ΦΦΦΦAbstract -- This paper presents a control model of squirrel

cage induction machine with broken rotor bars. The basis for model’s equations is the fact that this type of faults inevitably alters the rotor parameters. Both the resistance and inductance take a sinusoidal shape, being a function of rotor angle, almost regardless of the configuration of fault. Differing from previous work, in this paper a model has been developed in rotor field coordinates, being this the preferable reference frame for machine control and thus easier for control engineers to grasp, especially in closed-looped systems. The approach has been tested on laboratory models of two different induction machines connected to the grid with completely different severities of faults. The results show a very good agreement between measurements and simulations.

Index Terms-- Control engineering, fault diagnosis,

induction motors.

I. NOMENCLATURE

Symbols

L, R, τ inductance, resistance, time constant

v, i, ψ voltage, current, flux

β, ε, ρ slip, rotor and rotor flux angle

σ leakage factor

Indexes

ab stator reference frame (axes) dq rotor field reference frame (axes) DQ rotor reference frame (axes)

R, S rotor, stator

sl, mR slip, rotor magnetization

Abbreviations

FRF (rotor) field reference frame

FOC field oriented control

IM induction motor (squirrel cage)

RRF rotor reference frame

BRB broken rotor bar

II. INTRODUCTION

CCURATE and timely fault detection in electrical

machines is a prerequisite for their reliable operation.

The ability to model various faults is an important tool

when devising a new diagnostic method. The immediate

benefits of using model-based studies over experimental tests

All authors are with University of Ljubljana, Faculty of Electrical

Engineering, Trzaska 25, SI-1000 Ljubljana, Slovenia (e-mail:

vanjaa@fe.uni-lj.si, rastko.fiser@fe.uni-lj.si, mitja.nemec@fe.uni-lj.si,

klemen.drobnic@fe.uni-lj.si).

are reduced economical cost, greater flexibility and readily

available results. These models are mainly built upon healthy

motor representations.

Finite element analysis (FEA) offers a detailed insight

into field distribution of the induction machine (IM). FEA

takes into account various secondary effects which

accompany the primary fault expression. In case of broken

rotor bars (BRBs), the FEA can predict an increase of

magnetic field [1] and increased iron losses [2] in rotor

laminations in immediate vicinity of the fault. In addition, a

damping effect of skewing and local saturation onto typical

frequency components in stator current spectrum can be

ascertained. Computational burden of FEA is its major

drawback, especially in case of transients and where

asymmetric geometry has to be considered.

In multiple coupled circuits model (MCCM) N bars of

rotor squirrel cage are replaced by N + 1 magnetically

coupled meshes. A separate electrical equation is introduced

into each rotor mesh and end-ring. As stator windings are not

point of interest their phenomena are generalized into one

equation each. Then, mesh currents are calculated in order to

determine the behavior of the machine [3, 4]. Even though

basic MCCM does not take into account saturation and inter-

bar currents it still offers a very detailed insight into motor

operation [5].

Because of the conceptual simplicity a dq model of

induction machine is well suited for simple and

non-demanding drive analysis. The model itself enables

simple implementation of other drive components (power

converter, switching and modulation control, and mechanical

system) into coherent simulation unit. The immediate

advantage of the approach is fast execution time especially

when comparing to FEA or MCCM [6-8].

In this paper, a dq model of induction machine with BRBs

is presented. The reference frame is locked onto rotor

magnetizing current. Field oriented control (FOC), most

frequently used control scheme, depends on aforementioned

reference frame representation as well. In order to

understand behavior of closed-loop drive with faulty motor it

is propitious to grasp the changes in internal signals. In this

way the model becomes suitable for analysis of BRB fault in

closed-loop schemes.

