MODELING THE SKEWED ROTOR BARS IN INDUCTION MOTORS BASED ON WINDING FUNCTION

F. Karami1, J. Poshtan2, and M. Poshtan3

1, 2: Department of Electrical Engineering Iran University of Science and Technology, (E-mail: fkarami@ee.iust.ac.ir, jposhtan@iust.ac.ir).

3: Petroleum Institute, Abu Dhabi, UAE, (E-mail: mposhtan@pi.ac.ae).

Abstract

In this paper, a new method to model the skew effect of rotor bars in induction motors, based on winding function method (WFM), is presented. In this method a new turn function for rotor bars is considered. This approach has been used for simulating the machine behavior under healthy and broken-rotor conditions. The proposed method is applied to rotor bars fault detection based on Park's vectors approach. This technique is based on spectral analysis of the motor current Park's vector modulus for the detection of broken rotor bars.

Introduction

Electrical machines are widely used in industry. During operation, they are subjected to various operational stresses that degrade their performance with time [1]–[3]. In general, squirrel-cage induction-motor faults can be categorized into electrical and mechanical faults. Electrical faults can be also categorized into rotor and stator faults [4]. All these possible faults in induction motors and their associated subsets are summarized as depicted in the block diagram schematic of Fig. 1

Broken rotor bars faults now account for a large percentage of total induction motor faults [5]. In general, modeling and simulation of machine operation will provide economical and useful information for fault prediction and identification. Computer simulation of motor operation can be especially useful in attaining a close insight into the dynamic behavior of machines. Corresponding to the above-mentioned faults, many techniques have been proposed for motor faults detection and diagnosis. These techniques include vibration monitoring, motor current signature analysis (MCSA), electromagnetic field monitoring... Among these methods, vibration analysis and current analysis are the most popular due to their easy measurability, high accuracy, and reliability. Broken rotor bar faults can be a serious problem, although they do not initially cause an induction motor to fail. Therefore, there can be serious secondary effects. Several authors have addressed the typical causes behind broken rotor (see [5-9] and the references therein).

According to MCSA, the current drawn by an induction motor should have a single component of supply frequency. However, changes in load will modulate the amplitude of the current to produce sidebands. According to the basic theory when a broken rotor bar fault develops, it will result in modulation of the air gap flux. This will consequently affect the current in the stator-winding conductors. A completely broken bar can carry no significant current so the current is re-distributed in the adjacent bars. The induced currents will enhance load-dependent sideband components around the main supply frequency component at multiple components of twice the per unit slip multiplied by the supply frequency 2sf in the stator windings [5], where f is the supply

frequency, and s is slip. The stator current on-line monitoring technique is one of the popular techniques for fault diagnosis of induction motors. Among several techniques, stator currents’ Park pattern or Concordia pattern has been demonstrated to be indicative of the state of motor health and any incipient fault. These Concordia patterns, however, are also affected by operating conditions of the motor such as supply voltage and mechanical loading. In order to provide a more detailed insight into the results obtained by the Park’s Vector, the present work is concerned with the use of a technique based on the spectral analysis of the motor current Park's vector modulus for the detection of rotor cage faults in a squirrel cage induction motor. The objective of this paper is to develop a model which is capable to predict the performance of induction machines under broken rotor bars and then spectral analysis of the motor currents park vector modulus is applied to rotor fault detection.

Induction machine modelling

This model follows the coupled magnetic approach by treating the current in each rotor bar as an independent variable. The effect of non-sinusoidal air-gap MMF produced by both the stator and the rotor currents have been incorporated into the model. This is done by use of the winding function approach. Our analysis is based on the following assumptions [10]: Symmetric machine, uniform air-gap, negligible saturation and insulated rotor bar. The stator comprises of three phase concentric winding. Each of these windings is treated as a separate coil. The cage rotor consists of n bars can be described as n identical and equally spaced rotor loop [11]. Each loop is formed by two adjacent rotor bars and the connecting portions of the end-rings between them. Hence, the rotor circuit has n+1 independent current as variables. The n rotor loop currents are coupled to each other and to the stator windings through mutual inductances. The end ring loop does not couple with the stator windings [11]; it however couples the rotor currents only through the end leakage inductance and the end-ring resistance.

