**Subhasis Nandi Ph.D.**

*University of Victoria*

Fault detection implies a two-valued outcome depending upon the normal or abnormal operating characteristics of a system [1]. Fault diagnosis is the process that actually decides the cause, nature, and location of a fault. Incipient fault diagnosis may even be the preemptive process to minimize damage due to faults. To make a fault diagnosis scheme incipient, it requires monitoring the system at every instant. The most logical way to implement this is to compare the system outputs with set reference values. This could be based on three possible ways: (1) signal, (2) knowledge, and (3) model. In a signal-based approach, the outputs are compared with average or limit values. It is very simple to apply. However, its use for early detection or trend monitoring is very limited.

Knowledge-based methods usually depend upon qualitative process structure, functions, and qualitative models to predict fault. Model-based techniques use analytical models of the process to generate “normal outputs” that are compared with the actual process outputs to generate “residuals” that are ultimately used for fault detection. A very simple model-based fault detection scheme is shown in Figure 7.1 [2]. The analytical models can be mathematical models, or generic models using neural networks as shown in Figure 7.2 [1], fuzzy logic presented in Figure 7.3 [3], or genetic algorithm. These generic models are then trained with healthy and faulty data obtained from real systems. Once trained, they can generate the residuals reliably to detect faults. As an example, the fault diagnosis and detection scheme for induction motor faults shown in Figure 7.2 uses three-phase line voltages *V*^{NS}(*t*), currents *I*^{NS}(*t*), and speed ω^{NS}(*t*) that are essentially nonstationary. Using the present values of voltage, speed, and past current predictions, current predicted values of current *Î*^{NS}(*t*) are generated using a multistep ahead neural network predictor that has been trained to emulate a healthy motor. The residuals *r*^{NS}(*t*) are then formed by comparing the actual values with the predicted values. The residuals *r*^{NS}(*t*) are further processed along with the currents *I*^{NS}(*t*) to separate the currents and residuals into their fundamental (INSf(t),rNSf(t))*S*(·), the root mean square value of the normalized harmonics of the residual to detect mechanical faults and (2) *r*^{−}(·), the negative sequence component of the residual to detect electrical faults. This also provides a broad classification of fault category.

**FIGURE 7.1**

A general model-based fault detection scheme. (From F. Fischer et al., “Explicit modeling of the stator winding bar water cooling for model-based fault diagnosis of turbogenerators with experimental verification,” Proceedings of the 3rd IEEE Conference on Control Applications, pp. 1403–1408, August 1994. With permission.)

Model-based fault diagnosis techniques are finding increasing importance for condition-based maintenance (CBM) rather than scheduled or preventive maintenance. CBM is perceived as the preferred technique when scheduled maintenance or routine machine replacement is not required. Even for systems where scheduled maintenance is desirable, early detection using model-based techniques provides the flexibility to stop operation anytime for preventing catastrophic failures and subsequent damages, fatalities, economic, and legal fallout.

In this chapter we will describe simple linear circuit theory based mathematical models used to predict electrical machine faults. Other types of models using finite element (FE) magnetic circuit equivalents and artificial intelligence (AI) have been already described in other related chapters. Although the models may not always be strictly used the way model-based fault diagnosis systems are designed, the insight and information available from studying these models can be immense. The inferences derived from them have been extensively used to fine-tune signal-based, knowledge- based, and other types of model-based fault diagnosis techniques. The accuracy of these models are usually not very good; hence the users should be well aware of their limitations and to what extent they are being used.

Before we deal with electric motors with fault, it is essential that we deal with healthy motors. We will start with a discussion about the healthy induction motor model and then describe how the different types of faults can be implemented in them. We will also discuss synchronous machine fault models later in the chapter.

**FIGURE 7.2**

Neural-network-model-based electric motor fault detection. (From K. Kim and A. Parlos, “Induction motor fault diagnosis based on neuropredictors and wavelet signal processing,” IEEE/ASME Transactions on Mechatronics, vol. 7, no. 2, pp. 201–219, June 2002. With permission.)

**FIGURE 7.3**

Fuzzy-logic-based fault diagnosis. (From K. Kim and A. Parlos, “Induction motor fault diagnosis based on neuropredictors and wavelet signal processing,” IEEE/ASME Transactions on Mechatronics, vol. 7, no. 2, pp. 201–219, June 2002. With permission.)