III. BASIS FOR TWO-PHASE FAULTY ROTOR MODELING

Two-phase models of the IM with BRB can be modeled

on the fact that deviations in faulty rotor are expressed in

both rotor resistance and inductance [9]. In that sense faulty

rotor differs from the healthy one which preserves uniform

Dynamic Model of Induction Machine with Faulty Rotor in Field Reference Frame

Vanja Ambrožič, Rastko Fišer, Mitja Nemec, and Klemen Drobnič

A

978-1-4799-0025-1/13/$31.00 ©2013 IEEE

142142

parameters in both axes. In general, regardless of the

configuration of the fault, both rotor parameters take a quasi-

sinusoidal shape depending on rotor angle.

Fig. 1 shows several configurations of rotors with BRBs

for Motor 1 (see Appendix) and corresponding dependence

of resistance on rotor angle. This relation is obvious for

faulty rotor, being its amplitude dependent on fault

configuration. The measurement was made using pulsating

(non-rotational) magnetic field in which successive rotor

angular positions were set manually. At every angular

position the input impedance was calculated using U-I

method [9]. In this way, an angular dependence of rotor

resistance is calculated. The shape of the graph is well

defined for faulty rotor, being its amplitude dependent on

fault configuration. An interesting additional conclusion can

be drawn from the measurements: a healthy rotor (denoted as

“rotor I”) also exhibits a non-constant resistance. This fact

can be explained as consequence of manufacturing

imperfection and is usually neglected in traditional modeling

of a healthy machine. However, this manifestation is still

small compared to faulty rotors with one or two BRBs (here

in three different configurations).

This saliency effect allows for determining two

distinctively different values in rotor’s orthogonal D and Q

axes. Consequently, models have been developed in stator

coordinates [10]. However, these models suffer from usually

complicated terms for transformation into rotor reference

frame (RRF), in which the fault is best described.

The easiest way of modeling such a motor for control

purposes is to develop the equations in RRF [11], as shown

in Fig. 2. One of the benefits of this model is that instead of

separate rotor parameters (resistance and inductance) these

two merged into one parameter – rotor time constant τR. Of

course, as said before, unlike the healthy motor, in faulty

rotor actually two different rotor time constants exist – one

for each rotor axis: τRD and τRQ. Additional influence of rotor

inductance deviation and its different values in rotor

coordinates is expressed through different rotor leakage

inductances. Analysis of measurement procedure and

sensitivity to measurement error of these parameters is not

addressed in this paper.

Although simple, the mere fact that the model is

developed in RRF cannot be much of a use for understanding

the behavior of faulty machine for control purposes. Namely,

the control engineers better understand the machine’s

behavior in rotor field reference frame (FRF), as field

orientation control (FOC) is still the prevailing technique for

machine control. Therefore, this paper proposes a model of

IM developed in FRF showing how the equations of healthy

motor change and what impact these changes have on the

model itself. Consequently, in the next phase, this altered

model can be used in order to understand the machine’s

behavior in a closed-loop system and offer an insight to

alteration of control variables thus giving the opportunity for

machine diagnosis.

Since in closed-loop systems the diagnostic footprint is

somewhat compensated by the control loop [12], which

intrinsically cause some complicated interactions, this paper,

as the first step, will focus on the analysis of IM fed by the

grid.

Fig. 2. Control model of IM in rotor reference frame.

Mechanical angle [10°/div]

Res

ista

nce

[0

.01

Ω/d

iv]

rotor I

rotor II

rotor III

rotor IV

rotor V

Fig. 1. Resistance vs. rotor angle (below) for different types of faults

(above) – BRBs are shaded.

143143

IV. MODEL OF FAULTY MOTOR IN FIELD REFERENCE FRAME

Due to different nomenclature presented in literature, in

this paper the notation in original works presenting field

oriented control [13, 14] has been used. Voltage equations

for stator (1) and rotor (2) winding of a 2-pole IM are

defined in their own reference frames; DQ for rotor reference

frame, as marked by the superscript and stator ones in stator

reference frame ab (no superscript).

( )DQ jS

S S S S m R

d dR L L e

dt dt

ε= + +

iv i i (1)

( )0

DQ

DQ DQ jR

R R R R m S

d dR L L e

dt dt

ε−= = + +

iv i i (2)

A. Current model in FRF

Rotor current in its own reference frame is defined as (3)

[13].

