**2. The mathematical model of the three-phase induction motor in the case of non-sinusoidal supply**

In the literature there are known various mathematical models associated to induction machines fed by static frequency and voltage converters. The majority of these models are based on the association between an induction machine and an equivalent scheme corresponding to the fundamental and a lot of schemes corresponding to the various ν frequencies, corresponding to the Fourier series decomposition of the motor input voltage see Fig. 1 (Murphy & Turnbull, 1988). In this model the skin effect is not considered.

**Figure 1.** Equivalent scheme of the machine supplied through frequency converter: a) for the case of fundamental; b) for the order harmonics (positive or negative sequence).

For the equivalent scheme in Fig.1.a, corresponding to the fundamental, the electrical parameters are defined as:

$$
\mathbf{R}\_{\mathbf{1}(1)} = \mathbf{R}\_1 = \mathbf{R}\_{1n}; \ \mathbf{X}\_{\mathbf{1}(1)} = \mathbf{X}\_1 = \mathbf{a} \mathbf{X}\_{1n};
$$

$$
\mathbf{R}\_{\mathbf{2}(1)} = \mathbf{R}\_2^\cdot = \mathbf{R}\_{2n}^\cdot; \ \mathbf{X}\_{\mathbf{2}(1)}^\cdot = \mathbf{X}\_2^\cdot = \mathbf{a} \mathbf{X}\_{2n}^\cdot;
$$

$$
\mathbf{R}\_{\mathbf{m}(1)} = \mathbf{R}\_m = \mathbf{a}^2 \mathbf{R}\_{\mathbf{m}n}; \ \mathbf{X}\_{\mathbf{m}(1)} = \mathbf{X}\_m = \mathbf{a} \mathbf{X}\_{\mathbf{m}n}; \tag{4}
$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 47

$$\frac{\overset{\text{R}^{\cdot}}{\text{s}^{\cdot}}\_{\text{2}^{\text{(1)}}} = \frac{\overset{\text{R}^{\cdot}}{\text{s}^{\cdot}}\_{\text{2}}}{\text{s}} = \frac{\overset{\text{a}}{\text{a}}}{\text{c}} \overset{\text{R}^{\cdot}}{\text{c}}\_{2n}$$

46 Induction Motors – Modelling and Control

**of non-sinusoidal supply** 

I1(1) I'2(1)

R1(1) X1(1) X'2(1)

I01(1)

Rm(1)

Xm(1)

a) b)

fundamental; b) for the order harmonics (positive or negative sequence).

U1(1) Ue1(1)

parameters are defined as:

<sup>f</sup> Jm k (3)

In the above relations, mf represents the frequency modulation factor, f1 is the fundamental's frequency and fc is the frequency of the control modulating signal. Whereas the harmonic spectrum contains only ν order odd harmonics, in order that (Jmf±k) is odd, an odd J determines an even k and vice versa. The present chapter aims to analyze the behavior of the induction motor when it is supplied through an inverter. The purpose of this study is to develop the theory of three-phase induction machine with a squirrel cage, under the conditions of the non-sinusoidal supply regime to serve as a starting point in improving the methodology of

**2. The mathematical model of the three-phase induction motor in the case** 

In the literature there are known various mathematical models associated to induction machines fed by static frequency and voltage converters. The majority of these models are based on the association between an induction machine and an equivalent scheme corresponding to the fundamental and a lot of schemes corresponding to the various ν frequencies, corresponding to the Fourier series decomposition of the motor input voltage -

see Fig. 1 (Murphy & Turnbull, 1988). In this model the skin effect is not considered.

 2 1 1

**Figure 1.** Equivalent scheme of the machine supplied through frequency converter: a) for the case of

For the equivalent scheme in Fig.1.a, corresponding to the fundamental, the electrical

R R R ; X X aX ; 1 1 1 1n 1 1 1 1n

' '' ' ' ' R R R ; X X aX ; 2 1 2 2n 2 1 2 2n

U1() Ue1()

<sup>2</sup> R R a R ; X X aX ; (4) m 1 m mn m 1 m mn

I1() I'2()

R1() X1() X'2()

I01()

Rm()

 R '2 s

Xm()

R ' s

its constructive-technological design as advantageous economically as possible.

