**4.2. Motor Current Signature Analysis (MCSA)**

MCSA is the technique used to analyze and monitor the trend of dynamic energized systems. The appropriate analysis of the results of applying predictive technique helps in identifying problems in stator winding, rotor problems, problems in the coupling, problems in attached load, efficiency and system load, problems in the bearing, among others.

This technique uses the induction motor as a transducer, allowing the user to evaluate the electrical and mechanical condition from the panel and consists primarily in monitoring of one of the three phases of the supply current of the motor. A simple and sufficient system for the implementation of the technique is presented in the Figure 3a.

**Figure 3.** Basic System for Spectral Analysis of the Current

Thus, the current signal of one of the phases of the motor is analyzed to produce the power spectrum, usually referred to as motor signature. The goal is to get this signature to identify the magnitude and frequency of each individual component that integrates the motor current signal. This allows that patterns in current signature be identified to differentiate "healthy" motors from "unhealthy" ones and even detect in which part of machine failure should occur.

494 Induction Motors – Modelling and Control

might occur.

astonishment that the electrical signals contain information in addition to the electrical characteristics of the machine under supervision, but they work for mechanical defects as a transducer, allowing the electrical signals (voltage and/or current) can carry information of

The signs of current and/or voltage of one or three phases of the machine produce, after analyzed, the *signature of machine*, i.e., its operating pattern. This signature is composed of magnitudes of frequencies of each individual component extracted from their signals of current or voltage. This isolated fact itself is an advantage, as it allows the monitoring of the evolution of the magnitudes of the frequencies, which can denote some sort of evolution of operating conditions of the machinery. The response that the user of such a system needs to know is whether your machine is "healthy" or not, and that part of the machine the failure

This analysis (diagnosis) is not something easy to be done, because it involves a set of comparisons with previously stored patterns and own "history" of the machine under analysis. In this instant, normally a specialist is called to produce the final diagnosis,

MCSA is the technique used to analyze and monitor the trend of dynamic energized systems. The appropriate analysis of the results of applying predictive technique helps in identifying problems in stator winding, rotor problems, problems in the coupling, problems in attached

This technique uses the induction motor as a transducer, allowing the user to evaluate the electrical and mechanical condition from the panel and consists primarily in monitoring of one of the three phases of the supply current of the motor. A simple and sufficient system

**Motor**

**3-phase supply**

(a) (b)

**CT**

**Signal Conditioner** **Spectrum Analyzer**

**Expert Knowledge Base**

**Expert**

Thus, the current signal of one of the phases of the motor is analyzed to produce the power spectrum, usually referred to as motor signature. The goal is to get this signature to identify the magnitude and frequency of each individual component that integrates the motor

electrical and mechanical problems until the power panel of the machine.

generating the command when stopping the machine.

**4.2. Motor Current Signature Analysis (MCSA)** 

**Figure 3.** Basic System for Spectral Analysis of the Current

**Signal Conditioner**

**Motor**

**3-phase supply**

**CT**

load, efficiency and system load, problems in the bearing, among others.

for the implementation of the technique is presented in the Figure 3a.

**Spectrum Analyzer** However, it is important to note that the diagnosis is something extremely complicated, i.e. the definition of stopping or not the production process in view of the indications of the power spectrum is always difficult and requires experience and knowledge of the process. This time, it is important to consider the expert knowledge and the data history of the behavior of the set (motor, transmission system and load). For this reason, an automatic diagnostic system that combines the data history of the motor to the attention of specialist is a niche market quite promising. This way, the automatic diagnosis and analysis system is no longer as simple as the model shown in Figure 3a and can be represented by the new elements in Figure 3b.

The Fast Fourier Transform (FFT) is the main tool employed, however some systems employ in conjunction with other techniques to increase the ability of fault detection since signal acquisition, through processing, up to the diagnostic step. Among the most important issues related to acquisition of signals and the FFT include:


$$
\Delta f = \frac{f\_s}{N} \tag{1}
$$

Where *f* is the spectral resolution, *fs* is the sampling frequency used, and *N* is the number of samples.

Other important issues are related to the own operation of induction motors. The first one is the induction motor synchronous speed that is given by (2):

$$N\_S = \frac{f\_1}{p} \tag{2}$$

Where *f*1 represents the power frequency, *Ns* is the velocity of the rotating field, and *p* is the number of motor pole pairs.

From the synchronous speed, two important concepts for the current signature analysis can be presented: the slip speed and the slip. In MCSA is important to note that the rotor speed is always less than the synchronous speed. The frequency of the induced currents in the rotor is a function of frequency and power slip. When operating without load, the rotor rotates at a speed close to the synchronous speed. In this case, torque should be just sufficient to overcome friction and ventilation. The difference between the rotor speed (*Nr*) and the synchronous speed (*Ns*) is named as slip speed (*NSlip*):

$$N\_{Slip} = N\_s - N\_r \tag{3}$$

Predictive Maintenance by Electrical Signature Analysis to Induction Motors 497

(7)

(8)

(10)

*<sup>r</sup>*) can be

is the

specific failure, and more, the severity of this failure. The frequency component that appears

*lf* <sup>1</sup> *<sup>r</sup> f f f* 

constant resulting from the drive train constructive characteristics and *fr* is the motor

It is known that when a mechanical failure has developed in the load, it generates an additional torque (*T1f*). Thus, the overall load torque (*Tload*) can be represented by an invariable component (*Tconst*) plus this additional variable component which varies

cos( ) *load const lf lf T tT T t*

Where *Tlf* is the amplitude of the load torque oscillation caused by the load mechanical

*lf* = 2*πflft*. Also, the torque relates to the rotational frequency (

 ( ) *<sup>r</sup> motor load*

 cos *<sup>r</sup> motor const lf lf*

*J T tT T t dt*

*r lf r lf*

which varies according to a sinusoidal signal. Then, the integration of mechanical speed

