**1. Introduction**

Generally, the electric induction motors are designed for supply conditions from energy sources in which the supply voltage is a sinusoidal wave. The parameters and the functional sizes of the electric motors are guaranteed by designers only for it. If the electric motor is powered through an inverter, due to the presence in the input voltage waveform of superior time harmonics, both its parameters and its functional characteristic sizes will be more or less different from those in the case of the sinusoidal supply. The presence of these harmonics will result in the appearance of a deforming regime in the machine, generally with adverse effects in its operation. Under loading and speed conditions similar to those in the case of the sinusoidal supply, it is registered an amplification of the losses of the machine, of the electric power absorbed and thus a reduction in efficiency. There is also a greater heating of the machine and an electromagnetic torque that at a given load is not invariable, but pulsating, in rapport with the average value corresponding to the load. The occurrence of the deforming regime in the machine is inevitable, because any inverter produces voltages or printed currents containing, in addition to the fundamental harmonic, superior time harmonics of odd order. The deforming regime in the electric machine is unfortunately reflected in the supply power grid that powers the inverter. Generalizing, the output voltage harmonics are grouped into families centered on frequencies:

$$\mathbf{f}\_{\parallel} = \mathbf{J} \mathbf{m}\_{i} \mathbf{f}\_{c} = \mathbf{J} \mathbf{m}\_{i} \mathbf{f}\_{1} \quad \left(\text{J} = 1, \, \text{\textquotedblleft}, \, \text{\textquotedblright} \dots \text{\textquotedblright} \right), \tag{1}$$

and the various harmonic frequencies in a family are:

$$\mathbf{f}\_{\left(\mathbf{v}\right)} = \mathbf{f}\_{\right]} \pm \mathbf{k} \mathbf{f}\_{\mathbf{c}} = \left(\mathbf{J} \mathbf{m}\_{\mathbf{i}} \pm \mathbf{k}\right) \mathbf{f}\_{\mathbf{c}} = \left(\mathbf{J} \mathbf{m}\_{\mathbf{i}} \pm \mathbf{k}\right) \mathbf{f}\_{\mathbf{i}} \,\prime \,\tag{2}$$

© 2012 Muşuroi, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Muşuroi, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

with

$$\mathbf{v} = \mathbf{J}\mathbf{m}\_t \pm \mathbf{k} \tag{3}$$

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

2n

 1 11 1n 1 1n

nn a , (6)

R R R R , 1 11 1 1n (7)

LLL 1 11 <sup>1</sup> (9)

X L X aX 1 1 1 1 1n (10)

' ' '' R R RR 2 21 2 2n , (11)

'' ' X X aX 2 <sup>2</sup> 2n (12)

X L L , 1 11 1 <sup>1</sup> (8)

(5)

n n n nn c s a n nn

 

1

111 1n 1n 1n f n a ; f n

are not practically affected by the skin effect. In this case we can write:

both reduced to the stator the following expressions were established:

voltage frequency and load) and

can say that:

' ' 2 1 <sup>2</sup> '

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

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:

> 

 <sup>1</sup> 1 1 n n n 1 c1 s 11

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

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

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

For the rotor resistance and rotor leakage reactance, corresponding to the order harmonic,

<sup>R</sup> <sup>R</sup> <sup>a</sup> <sup>R</sup> s sc

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 its constructive-technological design as advantageous economically as possible.