( )

( )( )

( )

( )

1

1

1

DQ jmR S

R RD RQ

R

j j

jmR Sd Sq

R

jmR Sd Sq

R

i ji e

i e i ji ee

i i jie

ε

ρ ρ

ε

β

σ

σ

σ

−

−

−= + = =

+

− += =

+

− −=

+

i ii

(3)

where ρ, ε, and β are rotor field, rotor and slip angle with

respect to stator reference frame ab, as shown in Fig. 3.

Here, the rotor magnetizing current and stator current are

already presented in (rotor) field reference frame dq, as this

is the system in which FOC is performed. Of course,

magnetizing current has only a direct component in these

coordinates.

As already mentioned in Chap. III, the influence of BRBs

is manifested through different parameters in both rotor co-

ordinates. For rotor current, this means that (3) has to be

arranged properly by separating real and imaginary part and

then consider different parameters (here, rotor leakage

factor). Thus the equation for rotor current in RRF for a

faulty motor is obtained

( )( )

( )

( )cos sin sin cos

1 1

RD RQ

mR Sd Sq mR Sd SqDQ

R

RD RQ

i i

i i i i i ij

β β β β

σ σ

− + − −= +

+ +i

(4)

When inserting (4) into rotor voltage equation (2), influence

of different rotor resistances also has to be considered (5)

( )( )

( )( )

( )( )

( )

( )

( )( )

0 cos sin1

sin cos1

cos sin

1

sin cos

1

RDmR Sd Sq

RD

RQmR Sd Sq

RQ

mR Sd SqRD

RD

mR Sd SqRQ

RQ

jm Sd Sq

Ri i i

Rj i i i

i i idL

dt

i i ijL

dL i ji e

dt

β

β βσ

β βσ

β β

σ

β β

σ

= − + +

+

+ − − +

+

− ++ + +

− −++

+ +

(5)

For future manipulation we will consider different terms for

rotor inductances in both axes

( ) ( )1 ; 1RD m RD RQ m RQL L L Lσ σ= + = + (6)

After dividing the equation (5) by Lm, the only parameters

remaining are rotor time constants in the first term

;RQRD

RD RQ

RD RQ

LL

R Rτ τ= = (7)

The resulting equation

( )( )

( )( )

10 cos sin

1sin cos

cos sin

sin cos

mR Sd SqRD

mR Sd Sq

RQ

mRsl mR

mRsl mR

i i i

j i i i

dii

dt

dij i

dt

β βτ

β βτ

β ω β

β ω β

= − + +

+ − − +

+ − +

+ +

(8)

is still defined in RRF. Note the definition of slip angular

frequency

sl

d

dt

βω = (9)

In order to transform (8) to FRF (in which FOC is

performed), the equation has to be multiplied by e-jβ

.

Consequently, after separating real and imaginary part, we

get somewhat familiar terms that define rotor magnetizing

current (10) and slip frequency (11)

( ) ( ) ( )mRSd mR Sq

dif i i h i

dtβ β= − − (10)

( ) ( ) 1Sq Sd

slmR mR

i ig h

i iω β β

= + −

(11)

where

d

q

Fig. 3. Definition of space vectors and reference frames.

144144

( ) 2 21 1cos sin

RD RQ

f β β βτ τ

= + (12)

( ) 2 21 1sin cos

RD RQ

g β β βτ τ

= + (13)

( ) ( )1 1 1

sin 22 RD RQ

h β βτ τ

= −

(14)

Functions f(β), g(β) and h(β) depend on parameters (rotor

time constants) and slip angle. Note that in the healthy rotor,

where τRD = τRQ = τR, both equations change into a known

form

( )1mR

Sd mRR

dii i

dt τ= − (15)

Sqsl

R mR

i

iω

τ= (16)

since f(β) = g(β) =1/τR and h(β) = 0.