*t tt*

 

*T t t J*

*lf*

*lf r lf r*

*T*

*J*

*Tt T t T t J dt*

*d t*

(9)

*t dt Const*

 

(12)

(13)

(11)

*<sup>r</sup>*0 and a component

 cos . *lf r lf T*

*J*

 

Where *flf* is the characteristic frequency of the load fault, *f*1 is the supply frequency,

*lf* in (8)

in the stator current spectrum can be expressed by:

periodically at a characteristic frequency

Where *J* is the total inertia of machine and load. Thus:

*d t*

 <sup>1</sup> cos and *<sup>r</sup> lf lf*

<sup>0</sup> sin *lf*

<sup>2</sup> <sup>0</sup> cos

Observing (12), the mechanical speed consists of a constant component

*<sup>r</sup>*(*t*):

*T t*

rotational frequency.

failure and

Then

expressed by:

In steady state, *Tmotor* = *Tconst* and:

*d t*

results in the mechanical rotor position

*dt J*

When mechanical load is attached to the rotor demanding torque the rotor speed decreases. In this turn, the slip speed increases and also the current in the rotor to provide more torque. As the load increases, the rotor continues having reduced its speed relative to synchronous speed. This phenomenon is known as motor slip, denoted by *s*.

$$s = \frac{\left(N\_s - N\_r\right)}{N\_s} \tag{4}$$

Another important definition refers to slip frequency. The frequency induced in the rotor is correctly set to slip frequency and is given by:

$$f\_2 = \left(N\_s - N\_r\right) \cdot p \tag{5}$$

As noted, the rotor frequency is directly proportional to the slip speed and the number of pair of poles. Thus:

$$s \cdot N\_S = N\_S - N\_r \text{ and } \ p \cdot N\_S = f\_1 \\ \text{then } f\_2 = s \cdot f\_1 \tag{6}$$

This is a very important result for MCSA once the current frequency is rotor slip function. The characteristic frequencies are well known. The patterns of these failures are presented below.

The stator line current spectral analysis has been widely used recently for the purpose of diagnosing problems in induction machine. This technique is known as MCSA and the current signal can be easily acquired from one phase of the motor supply without interruption of the machine operation. In MCSA the current signal is processed in order to obtain the frequency spectrum usually referred to current signature. By means of the motor signature, one can identify the magnitude and frequency of each individual component that constitutes the signal of the motor. This characteristic allows identifying patterns in the signature in order to differentiate healthy motors from unhealthy ones. Mechanical failures such as rotor imbalance, shaft misalignment, broken bars and bearing problems are common in induction machines applications and commonly discussed or presented when talking about MCSA. Another very important cause of poor functioning of induction motor is load mechanical failure. When a mechanical failure is present either in the motor, or in the transmission system or in the attached load, the frequency spectrum of the line current, in other words, the motor signature, becomes different from that of a non-faulted machine.

When a mechanical failure occurs in the attached load of an induction motor, multiples rotational frequencies appear in the stator current due to the load torque oscillation (Benbouzid, 2000). These frequencies are related to the constructive characteristics of the load and the transmission system, and an abnormal value of a given frequency expresses a specific failure, and more, the severity of this failure. The frequency component that appears in the stator current spectrum can be expressed by:

$$f\_{\rm lf} = f\_1 \pm \kappa f\_r \tag{7}$$

Where *flf* is the characteristic frequency of the load fault, *f*1 is the supply frequency, is the constant resulting from the drive train constructive characteristics and *fr* is the motor rotational frequency.

It is known that when a mechanical failure has developed in the load, it generates an additional torque (*T1f*). Thus, the overall load torque (*Tload*) can be represented by an invariable component (*Tconst*) plus this additional variable component which varies periodically at a characteristic frequency *lf* in (8)

$$T\_{load}\left(t\right) = T\_{const} + T\_{lf} \cos(\alpha\_{lf} t) \tag{8}$$

Where *Tlf* is the amplitude of the load torque oscillation caused by the load mechanical failure and *lf* = 2*πflft*. Also, the torque relates to the rotational frequency (*<sup>r</sup>*) can be expressed by:

$$T\left(t\right) = T\_{motor}\left(t\right) - T\_{load}(t) = J \frac{d o\_r\left(t\right)}{dt} \tag{9}$$

Where *J* is the total inertia of machine and load. Thus:

$$J\frac{d\phi\_r(t)}{dt} = T\_{motor}\left(t\right) - T\_{const} - T\_{lf}\cos\left(\alpha\_{lf}t\right) \tag{10}$$

In steady state, *Tmotor* = *Tconst* and:

$$\frac{d o o\_r(t)}{dt} = -\frac{1}{J} \left( T\_{\circ f} \cos \left( o\_{\circ f} t \right) \right) \\ \text{and} \quad o\_r(t) = -\frac{T\_{\circ f}}{J} \left[ \cos \left( o\_{\circ f} \right) dt + \text{Const.} \right. \tag{11}$$

Then

496 Induction Motors – Modelling and Control

sufficient to overcome friction and ventilation. The difference between the rotor speed (*Nr*)

When mechanical load is attached to the rotor demanding torque the rotor speed decreases. In this turn, the slip speed increases and also the current in the rotor to provide more torque. As the load increases, the rotor continues having reduced its speed relative to synchronous

> *s r s*

*N N*

*N*

Another important definition refers to slip frequency. The frequency induced in the rotor is

As noted, the rotor frequency is directly proportional to the slip speed and the number of

This is a very important result for MCSA once the current frequency is rotor slip function. The characteristic frequencies are well known. The patterns of these failures are presented