with

In relations (4), R1n, X1n, R'2n, X'2n, Rmn, Xmn represents the values of the parameters R1, X1, R'2, X'2, Rm and Xm in nominal operating conditions (fed from a sinusoidal power supply, rated voltage frequency and load) and

$$\mathbf{a} = \frac{\mathbf{f\_1}}{\mathbf{f\_{1n}}} = \frac{\mathbf{o\_1}}{\mathbf{o\_{1n}}} = \frac{\mathbf{n\_1}}{\mathbf{n\_{1n}}}; \quad \mathbf{c} = \frac{\mathbf{n\_1} - \mathbf{n}}{\mathbf{n\_{1n}}} = \frac{\mathbf{n\_1} - \mathbf{n}}{\mathbf{n\_1}} \cdot \frac{\mathbf{n\_1}}{\mathbf{n\_{1n}}} = \mathbf{s} \cdot \mathbf{a} \tag{5}$$

In the relations (5), f1 and f1n are random frequencies of the rotating magnetic field, and the nominal frequency of the rotating magnetic field respectively. For order harmonics, the scheme from Fig. 1.b is applicable. The slip s(), corresponding to the order harmonic is:

$$\mathbf{s}\_{\left(\mathbf{v}\right)} = \frac{\mathbf{v}\mathbf{n}\_1 \mp \mathbf{n}}{\mathbf{v}\mathbf{n}\_1} = \mathbf{1} \mp \frac{\mathbf{n}}{\mathbf{v}\mathbf{n}\_1} = \mathbf{1} \mp \frac{\mathbf{1}}{\mathbf{v}} \pm \frac{\mathbf{c}}{\mathbf{a}} \frac{\mathbf{1}}{\mathbf{v}} \,' \tag{6}$$

where sign (-) (from the first equality) corresponds to the wave that rotates within the sense of the main wave and the sign (+) in the opposite one. For the case studied in this chapter that of small and medium power machines – the resistances R1() and reactances X1() values are not practically affected by the skin effect. In this case we can write:

$$\mathbf{R}\_{1(\vee)} = \mathbf{R}\_{1(1)} = \mathbf{R}\_1 = \mathbf{R}\_{1\text{n}} \prime \tag{7}$$

$$\mathbf{X}\_{\mathbf{1}(\boldsymbol{\nu})} = \boldsymbol{\alpha}\_{\mathbf{1}(\boldsymbol{\nu})} \cdot \mathbf{L}\_{\mathbf{1}\boldsymbol{\alpha}(\boldsymbol{\nu})} = \mathbf{v} \boldsymbol{\alpha}\_{\mathbf{1}} \mathbf{L}\_{\mathbf{1}\boldsymbol{\alpha}(\boldsymbol{\nu})} \tag{8}$$

where L1σ() is the stator dispersion inductance corresponding to the order harmonic. If it is agreed that the machine cores are linear media (the machine is unsaturated), it results that the inductance can be considered constant, independently of the load (current) and flux, one can say that:

$$\mathbf{L}\_{1\sigma(\mathbf{v})} = \mathbf{L}\_{1\sigma(1)} = \mathbf{L}\_{1\sigma} \tag{9}$$

By replacing the inductance L1σ() expression from relation (9) in relation (8), we obtain:

$$\mathbf{X}\_{\mathbf{1}(\mathbf{v})} = \mathbf{v} \mathbf{o}\_1 \mathbf{L}\_{\mathbf{1}\sigma} = \mathbf{v} \mathbf{X}\_1 = \mathbf{v} \mathbf{a} \mathbf{X}\_{\mathbf{1}\mathbf{n}} \tag{10}$$

For the rotor resistance and rotor leakage reactance, corresponding to the order harmonic, both reduced to the stator the following expressions were established:

$$\overset{\text{R}}{\text{R}}\_{\text{2}(\text{v})}^{\cdot} = \overset{\text{R}}{\text{R}}\_{\text{2}(\text{1})}^{\cdot} = \overset{\text{R}}{\text{R}}\_{\text{2n}}^{\cdot} = \overset{\text{R}}{\text{R}}\_{\text{2n}}^{\cdot} \, \, \tag{11}$$

$$\boldsymbol{\mathcal{X}}\_{2\text{(v)}}^{\cdot} = \mathbf{v} \cdot \boldsymbol{\mathcal{X}}\_{2}^{\cdot} = \mathbf{v} \cdot \mathbf{a} \cdot \boldsymbol{\mathcal{X}}\_{2n}^{\cdot} \tag{12}$$

The magnetization resistance corresponding to the order harmonic, Rm, is given by the relation:

$$\mathbf{R}\_{\text{m}(\text{v})} = \mathbf{k}\_{\text{g}^{\circ}} \cdot \mathbf{v}^{2} \cdot \mathbf{a}^{2} \cdot \mathbf{R}\_{\text{mn}} \tag{13}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 49