B. Torque equation in FRF

In order to grasp the influence of the rotor fault,

electromagnetic torque will be defined through rotor

currents, using generally known equation (17). Note that in

that case, torque is negative. Vectors can be defined in

arbitrary reference frame but in our case, for the reasons

described before, RRF has been chosen

*2 2Im

3 3

DQ DQ

el R R R RT p p= − × = −ψ i ψ i (17)

In healthy motor, rotor flux in RRF is defined as

( ) ( )0

DQ j

R R RD RQ Sd SqL i ji L i ji eβ= + + +ψ (18)

In a faulty rotor, the complex conjugates of rotor flux, from

(4) and (18) becomes

( ) ( )( )*

*

0

DQ j

R RD RD RQ RQ Sd SqL i jL i L i ji e

β= − + +ψ (19)

After some mathematical manipulations, a final term for

electrical torque is obtained (20)

( )( ( ) ( ) )0

2

3el mR Sq mR Sd mRT pL l i i k i i iβ β= + − (20)

with

( )2 2sin cos

1 1RD RQ

lβ β

βσ σ

= ++ +

(21)

( ) ( )1 1 1

sin 22 1 1RD RQ

k β βσ σ

= − + +

(22)

Again, for the healthy rotor, the rotor leakage factors in both

axes become equal σRD = σRQ = σR, l(β) = (1+σR)-1

and

k(β) = 0, thus obtaining the known torque equation

02

3 1el mR Sq

R

LT p i i

σ=

+ (23)

Equations (10), (11) and (20) form the basis for the

current model of IM with BRB in rotor FRF, which is usually

used for explaining the behavior of a controlled machine

(Fig. 4).

C. Voltage model in FRF

Next, the voltage model is introduced, so as to encompass

the limitations of voltage supply. In order to write the

original stator voltage equation (1) in rotor field dq co-

ordinates, the equation has to be multiplied by e-jρ

.

Consequently, it takes the form of (24).

( ) ( ) ( )

( )

Sd Sq S Sd Sq mR S Sd Sq

DQDQ jR

S Sd Sq m R

v jv R i ji j L i ji

d dL i ji L j e

dt dt

β

ω

ω −

+ = + + + +

+ + + +

ii

(24)

After taking into account (4) and a definition of

synchronous or rotor flux angular frequency (25)

mR slω ω ω= + (25)

Fig. 4. Block scheme of current model of IM with faulty rotor in field reference frame.

145145

the final stator voltage equation, here split into real (26) and

imaginary parts (27), becomes

( ) ( )

( )

( )

( ) ( ) 2

( ) ( )

Sd mR SdSd S Sd mR S Sq S

SqmR sl mR Sd

Sq

di di div R i L i L p

dt dt dt

dir r i i

dt

s m t n i

ω β

β β ω ω

β β

= − + + − +

+ + − − +

+ +

(26)

( )

( ) ( )

( )

( ) ( ) 2

( ) ( )

Sq mR SdSq S Sq mR S Sd S

SqmR sl Sq

mR Sd

di di div R i L i L r

dt dt dt

diq r i

dt

t m s n i i

ω β

β β ω ω

β β

= + + − − −

− + − +

+ + −

(27)

Terms in (26) and (27) being depended on the slip angle are

( )

( )( )

2 2

2 2

2 2

2 2

( ) cos sin

( ) cos sin

1( ) sin 2

2

( ) sin cos

( ) sin cos

sl

sl

p m n

q n m

r m n

s

t

β β β

β β β

β β

β ω β ω β

β ω β ω β

= +

= +

= −

= +

= +

(28)

where

( ) ( )

;1 1

m m

RD RQ

L Lm n

σ σ= =

+ + (29)

Interesting terms emerge in both (26) and (27) equal to

twice the slip frequency 2ωsl. Namely, this characteristic

frequency component is known to be found in stator current

spectrum of a faulty rotor (frequently employed). Both

voltage components in FRF influence the stator current

components that eventually form the output stator vector.