The stator line current spectral analysis has been widely used recently for the purpose of diagnosing problems in induction machine. This technique is known as MCSA and the current signal can be easily acquired from one phase of the motor supply without interruption of the machine operation. In MCSA the current signal is processed in order to obtain the frequency spectrum usually referred to current signature. By means of the motor signature, one can identify the magnitude and frequency of each individual component that constitutes the signal of the motor. This characteristic allows identifying patterns in the signature in order to differentiate healthy motors from unhealthy ones. Mechanical failures such as rotor imbalance, shaft misalignment, broken bars and bearing problems are common in induction machines applications and commonly discussed or presented when talking about MCSA. Another very important cause of poor functioning of induction motor is load mechanical failure. When a mechanical failure is present either in the motor, or in the transmission system or in the attached load, the frequency spectrum of the line current, in other words, the motor signature, becomes different from that of a non-faulted machine.

When a mechanical failure occurs in the attached load of an induction motor, multiples rotational frequencies appear in the stator current due to the load torque oscillation (Benbouzid, 2000). These frequencies are related to the constructive characteristics of the load and the transmission system, and an abnormal value of a given frequency expresses a

*s*

*N NN Slip s r* (3)

(4)

<sup>2</sup> *s r f NNp* (5)

*f s f* (6)

and the synchronous speed (*Ns*) is named as slip speed (*NSlip*):

speed. This phenomenon is known as motor slip, denoted by *s*.

and *SSr sN N N* <sup>1</sup> then *<sup>S</sup> pN f* 2 1

correctly set to slip frequency and is given by:

pair of poles. Thus:

below.

$$
\alpha\_r(t) = -\frac{T\_{\text{lf}}}{J\alpha\_{\text{fl}}} \sin \left(\alpha\_{\text{fl}} t \right) + \alpha\_{r0} \tag{12}
$$

Observing (12), the mechanical speed consists of a constant component *<sup>r</sup>*0 and a component which varies according to a sinusoidal signal. Then, the integration of mechanical speed results in the mechanical rotor position *<sup>r</sup>*(*t*):

$$\theta\_r(t) = \frac{T\_{\!\!\!f}}{\!\!\!\!/\phi\_{\!\!\!f}^2} \cos\left(\alpha\_{\!\!\!\!/\!f}t\right) + \alpha\_{r0}t \tag{13}$$

The rotor position oscillations act on the magneto motive force (MMF). In normal conditions, the MMF referred to as the rotor (*Fr* (*R*) ) can be expressed by (14).

$$F\_r^{\{K\}}\left(\theta',t\right) = F\_r \cos(p\theta' - s\alpha\_1 t) \tag{14}$$

Predictive Maintenance by Electrical Signature Analysis to Induction Motors 499

 

 

(21)

(22)

 

> 

*t t tt <sup>s</sup>* cos 1 1 *s r* cos cos *lf* (20)

 

(*t*). It is important to explain

*,t*), then all phase

(23)

(24)

, , , and *s r Bt F tF t*

*BtB p t B p t t*

> 

 

*d t*

*dt* 

11 11

 

> 

> >

is null.

 (25)

(26)

1 1 sin sin cos *<sup>s</sup> <sup>s</sup> <sup>r</sup> lf*

 

 

 

*t tt*

*s s r lf*

, cos *<sup>s</sup>* 1 1 *s r* cos cos *lf* (19)

(*t*) is obtained by the integration of the flux density *B*(

that the winding structure affects only the flux amplitude and not its frequencies. Thus:

 

*<sup>s</sup>*

 <sup>1</sup> *s s*

11 11

sin cos sin

<<1. Finally:

Stator Rotor

 sin 1 1 sin cos *st rt st s rt lf i i It I t I t t*

Notice that the term *ist* results from stator MMF and it is not influenced by the torque oscillation, and the term *irt* results from the rotor MMF and presents phase modulation due

<sup>1</sup> cos *lf jt t*

 

Considering the component *irt* with phase modulation in (14) given in its complex form:

*rt rt i t Ie*

*lf r lf lf*

*I t t t t*

( ) sin sin cos

*d t t tt*

1

 

*Vt d t*

*R R dt*

 

 

> 

*I t*

 

*s s s*

 

*R R R*

The relationship between the flux and the current is given by the equation (21).

*V t RI t*

modulation existing in the flux density also exists in the flux

 

Where *Rs* is the stator resistance. Thus,

*dt*

 

to torque oscillations. And also, when the motor is healthy

With the last term being neglected once

*V t*

As the flux

And as:

Where *'* is the mechanical angle in the rotor reference frame, *p* is the number of pole pairs, 1 is the synchronous speed, *s* is the motor slip, and *Fr* is the rotor MMF.

Figure 4 shows a phasorial diagram for the rotor MMF (R axes) referred to the stator frame (S axes), the difference can be expressed by the angle '.

**Figure 4.** Phasorial diagram of the rotor MMF referred to the stator frame

According the Figure 4 and Equation (14) and replacing (13) in (15), it results in:

$$\theta = \theta' + \theta\_r \text{ and } \,\, F\_r\{\theta, t\} = F\_r \cos[p(\theta - \theta\_r) - s\alpha\_1 t] \tag{15}$$

S

$$F\_r\left(\theta, t\right) = F\_r \cos\left(p\theta - p o\_{r0} t - \frac{p T\_{\rm ff}}{J o\_{\rm ff}^2} \cos\left(o\_{\rm ff} t\right) - s o\_1 t\right) \tag{16}$$

Doing <sup>2</sup> *lf lf pT J* and using the relation 0 1 1 *<sup>r</sup> s p* , it produces:

$$F\_r\left(\theta, t\right) = F\_r \cos\left(p\theta - a\_1 t - \beta \cos\left(a\_{\parallel} t\right)\right) \tag{17}$$

Where is the modulation index and generally <<1.