 2 2 1(CSF) 1(1) 1( )

1

 <sup>A</sup> 1(CSF) 1 B 1(CSF) <sup>1</sup> <sup>C</sup> 1(CSF) <sup>1</sup> 2 2 u 2 U sin t ; u 2 U sin t ; u 2 U sin t 3 3 , (15)

U is the phase voltage su 1( ) pply corresponding to the order harmonic. Corresponding to the system supply voltages, the current system which go through the stator phases is as follows:

i 2 I sin t

A 1 1 CSF 1 CSF

B 1 1 CSF 1 CSF

C 1 1 CSF 1 CSF

<sup>2</sup> i 2 I sin t

<sup>4</sup> i 2 I sin t

**Figure 2.** The equivalent scheme of the asynchronous motor powered by a static frequency converter.

 

1 CSF 1 CSF

CSF

Power factor in the deforming regime is defined as the ratio between the active power and

P P

If we consider the non-sinusoidal regime, the active power absorbed by the machine P1(CSF) is defined, as in the sinusoidal regime, as the average in a period of the instantaneous power.

 

S UI (19)

1 CSF 1 CSF 1 CSF

where,

where I1(CSF) is given by:

the apparent power, as follows:

The following expression is obtained:

U UU (16)

3

, (17)

3

I II (18)

 2 2 1(CSF) 1(1) 1( )

1

kK" is a coefficient dependent on iron losses and on the magnetic field variation. The magnetization reluctance corresponding to the magnetic field produced by the order harmonic is:

$$\mathbf{X}\_{\rm m(v)} = \mathbf{k}\_{\rm \chi} \mathbf{v} \cdot \mathbf{a} \cdot \mathbf{X}\_{\rm mn} \tag{14}$$

Further the author intends to establish a single mathematical model associated to induction motors, supplied by static voltage and frequency converter, which consists of a single equivalent scheme and which describes the machine operation, according to the presence in the input power voltage of higher time harmonics. For this, the following simplifying assumptions are taken into account:


Under these conditions of non-sinusoidal supply, the asynchronous motor may be associated to an equivalent scheme, corresponding to all harmonics. The scheme operates in the fundamental frequency *f*1(1) and it is represented in Fig. 2. According to this scheme, it can be formally considered that the motors, in the case of supplying through the power frequency converter (the corresponding parameters and the dimensions of this situation are marked with index "CSF") behave as if they were fed in sinusoidal regime at fundamental's frequency, *f*1(1) with the following voltages system:

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 49

$$\mathbf{u}\_{\rm A} = \sqrt{2} \cdot \mathbf{U}\_{\rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm$$

48 Induction Motors – Modelling and Control

assumptions are taken into account:

and isotropic media;

abruptly;

relation:

harmonic is:

The magnetization resistance corresponding to the order harmonic, Rm, is given by the

"

kK" is a coefficient dependent on iron losses and on the magnetic field variation. The magnetization reluctance corresponding to the magnetic field produced by the order

Further the author intends to establish a single mathematical model associated to induction motors, supplied by static voltage and frequency converter, which consists of a single equivalent scheme and which describes the machine operation, according to the presence in the input power voltage of higher time harmonics. For this, the following simplifying






Under these conditions of non-sinusoidal supply, the asynchronous motor may be associated to an equivalent scheme, corresponding to all harmonics. The scheme operates in the fundamental frequency *f*1(1) and it is represented in Fig. 2. According to this scheme, it can be formally considered that the motors, in the case of supplying through the power frequency converter (the corresponding parameters and the dimensions of this situation are marked with index "CSF") behave as if they were fed in sinusoidal regime at fundamental's

current density is considered as constant throughout the cross section of the bar; - the passing from the constant density zone into the variable density zone occurs

The electromagnetic fields are considered, in this case plane-parallels;

sinusoidal in time, both for the fundamental and for each harmonic; - one should take into account only the fundamental space harmonic of the EMF.

establishing the relationships for equivalent parameters;

frequency, *f*1(1) with the following voltages system:

2 2 R k aR <sup>m</sup> <sup>K</sup> mn (13)

X k aX m <sup>K</sup>' mn (14)

$$\text{where,}\\
\text{where}\\
\qquad \qquad \qquad \mathbf{U}\_{1(\text{CSF})} = \sqrt{\mathbf{U}\_{1(1)}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{U}\_{1(\mathbf{v})}^2} \tag{16}$$

