BROKEN BAR DETECTION IN INDUCTION MACHINES: COMPARISON BETWEEN CURRENT SPECTRUM APPROACH AND PARAMETER ESTIMATION APPROACH

F. Filippettj Istituto di Elettrotecnica, Universita di Bologna, Italy

G. Franceschini, C. Tassoni, senior member,IEEE Dipartimento di Ingegneria dell' Informazione, Universid & Parma, Italy

P. Vas Department of Engineering, University of Aberdeen, United Kingdom

ABSTRACT: In this paper the diagnosis of induction machine rotor electrical faults is considered. Two approaches are compared: the current spectrum analysis and the apparent rotor resistance estimation. For the first approach the authors have developed several procedures based on dfferent fault models of the machine. Their experience is used to approach the parameter estimation method @om a theoretical point of view: the resistance variation of the balanced per-phase model is computed using the faulted machine model. It is possible to obtain, by the simulation, the expected resistance variation when some bars break in a specific machine. Moreover the numerical results can be generalized. Using a simplijied model of faulted machine a relationship is obtained, which correlates the apparent resistance variation with the number of broken bars. This relationship needs several assumptions and therefore it is an approximate one, but can be used to define the threshold level for the apparent resistance variation expected in the case of one broken bar. By this relationship it is possible to have indication on the sensttivitl, of the parumeter esrimution approach. The siiperiority of the current spectrum approach over with the parameter estimation approach is shown.

1. INTRODUCTION

One of the most well-known approaches to the diagnosis of broken rotor bars in induction machmes is based on the on-line monitoring and the processing of the stator currents, to detect typical spectrum lines. In general t h s approach is based on the detection of effects whch arise in an asymmetrical system and whch are not present in the healthy one.

A solid theory of the phenomenon involved allows the modelling of the asymmetrical conditions and by using simulation a data-set containing the different variables can be constructed. These can then be used for the determination of &agnostic indexes, which are suitable for the characterization of the various faults. References [1,2.3.4,51 contain examples of the application of ths approach. The current spectrum obtained by using a model of the asymmetrical machine (in particular the amplitude and location of the specific spectrum lines) allows the Qagnosis of the asymmetries and also the

95 0-7803-1993-1/94 $4 00 0 1994 IEEE

determination of the correlation between the degree of asymmetry and line current amplitude. Ths approach fully takes account of the operating conditions of the machine. However, it has some disadvantages, these are the complexity of the system model, which requires various design parameters, and numerous simplifylng assumptions, whch decrease the accuracy of the model.

Another approach based on parameter estimation may be a suitable alternative. In references [6,7,8,9] th~s approach has been followed for the detection of asymmetries in induction machines. In particular [6] and [q discuss the detection of brokenrotorbars. From a general point of view t h s approach is b a d on the variations of the machme parameters from their normal values to detect failures. The detection of broken bars is based on the increase of the apparent rotor resistance due to the break of a rotor bar. The apparent rotor resistance is present in the conventional per-phase steady-state equivalent circuit of the balanced induction maclune. From the measurements of the input and output variables this resistance is estimated and compared to its nominal value, but for greater accuracy the thermal variation of the rotor resistance must also be considered. Some experimental results which confirm the possibility to detest the presence of a single broken bar have also been presented. However, analytical considerations about the entity of the resistance variation, its dependence on operating conditions and other parameters, such as machme size. have not been considered previously.

In this paper the parameter estimation is approached from a theoretical point of view. The resistance variation of the balanced single-phase model is computed both using a complete model of the asymmetrical squirrelcage machine, which considers all the rotor loops [lO,ll] and also a simplified model [3,5] which is suitable for the analysis of adjacent bar failures. In contrast to the first model, which can only provide numerical results, the simplified second model allows (under suitable assumptions) the development of analytical expressions between the number of broken rotor bars, the current components and the equivalent rotor resistance. In the present paper numerical results for Merent machines are given. They show the special features of the different approaches used for &agnostic aims.