At this point, it is important to notice that the term cos *lft* means a phase modulation. The failure does not have direct effect on stator MMF which can be expressed by:

$$F\_s\left(\theta, t\right) = F\_s \cos\left(p\theta - \alpha\_1 t - \varphi\_s\right) \tag{18}$$

Where *<sup>s</sup>* is the initial phase between rotor and stator MMFs.

Supposing for the sake of simplicity the value of the air gap permeance constant (because slotting effects and eccentricity were neglected), the air gap flux density B can be expressed by the product of total MMF and :

Predictive Maintenance by Electrical Signature Analysis to Induction Motors 499

$$B\left(\theta, t\right) = \left[F\_s\left(\theta, t\right) + F\_r\left(\theta, t\right)\right] \Lambda \text{ and}$$

$$B\left(\theta, t\right) = B\_s \cos\left(p\theta - \alpha\_1 t - \varphi\_s\right) + B\_r \cos\left(p\theta - \alpha\_1 t - \beta \cos\left(\alpha\_{l\_f} t\right)\right) \tag{19}$$

As the flux (*t*) is obtained by the integration of the flux density *B*(*,t*), then all phase modulation existing in the flux density also exists in the flux (*t*). It is important to explain that the winding structure affects only the flux amplitude and not its frequencies. Thus:

$$\rho \rho \begin{pmatrix} t \end{pmatrix} = \rho\_s \cos \left( o\_1 t - \phi\_s \right) + \rho\_r \cos \left( o\_1 t - \beta \cos \left( o\_{\uparrow} t \right) \right) \tag{20}$$

The relationship between the flux and the current is given by the equation (21).

$$V\left(t\right) = R\_s I\left(t\right) + \frac{d\rho\left(t\right)}{dt} \tag{21}$$

Where *Rs* is the stator resistance. Thus,

$$I(t) = \frac{V\left(t\right)}{R\_s} - \frac{1}{R\_s} \frac{d\phi\left(t\right)}{dt} \tag{22}$$

And as:

498 Induction Motors – Modelling and Control

Where 

conditions, the MMF referred to as the rotor (*Fr*

(S axes), the difference can be expressed by the angle

( )

' and *<sup>r</sup>*

Doing <sup>2</sup>

*pT J*

Where 

Where  *lf lf*

by the product of total MMF and :

The rotor position oscillations act on the magneto motive force (MMF). In normal

(*R*)

<sup>1</sup> ', cos( ' ) *<sup>R</sup> r r F t F p st*

 

> '.

> > R

*'* is the mechanical angle in the rotor reference frame, *p* is the number of pole pairs,

Figure 4 shows a phasorial diagram for the rotor MMF (R axes) referred to the stator frame

*r*

<sup>1</sup> , cos[ ( ) ] *rr r F t F p st*

 0 1 <sup>2</sup> , cos cos *lf r r r lf*

*pT F t F p pt t st*

*F tF p t t <sup>r</sup>*

<sup>1</sup> , cos *ss s F tF p t*

Supposing for the sake of simplicity the value of the air gap permeance constant (because slotting effects and eccentricity were neglected), the air gap flux density B can be expressed

<<1.

 

*lf*

 

 *s p* , it produces:

 

*J*

 

 , cos *<sup>r</sup>* <sup>1</sup> cos *lf* (17)

> 

 

According the Figure 4 and Equation (14) and replacing (13) in (15), it results in:

 

and using the relation 0 1 1 *<sup>r</sup>*

The failure does not have direct effect on stator MMF which can be expressed by:

*<sup>s</sup>* is the initial phase between rotor and stator MMFs.

'

1 is the synchronous speed, *s* is the motor slip, and *Fr* is the rotor MMF.

**Figure 4.** Phasorial diagram of the rotor MMF referred to the stator frame

is the modulation index and generally

At this point, it is important to notice that the term

) can be expressed by (14).

S

 

> 

(14)

(15)

(16)

cos *lft* means a phase modulation.

(18)

$$\begin{split} \frac{d\rho(t)}{dt} &= -\alpha\_1 \rho\_s \sin\left(\alpha\_1 t + \rho\_s\right) - \alpha\_1 \rho\_r \sin\left(\alpha\_1 t + \beta \cos\left(\alpha\_\circ t\right)\right) \\ &+ \alpha\_\circ \beta \rho\_r \sin\left(\alpha\_1 t + \beta \cos\left(\alpha\_\circ t\right)\right) \sin\left(\alpha\_\circ t\right) \end{split} \tag{23}$$

With the last term being neglected once <<1. Finally:

$$I(t) = \underbrace{\frac{V(t)}{R\_s} + \frac{1}{R\_s} \alpha\_1 \wp\_s \sin\left(\alpha\_1 t + \wp\_s\right)}\_{\text{Stator}} + \underbrace{\frac{1}{R\_s} \alpha\_1 \wp\_r \sin\left(\alpha\_1 t + \beta \cos\left(\alpha\_{\left[t\right]} t\right)\right)}\_{\text{Rotor}}\tag{24}$$

$$I\left(t\right) = \underbrace{I\_{st}\sin\left(\alpha\_1 t + \rho\_s\right)}\_{i\_{st}} + \underbrace{I\_{rt}\sin\left(\alpha\_1 t + \beta\cos\left(\alpha\_{l\uparrow}t\right)\right)}\_{i\_{st}}\tag{25}$$

Notice that the term *ist* results from stator MMF and it is not influenced by the torque oscillation, and the term *irt* results from the rotor MMF and presents phase modulation due to torque oscillations. And also, when the motor is healthy is null.