U is the phase voltage su 1( ) pply corresponding to the order harmonic. Corresponding to the system supply voltages, the current system which go through the stator phases is as follows:

$$\begin{cases} \mathbf{i}\_{\rm A} = \sqrt{2} \cdot \mathbf{I}\_{\rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm$$

where I1(CSF) is given by:

$$\mathbf{I}\_{1(\text{CSF})} = \sqrt{\mathbf{I}\_{1(1)}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{I}\_{1(\mathbf{v})}^2} \tag{18}$$

**Figure 2.** The equivalent scheme of the asynchronous motor powered by a static frequency converter.

Power factor in the deforming regime is defined as the ratio between the active power and the apparent power, as follows:

$$\Delta\_{\text{(CSF)}} = \frac{\mathbf{P}\_{\text{1(CSF)}}}{\mathbf{S}\_{\text{1(CSF)}}} = \frac{\mathbf{P}\_{\text{1(CSF)}}}{\mathbf{U}\_{\text{1(CSF)}}\mathbf{I}\_{\text{1(CSF)}}} \tag{19}$$

If we consider the non-sinusoidal regime, the active power absorbed by the machine P1(CSF) is defined, as in the sinusoidal regime, as the average in a period of the instantaneous power. The following expression is obtained:

$$P\_{\mathbf{i}\{\text{CSF}\}} = \frac{1}{T} \Big| \frac{\mathbf{r}}{\mathbf{p}} \cdot \mathbf{dt} = \sum\_{\mathbf{v}=1} \mathbf{U}\_{\mathbf{i}\{\mathbf{v}\}} \mathbf{I}\_{\mathbf{i}\{\mathbf{v}\}} \cos \boldsymbol{\uprho}\_{\mathbf{i}\{\mathbf{v}\}} = \mathbf{U}\_{\mathbf{i}} \mathbf{I}\_{\mathbf{i}} \cos \boldsymbol{\uprho}\_{\mathbf{i}} + \sum\_{\mathbf{v}\neq 1} \mathbf{U}\_{\mathbf{i}\{\mathbf{v}\}} \mathbf{I}\_{\mathbf{i}\{\mathbf{v}\}} \cos \boldsymbol{\uprho}\_{\mathbf{i}\{\mathbf{v}\}} \tag{20}$$

Therefore, the active power absorbed by the motor when it is supplied through a power static converter is equal to the sum of the active powers, corresponding to each harmonic (the principle of superposition effects is found). In relation (20), cos(1) is the power factor corresponding to the order harmonic having the expression:

$$\cos \boldsymbol{\upalpha}\_{\mathbf{1}\_{\{\boldsymbol{\upnu}\}}} = \frac{\mathbf{R}\_{\mathbf{1}\_{\{\boldsymbol{\upnu}\}}} + \frac{\mathbf{R}\_{\mathbf{2}\_{\{\boldsymbol{\upnu}\}}}^{\cdot}}{\mathbf{s}\_{\{\boldsymbol{\upnu}\}}}}{\sqrt{\left(\mathbf{R}\_{\mathbf{1}\_{\{\boldsymbol{\upnu}\}}} + \frac{\mathbf{R}\_{\mathbf{2}\_{\{\boldsymbol{\upnu}\}}}^{\cdot}}{\mathbf{s}\_{\{\boldsymbol{\upnu}\}}}\right)^{2} + \left(\mathbf{X}\_{\mathbf{1}\_{\{\boldsymbol{\upnu}\}}} + \mathbf{X}\_{\mathbf{2}\_{\{\boldsymbol{\upnu}\}}}^{\cdot}\right)^{2}}}\tag{21}$$

The apparent power can be defined in the non-sinusoidal regime also as the product of the rated values of the applied voltage and current:

$$\mathbf{S}\_{\rm 1(CSF)} = \mathbf{U}\_{\rm 1(CSF)} \cdot \mathbf{I}\_{\rm 1(CSF)} \tag{22}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 51

1 1 U cos cos f x U

1 \* 1 1 1r sc 1 1 1 CSF 2 2

U U 1 1 1 1 U U f x

1 1 1 1 1r sc 1 1

 

**3. The determination of the equivalent parameters of the stator winding** 

The equivalent parameters of the scheme have been calculated at the fundamental's frequency, under the presence of all harmonics in the supply voltage. Under these conditions, we note by pCu1(CSF) the losses that occur in the stator winding when the motor is supplied through a power frequency converter. These losses are in fact covered by some active power absorbed by the machine from the network, through the converter, P1(CSF). According to the principle of