2. FAULT MODEL APPROACH

For the development of an induction motor model which takes account of various rotor asymmetries, a machine with a symmetrical three-phase stator winding is assumed which induces a sinusoidal mmf in the air-gap. Steady-state condtions are assumed (it should be noted that transient conditions can also be considered [ll], but for diagnostic purposes steady-state analysis can provide satisfactory information).

For this purpose generalized rotating-field theory can be applied: forward and backward voltage and current components can be used for machines with symmetrical stator windings. By assuming balanced supply voltages, only the forward voltage component will be present. When tlie rotor cage is Sqmmetrical (healthy condition), the currents in the bars have equal amplitude and regular phase sequence: one bar current is sufficient to describe the entire rotor and an equivalent per-phase steady-state equivalent circuit can be used When a break occurs, the bar current amplitudes become different and the phase sequence is altered

Several asymmetrical machines are described in the literature and the model presented in [ 101 is used because of its clarity. The rotor mesh currents are assumed to be the unknown rotor variables (see Fig. 1). Each mesh can be considered as a single-turn winding where the stator forward air-gap nmf induces an emf and consequently a current, at slip frequencysf. The air-gap field due to the slip frequency current flowing in the a rotor mesh can be considered as the superposition of a component which rotates at slip frequency in the forward direction with respect to the rotor and another component of equal amplitude which rotates in the backward direction. The first component (the positive sequence component) induces in the stator an emf at supply frequency f, the second one induces an emf at frequency fA=( 1-2s)f, which is usually called the negative sequence component. Therefore to describe the stator behaviour, another equation related to an emf and consequently to a current 1" at frequency f", must be added to the equation at the supply frequency.

Fig. 1 Rotor cage system with rotor loops currents

It follows that the physical system is described by two stator equations, N equations for the rotor meshes , where N is the

number of rotor bars and also an equation for one of the end- rings. In total there are N+3 equations. The emfs can be expressed by using coupling impedances. To show the parameters required for the computations, here the espression of the mutual reactance between two rotor meshes is recalled:

Xr = po 2 d (L?rD)/(N26) where L and D are the rotor length and diameter respectively and 6 is the air-gap-length.

The model neglects saturation, therefore to obtain the same results by using the full cage mesh-model and the stator referred per- phase equivalent circuit, the magnetizing reactance Xm and Xr must be related to the same air-gap length vahe. Using the rated voltage, ie assuming saturated condtions, the effective air-gap length can be derived by using the measured value of Xm:

The air-gap length, 6 is a critical parameter.

6- 6 f po N2LD/P2 X,.

The design parameters L, D, N, P (pole pairs) and N' (number of stator turns) must be available. Moreover, the distribution of the resistance and leakage inductance between bars and end-rings should be known. It has been found by computation that their Merent subdwisions yield rather similar current values. Thus if unknown, the bar and end- ring parameters can be considered to be equal and their assumed value can be expressed as follows, by using the relationship between the cage and stator referred equivalent circuit parameters:

Rt, =R, = R2/[ 12W2/N)( 1+2/N(2sin(c~/2)~] where CY= 2PdN. Similar considerations hold for the bar

and end-ring leakage reactances. When a rotor fault OCCU~~, it can be introduced in the mesh-

model through an additional resistance with a value of several orders of magnitude greater than the bar resistance. The model allows the simulation of every kind of electrical failure of the rotor cage (broken bars and end-ring segments at any position). Extensive simulation work has shown that there exists a correspondence bemen the fault entity and the stator current components only if particular events are considered: only failure of adjacent bars must be considered. However, this kind of failure is the most probable: the failure of a bar causes the increase of the currents in the adjacent bars, causing them to break.

Assuming n adjacent broken rotor bars, simulation allows the construction of a data set which contains the forward and backward stator current components, I and I" as a function of the operating conditions (slip) and n. It can be shown that IVI is a sigmiicant variable and it is approximately constant for a slip range close to the rated value. This ratio was therefore considered to be a diagnostic index of the rotor failures [3,4]. Fig. 2 shows the ratio IA/I versus the slip for 1, 2..and 5 rotor bars and Fig. 3 shows the IA/I characteristics versus n for different values of the slip. Three machines (AB and C) of different rating were used to show the effects of machine rating. Table 1 shows their parameters.