Considering the component *irt* with phase modulation in (14) given in its complex form:

$$i\_{rt}\left(t\right) = I\_{rt}e^{j\left(\alpha\_l t + \beta \cos\left(\alpha\_{l\circ} t\right)\right)}\tag{26}$$

Applying a Discrete Fourier Transform (DFT) in (26), as well known from communications theory, it can be expressed by (27).

$$I\_{rt}(f) = I\_{rt} \sum\_{n = -\infty}^{\infty} j^n I\_n \left(\beta \right) \delta \left( f - \left( f\_1 + nf\_{lf} \right) \right) \tag{27}$$

Predictive Maintenance by Electrical Signature Analysis to Induction Motors 501

(31)

 

(32)

is the angular frequency

 

> 

 ± 

  (33)

(34)

*<sup>f</sup>*)/2*π*, the

 

 

Let's consider now the presence of a mechanical fault in the drive train, resulting in the appearance of motor torque oscillations accompanied by surges of speed and slip, which in

For simplicity, it is considered that the failure cause only an amplitude modulation on the stream of the stator by deleting the effect on stage. It could also prove that phase modulations, in function of torque oscillations, appear in current as amplitude modulations by processed result from similar functions to the Bessel functions. The modulated current *iL*

*Lf f*

can be used as information for the diagnosis of the condition of the machine.

*pt p t t t*

<sup>0</sup> cos 2 cos 2

*MI i t <sup>t</sup> <sup>t</sup>*

cos cos

 

2 6 6

2 66

 

*f f*

2cos cos 6

 

spectrum of instantaneous power contains an additional component directly related to the modulation caused by failure. This component is named as **Characteristic Component** and

The following simulation which considers a motor current modulation originated by an alleged mechanical failure whose frequency characteristic is of 15 Hz. Note the Figure 5 that the spectrum of voltage does not have any type of modulation, since the current spectrum has lateral bands apart from 15 Hz fundamental's (located at 60 Hz). The instantaneous power spectrum has the fundamental frequency in 120 Hz with modulations of 15 Hz at 105 and 135 Hz, besides presenting the fault feature component

 

*<sup>f</sup>* is the angular frequency of the failure.

 

/2*π* and the lateral bands in (2

*f*

*t*

 ,0 2 cos <sup>6</sup> *L L it I t*

 <sup>0</sup> ,0 cos 2 cos 6 6 *LL L LL L p v ti t V I t*

Where *VLL* and *IL* are the RMS values of voltage and current line,

is the phase angle of the motor load.

turn result in modulations in the current spectrum.

1 cos

*L*

*LL L*

*MV I*

*L L f*

*iit M t*

,0

,0

Where *M* is the index modulation and

The expression of instant power results in:

Besides the fundamental component 2

and 

can be expressed by:

in isolated 15 Hz.

Where *Jn* denotes the *n*th-order Bessel function of first kind and (*f*) is the Dirac delta function. Since is so small, the Bessel functions of order *n* ≥ 2 can be neglected.

Finally, the Power Spectral Density (PSD) of the stator current, considering the approximations used, is given by:

$$\left| I\left( f \right) \right| = \left( I\_{st} + I\_{rt} I\_0 \left( \beta \right) \right) \delta \left( f - f\_1 \right) + I\_{rt} I\_1 \left( \beta \right) \delta f - \left( f\_1 \pm f\_{\left| f \right|} \right) \tag{28}$$

It is clear that the phase modulation leads to sideband components of the fundamental at *f*<sup>1</sup> *flf* as it happens in an amplitude modulation. Considering all the development accomplished in this section and the result in (28), the load failure patterns can be presented.

### **4.3. Voltage Signature Analysis (VSA)**

The technique of Voltage Signature Analysis follows the same strategy of analysis of the current signature; however the signal is analyzed from the voltage supply of the motor. This technique is most often used in analysis of generating units. In the case of motors, it can be usefully employed in cases of problems from the motor power and the analysis of electric stator imbalance in conjunction with the analysis of the current signature. It can be used also to know the origin of certain components in the power spectrum, that is, it can be used to infer if the source of the component comes from the mains or has its origin in the array itself.

### **4.4. Instantaneous Power Signature Analysis (IPSA)**

The analysis of the instantaneous power is another failure analysis technique based on spectral analysis. The big difference between this technique and MCSA and VSA is that it considers the information present in voltage and current signals of a motor phase concurrently and demodulated fault component appears under the name of Characteristic Frequency. Considering an ideal three phase system, instant power *p*(*t*) is given by:

$$p(t) = v\_{LL}(t)i\_L(t) \tag{29}$$

Where *vLL* is the voltage between two terminals of the motor and *iL* is the current entering one of these terminals. And, a motor under normal conditions, i.e. without breakdowns, and constant velocity, one has:

$$
v\_{LL}\left(t\right) = \sqrt{2}V\_{LL}\cos\left(\alpha t\right)\tag{30}$$

Predictive Maintenance by Electrical Signature Analysis to Induction Motors 501

$$i\_{L,0}\left(t\right) = \sqrt{2}I\_L \cos\left(\alpha t - \varphi - \frac{\pi}{6}\right) \tag{31}$$

$$p\_0 = v\_{LL}\left(t\right)i\_{L,0}\left(t\right) = V\_{LL}I\_L\left[\cos\left(2\alpha t - \varphi - \frac{\pi}{6}\right) + \cos\left(\varphi + \frac{\pi}{6}\right)\right] \tag{32}$$

Where *VLL* and *IL* are the RMS values of voltage and current line, is the angular frequency and is the phase angle of the motor load.

