRq (s)

0.155 0.142 0.115

79

V. RESULTS

In order to check the validity of the model, simulation

results have been compared with experimental

measurements on a 3 kW induction machine (data in the

Appendix). Rotor cage has been used for previous tests,

when the bars have been gradually broken until reaching

seven. Therefore, the tests were performed only with this

severely damaged rotor.

The machine was fed by three-phase sinusoidal voltages

in an open loop to avoid suppression of oscillations that

occur under closed loop operation. Thus, the dynamic

behavior of the model can be examined.

Fig. 7 shows the speed response in steady state under

constant 10 Nm and 20 Nm load (50% and 100% of the

rated torque).

Time (0.1 s/div)

Speed (

2 r

pm

/div

)

measurement

model

Time (0.1 s/div)

Speed (

5 r

pm

/div

)

measurement

model

Fig. 7: Speed ripple at 10 Nm (above) and 20 Nm load (below).

As expected, speed oscillates whereas the magnitude

and frequency of the ripples show very good agreement.

Next, the torque ripple has been derived out of speed

measurement and compared to the simulation (Fig. 8).

Again, the shape, magnitude and frequency match very

well and in both cases differ from the purely sinusoidal

one.

Time (0.1 s/div)

Torq

ue (

0.2

Nm

/div

)

measurement

model

Time (0.1 s/div)

Torq

ue (

1 N

m/d

iv)

measurement

model

Fig. 8: Torque ripple at 10 Nm (above) and 20 Nm load (below).

In Figs. 9 and 10, the ripples obtained from

measurement and simulations are compared numerically.

Again, Fig. 9 shows rotor speed at two different loads as

above. Model and simulations were compared regarding

the average value and the peak value of the speed ripple. It

is clear that the difference is minimal.

Measurement Model1460

1465

1470

1475

1480

1485

1490

1495

1500

Speed [

rpm

]

average

peak ripple

Measurement Model1440

1445

1450

1455

1460

1465

1470

1475

1480

Speed [

rpm

]

average

peak ripple

Fig. 9: Speed: quantification of average value and peak ripple at 10 Nm

(above) and 20 Nm load (below).

The same form of numerical analysis has been

performed on the machine torque at different loads (Fig.

10). Again, the values match very well.

80

Measurement Model9

9.5

10

10.5

11

Torq

ue [

Nm

]

average

peak ripple

Measurement Model19

19.5

20

20.5

21

21.5

22

22.5

23

Torq

ue [

Nm

]

average

peak ripple

Fig. 10: Torque: quantification of average value and peak ripple at 10 Nm

(above) and 20 Nm load (below).

In the end, the response of the model to both torque

changes has been simulated throughout the whole time

range (Fig. 11). Again, the presence of torque ripple in a

faulty machine, as well as its absence in a healthy one,

show the validity of the approach.

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

Torque (Nm)

Fig. 11: Torque response of the proposed model at various loads (0 Nm,

10 Nm, and 20 Nm) for faulty (above) and healthy rotor (below).

Since the behavior of the mathematical model of the

faulty machine depends solely on the time constants

measured in both coordinates (assuming correct values of

the other, fault-independent parameters), it is interesting to

investigate how their erroneous determination influence

the comparison with measured data. Fig. 12 shows the

speed error between the simulations and measurements for

various values of both model rotor time constants. Value

of 1 p.u. corresponds to the measured time constants of a

faulty rotor summarized in Table I. In this case, a certain,

although very small error, as presented in previous figures,

exists. This is due to several causes: mismatch of the other

machine parameters, simplifications in the rotor time

constant measurement model or measurement error.

Important conclusion is that the error in either one or both

rotor time constants introduces a considerable speed error.

Fig. 12: Speed difference between measurements and simulation

depending on the rotor time constants’ error. Percents (%) are expressed

relatively to the rated slip (44 rpms).

VI. CONCLUSION

In this paper, a dynamic model of an IM with broken

bars has been presented. The fault causes parameter

asymmetry in the rotor circuit and is best represented

through different time constants in both rotor co-ordinates.

Consequently, current components in rotor reference

frame, as function of stator voltages, are calculated. The

response of the system shows good agreement with the

dynamics of the laboratory model.

The model could be used to analyze the behavior of

controlled IM for different states of rotor damage. Future

work involves modeling of various spatial combinations of

broken rotor bars, thus allowing for the development of

method for their early detection in servo drives. It is also

expected to retrieve the machine parameters using FEM at

different stages of faults and combinations of broken bars.

The simulation model is intended for analysis in closed-

loop systems, in which determination of rotor faults, due to

the intrinsic speed correction performed by the controller

is much harder to achieve.

81

APPENDIX

TABLE II

MACHINE PARAMETERS

Pmeh 3 kW RS 0.214 Ω

VS 177 V LS 35.4 mH

IS 14.8 A RR 0.231 Ω

ω 153,4 s-1 LR 35.0 mH

Tel 20 Nm M 34.1 mH

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