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Tab.1 machine B; c) machine C, in a slip range around the rated one (marked on the axis).

Machine parmeters A B C

Rated power output Rated line voltage Ratedcurrent Rated slip

Number of pole pairs Mean diameter airgap

Effective airgap length Number of rotor slots Number of effective stator turndphase Stator resistance

reactance h&pe&mgreactance Rotor eq. resistance Rotor eq. reactance Rotor bar-end ring resistance Rotor slotend ring leakage reactance

Supply frequency

Length

Stator leakage . .

0.15 i

1700 kW 3464 v 200 A 0.007 50 Hz 3 777 mm 800 mm 4.6 m 74 110

0.110 1.940

69.3 0 0.0780 2.80 28pU

1006 p0

15kW 220v 20A 0.022 5OHZ 2 205 mm 195 mm 0.77 rmn 38 78

0260 0.73

300 0.2 0.570 70p0

200pQ

450W 127 V 1.7A 0.035 5OHZ 1 75 mm 60 mm 0.38 mm 27 193

4.10 5 .50

1670 2.90 4.10 74p0

IO4p0

/------4 0.1 +

a 3

- 2

I"!I /-- 0.05

I 1 0- i 0.3 0.5 0.7 0.9 S%

o.2 I

I /------ 0.15 1 /

0.2 7

/----

/--

0.15 I I

0.1 I C

O''' T a

0 1 2 3 4 n 5

0.15 1

0 1 2 3 4 n 5

0.2 T C

0.15 - -

I"A 0.1 -.

0.05 --

0 1 2 3 4 n 5

Fig. 3 IVI versus the number of broken bars at different slips in the range of Fig. 2: a) machine A, b) machine B; c) machine C, .

In [12] a general steady state equivalent circuit has been shown for a cage machine with rotor asimmetry. In this model the positive-sequence and negativesequence impedances of the machine are coupled via a coupling circuit, which in general, can be an asymmetrical T-type or PI-type network. It should be noted that for the case of adjacent broken rotor bars, the coupling circuit does not contain any controlled generators (which would be present for general asymmetries). Fig. 4 shows the general steady-state equivalent circuit of the machine with general cage asymmetry, where the coupling network is realized by impedance parameters and a controlled

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sequence rotor impedance and %+j% is the referred value of the negative-sequence rotor impedance. A similar equivalent circuit has been described in [SI for slipring induction motors with general rotor asymmetry, but due to exterrnal rotor resistance asymmetry. the coupling network has been realized by a resistive network (which in general contains the zero-, positive- and negative-sequence components of the external rotor resistances respectively) and also a controlled generator. For special rotor resistance asymmetry, the controlled generator dtsappears from the equivalent circuit. Such a special case will be considered below.

f

Fig. 4 General steady-state equivalent circuit of a three-phase induction motor with cage asymmetry

Fig. 4 can be used to obtain the machine currents (e.g. forward and backward stator current components etc), electromagnetic torque (forward and backward components), stator and rotor losses etc. It can also be used to obtain the amplitude of the pulsating torque which is solely due to rotor asymmetry.

This model can be further simplified if the asymmetry is localized to a single-phase of the slipring rotor obtaing a simple expression for the ratio of I"/I that can be extended to asymmetrical cage machines [ 3 ] . For this purpose the following assumptions are used:

-the number of broken rotor bars is small (much smaller than the badpole number), -the rotor impedance is localized in the bars, -the slip is close to the rated one, so the effective rotor

reactances are small compared to the resistances. With these assumptions a cage rotor with n broken rotor

bars can be analyzed similarly to a three-phase slipring induction motor with a single-phase affected by an increased resistance AR. It should be noted that the difference in the mutual linkage reactance coefficients between different windmgs is neglected. AR is linked to the number of broken rotor bars, n, by the expression [ 3 ] :

AR=3 Rzd(N-3 n)

The model of the symmetrical three-phase machine with a resistance inserted in a rotor phase leads to the equivalent

circuit shown in Fig.5, which can be further simplified, if the magnetizing current is neglected and the resistance terms are considered to be dominant. These further assumptions lead to the ratio between I" and I:

IA/I = ( h R / 3 S ) / (R2/S+hR/3S)

Substitution of AR gives

IA/I = n/(N-2n) = n/N

Fig. 5 Equivalent circuit containing AR in a slipring rotor phase

The above expression leads to an important conclusion: the ratio I"/I depends mainly on the ratio between the equivalent rotor fault resistance AR and the rotor resistance, thus an exact determination of the rotor temperature and slip is not required. The validity of the simplified model decreases as the number of broken bars increases and also for slip values whch are much different from the rated value. This can be seen by comparing the ratio I"/I versus slip for one and two broken rotor bars respectively in case of machines A, B and C performing computation using the two models discussed above (see Fig.6).

B C

1 / __-- ___-- 002 I // /---

0 1 - - - - 0 1 2 3 4 SYO 5

Fig. 6 IA/I versus slip for one and two broken rotor bars for machmes A, B and C (continuous line-complete model, dashed line- sempllfied model)

It should be noted that when only one bar is broken or one end-ring segment is broken, it is possible to use the symmetrical component fault location technique to obtain simple steady-state equivalent circuits. For the case when a single bar is broken, this will lead to the same equivalent circuits as discussed above. llus can be Seen by assuming that only the resistance of y e rotor bar differs to those of the other bars, by the value AR , and thus at the place of the rotor fault the current is AIa=AVa/ARo, where AIa is the current at the

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fault, AVa is the voltage at the fault location, which is at phase one (phase a), By this way the effect of the extended rotor resistances can be incorporated into the line-to-line neutral rotor voltage which, due to asymmetries, is not zero.

For the other M-1 rotor phases Avb=Avc ... AVMO. Since the system can be considered as being wyeannected,

thus there m no zero sequence currents. It follows that the symmetrical component voltages can be erqrressed as

AVo=AVl= ... AVM-1 =AVa/M thus

The last two equations can be used to construct the steady- state equivalent circuit of the asymmetrical machine. It follows from this circuit that if the number of rotor bars is increased, the amplitude of the non-positive sequence stator current components decreases, thus resulting in the &crease of the torque cusps due to the negative sequence stator current and also to the decrease of the amplitudes of the pulsating torques. A full development of this technique will be described in another paper by the authors but it can be easily shown that ifM=3 and ARo is infinite, then the well-known equivalent circuit is obtained when one of the rotor phases of a slipring machine is open circuited If the rotor contains an end-ring segment whose resistance is different from those of the other end-ring segments, then it is again possible to use the symmetrical component fault location technique described *e.

Ava=da ARo = ( d 1 d 2 + . . . d ~ - 1 ) mo/hd

3 PARAMETER ESTIMATION APPROACH

Concerning the parameter estimation approach for induction machines, it is usual to make reference to estimate parameters of already dedined models. These models are obtained on the basis of physical considerations and allow to obtain equations linking the input variables with the output variables.

The choice of the model is a trade-off between complexity and effectiveness.

The random noise that affects the experimental measurements is reduced using many sets of data (greater than those necessary) and the data set is then processed for example by the application of least-square-error techniques. Therefore attention is usually focused on model formalization and on the difficulties of taking into account those phenomena which cause discrepancies between the ideal machine performance and the actual one: magnetic saturation, temperature and skin effects and also additional losses.

Many investigators have followed similar arguments [5,14,15,16,17,18].

Rotor parameters, and overall rotor resistance, must be carefully considered, because assumed constant values could not accurately represent the machine under all operating conditions.

Among the different approaches we recall the possibility of using the classical per-phase equivalent circuit introducing

analytical relationships for resistance and reactance variations with the slip, so the problem becomes the identification of the relationship parameters [14,15]. This procedure is tipically applied to large machines. Another classical approach is the representation of the cage with several branches: a rotor bar is divided into a finite number of sections and a lumped parameter circuit is USBd to represent each Section [13,16]. In particular a QuMe-cage can be modelled by using this aPP=ch.