500 Induction Motors – Modelling and Control

function. Since

presented.

array itself.

constant velocity, one has:

theory, it can be expressed by (27).

approximations used, is given by:

**4.3. Voltage Signature Analysis (VSA)** 

**4.4. Instantaneous Power Signature Analysis (IPSA)** 

Applying a Discrete Fourier Transform (DFT) in (26), as well known from communications

( ) <sup>1</sup> *<sup>n</sup> rt rt <sup>n</sup> <sup>n</sup> lf I f I j J f f nf* 

is so small, the Bessel functions of order *n* ≥ 2 can be neglected.

Finally, the Power Spectral Density (PSD) of the stator current, considering the

*If I IJ f f IJ f f f st rt* 0 11 1

It is clear that the phase modulation leads to sideband components of the fundamental at *f*<sup>1</sup> *flf* as it happens in an amplitude modulation. Considering all the development accomplished in this section and the result in (28), the load failure patterns can be

The technique of Voltage Signature Analysis follows the same strategy of analysis of the current signature; however the signal is analyzed from the voltage supply of the motor. This technique is most often used in analysis of generating units. In the case of motors, it can be usefully employed in cases of problems from the motor power and the analysis of electric stator imbalance in conjunction with the analysis of the current signature. It can be used also to know the origin of certain components in the power spectrum, that is, it can be used to infer if the source of the component comes from the mains or has its origin in the

The analysis of the instantaneous power is another failure analysis technique based on spectral analysis. The big difference between this technique and MCSA and VSA is that it considers the information present in voltage and current signals of a motor phase concurrently and demodulated fault component appears under the name of Characteristic

Where *vLL* is the voltage between two terminals of the motor and *iL* is the current entering one of these terminals. And, a motor under normal conditions, i.e. without breakdowns, and

2 cos *LL LL vt V t*

*LL L pt v ti t* (29)

(30)

Frequency. Considering an ideal three phase system, instant power *p*(*t*) is given by:

*rt*

 

(27)

  *lf* (28)

(*f*) is the Dirac delta

Where *Jn* denotes the *n*th-order Bessel function of first kind and

Let's consider now the presence of a mechanical fault in the drive train, resulting in the appearance of motor torque oscillations accompanied by surges of speed and slip, which in turn result in modulations in the current spectrum.

For simplicity, it is considered that the failure cause only an amplitude modulation on the stream of the stator by deleting the effect on stage. It could also prove that phase modulations, in function of torque oscillations, appear in current as amplitude modulations by processed result from similar functions to the Bessel functions. The modulated current *iL* can be expressed by:

$$\begin{split} \dot{i}\_{L} &= i\_{L,0} \left( t \right) \Big[ 1 + M \cos \left( \alpha\_{f} t \right) \Big] \\ &= i\_{L,0} \left( t \right) + \frac{M I\_{L}}{\sqrt{2}} \left\{ \cos \left[ \left( \phi + \alpha\_{f} \right) t - \phi - \frac{\pi}{6} \right] + \cos \left[ \left( \phi + \alpha\_{f} \right) t - \phi - \frac{\pi}{6} \right] \right\} \end{split} \tag{33}$$

Where *M* is the index modulation and *<sup>f</sup>* is the angular frequency of the failure. The expression of instant power results in:

$$p(t) = p\_0(t) + \frac{MV\_{LL}I\_L}{2} \left\{ \cos\left[ \left( 2\alpha + \alpha\_f \right)t - \varphi - \frac{\pi}{6} \right] + \cos\left[ \left( 2\alpha - \alpha\_f \right)t - \varphi - \frac{\pi}{6} \right] + \dotsb \right\} \tag{34}$$

$$\begin{split} +2\cos\left(\varphi + \frac{\varphi}{6}\right)\cos\left(\alpha\_f t\right) \end{split} \tag{35}$$

Besides the fundamental component 2/2*π* and the lateral bands in (2 ± *<sup>f</sup>*)/2*π*, the spectrum of instantaneous power contains an additional component directly related to the modulation caused by failure. This component is named as **Characteristic Component** and can be used as information for the diagnosis of the condition of the machine.

The following simulation which considers a motor current modulation originated by an alleged mechanical failure whose frequency characteristic is of 15 Hz. Note the Figure 5 that the spectrum of voltage does not have any type of modulation, since the current spectrum has lateral bands apart from 15 Hz fundamental's (located at 60 Hz). The instantaneous power spectrum has the fundamental frequency in 120 Hz with modulations of 15 Hz at 105 and 135 Hz, besides presenting the fault feature component in isolated 15 Hz.

**Figure 5.** Fault Simulation in 15 Hz and the respective spectra of voltage, current and instant power

### **4.5. Enhanced Park's Vector Approach (EPVA)**

The first research involving the use of Park's vector method for the diagnosis of failures in motors such as short circuit between turns, airgap eccentricity and broken bars, etc.(Cardoso & Saraiva, 1993). At first, the proposed damage detection was based only on the distortion suffered by circle of Park on the emergence and on the aggravation of the damage. More recently, the technique has been improved (now named EPVA) and may be described as following steps. The three phases of currents in a motor can be described by:

$$\dot{\mathbf{u}}\_A = \mathbf{i}\_M \cos(\alpha t - \alpha) \tag{35}$$

Predictive Maintenance by Electrical Signature Analysis to Induction Motors 503

<sup>6</sup> ( ) <sup>2</sup>

 (39)

*i i sen t Q M*

1 1

2 2 *Q BC i ii* (38)

21 1 and

<sup>6</sup> cos( )and <sup>2</sup>

 

Graphically, ideal conditions generate a perfect Park circle centered at the origin of

Under abnormal conditions of operation, i.e. when the emergence of mechanical or electrical failure, the previous equations are no longer valid and the circle of Park passes to suffer distortions. As these changes in the circle of Park are difficult to be measured, was proposed by EPVA method of observation of spectrum of Park's vector module. The advantage of EPVA technique combines the simplicity of the previous method (analysis of the Park's circle) with spectral analysis capability. In addition, the fundamental component of the motor power is automatically subtracted from the spectrum by Park transformation, causing the failure characteristics components appear prominently. The most important point is the fact that the technique considers the three phases of current, generating a more significant spectrum by encompass information from three phases. This feature is extremely useful in cases where failure can only be detected if considered the three phases. This is the case of

When there is an unbalanced voltage supply, the motor currents can be represented by:

cos cos 3 3 *Bd d i i ii t i t* 

cos cos 3 3 *Cd d i i ii t i t* 

 

 

cos( ) cos( ) *<sup>A</sup> d di i ii t i t*

 

2 2

2 2

 

 

(41)

(42)

(40)

<sup>366</sup> *D ABC i iii* 

*i it D M*

coordinates, as shown in Figure 6.