A squirrel-cage induction motor consists of a stator with a symmetrical multiphase winding and a squirrel-cage with many bars placed at equal distance from one another and shorted at the two ends by two circular slip rings [4]. For the present case we will consider a three-phase, star-connected, single-circuit stator winding with *n* rotor bars. The model also makes the following assumptions:

The motor is unsaturated.

It has negligible eddy current, hysteresis, friction, and windage losses.

It has insulated rotor bars.

Following simple circuit theory principles, with the star junction voltage as *v*_{s}, the instantaneous phase voltage and current relationship for the three-stator phases can then be written as

υa=Rasias+dλasdt+υs(7.1)

υb=Rbsibs+dλbsdt+υs(7.2)

υc=Rcsics+dλcsdt+υs(7.3)

where the flux linkages are given by

λas=Lasˉis+Larˉir(7.4)

λbs=Lbsˉis+Lbrˉir(7.5)

λcs=Lcsˉis+Lcrˉir(7.6)

and inductance matrices are defined by

Las=[LaaLabLac](7.7)

Lbs=[LbaLbbLbc](7.8)

Lcs=[LcaLcbLcc](7.9)

Lar=[Lar1Lar2⋯Larn+1](7.10)

Lbr=[Lbr1Lbr2…Lbrn+1](7.11)

Lcr=[Lcr1Lcr2….Lcrn+1](7.12)

and the stator and rotor currents vectors are given by

ˉis=[iasibsics]′(7.13)

ˉir=[ir1ir2⋯irn+1]′(7.14)

The subscript *n* + 1 in Equations (7.10) to (7.14) refers to the end-ring. Since there are *n* rotor bars, 2*n* end-ring segments and 2*n* nodes, *n* + 2*n* − 2*n* + 1 or *n* + 1 independent voltage loop equations can be written. Also, since in a star-connected machine

ias+ibs+ics=0(7.15)

we can eliminate *i*_{cs} from Equations (7.1) to (7.3) and write

υab=υaυb=Rasias−Rbsibs)ibs+(Laa−Lac−Lba+Lbc)diasdt+(Lab−Lac−Lbb+Lbc)dibsdt+ω∂(LarLbr)∂θˉir+(Lar−Lbr)dˉirdt(7.16)

and

υbc=υb−υc=Rcsias+(Rbs+Rcs)ibs+(Lba−Lbc−Lca+Lcc)diasdt+(Lbb−Lbc−Lcb+Lcc)dibsdt+ω∂(Lbr−Lcr)∂θˉir+(Lbr−Lcr)dˉirdt(7.17)

where θ and ω are the angular position and the speed of the rotor, respectively. This way one state variable and *v*_{s} can be eliminated in the final solution. Similarly, for the rotor loops (comprising two bars and two portions of the end-rings) we can write

ˉυr=Rrˉir+dλrdt(7.18)

where

λr=Lrˉir+Lrsˉis(7.19)

and

Rr=[2(Rb+Re)−Rb0⋯0−Re−Re−Rb2(Rb+Re)−Rb⋯.00−Re⋮⋮⋮⋯⋮⋮⋮⋮⋮⋮⋯⋮⋮⋮000⋯2(Rb+Re)−Rb−Re−Rb00⋯−Rb2(Rb+Re)−Re−Re−Re−Re⋯−Re−RenRe](7.20)

Lr=[Lmr+2(Lb+Le)Lr1r2−LbLr1r3⋯Lr1rn−1Lr1rn−Lb−LeLr2r1−LbLmr+2(Lb+Le)Lr2r3−Lb⋯.Lr2rn−1Lr2rn−Le⋮⋮⋮⋯⋮⋮⋮⋮⋮⋮⋯⋮⋮⋮Lrn−1r1Lrn−1r2Lrn−1r3⋯Lmr+2(Lb+Le)Lrn−1rn−Lb−LeLrnrn−LbLrnr2Lrnr3⋯Lrnrn−1−LbLmr+2(Lb+Le)−Le−Le−Le−Le⋯−Le−LenLe](7.21)

Lrs=[LraLrbLrc](7.22)

Lra=[Lr1aLr2a⋮Lrn+1a](7.23)

Lrb=[Lr1bLr2b⋮Lrn+1b](7.24)

Lrc=[Lr1cLr2c⋮Lrn+1c](7.25)

Using Equation (7.15), *i*_{cs} can be eliminated from Equation (7.18) and the resulting equation can be written as

ˉυr=Rrˉir+(Lra−Lrb)diasdt+(Lrb−Lrc)dibsdt+ω∂(Lra−Lrc)∂θias+ω∂(Lrb−Lrc)∂θibs+Lr+dirdt(7.26)