Stator voltage equations

The stator equations for the induction machine can be written in vector matrix form as:

Rs  s dtd 

VsI s {1}

Where

sss



 s Lss  Is Lsr Ir {2} [I ] [iii ] {3}

S abc

rrr s

[Ir ] i1i2 ... in ier {4} [VS ] [vas vbs v ] {5}

c

The matrix  is a diagonal 3 by 3 matrix which consists of resistances of each coil. Due to

Rsconservation of energy, the matrix Lss is a symmetric 3 by 3 matrix. The mutual inductance matrix is a 3 by n matrix comprised of the mutual inductances between the stator coils and the

Lsr

rotor loops.

Where Lsrij is the mutual inductance between the Lsr11 Lsr12 Lsr13 ... Lsr1e

stator phase i (i=1,2 or 3) and the rotor loop j (j=1,2 … [Lsr ] Lsr21 Lsr22 Lsr23 ... Lsr2e {6} n) and Lsrie the mutual inductance between the stator Lsr31 Lsr32 Lsr33 ... Lsr3e phase I (i=1,2 or 3) and the end-ring.

Rotor voltage equations

Given the structural symmetry of the rotor, it is convenient to model the cage as identical magnetically coupled circuits. For simplicity, we assume that each loop is defined by two adjacent rotor bars and the connecting portions of the end-rings between them [11]. For the purpose of analysis, each rotor bar and segment of end-ring is substituted by an equivalent circuit representing the resistive and inductive nature of the cage. Such an equivalent circuit is shown in Fig. 2.

From Fig. 2, the voltage equations for the rotor loops can be written in vector matrix form as:

R  d  {7}

VI

rrr r

dt

Where

rr rr

[vv ... vv ] 0 L I L Is {9}

V {8} r rss s

r 12 ne

In case of a cage rotor, the rotor end ring voltage is Ver 0 , and the rotor loop voltages areVkr 0, k = 1,2…n. The loop equation for 1th rotor circuit and the voltage equation for the end-ring are:

rr

r rrr rr rrd1 r rr rr rrde

V1 Rb (i1 in ) Rei1 Rb (i1 i2) Re (i1 ie ) 0 Ve Re (ie i1) Re (ie i2) ...Re (ie in ) 0

dt dt

{10} {11}

Where Rb is the rotor bar resistance, Re is

the end-ring segment resistance and kr is the total flux linked by the kth loop. Since each loop is assumed to be identical, the equation

{10} is valid for every loop. Therefore the resistance matrix  is a symmetric (n+1) by

Rr(n+1) matrix .Due to the structural symmetry of the rotor,  can be written in matrix form.

Lrr

Fig. 2. Rotor cage equivalent circuit showing rotor loop currents

Where Lm is the self inductance of the kth rotor

and circulating end ring current. loop, Lb is the rotor bar leakage inductance, Le

is the rotor end ring leakage inductance and Lrirj is the mutual inductance between two rotor loops.

R Lrr

r

2(Rb Re ) Rb 0.. 0 Rb Re Lm2(Lb Le) Lr1r2Lb Lr1r3 .. Lr1rn1 Lr1rnLb Le  

Rb 2(Rb Re ) Rb ..0 0 Re Lr2r1Lb Lm2(Lb Le) Lr2r3Lb .. Lr2rn1 Lr2rn Le

   : ::..: ::  : : :..: :: 

 

: ::..: :: : : :..: ::

   0 0 0 ..2(R R ) R R L LL .. Lm2(L L ) L L L

bebe rn1r1 rn1r2 rn1r3 be rn1rnb e

 

Rb 0 0.. Rb 2(Rb Re ) Re Lrnr1Lb Lrnr2 Lrnr3 .. Lrnrn1Lb Lm2(LbLe) Le  

R R R .. R R nR L L L .. L L nL

eee eee eee eee

{12} {13}

Calculation of torque

The mechanical equation of the machine is {14}. Where Te is the electromagnetic torque, Tl is the load torque, J is the inertia of the rotor and r is the mechanical speed.