**Figure 6.** Signals in time and Park circle

unbalanced electric motor fuelled in open loop.

Ideally:

$$\dot{i}\_B = \dot{i}\_M \cos\left(\alpha t - \alpha - \frac{2\pi}{3}\right) \tag{36}$$

$$\dot{i}\_{\mathcal{C}} = \dot{i}\_{M}\cos\left(\alpha t - \alpha + \frac{2\pi}{3}\right) \tag{37}$$

Where *iM* is the peak value of the supply current, is the angular frequency in rad/s, the is the initial phase angle in rad, *t* is the time variable; and *iA*, *iB* and *iC* are respectively the currents in the phases A, B and C. The current components of the Park's vector are given by:

Predictive Maintenance by Electrical Signature Analysis to Induction Motors 503

$$i\_D = \left(\frac{\sqrt{2}}{\sqrt{3}}\right)i\_A - \left(\frac{1}{\sqrt{6}}\right)i\_B - \left(\frac{1}{\sqrt{6}}\right)i\_C \text{ and } i\_Q = \left(\frac{1}{\sqrt{2}}\right)i\_B - \left(\frac{1}{\sqrt{2}}\right)i\_C\tag{38}$$

Ideally:

502 Induction Motors – Modelling and Control

**Figure 5.** Fault Simulation in 15 Hz and the respective spectra of voltage, current and instant power

The first research involving the use of Park's vector method for the diagnosis of failures in motors such as short circuit between turns, airgap eccentricity and broken bars, etc.(Cardoso & Saraiva, 1993). At first, the proposed damage detection was based only on the distortion suffered by circle of Park on the emergence and on the aggravation of the damage. More recently, the technique has been improved (now named EPVA) and may be described as

following steps. The three phases of currents in a motor can be described by:

*ii t A M* cos

cos

cos

*ii t C M*

 

 

the initial phase angle in rad, *t* is the time variable; and *iA*, *iB* and *iC* are respectively the currents in the phases A, B and C. The current components of the Park's vector are given by:

*ii t B M*

2

3

2

 

3

(35)

(36)

(37)

is the angular frequency in rad/s,

the is

**4.5. Enhanced Park's Vector Approach (EPVA)** 

Where *iM* is the peak value of the supply current,

$$i\_D = \left(\frac{\sqrt{6}}{2}\right) i\_M \cos(\alpha t - \alpha) \text{and } i\_Q = \left(\frac{\sqrt{6}}{2}\right) i\_M \sin(\alpha t - \alpha) \tag{39}$$

Graphically, ideal conditions generate a perfect Park circle centered at the origin of coordinates, as shown in Figure 6.

**Figure 6.** Signals in time and Park circle

Under abnormal conditions of operation, i.e. when the emergence of mechanical or electrical failure, the previous equations are no longer valid and the circle of Park passes to suffer distortions. As these changes in the circle of Park are difficult to be measured, was proposed by EPVA method of observation of spectrum of Park's vector module. The advantage of EPVA technique combines the simplicity of the previous method (analysis of the Park's circle) with spectral analysis capability. In addition, the fundamental component of the motor power is automatically subtracted from the spectrum by Park transformation, causing the failure characteristics components appear prominently. The most important point is the fact that the technique considers the three phases of current, generating a more significant spectrum by encompass information from three phases. This feature is extremely useful in cases where failure can only be detected if considered the three phases. This is the case of unbalanced electric motor fuelled in open loop.

When there is an unbalanced voltage supply, the motor currents can be represented by:

$$\dot{\mathbf{i}}\_A = \dot{\mathbf{i}}\_d \cos(\alpha t - \alpha\_d) + \dot{\mathbf{i}}\_i \cos(\alpha t - \beta\_i) \tag{40}$$

$$\dot{\mathbf{u}}\_B = \dot{\mathbf{u}}\_d \cos\left(\alpha t - \alpha\_d - \frac{2\pi}{3}\right) + \dot{\mathbf{u}}\_i \cos\left(\alpha t - \beta\_i + \frac{2\pi}{3}\right) \tag{41}$$

$$i\_C = i\_d \cos\left(\alpha t - \alpha\_d + \frac{2\pi}{3}\right) + i\_i \cos\left(\alpha t - \beta\_i - \frac{2\pi}{3}\right) \tag{42}$$

Where *id* is the maximum value of the current direct sequence, *ii* is the maximum value of reverse sequence current, *d* is the current initial phase angle direct sequence in rad, and *<sup>i</sup>* is the initial phase angle reverse sequence current in rad. In the Park's vector:

$$i\_D = \left(\frac{\sqrt{3}}{\sqrt{2}}\right) \left(i\_d \cos\left(\alpha t - \alpha\_d\right) + i\_i \cos\left(\alpha t - \beta\_i\right)\right) \text{and} \quad i\_Q = \left(\frac{\sqrt{3}}{\sqrt{2}}\right) \left(i\_d \sin\left(\alpha t - \alpha\_d\right) - i\_i \sin\left(\alpha t - \beta\_i\right)\right) \text{(43)}$$