The electromechanical equation can be written as

Jdωdt=Tm−Tl(7.27)

where

Tm=0.5ˉits∂Lss∂θˉis+0.5ˉitr∂Lsr∂θˉis+0.5ˉitr∂Lrs∂θˉis+0.5ˉitr∂Lrr∂θˉir(7.28)

and

Lsr=[LarLbrLcr](7.29)

dθdt=ω(7.30)

Equations (7.16), (7.17), (7.26), and (7.28) then can be combined in the statespace form as

•x=Ax+Bu(7.31)

where

x=[iasibsˉirωθ]t(7.32)

u=[υabυbc0⋯0(Tm−Tl)J0]t(7.33)

Equation (7.33) assumes that ˉυr

A=−A−11A2(7.34)

B=A−11(7.35)

A1=[(Laa−Lac−Lba+Lbc)(Lab−Lac−Lbb+Lbc)(Lar−Lbr)00(Lba−Lbc−Lca+Lcc)(Lbb−Lbc−Lcb+Lcc)(Lbr−Lcr)00(Lra−Lrb)(Lrb−Lrc)Lr000001000001](7.36)

A2=[Ras−Rbsω∂(Lar−Lbr)∂θ00Rcs(Rbs+Rcs)ω∂(Lbr−Lcr)∂θ00ω∂(Lra−Lrc)∂θω∂(Lrb−Lrc)∂θRr0000000000−10](7.37)

Additionally, the following assumptions simplify the solution of Equation (7.31).

Application of reciprocity theorem and symmetry considerations reduces the number of mutual inductance calculations, since all the magnetic and electric circuits are considered linear. For a healthy machine the stator phase to rotor loop inductances are also similar except for the phase shift. Also for multipolar machines all the inductances repeat after every 360° electrical.

The resistances and leakage inductances of the stator phases are considered identical. The same is true for rotor bars and end-ring segments. This is not true, however, in any practical motor. For example, the rotor bars, due to blow holes caused during the manufacturing process, will show variation in resistance.

Depending upon the solvers used,

*A*_{1}and parts of*A*_{2}can be precomputed and stored in memory for different rotor positions. This way computation time can be saved, especially when each of the inductance computations involve several terms.Since the rotor loop inductances and resistances are small, computational errors can be minimized by scaling them.

The mutual and magnetizing inductances used in the solution of Equation (7.29) can be computed by using either the winding function approach (WFA) or FE method. However, the FE method of computing the inductance is only good if stored inductance data is used, since it is very time consuming. The WFA can be used to compute inductances at every iteration if the computations are fast enough.

**FIGURE 7.4**

Simulated starting transients of an unloaded 4 pole, 3 hp induction motor. Speed (top), stator line current (bottom).

A few simulation results of an unskewed 3 hp, 4 pole, 28 rotor bar machine are presented by solving Equation (7.31) using MATLAB. The inductances have been recomputed at every iteration. Figure 7.4 shows the unloaded starting transient of the motor. The steady state current and speed under full load condition is shown in Figure 7.5. One cycle of rotor current is also shown in the same figure. The computed current for the end-ring is zero under steady-state. The computation time for the unloaded motor was around 32 seconds (for 0.8 seconds simulation time), and 432 seconds (for 10.5 seconds of simulation time) on a 3.4 GHz, 1 GB, Pentium 4 machine running on Windows XP.

**FIGURE 7.5**

Simulated steady state performance of a loaded 4 pole, 3 hp induction motor. Speed (top), stator line current (middle), and rotor loop 1 current (bottom).

The stator inter-turn fault can be modeled by considering another additional stator circuit [5,6]. This additional circuit *f* can be represented by the following equation

0=Rfsifc+dΛfsdt(7.38)

with

Λf=Lfsˉifs+Lfrˉir(7.39)

where

Lfs=[LffLfaLfbLfc](7.40)

ˉis=[ifsiasibsics]′(7.41)

Lfr=[Lfr1Lfr2.⋯Lfrr1+](7.42)

All the stator–stator and stator–rotor mutual inductances will have an extra term due to this shorted loop. Thus

Las=[LafLaaLabLac](7.43)

Lbs=[LbfLbaLbbLbc](7.44)

Lcs=[LcfLcaLcbLcc](7.45)

Equations (7.16), (7.17), and (7.26) are now rewritten as

υab=υa−υb=Rasias−Rbsi