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

Written By

Sorin Muşuroi

Submitted: June 10th, 2011 Published: November 14th, 2012

DOI: 10.5772/38009

From the Edited Volume

## Induction Motors

Edited by Prof. Rui Esteves Araújo

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## 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:

fj=Jmffc=Jmff1   (J=1, 2, 3,...)E1

and the various harmonic frequencies in a family are:

f(ν)=fj±kfc=(Jmf±k)fc=(Jmf±k)f1E2

with

ν=Jmf±kE3

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.

## 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:

R1(1)=R1=R1n;X1(1)=X1=aX1n;E4
R2(1)'=R2'=R2n';X2(1)'=X2'=aX2n';E5
Rm(1)=Rm=a2Rmn;Xm(1)=Xm=aXmn;E6
R2(1)'s(1)=R2's=acR2n'E7

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

a=f1f1n=ω1ω1n=n1n1n;c=n1nn1n=n1nn1n1n1n=saE8

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:

s(ν)=νn1nνn1=1nνn1=11ν±ca1νE9
,

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:

R1(ν)=R1(1)=R1=R1nE10
(7)
X1(ν)=ω1(ν)L1σ(ν)=νω1L1σ(ν)E11

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:

L1σ(ν)=L1σ(1)=L1σE12

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

X1(ν)=νω1L1σ=νX1=νaX1nE13

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

R2(ν)'=R2(1)'=R2'=R2n'E14
X2(ν)'=νX2'=νaX2n'E15

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

Rm(ν)=kK"ν2a2RmnE16

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:

Xm(ν)=kK'νaXmnE17

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:

• the permeability of the magnetic core is considered infinitely large comparing to the air permeability and the magnetic field lines are straight perpendicular to the slot axis;

• both the ferromagnetic core and rotor cage (bar + short circuit rings) are homogeneous and isotropic media;

• the marginal effects are neglected, the slot is considered very long on the axial direction. The electromagnetic fields are considered, in this case plane-parallels;

• the skin effect is taken into account in the calculations only in bars that are in the transverse magnetic field of the slot. For the bar portions outside the slot and in short circuit rings, 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 abruptly;

• in the real electric machines the skin effect is often influenced by the degree of saturation but the simultaneous coverage of both phenomena in a mathematical relationships, easily to be applied in practice is very difficult, even precarious. Therefore, the simplifying assumption of neglecting the effects of saturation is allowed as valid in establishing the relationships for equivalent parameters;

• the local variation of the magnetic induction and of current density is considered sinusoidal in time, both for the fundamental and for each harmonic;

• one should take into account only the fundamental space harmonic of the EMF.

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 f1(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, f1(1) with the following voltages system:

uA=2U1(CSF)sinω1t;uB=2U1(CSF)sin(ω1t2π3);uC=2U1(CSF)sin(ω1t+2π3)E18

where,

U1(CSF)=U1(1)2+ν1U1(ν)2E19
U1(ν)is the phase voltage supply corresponding to the order harmonic. Corresponding to the system supply voltages, the current system which go through the stator phases is as follows:
{iA=2I1(CSF)sin(ω1tφ1(CSF))iB=2I1(CSF)sin(ω1tφ1(CSF)2π3)iC=2I1(CSF)sin(ω1tφ1(CSF)4π3)E20

where I1(CSF) is given by:

I1(CSF)=I1(1)2+ν1I1(ν)2E21

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

Δ(CSF)=P1(CSF)S1(CSF)=P1(CSF)U1(CSF)I1(CSF)E22

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:

P1(CSF)=1T0Tpdt=ν=1U1(ν)I1(ν)cosφ1(ν)=U1I1cosφ1+ν1U1(ν)I1(ν)cosφ1(ν)E23

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:

cosφ1(ν)=R1(ν)+R2(ν)'s(ν)(R1(ν)+R2(ν)'s(ν))2+(X1(ν)+X2(ν)')2E24

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:

S1(CSF)=U1(CSF)I1(CSF)E25

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

Δ(CSF)=U1I1cosφ1+ν1U1(ν)I1(ν)cosφ1(ν)U1(1)2+ν1U1(ν)2I1(1)2+ν1I1(ν)2E26

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:cosφ1(CSF)=Δ(CSF). With this, the relation (23) can be written:

cosφ1(CSF)=cosφ1+ν1U1(ν)U1(1)I1(ν)I1cosφ1(ν)1+ν1(U1(ν)U1(1))21+ν1(I1(ν)I1(1))2E27

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

I1(ν)I1(1)=1ν1f1rxsc*U1(ν)U1(1)E28

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

cosφ1(CSF)=cosφ1+ν11ν1f1rxsc*(U1(ν)U1(1))2cosφ1(ν)[1+ν1(U1(ν)U1(1))2][1+ν1(1ν1f1rxsc*U1(ν)U1(1))2]E29

## 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:

pCu1(CSF)=pCu1(1)+ν1pCu1(ν)=3R1(1)I1(1)2+3ν1R1(ν)I1(ν)2E30

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:

pCu1(CSF)=3R1(CSF)I1(CSF)2=3R1(CSF)(I1(1)2+ν1I1(ν)2)E31

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

3R1(CSF)(I1(1)2+ν1I1(ν)2)=3R1(1)(I1(1)2+ν1I1(ν)2)=3R1(ν)(I1(1)2+ν1I1(ν)2)E32

from which:

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

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

QCu1(CSF)=QCu1(1)+ν1QCu1(ν)=3X1(1)I1(1)2+3ν1X1(ν)I1(ν)2E33

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:

QCu1(CSF)=3X1(CSF)I1(CSF)2=3X1(CSF)(I1(1)2+ν1I1(ν)2)E34

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

X1(CSF)(I1(1)2+ν1I1(ν)2)=X1(1)I1(1)2+ν1νX1I1(ν)2=X1(I1(1)2+ν1νI1(ν)2)E35

One can notice the following:

kX1=X1(CSF)X1E36

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:

kX1=X1(CSF)X1=1+ν1ν(1f1rxsc*)21ν2(U1(ν)U1(1))21+ν1(1f1rxsc*)21ν2(U1(ν)U1(1))2=1+ν11ν(1f1rxsc*)2(U1(ν)U1(1))21+ν11ν2(1f1rxsc*)2(U1(ν)U1(1))2E37

where:

X*sc=XscZ(1)E38

- is the short circuit impedance reported, corresponding to the frequency f1=f1n and f1r is the reported frequency. One can notice that: kX1>1. With the equivalent resistance given by (30) and the equivalent reactance resulting from the relationship (34) we can now write the relation for the equivalent impedance of the stator winding, Z1(CSF) covering all frequency harmonics and including the fundamental:

Z_1(CSF)=R1(CSF)+jX1(CSF)=R1(CSF)+jkX1X1E39

## 4. Determining the equivalent global change parameters for the power rotor fed by the static frequency converter

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

krδ(1)=Rδ(1)~Rδ=1Icδ(1)2ε=NδiNδsIε(1)2bεε=NδiNδsbεE40
kxδ(1)=Lδnσ(1)~Lδnσ=|Re[Ψ_δnσ(1)]|(ε=NδiNδsbε)22μδLhsIcδ(1)λ=NδiNδs1bλ[(ε=Nδiλ1bε)(ε=Nδiλbε)+bλ23]E41

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 density on the strip.

If in the motor power supply one considers only the order harmonic which corresponds to the supply pulsation 1()=1, the relations (36) and (37) remain valid with the following considerations: index "1" is replaced by index "" and the rotor phenomena are with the pulsation 2() given by the relation:

ω2(ν)=s(ν)ω1(ν)=(11ν±sν)νω1E42

Subsequently we shall consider the real case, where in the bar both the fundamental and order time harmonics are present. For this, the equivalent d.c. global factor of the bar resistance modification is calculated with the relation:

krδ(CSF)=pδ(CSF)~pδ(CSF)=Rδ(CSF)~Rδ(CSF)E43

where p(CSF)~ represents the total a.c. losses in bar (considering the appropriate skin effect for all harmonics) and p(CSF)- represents the bar total losses, without considering the repression phenomenon. The a.c. total losses in the bar are obtained by applying the effects superposition principle by adding all the bar a.c. losses caused by each order time, including the fundamental. Therefore one can obtain:

pδ(CSF)~=pδ(1)+ν1pδ(ν)~E44

The a.c. loss in bar, corresponding to the fundamental, p(1)~, is calculated with the following relation:

pδ(1)~=Icδ(1)2krδ(1)RδE45

In the same way, the expression of the bar a.c. losses produced by some order time harmonic is obtained:

pδ(ν)~=Icδ(ν)2Rδ(ν)~=Icδ(ν)2krδ(ν)RδE46

By replacing the relations (41) and (42) in relation (40), it results:

pδ(CSF)~=Icδ(1)2krδ(1)Rδ+ν1Icδ(ν)2krδ(ν)Rδ=Rδ(Icδ(1)2krδ(1)+ν1Icδ(ν)2krδ(ν)).E47

The bar losses without considering the repression phenomenon in the bar are calculated using the following relationship:

Pδ(CSF)=Icδ(CSF)2RδE48

where:

Icδ(CSF)=Icδ(1)2+ν1Icδ(ν)2E49

is the rated value of the current which runs through the bar, in the case of a motor supplied by a frequency converter. By replacing the relation (45) in relation (44):

pδ(CSF)=Rδ(Icδ(1)2+ν1Icδ(ν)2)E50

By replacing the relations (43) and (46) in (39) one obtains the expression for the global equivalent factor of the a.c. increasing resistance in the bar , kr (CSF), in case of the presence of all harmonics in the motor power:

krδ(CSF)=pδ(CSF)~pδ(CSF)=Rδ(Icδ(1)2krδ(1)+ν1Icδ(ν)2krδ(ν))R(Icδ(1)2+ν1Icδ(ν)2)=krδ(1)+ν1krδ(ν)(Icδ(ν)Icδ(1))21+ν1(Icδ(ν)Icδ(1))2E51

The global equivalent change of a.c. bar inductance modification has the expression:

kxδ(CSF)=qδ(CSF)~qδ(CSF)E52

where q(CSF)~ is the a.c. total reactive power, in the bar, and q(CSF)- is the total reactive power for a uniform current distribution in the bar. Applying the superposition in the case of a.c. total reactive power, the following relationship is obtained:

qδ(CSF)~=qδ(1)~+ν1qδ(ν)~E53

A.c. reactive power corresponding to the fundamental is calculated using the following relation:

qδ(1)~=ω1kxδ(1)LδnσIcδ(1)2E54

In the same way, the expression of the a.c. reactive power in the bar corresponding to the order harmonic is obtained:

qδ(ν)~=ω1(ν)Lδnσ(ν)~Icδ(ν)2=νω1kxδ(ν)LδnσIcδ(ν)2E55

By replacing the relations (50) and (51) in the relation (28), the expression for calculating the total a.c. reactive power in the bar is obtained:

qδ(CSF)~=ω1kxδ(1)LδnσIcδ(1)2+(kxδ(1)Icδ(1)2+ν1νkxδ(ν)Icδ(ν)2)==ω1Lδnσ(kxδ(1)Icδ(1)2+ν1νkxδ(ν)Icδ(ν)2)E56

The total reactive power for an uniform current repartition in the bar, in the case of a motor supplied through a frequency converter, is calculated by the relation:

qδ(CSF)=qδ(1)+ν1qδ(ν)E57

where q(1)- is the reactive power corresponding to the fundamental, in case of an uniform current distribution Ic(1) in the bar, while q()- is the reactive power corresponding to the harmonic in case of a uniform current distribution Ic() in the bar:

qδ(1)=ω1(1)LδnσIcδ(1)2=ω1LδnσIcδ(1)2E58

Similarly, for the reactive power corresponding to the harmonic, in the case of an uniform current Ic() repartition in the bar, the following relation is obtained:

qδ(ν)=ω1(ν)LδnσIcδ(ν)2=νω1LδnσIcδ(ν)2E59

By replacing the relations (54) and (55) in relation (53), the expression for the total reactive power for a uniform current distribution in the bar becomes:

qδ(CSF)=ω1LδnσIcδ(1)2+ν1νω1LδnσIcδ(ν)2=ω1Lδnσ(Icδ(1)2+ν1νIcδ(ν)2)E60

By replacing the relations (52) and (56) in relation (48), the expression for the global equivalent factor of the a.c. modifying inductance is obtained:

kxδ(CSF)=qδ(CSF)~qδ(CSF)=ω1Lδnσ(kxδ(1)Icδ(1)2+ν1νkxδ(ν)Icδ(ν)2)ω1Lδnσ(Icδ(1)2+ν1νIcδ(ν)2)=kxδ(1)+ν1[ν(Icδ(ν)Icδ(1))2kxδ(ν)]1+ν1[ν(Icδ(ν)Icδ(1))2]E61

## 5. Determining the equivalent parameters of the winding rotor, considering the skin effect

The rotor winding’s parameters are affected by the skin effect, at the start of the motor and also at the nominal operating regime. For establishing the relations that define these parameters, considering the skin effect, the expression of the rotor phase impedance reduced to the stator is used. For this, the rotor with multiple bars is replaced by a rotor with a single bar on the pole pitch. Initially only the fundamental present in the power supply of the motor is considered. The rotor impedance reduced to the stator has the equation:

Z2(1)'=R2(1)'s(1)+jX2(1)'E62

Knowing that the induced EMF by the fundamental component of the main magnetic field from the machine in the pole pitch bars is:

U_e(1)=I_2(1)'Z_2(1)'E63
,

where, for the general case of multiple cages is valid the relation:

I_2(1)'=δ=1cI_cδ(1)'=U_e(1)Δ_(1)δ=1cΔ_δ(1)E64

In the relation (60), the number of the cages and respectively the rotor bars/ pole pitch is equal to “c”. In the case of motors with the power up to 45 [kW], c=1 (simple cage or high bars) or c=2 (double cage). Δ(1) is the determinant corresponding to the equation system:

U_e(1)=ε=1cR_δε(1)I_cε(1),e=1, 2, , c ,E65

having the expression:

Δ_(1)=|R_11(1)...R_1n(1)....R_n1(1)...R_nn(1)|E66

Δδ(1) is the determinant corresponding to the fundamental obtained from Δ(1), where column δ is replaced by a column of 1:

Δ_δ(1)=|R_11(1)...R_1,δ1(1)1R_1,δ+1(1)...R_1n(1)....R_n1(1)...R_n,δ1(1)1R_n,δ+1(1)...R_nn(1)|E67

Because in the first phase the steady-state regime is under focus, the phenomenon in the rotor corresponding to the fundamental has the pulsation ω2(1)=sω1, where s is the motor slip for the sinusoidal power supply in the steady-state regime. If the relation (63) is introduced in (60), the expression of the equivalent impedance of the rotor phase reduced to the stator, corresponding to the fundamental valid when considering the skin effect is obtained:

Z_2(1)'=Δ_(1)δ=1cΔ_δ(1)E68

Thus, the expressions for the rotor phase resistance and inductance reduced to the stator, corresponding to the fundamental, both affected by the skin effect can be written.

R2(1)'s(1)=e[Z_2(1)']E69
X2(1)'=m[Z_2(1)']E70

By considering in the motor power supply the ν harmonic only, similar expressions are obtained for the corresponding rotor parameters. Thus:

Z_2(ν)'=Δ_(ν)δ=1cΔ_δ(ν)E71
R2(ν)'s(ν)=e[Z_2(ν)']E72
X2(ν)'=m[Z_2(ν)']E73

Further on we consider the real case of an electric induction machine fed by a frequency converter. For the beginning, the case of simple cage respectively high bars induction motors will be analyzed. Thus, a rotor phase resistance corresponding to the fundamental, R’2(1), and rotor phase resistance corresponding to higher order harmonics R’2(ν) are replaced by an equivalent resistance R’2(CSF), which dissipates the same part of active power as in the case of “ν” resistances. This equivalent resistance is defined at the fundamental’s frequency and it is traversed by the I’2(CSF) current:

I2(CSF)'=I2(1)'2+ν1I2(ν)'2E74

For the rotor phase equivalent resistance reduced to the stator, corresponding to all harmonics, defined at the fundamental’s frequency, one can write:

R2(CSF)'=kr(CSF)R2c'+R2i'E75

where: R’2c is the resistance, considered at the fundamental’s frequency of a part from the rotor phase winding from notches and reported to the stator, R’2i is the resistance of a part of the rotoric winding, neglecting skin effect reported to the stator, kr(CSF) is the global modification factor of the rotor winding resistance, having the expression given by the relation (47). To track the changes that appear on the resistance of the rotor winding when the machine is supplied through a frequency converter, comparing to the case when the machine is fed in the sinusoidal regime, the kR’2 factor is introduced:

kR2'=R2(CSF)'R2'E76

where R’2 is the rotor winding resistance reported to the stator, when the machine is fed in the sinusoidal regime:

R2'=krR2c'+R2i'E77

where kr is the modification factor of the a.c. rotor resistance, in the case of sinusoidal: krkr(1). It is obtained:

kR2'=kr(CSF)R2c'+R2i'krR2c'+R2i'E78

If both the nominator and the denominator of the second member on the relation (74) are divided by kr and then by R’2c, the following expression is obtained:

kR2'=kr(CSF)kr+R2i'R2c'1kr1+R2i'R2c'1kr=kkr+r21kr1+r21krE79

where:

r2=R2i'R2c'const.E80

which is constant for the same motor, at a given fundamental’s frequency. For c=1, kkr>1, it results that kR’2 >1, which means that R’2(CSF)>R’2 also. The procedure is similar for the reactance. The rotor phase reactance, corresponding to the fundamental, X’2(1), and also the reactance corresponding to the higher harmonics, X’2(), are replaced by an equivalent reactance X’2(CSF). As in the case of the rotor resistance, we can write:

kX2'=X2(CSF)'X2'E81

where X’2(CSF) is the equivalent reactance of the rotor phase, reduced to the stator, corresponding to all harmonics, including the fundamental, on the fundamental’s frequency:

X2(CSF)'=kX(CSF)X2c'+X2i'E82

and X’2 is the reactance of the rotor phase reduced to the stator which characterizes the machine when it is fed in the sinusoidal regime:

X2'=kXX2c'+X2i'E83

In relation (77) and (78), we noted: X’2c -the reactance of the rotor winding part from the notches, reduced to the stator, in which the skin effect is present, X’2i- the reactance of the rotor winding phase where the skin effect can be neglected. kX(CSF) is defined in relation (57), where c1. Taking into account the relations (77) and (78), the relation (76) becomes:

kX2'=kX(CSF)X2c'+X2i'kXX2c'+X2i'=kX(CSF)kX+X2i'X2c'1kX1+X2i'X2c'1kX=kkX+x21kX1+x21kXE84

where:

x2=X2i'X2c'E85

is a constant for the same motor at a given fundamental’s frequency kkX<1, with the consequences kX’2<1 and X’2(CSF)<X’2. With this, the impedance of a rotor phase reported to the stator in the case of a machine supplied by a power converter, receives the form:

Z_2(CSF)'=R2(CSF)'s(CSF)+jX2(CSF)'E86

where:

s(CSF)=R2(CSF)'I2(CSF)'Ue1(CSF)E87

and

:
Ue1(CSF)=Ue1(1)2+ν1Ue1(ν)2E88

In the case of double cage induction motors, the rotor parameters are necessary to be determined for both cages. The principle of calculation keeps its validity from the above presented case, the induction motors with simple cage, respectively cage with high bars, with one remark: in the relations for determining kr(CSF) respectively kx(CSF),it is considered that c=2 (for δ=1 the working work cage results and for δ=c=2 the startup cage results). The complex structure of the used algorithm and its component computing relations synthetically presented in the paper, request a very high volume of calculation. Therefore the presence of a computer in solving this problem is absolutely necessary. In the Laboratory of Systems dedicated to control the electrical servomotors from the Polytechnic University of Timişoara the software calculation CALCMOT has been designed. It allows the determination and the analysis of the factors kr(CSF), kx(CSF) and the parameters of the equivalent winding machine induction in the non-sinusoidal regime. Further on, the expressions of the equivalent parameters for the magnetic circuit will be set (corresponding to all harmonics). Thus, to determine the equivalent resistance of magnetization R1m(CSF), we have to take into account that this is determined only by the ferromagnetic stator core losses which are covered directly by the stator power without making the transition through the stereo-mechanical power. By approximating that I01(CSF) Iμ(CSF), for R1m(CSF) it is obtained:

R1m(CSF)=pz1(CSF)+pj1(CSF)3Iμ(CSF)2E89

where pz1(CSF) and pj1(CSF) are global losses occurring respectively in the stator teeth and in the yoke due to the supplying of the motor through the frequency converter. In determining the total magnetization current Iμ(CSF), the principle of the superposition effects is applied:

Iμ(CSF)=Iμ(1)2+ν1Iμ(ν)2E90

For the equivalent magnetizing reactance, corresponding to all harmonics, determined at the fundamental’s magnetization frequency f1(1), we obtain:

X1m(CSF)(U1(CSF)Iμ(CSF))2(R1(CSF)+R1m(CSF))2E91

For the equivalent impedance of the magnetization circuit it can be written:

Z_1m(CSF)=R1m(CSF)+jX1m(CSF)E92

Given these assumptions and considering that the equivalent parameters were calculated reduced to the fundamental’s frequency (in the conditions of a sinusoidal regime), one may formally accept the calculation in complex quantities. Corresponding to the unique scheme shown in Fig. 2, the motor equations are:

U_1(CSF)=Z_1(CSF)I_1(CSF)U_e1(CSF);E93
U_e2(CSF)'=Z_2(CSF)'I_2(CSF)'=U_e1(CSF);E94
U_e1(CSF)=Z_m(CSF)I_01(CSF);E95
I_01(CSF)=I_1(CSF)+I_2(CSF)'E96

## 6. Experimental validation

The induction machines which have been tested are: MAS 0,37 [kW] x 1500 [rpm] and MAS 1,1 [kW] x 1500 [rpm]. To validate the experimental studies of the theoretical work, tests were made both for the operation of motors supplied by a system of sinusoidal voltages, and for the operation in case of static frequency converter supply. In Tables 1 and 2 are presented theoretical values (obtained by running the calculation program) and the results of measurements, for kR’2 and kX’2, factors, respectively the calculation errors of, for both motors tested.

### Table 1.

The theoretical and experimental values of factors kR'2 and kX'2, respectively the errors of calculation, corresponding to 0.37 [kW] x 1500 [rpm] MAS.

### Table 2.

The theoretical and experimental values of factors kR’2 şi kX’2, respectively the errors of calculation, corresponding to 1.1 [kW] x 1500 [rpm] MAS.

Parameters of the winding machine supplied by the power converter can be calculated with errors less than 10 [%]. The main cause of errors is the assumption of saturation neglect. Even in this case the results can be considered satisfactory, which leads to validate the theoretical study carried out in the paper.

## 7. Theoretical analysis of the magnetic losses

### 7.1. Statoric iron losses

#### 7.1.1. The main stator iron losses

A. The main stator teeth losses

In the teeth, the magnetic field is alternant and generates this type of losses. In the case of the direct supplying system the total losses from the stator teeth pzl are being composed by the magnetic hysteresis losses, pzlh and the eddy currents losses, pzlw:

pz1=(kzhσhf1+kzwσwf12Δ2)Bz1m2Gz1E97

where: h is a material constant depending on the thickness and the quality of the steel sheet, f1 is the supplying frequency, Bzlm represents the magnetic induction in the middle of the stator tooth, Gzl represent the weight of the stator teeth, w is a material constant similar to h, depending on the sheet thickness and quality and represents the thickness of the sheet. kzh and kzw are two factors which have the mission of underlining respectively the hysteresis losses increment and the eddy currents losses increment due to the mechanical modifications of the stator’s sheets. In the case of converters-mode supplying system, at the total losses from the stators teeth caused by the fundamental the losses induced by the higher time harmonics must be taken into account. For an exact analytic expression in the following it is proposed an analysis method of the iron losses based upon the equalization of the hysteresis losses with the eddy currents ones. For the start, only the fundamental is considered present in the supplying system. Distinct from the sine-mode supplying system, when in most cases the supplying frequency is f1=f1n=50 [Hz], is the fact that in the case of the inverter based supplying system the fundamental frequency can take values higher than 50 [Hz]. At very high magnetization frequencies the influence of the skin effect must be taken in consideration. In the following, the minimum value of the magnetization frequency is being determined and for that the skin effect must be considered. The computing relation for the magnetization frequency f1 is the following:

f1=(ξΔ)2ρμπE98

where ξ is the refulation factor.

The minimum magnetization frequency fmin, computed with the relation (89), from which the skin effect must be considered is 140[Hz]. Consequently, in the fundamental - wave supplying mode, at which usually we have f1≤120 [Hz], the principal losses from the stators teeth, can be written as following:

pz1(1)=(kzhσhf1+kzwσwf12Δ2)2Bz1m(1)2Gz1E99

where Bz1m(1) represents the magnetic induction from the middle of the tooth,Bz1m(1)=Bz1m. In order to be able to apply the principle of over position effects, the machine is being considered as being ideal; therefore we neglect the hysteresis phenomenon. For this, we proposed the equalization of the hysteresis losses with the eddy current losses, an assumption that allows the linearization of the machines’ equations. Through this equalization, the real machine – that is practically non-linear and in which the principal losses are made of a sum of two components: the one of eddy currents losses and the one of hysteresis losses - is being replaced with a theoretical linear machine, characterized only by its eddy currents losses. Energetically speaking, the two machines must be equivalent. As a following, if we take p*z1w(1) as the eddy currents losses corresponding to the fundamental, which appear in the theoretical model of the machine adopted, than these losses must be equal to the main losses from the stator teeth characteristic to the real machine, losses given through the relation:

pz1w(1)*=pz1(1)E100

We consider these equivalent losses, p*z1w(1), equal to the real losses through the eddy currents corresponding to the fundamental, pz1w(1), multiplied with a kz1e(1) factor. This is an equalization factor of the real losses from the stators teeth with losses resulted only from ”pz1w(1)” – fundamental-mode supplying state:

pz1w(1)*=kz1e(1)pz1w(1)E101

We consider that through this equalization factor a covering value of the principal stator teeth losses is obtained. The relation (91) made explicit becomes:

(kzhσhf1+kzwσwf12Δ2)Bz1m(1)2Gz1=kz1e(1)kzwσwf12Δ2Bz1m(1)2Gz1E102

Because of the fact that the usually used sheets have the thickness =0.5 [mm]=const, one can consider that:

kz1e(1)=1+KzΔf1E103

where we have

KzΔ=Kz/Δ2E104

with

Kz=σhkzhσwkzwE105

In the following part we consider that only the order harmonic is present in the supplying wave, characterized by the magnetization frequency f1()=f1. Therefore, the principal losses in the stator teeth occurring in the real machine corresponding to the order time harmonic must be corrected through the two factors kh() and kw(), which are a function of the reaction of the eddy currents:

pz1(ν)=(kzhkh(ν)σhνf1+kzwkw(ν)σwν2f12Δ2)Bz1m(ν)2Gz1E106

In the relation (95), Bz1m() represents the magnetic induction according to the order time harmonic from the middle of the tooth. The factors kh() and kw() have the expressions:

kh(ν)=ξ(ν)2shξ(ν)+sinξ(ν)chξ(ν)cosξ(ν);kw(ν)=3ξ(ν)shξ(ν)sinξ(ν)chξ(ν)cosξ(ν);E107