Equations (26) and (27) denoting components of stator

voltage components in FRF are indeed complicated, but they

comprise all the influences of rotor asymmetry due to BRB

on stator voltage. Fig. 5 shows the block scheme of voltage

model in FRF.

Again, in healthy rotor, with equal rotor leakage factors in

both axes, the equations gets a much known form (30) [13].

( )

( )

1

1

Sd mRSd S Sd S S mR Sq S

Sq

Sq S Sq S S mR Sd S mR mR

di div R i L L i L

dt dt

div R i L L i L i

dt

σ σ ω σ

σ σ ω σ ω

= + − + −

= + + + −

(30)

after considering the definition of total leakage factor σ

( ) ( )

21

1 11 1

S R

m

S R

L

L Lσ

σ σ= − = −

+ + (31)

since

( )

( )

( ) ( )1

( ) 0

( ) ( ) ( ) ( )1

m

R

mmR

R

Lp q

r

Ls m t n s n t m

β βσ

β

β β β β ωσ

= =+

=

+ = + =+

V. MODEL VERIFICATION

In order to validate the model, it has been compared to the

measurement made on laboratory set-ups with two different

motors and different severity of rotor faults (number of

BRBs). The first one (Motor 1) had only two BRBs out of 30

(rotor III from Fig. 1 – plausible scenario) while the second

(Motor 2) has 7 consecutive BRBs out of 44. Although this

severe type of fault is rather exaggerated, it has been used

solely for purpose of demonstration the validity of the model.

Data for both motor are given in the Appendix.

The proposed (voltage) model requires rotor parameters

which describe the asymmetry of the cage. Rotor time

constants τRD,Q and rotor leakage factors σRD,Q determination

is based on equivalent circuit of IM [10].

The results obtained from the simulation (voltage model of

IM) have been compared with measurement results. In this

paper, only behavior of both motors connected directly to the

grid is presented. The torque has been measured with

dynamometer whose signals have been filtered to eliminate

the inherit noise.

First, the dynamic behaviors of torque and speed have

been tested in “quasi” steady state, after speed transient.

vSa

vSb

vS1

vS2

vS3 Tel3

2

ab

dq

J -1J

iSq

iSd imR

Tl

_+

mR

+1

+c

sl

p

1

1

_

_+

1

_

+

+

+

+

h( )

f( )

k( )

+

l( ) h( )

g( )

q( )

1τS_

1τS_

iSq

iSd

RS-1

τS

_

s( )m+t( )n

s( )n+t( )m

r( )p( )

_

2_

+

_

r( ) r( )

RS-1

__

__

__

Fig. 5. Block scheme of voltage model of IM with faulty rotor in field reference frame.

146146

However, as known, with faulty rotor, actual steady state

(constant speed) never occurs, as even with constant

(frequency and amplitude) supply voltage and constant load

torque and speed oscillate.

Fig. 6 and Fig. 7 show the torque and speed of Motor 1

(with rotor III – two BRBs) for different load torques (33%,

66% and 100% of rated torque). In order to compare the

model and experimental results more accurately, both the

simulation and measurement speed and torque are depicted

in detail. Please note, that in these two figures curves are

separated on purpose (and not partially superimposed, as in

reality) in order to better compare the frequency and

amplitudes of the ripple.

Both comparisons show that the model has reconstructed

the behavior of both quantities very well. As generally

known [15], when fed with pure sinusoidal voltages and

under constant load torque, machines with faulty rotor

exhibit oscillations in both the speed and torque. Frequency

of these oscillations equals twice the slip frequency. This

effect is clearly visible in all of the presented figures.

Especially interesting is the behavior of speed and torque

of a heavily damaged motor (Motor 2; seven BRBs) in Fig. 8

and Fig. 9, respectively. Again, measured and simulation

results are intentionally separated for better view. Shapes and

frequency of both variables match very well as before, thus

proving the general validity of a model.

As in the previous case the rotor speed ripple has almost

ideal sinusoidal form. However, the torque (Fig. 9) tends to

have a distorted sinusoidal shape, which is reconstructed by

the model in all details.