If the scheme having only a rotor branch with constant parameters is used, additional parameters can be added to consider additional effects: an example can be found in [17] where the effects of rotor stray-losses are taken into account by an additional resistance dependent on the slip value. An improvement in the parameter estimation of the classical

equivalent circuit is obtained applying sensitivity analysis [ 181. Starting from approximate values for circuit parameters, a set of djustment factors obtained from sensitivity analysis (of the performance characteristics with respect to each circuit parameter) can be defined and used to estimate new values of parameters. After some iterations the performance characteristics calculated from the estimated circuit parameters become close to the assumed values.

It follows that the classical per-phase equivalent circuit, shown in Fig. 7, is adequate to describe the machine performance only for small machines and when the skin effect is neglected.

I R l x1 X1

T - -Am I

Fig. 7 Per-phase eqwvalent circuit of an induction machine

When the parameter estimation approach is used for diagnostic purposes [6,7], the detection of the rotor broken bars is based on the apparent parameter changes occurring when a rotor bar breaks.

The most important parameter to be estimated is the rotor resistance, which increases if there is a break of the rotor bars. The only phenomena which cannot be neglected for low power machinesisthetemperaturevariation. Therefore in [7l the thermal variation of the rotor resistance has been compensated by using a thermal model of the machine, which iterates the estimated values converging in a few iterations. Here analytical considerations on the entity of resistance variation, its dependence on operating conditions and on other m e t e r s , such as machine rating, are shown.

The resistance estimation can be approached from a theoretical point of view: the resistance variation of the balanced single-phase model can be computed using the results of the asymmetrical machine model without using experimentaldata. Thus the problems associated with measurement errors and temperature variation, are completely

99

avoided, The assumptions related to the asymmetrical model are the same as those for the per-phase model, thus the comparison is focused on the sensitivity of the two methods to a break of a rotor bar in order to find their suitability to diagnosticaimS.

The estimation of the apparent rotor resistance is performed by assuming that the rotor resistance is the only changing parameter in the circuit of Fig. 7. Thus only one pameter must be identified and only a single information is required. Only the rms value of the stator current I' is used, i.e. the rms

of the t o d stator current Jz2 +I" according to the faulted machine model of paragraph n.2.

Figs. 8 a, b, c and 9 a, b, c show the results obtained using the three machines of Tab. 1. The rotor resistance is computed as a fiulction of the number of broken rotor bars for different values of slip. The ratio between resistance variation AR2 and normal resistance R2 is reported as the most significative parameter.

_-

04 _c_ 0.3 0.5 0.7 0.9 s%

0.2 , 0.15 1 , / b

m2m2 0.1 i __c__

--

1.5 2 2.5 s?A 3 0.2 T

-/---------------

0.15 1 C

0.05 0 2.5 i 3 3.5 4 syo 4.5 Fig. 8 Variation of apparent rotor resistance versus slip for

1,2,..5 broken bars: a) machine A; b) machine B; and c) machine C.

/

55 I

0 w a 0 1 2 3 4 n 5

02 7

I 0.05 b

0 1

0 1 2 3 4 x 1 5

0.2 7

C

0 1 2 3 4 n 5

Fig. 9 Variation of apparent rotor resistance versus number of broken rotor bars: a) machme A; b) machine B and c) machine C for the slip range of Fig. 8.

This procedure can obviously yield only numerical results. However interesting results can be obtained by using the simplified equivalent circuit developed above. In fact this circuit allows to obtain an analytical expression for currents I and I" and consequently for the nns value of the corresponding input current of the per-phase equivalent circuit 1'. Therefore an analytical comparison can be performed among currents and circuit parameters, in particular resistances.