And the square of the Park's vector module is given by:

$$\left|\dot{i}\_D + j\dot{i}\_Q\right|^2 = \left(\frac{3}{2}\right)\left(\dot{i}\_d^{\;\;2} + \dot{i}\_i^{\;\;2}\right) + 3\dot{i}\_d\dot{i}\_i\cos(2\alpha t - \alpha\_d - \beta\_i) \tag{44}$$

Predictive Maintenance by Electrical Signature Analysis to Induction Motors 505

in which *taps* was inserted to the gradual introduction of imbalance in power depending on the insertion of short. Figure 9a presents the characteristics of the motor and the *taps* as to

Tests have been made in the conditions of non-faulted motor (no imbalance) and five severities of short circuit generating imbalances of 1.2 V, 1.8 V, V, V 5.4 6.7 and 8.5 V. Figure 9b shows the overlap of the spectra of the motor in normal condition (in red) and motor in the worst condition of imbalance (8.5 V) highlighting the component twice the power

**Figure 9.** Featured for the inserted short-circuit and spectrum of Park vector module

The current trend curve (shown in Figure 10) demonstrates a general growth of electric unbalance component EPVA with increasing the short circuit. Each three points of the curve represent a condition of normal severity, starting and advancing to severity 1 (1.2 V), 2 (1.8 V), 3 (5.4 V), 4 (6.7 v) and 5 (V 8.5). Severity 4 presents amplitude less than Severity 3 due to a change in the equilibrium condition of input voltage shown in the trend curve in tension (shown in Figure 10), being thus possible to separate the effects of those supply imbalances

(a) (b)

**Figure 10.** Trend curve to the imbalance component: (a) for current and (b) for voltage

the short are introduced.

frequency in the spectrum of Park vector module.

caused by short circuits and other anomalies.

Now, just applying the FFT to the square of the Park's vector module and observe that this is composed by a DC level plus one additional term located at twice the supply frequency. It is exactly this additional term that indicates the emergence and intensification of stator electrical asymmetries. Let's the example shown in Figure 7a which is considered an unbalanced feed; and also, the Park circle passes to resemble an ellipse and arises in the spectrum the component located at twice the supply frequency, as shown in Figure 7 b and c. Thus, the whole process can be represented by the elements of Figure 8.

**Figure 7.** Imbalance between the phases, Park circle distorted and presence of the component at twice the supply frequency

**Figure 8.** Block Diagram of the EPVA technique

This demonstrates the effectiveness of the component located at twice the supply frequency (in this case 120 Hz) of the EPVA monitoring to diagnosis short circuit between turns. The test procedure was the following: used the Marathon motor failures Simulator Spectra Quest in which *taps* was inserted to the gradual introduction of imbalance in power depending on the insertion of short. Figure 9a presents the characteristics of the motor and the *taps* as to the short are introduced.

504 Induction Motors – Modelling and Control

<sup>3</sup>

And the square of the Park's vector module is given by:

2 *D d di i i i t it* 

cos cos and

 

reverse sequence current,

Figure 8.

the supply frequency

**Motor Acquisition**

**Figure 8.** Block Diagram of the EPVA technique

Where *id* is the maximum value of the current direct sequence, *ii* is the maximum value of

<sup>2</sup> <sup>3</sup> 2 2 3 cos(2 ) <sup>2</sup> *D Q d i di d i i ji i i i i t*

Now, just applying the FFT to the square of the Park's vector module and observe that this is composed by a DC level plus one additional term located at twice the supply frequency. It is exactly this additional term that indicates the emergence and intensification of stator electrical asymmetries. Let's the example shown in Figure 7a which is considered an unbalanced feed; and also, the Park circle passes to resemble an ellipse and arises in the spectrum the component located at twice the supply frequency, as shown in Figure 7 b and c. Thus, the whole process can be represented by the elements of

**Figure 7.** Imbalance between the phases, Park circle distorted and presence of the component at twice

This demonstrates the effectiveness of the component located at twice the supply frequency (in this case 120 Hz) of the EPVA monitoring to diagnosis short circuit between turns. The test procedure was the following: used the Marathon motor failures Simulator Spectra Quest

**iA i B**

**i C**

**Park Transformation** **iD**

**iQ**

**Park Vector Computation**

**Module FFT**

 

the initial phase angle reverse sequence current in rad. In the Park's vector:

*d* is the current initial phase angle direct sequence in rad, and

  

(44)

<sup>3</sup> sin sin

 

2 *Q d di i i i t it* 

*<sup>i</sup>* is

(43)

Tests have been made in the conditions of non-faulted motor (no imbalance) and five severities of short circuit generating imbalances of 1.2 V, 1.8 V, V, V 5.4 6.7 and 8.5 V. Figure 9b shows the overlap of the spectra of the motor in normal condition (in red) and motor in the worst condition of imbalance (8.5 V) highlighting the component twice the power frequency in the spectrum of Park vector module.

**Figure 9.** Featured for the inserted short-circuit and spectrum of Park vector module

The current trend curve (shown in Figure 10) demonstrates a general growth of electric unbalance component EPVA with increasing the short circuit. Each three points of the curve represent a condition of normal severity, starting and advancing to severity 1 (1.2 V), 2 (1.8 V), 3 (5.4 V), 4 (6.7 v) and 5 (V 8.5). Severity 4 presents amplitude less than Severity 3 due to a change in the equilibrium condition of input voltage shown in the trend curve in tension (shown in Figure 10), being thus possible to separate the effects of those supply imbalances caused by short circuits and other anomalies.

**Figure 10.** Trend curve to the imbalance component: (a) for current and (b) for voltage