Of course, it has to be emphasized that the model of a

faulty machine in FRF (as well as in its generalized form for

the healthy rotor) is somewhat simplified and that rotor

resistance and inductance dependence on rotor angle is just

an approximation of sine function. Nevertheless the

agreement between simulation and measurement is very

good.

Now, the behavior of the simulation model and its internal

values in FRF can be observed. For this purpose, machine

has been first connected to the grid at no load. In order to

analyze effect shown in Fig. 9, first a heavily damaged Motor

2 has been simulated.

After reaching steady state all model current components

(stator current components in FRF and magnetizing current)

reach constant values, additionally being iSq = 0 (Fig. 10).

Thus, as known from the literature [16], no fault indicators

are present at no-load, and diagnostics becomes impossible.

The same condition can be observed on angular velocities

(Fig. 11), where at no-load the rotor is rotating smoothly.

After impressing constant load at t = 1 s all currents start

oscillating, as well as angular speeds. However, rotor speed

is oscillating almost sinusoidally as in Fig. 8, but field

velocity takes a form of distorted sine wave, analog to torque

ripple in Fig. 9. This effect is obviously a consequence of

slip frequency, which is related to torque. From the point of

view of stator reference frame, the rotor flux (or magnetizing

current) vector in steady-state will rotate with constant speed

and additional oscillating component.

0.33⋅Tel

rated

Time (500 ms/div)

Sp

eed

(1

rp

m/d

iv)

0.66⋅Tel

rated

Time (500 ms/div)

Sp

eed

(2

rp

m/d

iv)

1⋅Tel

rated

Time (500 ms/div)S

pee

d (

2 r

pm

/div

)

measurement

simulation

Fig. 6. Comparison of speed ripples for Motor 1 at different load torques.

0.33⋅Tel

rated

Time (500 ms/div)

To

rqu

e (0

.2 N

m/d

iv)

0.66⋅Tel

rated

Time (500 ms/div)

To

rqu

e (0

.2 N

m/d

iv)

1⋅Tel

rated

Time (500 ms/div)

To

rqu

e (0

.5 N

m/d

iv)

measurement

simulation

Fig. 7. Comparison of torque ripples for Motor 1 at different load torques.

147147

On the other side described effects are not to be seen on

current components (Fig. 12) and velocities (Fig. 13) of

Motor 1, where fault extend does not affect sinusoidal shape.

VI. CONCLUSION

In this paper a dynamic model of induction motor with

BRBs has been presented. Differing from the previous work,

this model has been developed in rotor field co-ordinates.

Thus, the motor dynamics becomes much clearer to control

engineers working with electrical drives. Model performance

has been compared with experimental results on two motors

with different data and severity of the fault. In both cases the

results are in a very good agreement thus showing a general

validity of the approach.

Motor 2

Time (200 ms/div)

Sp

eed

(5

rp

m/d

iv)

measurement

simulation

Fig. 8. Comparison of measured and simulated speed ripple at rated load

torque (Motor 2).

Motor 2

Time (200 ms/div)

To

rqu

e (1

Nm

/div

)

measurement

simulation

Fig. 9. Comparison of measured and simulated torque ripple at rated load

torque (Motor 2).

0 0.5 1 1.5 2 2.5 3-100

-80

-60

-40

-20

0

20

40

60

80

100

Time (s)

Cu

rren

t (A

)

1.2 1.4 1.6 1.8

12

14

16

18

20

start transient load transient

iSq

imR

iSd

Fig. 10. Stator current components and magnetizing current in FRF

(Motor 2).

0 0.5 1 1.5 2 2.5 3295

300

305

310

315

320

Time (s)

An

gu

lar

spee

d (

rd/s

)

start transient load transient

ωmR

pωR

Fig. 11. Rotor and synchronous speed for faulty (black) and healthy (grey)

case (Motor 2).