> 2 2 Cu1 CSF Cu1 1 Cu1 11 11 1 1

Further, the stator winding resistance corresponding to the fundamental, R1(1) and stator winding resistances corresponding to the all higher time harmonics R1(), are replaced by a single equivalent resistance R1(CSF), corresponding to all harmonics, including the fundamental. The equalization is achieved under the condition that in this resistance the same loss pCu1(CSF) occurs, given by relation (27), as if considering the "" resistances R1(), each of them crossed by the current I1(). This equivalent resistance, R1(CSF), determined at the fundamental's frequency, is traversed by the current I1(CSF) , with the expression given by

 22 22 22 1 CSF 1 1 1 11 11 1 1 11 1

R1(CSF) = R1(1) = R1 . (30)

 2 2 Cu1 CSF Cu1 1 Cu1 11 11 1 1

1 1

Q Q Q 3X I 3 X I (31)

Applying the principle of the superposition effects to the reactive power absorbed by the

Cu1 CSF 1 CSF 1 CSF 1 CSF 1 1 1

2 2 <sup>2</sup>

11 1 3R I I 3R I I 3R I I , (29)

1 1

p p p 3R I 3 R I (27)

 

p 3R I 3R I I (28)

1

1 1

2 1

\*

  (26)

the superposition effects, it can be considered:

Making the relations (27) and (28) equal, it results:

stator winding QCu1 (CSF), the following expression is obtained:

(18). Therefore:

from which:

cos

Taken into account the relations (20), (21) and (22), the relation (19) becomes:

$$\Delta\_{\text{(CSF)}} = \frac{\mathbf{U}\_1 \mathbf{I}\_1 \cos \phi\_1 + \sum\_{\mathbf{v} \neq 1} \mathbf{U}\_{1(\mathbf{v})} \mathbf{I}\_{1(\mathbf{v})} \cos \phi\_{1(\mathbf{v})}}{\sqrt{\mathbf{U}\_{\text{(\!{v}})}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{U}\_{\text{(\!{v}})}^2} \cdot \sqrt{\mathbf{I}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{I}^2}} \tag{23}$$

Because Δ(CSF)≤1, formally (the phase angle has meaning only in harmonic values) an angle 1(CSF) can be associated to the power factor Δ(CSF), as: 1 CSF CSF cos . With this, the relation (23) can be written:

$$\cos \boldsymbol{\upmu}\_{\mathbf{l}(\text{CSF})} = \frac{\cos \boldsymbol{\upmu}\_{\mathbf{l}} + \sum\_{\mathbf{v} \sim \mathbf{l}} \frac{\mathbf{U}\_{\mathbf{l}(\text{v})}}{\mathbf{U}\_{\mathbf{l}(\text{l})}} \frac{\mathbf{I}\_{\mathbf{l}(\text{v})}}{\mathbf{I}\_{\mathbf{l}}} \cos \boldsymbol{\upmu}\_{\mathbf{l}(\text{v})}}{\sqrt{1 + \sum\_{\mathbf{v} \sim \mathbf{l}} \left(\frac{\mathbf{U}\_{\mathbf{l}(\text{v})}}{\mathbf{U}\_{\mathbf{l}(\text{l})}}\right)^{2}}} \cdot \sqrt{1 + \sum\_{\mathbf{v} \sim \mathbf{l}} \left(\frac{\mathbf{I}\_{\mathbf{l}(\text{v})}}{\mathbf{I}\_{\mathbf{l}(\text{l})}}\right)^{2}}\tag{24}$$

If one takes into account the relation (Murphy&Turnbull, 1988):

$$\frac{\mathbf{I}\_{1(\text{v})}}{\mathbf{I}\_{1(\text{1})}} = \frac{1}{\mathbf{v}} \cdot \frac{\mathbf{1}}{\mathbf{f}\_{1r} \cdot \mathbf{x}\_{\text{sc}}^{\*}} \cdot \frac{\mathbf{U}\_{1(\text{v})}}{\mathbf{U}\_{1(\text{1})}} \,, \tag{25}$$

where x \* sc is the reported short-circuit impedance, measured at the frequency f1 = f1n , relation (24) becomes:

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 51

$$\cos \boldsymbol{\phi}\_{\mathbf{i(CSF)}} = \frac{\cos \boldsymbol{\phi}\_{\mathbf{i(CSF)}} + \sum\_{\mathbf{v} \sim \mathbf{l}} \frac{1}{\mathbf{v}} \cdot \frac{1}{\mathbf{f}\_{\mathbf{ir}} \cdot \mathbf{x}\_{\mathbf{sc}}^{\ast}} \cdot \left(\frac{\mathbf{U}\_{\mathbf{i(v)}}}{\mathbf{U}\_{\mathbf{i(l)}}}\right)^{2} \cos \boldsymbol{\phi}\_{\mathbf{i(v)}}}{\sqrt{1 + \sum\_{\mathbf{v} \sim \mathbf{l}} \left(\frac{\mathbf{U}\_{\mathbf{i(v)}}}{\mathbf{U}\_{\mathbf{i(l)}}}\right)^{2}}} \cdot \left[1 + \sum\_{\mathbf{v} \sim \mathbf{l}} \left(\frac{1}{\mathbf{v}} \cdot \frac{1}{\mathbf{f}\_{\mathbf{i}\tau} \cdot \mathbf{x}\_{\mathbf{sc}}^{\ast}} \cdot \frac{\mathbf{U}\_{\mathbf{i(v)}}}{\mathbf{U}\_{\mathbf{i(l)}}}\right)^{2}\right] \tag{26}$$

### **3. The determination of the equivalent parameters of the stator winding**

The equivalent parameters of the scheme have been calculated at the fundamental's frequency, under the presence of all harmonics in the supply voltage. Under these conditions, we note by pCu1(CSF) the losses that occur in the stator winding when the motor is supplied through a power frequency converter. These losses are in fact covered by some active power absorbed by the machine from the network, through the converter, P1(CSF). According to the principle of the superposition effects, it can be considered:

$$\mathbf{p}\_{\text{Cu1(CSF)}} = \mathbf{p}\_{\text{Cu1(1)}} + \sum\_{\text{v} \neq 1} \mathbf{p}\_{\text{Cu1(v)}} = 3\mathbf{R}\_{\text{1(1)}}\mathbf{I}\_{\text{1(1)}}^2 + 3\sum\_{\text{v} \neq 1} \mathbf{R}\_{\text{1(v)}}\mathbf{I}\_{\text{1(v)}}^2 \tag{27}$$

Further, the stator winding resistance corresponding to the fundamental, R1(1) and stator winding resistances corresponding to the all higher time harmonics R1(), are replaced by a single equivalent resistance R1(CSF), corresponding to all harmonics, including the fundamental. The equalization is achieved under the condition that in this resistance the same loss pCu1(CSF) occurs, given by relation (27), as if considering the "" resistances R1(), each of them crossed by the current I1(). This equivalent resistance, R1(CSF), determined at the fundamental's frequency, is traversed by the current I1(CSF) , with the expression given by (18). Therefore:

$$\mathbf{p}\_{\text{Cu}\text{1(CSF)}} = \mathbf{\mathcal{R}} \mathbf{R}\_{\text{1(CSF)}} \cdot \mathbf{I}\_{\text{1(CSF)}}^2 = \mathbf{\mathcal{R}} \mathbf{R}\_{\text{1(CSF)}} \left( \mathbf{I}\_{\text{1(1)}}^2 + \sum\_{\text{v} \neq \text{1}} \mathbf{I}\_{\text{1(v)}}^2 \right) \tag{28}$$

Making the relations (27) and (28) equal, it results:

$$\text{SNR}\_{\mathbf{1}\{\text{CSF}\}} \left( \mathbf{I}\_{\mathbf{1}\{\text{1\}}}^{2} + \sum\_{\mathbf{v} \neq \mathbf{1}} \mathbf{I}\_{\mathbf{1}\{\text{v\}}}^{2} \right) = \mathbf{3R}\_{\mathbf{1}\{\text{1\}}} \left( \mathbf{I}\_{\mathbf{1}\{\text{1\}}}^{2} + \sum\_{\mathbf{v} \neq \mathbf{1}} \mathbf{I}\_{\mathbf{1}\{\text{v\}}}^{2} \right) = \mathbf{3R}\_{\mathbf{1}\{\text{v\}}} \left( \mathbf{I}\_{\mathbf{1}\{\text{1\}}}^{2} + \sum\_{\mathbf{v} \neq \mathbf{1}} \mathbf{I}\_{\mathbf{1}\{\text{v\}}}^{2} \right) \tag{29}$$

from which:

50 Induction Motors – Modelling and Control

T

corresponding to the order harmonic having the expression:

If one takes into account the relation (Murphy&Turnbull, 1988):

 

cos

cos

rated values of the applied voltage and current:

relation (23) can be written:

where x \*

relation (24) becomes:

 

Therefore, the active power absorbed by the motor when it is supplied through a power static converter is equal to the sum of the active powers, corresponding to each harmonic (the principle of superposition effects is found). In relation (20), cos(1) is the power factor