d 1 d

J r Te Tl {14} r rm {15}

dt pdt

Where rm the angular position of the rotor and p is denotes the number of motor pole pairs. The electromagnetic torque is given by the following equations:

1 T T TTT W

co

Wco (Is LssIs Is LsrIr Ir LsrIs Ir LrrIr ) {16} Te  {17}

2

rm (I ,I consant)

sr

T Tsr Tsr

T 1(I LI I LI ) {18} Te p IsT Lsr Ir {19}

esrr s

2   2rm

rm rm

Calculation of inductances

It is apparent that the calculation of all the machine inductances as defined by the inductances matrices in the previous section is the key to the successful simulation of an induction machine. The model must take into account the geometric construction of the machine and then will include the entire space harmonic. These machine inductances are conveniently calculated by means of winding functions. According to winding function theory, If ns () is the turn function for stator winding and

nr () is the turn function for rotor bars, the winding function for stator and rotor which shown by Ns () and Nr () will be:

Ns () ns () 

ns ()

{20) Nr () nr () 

nr ()

{21}

Where

and

nr()

are the average of stator and rotor turn function one period

respectively. The mutual inductance between two windings i and j in any electric machine can be computed by the following equation [8, 9 and 10]:

ns ()

Lij is the coil j mutual inductance due to current 2 Lij 0lr Ni (, )Nj (, )g 1(, )d {22}ii in coil i. g1(, ) is a function representing

0

the air gap. If there is no eccentricity in rotor position and if non- uniformity of air gap due to stator and rotor shapes is neglected g 1(, ) would be a constant value of 1/ g where g is the air gap, r is the average radius of the air gap,

0  4.107 H / m , is the angular rotor position, l is the active length of the motor, is a particular point along the air gap and Nj (,) is called the winding function and represents the magneto

motive force (MMF) distribution along the air gap for a unit current flowing in winding j. The machine has a stator that comprises of three phases concentric winding with 36 slots, 28 rotor bars and two coils per phase. Each coil is constituted of three sections having NS turns in series. Fig. 3 shows the MMF distribution produced by 1A of current through the stator phase "1" obtained by summing all the winding function of the stator phase coils [12].

Note that the both "2" and "3"stator phases produce

like MMF distribution but shifted by /3, 2 /3

respectively. The MMF distribution produced by 1A

of current through a rotor loop can just take two

values that depend on whether we are inside or

outside the loop. The angle between two adjacent

rotor bars is   /14 .The MMF distribution

produced by 1A of current through the first rotor

loop, is shown in Fig. 4. The mutual inductance

between stator and rotor branches will be a function

of the rotor position angle . Fig. 5 gives the mutual

inductance Lsr11( ) between the stator phase

"1","2"and"3" and the rotor loop "1". Note that the Fig. 3. Turn and winding function of the stator phases

mutual Inductance between the phase "2" and the

rotor loop ″1″ is Lsr21( ) but shifted to the right by 6 ; where  is the angle between two stator

slots. Mutual inductance between the phase ″1″ and the rotor loop ″2″ is Lsr12( ) but shifted to the

left by ; where  is the angle between two rotor bars.

Fig. 4. Turn and winding function of the rotor loop ″1″ and loop Fig. 5. Mutual inductance between the stator phases and rotor ″2″without skew loop

Modified winding function method

In this paper the modified winding function has used to model the skew effect of rotor bars. We have simply changed the skewed bar to n direct bars of equal length ( l / n ). Where l is the total length of rotor and each of them are shifted by ( / n ) Fig. 6 shows the hypothesis and related turn function for n=1, 2, 3, and 8. In practice, for a real skew, it can be said that it approaches to infinitive and turn function for the ith rotor bar can be written as:

(1/) (i 1) (i1) i

ni() {23}

(1/) (i 1) i (i1)

Fig. 7 gives the mutual inductance Lsr11( ) between the stator phase "1" and the rotor loop "1" and its derivative to rotor position without skew and with skew effect.

Fig. 6.Modeling rotor bar skew with a step by step approach and rotor bar turn Function

Fig. 7.Mutual inductance of stator phase and rotor bar 1(normal line) and its derivative to rotor position (dashed line), without skew (a), with skew (b)

Simulation results

To validate the proposed model the machine is first simulated under healthy condition. Fig. 8 shows the simultaneous electromagnetic torque and the phase currents of the machine during a start up with a balanced sinusoidal voltage supply and the application of load at instant t=3s.