The apparent rotor resistance due to n broken rotor bars is denoted as R2' = R2 + AR2 (R2 is the normal value). Assuming the parameter R~' / s as the dominant one in the equivalent circuit (obviously the slip value must be close the rated one), the input current change will be:

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From the simplified circuit of Fig. 5 , again assuming that the resistance terms are the dominant ones, the positive sequence current in the case of n broken bars and consequently for a fault resistance AR = 3R2N/n becomes V/(I22/s+AR/3s) with a variation with respect to the healthy current: m = (AR/3s) / (R2/s) = n/N We have to compare the currents related to the circuits of

Figs 5 and 7 . i.e.: I'* J I2 +IA2 ZI (I"

between the precision related to the rotor resistance estimation through the input variables and the precision related to the detection of a spectral line of the instantaneous input current. The numerical results shown in Tab. 11 allow to consider approximately 1, 2 % as reference value for the first broken bar. But, considering considering that the faulted machine model overestimates the effect of broken bars [3,4], a reference value of 1% it is more suitable. A prooedure based on the computation of the difference between the rms current values, each of wluch is affected by measurement errors and depends on neglected phenomena can not have this precision even if a perfect correction of temperature variation is considered. In contrast to this, the precision in the computation of the ratio I*/I can be higher if the measurement system and the spectrum processing procedure show the characteristics described in [3]. Further considerations are related to the trigger threshold problem. The diagnostic system should be able to detect rotor asymmetries at a lower level than one broken bar: a desirable goal is the detection of the inevitable rotor asymmetries due to manufacture problems. This goal can be obtained detecting the anomalous spectrum lines in the healthy maclune and these data constitute the basis of a "healthy machine" diagnosis obtained in [4] with the application of a neural network. On the contrary, considering the rotor resistance estimation value, the manufacturing type of asymmetries cause only a dispersion of resistance values around the medium value for identical machines.

5. CONCLUSIONS

Two approaches for the detection of broken bars in an induction machme have been compared. The first is based on the spectrum computation of instantaneous current values to determine the anomalous line correspondmg to rotor asymmetries.

Thls method, which has been throughly investigated by the authors allows not only to detect rotor failure but also to diagnose "healthy rotors", with the application of artilicial intelligence techruques.

The equivalent models of faulty machines, which form the basis of this procedure, have been applied to determine the theoretical deviation of the rms current value in the per-phase equivalent circuit of the symmetrical machine in case of a rotor failure.

An approximate value for the case of one broken bar has also been derived, this can be used as a reference value to develop interesting considerations on the precision that can be reached through the two diagnostic approaches. It can be concluded that the spectrum analysis approach has the features required by an efficient diagnostic system for broken bar detection, while the parameter estimation approach give unambiguous results only in the case of more than a few broken bars

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[l I] F.Fihppetti, G.Franceschini, C.Tassoni, P.Vas: TRANSIENT MODELLING ORIENTED TO DIAGNOSTICS OF INDUCTION MACHDJES WITH ROTOR ASYMMETRIES. ICEM '94, Paris,

[12] P.Vas: STEADY-STATE AND TRANSIENT PERFORMANCE OF INDUCTION MOTORS WITH ROTORS ASYMMETRY. IEEE Trans on PAS, Vol.101, No 9, pp.3246-3251, Sept. 1982. [13] P.Vas: ELECTRICAL MACHINES AND DRNES: A SPACE VECTOR THEORY APPROACH. oxford University Press, 1992. [14) J.F.Lindsay, T.H.Barton: A MODERN APPROACH TO INDUCTION MACHINE PARAMETER JDENTIFICAlION. IEEE Trans on PAS, pp.1493-1500, July 1972. [ 151 J.F.Lindsay, T.H.Barton: PARAMETER JDENTIFICATION FOR SQUIRREL CAGE INDUCTION MACHINES. IEEE Trans. on PAS, pp.1287-1291, 1973. [I61 E.A.Klingshim, H.E.Jordan: SIMULATION OF POLYPHASE INDUCTION MACHINES WITH DEEP ROTOR BARS. IEEE Trans. on PAS, Vo1.89, No 6, pp.1038-1043, 1970. [ 171 A.Bellini, R.Miglio, U.Reggiani, C.Tassoni: VOLTAGE- FREQUENCY LAW IN FREQUENCY CONTROLLED INDUCTION MOTOR-DRIVES. IEEE, IAS'78, pp.690-693, Toronto 1978. [IS] S.Ansuj, F.Shokooh R.Schinzinger: PARAMETER ESTIMATION FOR INDUCTION MACHINES BASED ON SENSITIVITY ANALYSIS. IEEE Trans. on IAS, Vol. 25, No 6,

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