0 0.5 1 1.5 2 2.5 3-100

-80

-60

-40

-20

0

20

40

60

80

100

Time (s)

Cu

rren

t (A

)

1.2 1.4 1.6 1.8

12

14

16

18

20

start transient load transient

iSq

imR

iSd

Fig. 12. Stator current components and magnetizing current in FRF

(Motor 1).

0 0.5 1 1.5 2 2.5 3295

300

305

310

315

320

Time (s)

An

gu

lar

spee

d (

rd/s

)

start transient load transient

ωmR

pωR

Fig. 13. Rotor and synchronous speed for faulty (black) and healthy (grey)

case (Motor 1).

148148

Now it also become clearer how the machine model in

FRF changes and how this change influences the behavior of

internal motor quantities (stator current components in FRF,

magnetizing current, slip frequency etc.).

Results in this paper have been obtained on machines

connected to the grid. Following work will focus on analysis

of the machine connected to converter and controlled in a

closed-loop. In this way the mismatch between the machine

model used by controller (“healthy motor”) and distorted

machine model presented here (“faulty motor”) is expected

to show the transition of fault signature otherwise invisible

during the control.

VII. APPENDIX

MACHINE DATA

Motor 1 Motor 2

Rated power 4.7 kW 3 kW

Rated torque 30 Nm 20 Nm

Rated speed 1500 rpm 1458 rpm

Rated current 13.0 A 14.6 A

Rated voltage 215 V 177 V

Number of pole pairs 3 2

Number of stator slots 36 36

Number of rotor slots 30 44

Moment of inertia 0.038 kgm2 0.075 kgm

2

Stator resistance 0.687 Ω 0.214 Ω

Rotor resistance 0.550 Ω 0.231 Ω

Stator inductance 55.6 mH 35.4 mH

Mutual inductance 50.9 mH 31.3 mH

No. of broken rotor bars 2 7

VIII. REFERENCES

[1] N. M. Elkasabgy, A. R. Eastham, and G. E. Dawson, "Detection of

broken bars in the cage rotor on an induction machine," IEEE Trans.

Ind. Appl., vol. 28, no. 1, p. 165–171, Jan/Feb 1992.

[2] J. F. Bangura and N. A. Demerdash, "Effects of broken bars/end-ring

connectors and airgap eccentricities on ohmic and core losses of

induction motors in asds using a coupled finite element-state space

method," IEEE Trans. Ener. Conv., vol. 15, no. 1, pp. 40–47, Mar

2000.

[3] X. Luo, Y. Liao, H. A. Toliyat, A. El-Antably, and T. A. Lipo,

"Multiple coupled circuit modeling of induction machines," IEEE

Trans. Ind. Appl., vol. 31, no. 2, pp. 311–318, Mar/Apr 1995.

[4] S. D. Sudhoff, B. T. Kuhn, K. A. Corzine, and B. T. Branecky,

"Magnetic equivalent circuit modeling of induction motors," IEEE

Trans. Ener. Conv., vol. 22, no. 2, p. 259–270, Jun 2007.

[5] H. A. Toliyat and T. A. Lipo, "Transient analysis of cage induction

machines under stator, rotor bar and end ring faults," IEEE Trans.

Ener. Conv., vol. 10, no. 2, pp. 241–247, Jun 1995.

[6] H. Rodriguez-Cortes, C. N. Hadjicostis, and A. M. Stankovic,

"Model-based broken rotor bar detection on an ifoc driven squirrel

cage induction motor," in Proceedings of the American Control

Conference, Jul 2004.

[7] M. Stocks, F. Rodyukov, and A. Medvedev, "Idealized two-axis

model of induction machines under rotor fault," in IEEE Conference

on Industrial Electronics and Applications, May 2006.

[8] T. J. Sobczyk and W. Maciolek, "On reduced models of induction

motors with faulty cage," in IEEE International Symposium on

Diagnostics for Electric Machines, Power Electronics and Drives

(SDEMPED), Aug. 31.-Sept. 3. 2009.

[9] D. Makuc, K. Drobnič, V. Ambrožič, D. Miljavec, R. Fišer, and M.