1

 

The apparent power can be defined in the non-sinusoidal regime also as the product of the

R

R

Taken into account the relations (20), (21) and (22), the relation (19) becomes:

1 2 '

 

<sup>2</sup> <sup>2</sup> '

1 1 2

1 1 <sup>1</sup> 1 1 <sup>1</sup> 11 1 11 1 1 CSF 2 22 2

1 1

 

U I cos cos U I

1 1 1 1 1 1 1 1 1 CSF 2 2

 

1 1

 1 1 \* 1 1 1r sc 1 1

I U 1 1

1 1 1 1 1 1 1 1

> 

sc is the reported short-circuit impedance, measured at the frequency f1 = f1n ,

 

U I

U I

Because Δ(CSF)≤1, formally (the phase angle has meaning only in harmonic values) an angle 1(CSF) can be associated to the power factor Δ(CSF), as: 1 CSF CSF cos . With this, the

U I cos U I cos

R XX s

'

R

s

2

1 CSF 1 CSF 1 CSF S UI , (22)

U UI I (23)

 

I U f x , (25)

(21)

(24)

<sup>1</sup> P p dt U I cos U I cos U I cos <sup>T</sup> (20)

1 CSF 11 1 11 1 11 1 0 1 1

$$\mathbf{R}\_{\mathbf{l}\{\mathbf{C}\mathbf{S}\mathbf{r}\}} = \mathbf{R}\_{\mathbf{l}\{\mathbf{l}\}} = \mathbf{R}\_{\mathbf{l}}.\tag{30}$$

Applying the principle of the superposition effects to the reactive power absorbed by the stator winding QCu1 (CSF), the following expression is obtained:

$$\mathbf{Q\_{Cu1(CSF)}} = \mathbf{Q\_{Cu1(1)}} + \sum\_{\mathbf{v} \neq 1} \mathbf{Q\_{Cu1(v)}} = \mathbf{3} \cdot \mathbf{X\_{1(1)}} \mathbf{I\_{1(1)}^2} + \mathbf{3} \sum\_{\mathbf{v} \neq 1} \mathbf{X\_{1(v)}} \mathbf{I\_{1(v)}^2} \tag{31}$$

As in the previous case, the stator winding reactance corresponding to the fundamental, X1(1) (determined at the fundamental's frequency f1(1)) and the stator winding reactances, corresponding to all higher time harmonics X1() (determined at frequencies f1()=f1 where Jmf±k) are replaced by an equivalent reactance, X1(CSF), determined at fundamental's frequency. This equivalent reactance, traversed by the current I1(CSF), conveys the same reactive power, QCu1(CSF) as in the case of considering "" reactances X1(), (each of them determined at f1() frequency and traversed by the current I1()). Following the equalization, the following expression can be written:

$$\mathbf{Q}\_{\text{Cu1(CSF)}} = \mathbf{\mathcal{X}} \mathbf{\mathcal{X}}\_{1(\text{CSF})} \mathbf{I}\_{1(\text{CSF})}^2 = \mathbf{\mathcal{X}} \mathbf{\mathcal{X}}\_{1(\text{CSF})} \left( \mathbf{I}\_{1(1)}^2 + \sum\_{\text{v} \neq 1} \mathbf{I}\_{1(\text{v})}^2 \right) \tag{32}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 53

**4. Determining the equivalent global change parameters for the power** 

Further, it is considered a winding with multiple cages whose bars (in number of "c") are placed in the same notch of any form, electrically separated from each other (see Fig. 3). These bars are connected at the front by short-circuiting rings (one ring may correspond to several bars notch). This "generalized" approach, pure theoretically in fact, has the advantage that by its applying the relations of the two equivalent factors kr(CSF) and kx(CSF), valid for any notch type and multiple cages, are obtained. The rotor notch shown in Fig. 3 is the height hc and it is divided into "n" layers (strips), each strip having a height hs = hc/n. The number of layers "n" is chosen so that the current density of each band should be considered constant throughout the height hs (and therefore not manifesting the skin effect in the strip). The notch bars are numbered from 1 to c, from the bottom of the notch. The lower layer of each bar is identified by the index "i" and the top layer by the index "s". Thus, for a bar with index characterized by a specific resistance and an absolute magnetic permeability, the lower layer is noted with Ni and the extremely high layer with Ns. The current that flows through the bar is noted with ic (Ic - rated value). The length of the bar, over which the skin effect occurs, is L. For the beginning, let us consider only the presence of the fundamental in the power supply,