Fig. 8. Torque and stator currents (left to right). Normal machine

Modelling rotor bars faults

Broken bar faults can be incorporated in the healthy machine model by assuming that bars or end-ring segments are fully broken. So, it can remove the loop equations corresponding to the broken bars or the end-ring segments from {7} in order to model the broken –bar or end-ring faults. In the case of m fully broken bars or end-ring segments, m loop equations are removed. Fig. 9 shows the simultaneous electromagnetic torque and the phase current of the machine with one broken rotor bars, during a start up with a balanced sinusoidal voltage supply and the application of load at instant t=1s.

Fig. 9. Torque and stator currents(left to right).Machine with one broken rotor bar

Park vector approach

This approach is based on Park transformation that is the transformation of three-phase electrical machine parameters to a two phases machine, using the d and q axes. From this transformation results the Park's vector for voltage and currents. This technique makes use of the two stator current components ip ,iq [13].The parks vectors currents ips ,iqs expressed in terms of stator currents

(i ,i ,i ) are:

abc

211 6

i ias i ics ids Im sin(t)

ds 36 bs 6 2 {25}

{24}

11 6 

i ics iqs Im sin(t  )i

qs bs

2222

Under ideal conditions, the three-phase currents lead to a Park's Vector with the {25}. Where: Im : Maximum value of the supply phase current (A), : Angular supply frequency (rad/s), and t : Time variable (s). In the case of a healthy motor, the lissajou's curve ids f (iqs ) has a circular shape,

centered at the origin and having a diameter equal to the stator current corresponding to the state of the operation of the motor. In the case of faulty motor the Lissajou's curve changes in shape and in thickness because of the harmonics presence generated by the fault. The strategy of this method is to compare both curves of Lissajou in both cases of the motor that is with and without fault during its operating. Fig. 10 shows the Lissajou's curve of healthy and unhealthy motor in with and without load.Fig.11, shows simulation and experimental result for unhealthy motor with different number of broken bars.

(a)
(b)

green for experimental and simulation result( right to left)

The Extended Park Vector Approach (EPVA) discusses the mechanical fault signature in the current’s Park vector modulus. Given those conditions mentioned above , the current’s Park vector modulus is constant [14].In the presence of rotor cage faults, such as broken rotor bars, the motor current spectrum will contain sideband components. These additional components at frequencies of (1  2s) f and (1  2s) f will also appear in both of the motor current’s Park vector components In fact,

in the presence of a rotor fault, motor supply currents can be expressed as

ia (t)  if cos(t 0)  i cos[(1 2sf )t l ]  idr cos[(1 2sf )t r ]

dl

2 2 2

ib(t)  if cos(t 0  )  idl cos[(1 2sf )t l  ]  idr cos[(1 2sf )t r  ] {26}

33 3 2 2 2

ic(t)  if cos(t 0  )  idl cos[(1 2sf )t l  ]  idr cos[(1 2sf )t r  ]

33 3

Where “ if ,idr and idl ” are the peak values of, respectively, the fundamental supply phase current,

the current’s lower sideband component at frequency (1  2s) f , and the current’s upper sideband component at frequency (1  2s) f ; three angles “ 0,l and r ” denote the initial phase angle, respectively, for fundamental supply current, the current’s lower sideband component and, finally,

the current’s upper sideband component with these conditions the square of the current’s Park vector modulus will be given by:

2222 2

| ids jiqs |  (if idl idr )  3if idl cos(2st  l )  3if idr cos(2st  r )  3idlidr cos(4st l r ) {27}

3

Consequently, it can be shown that the spectrum of the current’s Park vector modulus is the sum of a dc level, generated mainly from the fundamental component of the motor supply current, plus two additional terms, at frequencies of( 2sf ) and( 4sf ) , with no component at the fundamental

supply frequency. Fig. 12 shows the motor current’s Park vector modulus spectrum for healthy and unhealthy motor.