Nemec, "Parameters estimation of induction motor with faulty rotor,"

Przeglad Elektrotehniczny, vol. 88, no. 1a, 2012.

[10] K. Drobnič, M. Nemec, D. Makuc, R. Fišer and V. Ambrožič, "Pseudo-salient model of induction machine with broken rotor bars,"

in IEEE International Symposium on Diagnostics for Electric

Machines, Power Electronics and Drives (SDEMPED), 2011.

[11] V. Ambrožič, K. Drobnič, R. Fišer, and M. Nemec, "Dynamic model

of induction machine with faulty cage in rotor reference frame," in

2011 IEEE Ninth International Conference on Power Electronics and

Drive Systems (PEDS), Singapore, 2011.

[12] K. Drobnič, M. Nemec, R. Fišer, and V. Ambrožič, "Simplified

detection of broken rotor bars in induction motors controlled in field

reference frame," Control Engineering Practice, vol. 20, no. 8, pp.

761-769, Aug. 2012.

[13] W. Leonhard, Control of electrical drives. Springer Verlag, 2001.

[14] F. Blaschke, "Das Prinzip der Feldorientierung, die Grundlage fuer die

Transvektor-Regelung von Drehfeldmaschinen," Siemens Zeitschrift,

vol. 45, no. 10, pp. 757-760, 1971.

[15] M. Nemec, K. Drobnič, D. Nedeljković, R. Fišer, and V. Ambrožič, "Detection of broken bars in induction motor through the analysis of

supply voltage modulation," IEEE Trans. Ind. Electr., vol. 57, no. 8,

pp. 2879-2888, Aug. 2010.

[16] B. Mirafzal and N. A. O. Demerdash, "Effects of load magnitude on

diagnosing broken bar faults in induction motors using the pendulous

oscillation of the rotor magnetic field orientation," IEEE Trans. Ind.

Appl., vol. 41, no. 3, pp. 771-783, May-June 2005.

IX. BIOGRAPHIES

Vanja Ambrožič (M'92) received the B.S., M.S. and Ph.D. degrees in

1986, 1990, and 1993, respectively, from Faculty of Electrical Engineering,

University of Ljubljana, Slovenia. In 1986 he joined the Laboratory of

Control Engineering at the Faculty of Electrical Engineering, first as a

Junior Researcher, then as Assistant and Assistant Professor at the

Department of Mechatronics. He is currently Full Professor and head of the

same department. His main research interests include control of electrical

drives and power electronics.

Rastko Fišer (M'96) received the B.Sc., M.Sc., and Ph.D. degrees in

electrical engineering from the Faculty of Electrical Engineering, University

of Ljubljana, Slovenia, in 1984, 1989 and 1998, respectively. In 1986, he

joined the Laboratory of Electrical Drives at the same institution, first as a

Junior Researcher, then Assistant, Assistant Professor, and since 2009 he

has been working as an Associate Professor at the Department of

Mechatronics. He lectures on electrical machines, electrical drives and

power electronics in undergraduate and postgraduate studies. Currently he

is the Head of the Laboratory of Electrical Drives. His main research

interests include condition monitoring and diagnostics of electrical drives,

modeling, simulation, testing and control of electrical machines, power

electronic converters and electrical traction systems.

Mitja Nemec (M'04) received the B.S. and Ph.D. from the Faculty of

Electrical Engineering, University of Ljubljana, Slovenia, in 2003 and

2008, respectively. He is currently Senior Researcher at the same faculty in

the area of power electronics and motion control. His main research

interests include control of electrical drives, active power filters and

application of power electronics in automotive industry.

Klemen Drobnič (M'08) received the B.S. and Ph.D. in electrical

engineering from the Faculty of Electrical Engineering, University of

Ljubljana in 2007 and 2012, respectively. In 2007, he joined the

Department of Mechatronics as Junior Researcher and subsequently

become an Assistant in 2013. His research interests include diagnostics of

electrical machines and control of electrical drives.

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