**rotor fed by the static frequency converter** 

which corresponds to the supply pulsation, ω1(1)=ω1=2πf1. In this case:

s c 1

n 1~ N x 1 <sup>N</sup> <sup>1</sup> <sup>2</sup> <sup>n</sup>

Re <sup>b</sup> <sup>L</sup>

r 1 2

s

<sup>L</sup> <sup>1</sup> <sup>b</sup> 2 Lh I b b

n 1

where b and bε are the width of and ε order strips and Ψδnσ(1) is the bar flux corresponding to the fundamental of the own magnetic field, assuming that for the order strip, the magnetic linkage corresponds to a constant repartition of the fundamental current

 s s

R I <sup>1</sup> k b R b I

<sup>2</sup> N N 1 ~ <sup>1</sup>

 

c 1 N N

 

i i

s

<sup>2</sup> <sup>N</sup>

hs hs

Ncs Nci

N<sup>s</sup> N<sup>i</sup>

hc

N1s N1i

hs hs

hs hs

i

b 3

i ii

N NN

, (36)

(37)

c

1

**Figure 3.** Notch generalized for multiple cages.

k

density on the strip.

Making the relations (31) and (32) equal, it results:

$$\mathbf{X}\_{1(\text{CSF})} \left( \mathbf{I}\_{1(1)}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{I}\_{1(\mathbf{v})}^2 \right) = \mathbf{X}\_{1(1)} \mathbf{I}\_{1(1)}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{v} \mathbf{X}\_1 \mathbf{I}\_{1(\mathbf{v})}^2 = \mathbf{X}\_1 \left( \mathbf{I}\_{1(1)}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{v} \mathbf{I}\_{1(\mathbf{v})}^2 \right) \tag{33}$$

One can notice the following:

$$\mathbf{k}\_{\chi\_1} = \frac{\mathbf{\chi}\_{1(\text{CSF})}}{\mathbf{\chi}\_1}$$

the factor that highlights the changes that the reactants of the stator phase value suffer in the case of a machine supplied through a power frequency converter, compared to sinusoidal supply, both calculated at the fundamental's frequency. From relations (25) and (33) it follows:

$$\mathbf{k}\_{\chi1} = \frac{\mathbf{X}\_{\mathbf{l}(\text{CSF})}}{\mathbf{X}\_{\mathbf{l}}} = \frac{1 + \sum\_{\mathbf{v} \neq 1} \mathbf{v} \left(\frac{1}{\mathbf{f}\_{\text{tr}} \mathbf{x}\_{\text{sc}}^{\circ}}\right)^{2} \cdot \frac{1}{\mathbf{v}^{2}} \left(\frac{\mathbf{U}\_{\text{l}(\text{v})}}{\mathbf{U}\_{\text{l}(\text{l})}}\right)^{2}}{1 + \sum\_{\mathbf{v} \neq 1} \left(\frac{1}{\mathbf{f}\_{\text{tr}} \mathbf{x}\_{\text{sc}}^{\circ}}\right)^{2} \cdot \frac{1}{\mathbf{v}^{2}} \left(\frac{\mathbf{U}\_{\text{l}(\text{v})}}{\mathbf{U}\_{\text{l}(\text{l})}}\right)^{2}} = \frac{1 + \sum\_{\mathbf{v} \neq 1} \frac{1}{\mathbf{f}\_{\text{l}} \mathbf{x}\_{\text{sc}}^{\circ}} \left(\frac{1}{\mathbf{U}\_{\text{l}(\text{l})}}\right)^{2}}{1 + \sum\_{\mathbf{v} \neq 1} \frac{1}{\mathbf{v}^{2}} \left(\frac{1}{\mathbf{f}\_{\text{l}} \mathbf{x}\_{\text{sc}}^{\circ}}\right)^{2} \left(\frac{\mathbf{U}\_{\text{l}(\text{v})}}{\mathbf{U}\_{\text{l}(\text{l})}}\right)^{2}}\tag{34}$$

where:

$$X^\*\_{\
sc} = \frac{X\_{\
sc}}{Z\_{(1)}}$$


$$\underline{\mathbf{Z}\_{\rm 1(CSF)}} = \mathbf{R}\_{\rm 1(CSF)} + \mathbf{j}\mathbf{X}\_{\rm 1(CSF)} = \mathbf{R}\_{\rm 1(CSF)} + \mathbf{j}\mathbf{k}\_{\times 1}\mathbf{X}\_{\rm 1} \tag{35}$$