Conclusion

A new method for considering the rotor bar skew effects in modeling of induction machines based on winding function method was introduced the result shows that spotting skew in the rotor has better Simulation results. The result based on proposed method at first is used for rotor bar fault detection based on Lissajou's curve thickness (bars' fault detection can be easily obtained by observing the thickness of the Lissajou's curve. This is possible even in the case of one broken rotor bar of a loaded motor. The increase of the thickness of the Lissajou's curve is note a linear function of the degree of fault (number of broken bars) as it has been proven by AJ. M. Cardoso,), also an approach based on the spectral analysis of the motor current Park's vector modulus for the monitoring of induction motor by computer is being

effectively detected by the spectral analysis of the motor without imperfections (b) for two load levels (red)

full, and (blue) low.

current Park's vector modulus technique.

References

[1] W. Deleroi, “Squirrel-cage motor with broken bar in rotor—Physical phenomena and their experimental assessment,” in Proc. ICEM, Budapest, Hungary, Sep. 1982, vol. 3, pp. 767–771.

[2] A. J. Penman, J. C. Tait, and W. E. Bryan, “A software-based approach for machine condition monitoring,” in Proc. Int. Conf. Electr. Mach. Des.and Appl., London, U.K., Sep. 1985.

[3] G. B. Kliman and R. A. Koegl, “Non-invasive detection of broken bars in operating induction motors,” IEEE Trans. Energy Convers., vol. 3, no. 4, pp. 873–879, Dec. 1988.

[4] B. Mirafzal, N. A. O. Demerdash," On Innovative Methods of Induction Motor Interterm and Broken-Bar Fault Diagnostics," IEEE Trans Industry Applications, vol. 42, no2, pp. 405-414, Mar/Apr 2006.

[5] A. Y. Ben Sasi, F. Gu, Y. Li, A. D. Ball," A validated model for the prediction of rotor bar failure in squirrel-cage motors using instantaneous angular speed," Mechanical Systems and Signal Processing, vol. 20,pp. 1572–1589, 2006.

[6] A.H. Bonnett, G.C. Soukup, Rotor failures in squirrel cage induction motors, IEEE Trans Industry Applications IA vol. 22 {6}, pp.1165–1173, 1986.

[7] A.H. Bonnett, G.C. Houkup, Cause and analysis of stator and rotor failures in three-phase squirrel-cage induction motors, IEEE Trans Industry Applications vol. 28 {4}, pp.921–937, 1992.

[8] W.R. Finley, M.M. Hodowanec, Selection of copper versus aluminum rotors for induction motors, IEEE Trans Industry Applications vol. 37 {6}, pp. 1563–1573, 2001.

[9] B. Payne, B. Liang, A. Ball, Modern condition monitoring techniques for electric machines, in: Proceedings of the International Conference on the Integration of Dynamics Monitoring and Control, pp. 325–330, 1999.

[10] H.A. Toliyat, T.A. Lipo, and J.C. White," Analysis of a concentrated winding induction machine for adjustable speed drive application part 1 (Motor analysis)," IEEE Transaction on Energy Conversion, vol. 6, no 4, pp 679-684, Dec 1991.

[11] A. R. Munoz and T.A. Lipo. Complex vector model of the squirrel-cage induction machine including instantaneous rotor bar currents. IEEE-IAP, vol. 35, no.6, 1999.

[12] G.Houdouin, G. Baraket, B. Dakyo, H. Henao and G.A. Capolino. Coupled Magnetic Circuit Modeling of the Stator Windings Faults of Induction Machines Including Saturation Effect. In proceedings of the IEEE International Conference on industrial Technology (ICIT), pp 148-153, 2004.

[13] S. M. A. Cruz and A. J. M. Cardoso, “Rotor cage fault diagnosis in three-phase induction motors by extended Park’s vector approach,” Elect. Mach. Power Syst., vol. 28, pp. 289–299, 2000.

[14] M. Eltabach, A. Charara, , and I. Zein," A Comparison of External and Internal Methods of Signal Spectral Analysis for Broken Rotor Bars Detection in Induction Motors," , IEEE Trans Industry electronic, vol. 51, no.1, pp 107-121, February 2004.