As in the case of the fundamental-wave supplying case, the real machine is replaced by a theoretical linear machine which has only losses given by the eddy currents. Reasoning as in the case of the fundamental, we obtain:

kz1e(ν)=1+KzΔ21νf1kh(ν)kw(ν)==1+KzΔνf1kh(ν)kw(ν)E108
pz1(ν)=pz1w(ν)*=kz1e(ν)pz1w(ν)=kz1e(ν)kzwkw(ν)σwν2f12Δ2Bz1m(ν)2Gz1E109

where p*z1w(ν) are the equivalent losses corresponding to the ν harmonic. If we have pz1(CSF) for the losses from the stators teeth with the machine supplied by inverters, by applying the principle of over position effects for the theoretical linear model of the machine, it will be written:

pz1(CSF)=kzwσwf12Δ2Bz1m(1)2Gz1[kz1e(1)+ν1kz1e(ν)kw(ν)ν2(Bz1m(ν)Bz1m(1))2]E110

In order to analyze the modifications suffered by the main losses in the stators teeth while the motor is supplied by an inverter versus the sine-mode supplying system, we analyze the ratio between the relations (99) and (88). After making the intermediary computations in which the relations (93), (94) and (99) are taken into account we obtain:

kpz1=pz1(CSF)pz1=1+ν1(kz1e(ν)kz1e(1)kw(ν)ν2kBz1(ν,1)2)E111

where kBz(ν,1) = Bz1m(ν) / Bz1m(1).

B. The principal losses in the stator yoke

In the case of the direct – mode supplying system of the machine, the principal yoke losses consist of the hysteresis losses, pj1h and eddy currents losses, pj1w:

pj1=(σhf1kj1h+σwΔ2f12kj1w)Bj12Gj1E112

where: Bjl is the magnetic induction in the stator yoke, Gjl represents the weight of the stator yoke, kj1w=kj1w1kj1w2, where kj1w1 is a coefficient that corresponds to the non uniform repartition of the magnetic induction in the yoke and kj1w2 is a coefficient that corresponds to the currents closing perpendicular to the sheets, through the places with imperfections in the sheets isolation layer and also in the wholes made in the cutting process. In the case on an inverter supplying system at the total losses from the stator yoke caused by the fundamental, the superior time harmonics losses must be added. In order to apply the principle of over-position effect the method is similar to the one used in the case of the principal losses in the teeth. We equalize energetically the real machine with the linear theoretical one where we consider only the eddy currents losses. As a following, for the fundamental supplying mode, the principal losses in the stator yoke for a real machine, pj1(1) are:

pj1(1)=(σhf1kj1h+σwΔ2f12kj1w)Bj1(1)2Gj1E113

If we have p*j1w(1) as losses in eddy currents, than these must be equalized with the principal losses from the stator yoke described with the relation (102):

pj1w(1)*=pj1(1)E114

These equivalent losses, p*j1w(1) are considered equal to the real eddy currents losses pj1w(1), multiplied with an equalizing factor of the real yoke losses with “pj1w(1)” type losses, kj1e(1):

pj1w(1)*=kj1e(1)pj1w(1)E115

Similarly to point A, as a following of the equalization we obtain the relation:

kj1e(1)=1+KwΔ2f1=1+KwΔf1E116

where we have:

Kw=σhkj1hσwkj1wE117

and

KwΔ=KwΔ2E118

As a following we consider present in the supplying system of the machine only the order superior time harmonic. Because of the fact that the magnetization frequency f1() is the fundamental one multiplied with , the principal losses from the stator yoke which appear in the fundamental must be adjusted with the two coefficients: kh() and kw(). These factors take into account respectively the skin effect and the eddy currents reaction.

pj1(ν)=(kh(ν)σhνf1kj1h+kw(ν)σwΔ2ν2f12kj1w)Bj1(ν)2Gj1E119

In the relation (106), Bj1() represents the magnetic induction accordingly to the order harmonic. Through the energetically equalization realized from the replacement of the real machine with the linear model, we obtain the equalizing factor of the stator yoke losses, with the “pj1w()” type losses:

kj1e(ν)=1+KwΔ21νf1kh(ν)kw(ν)=1+KwΔνf1kh(ν)kw(ν)E120

In conclusion, the principal losses in the stator yoke, corresponding to the order time harmonic can be written by equalizing as:

pj1(ν)=pj1w(ν)*=kj1e(ν)pj1w(ν)E121

where:

pj1w(ν)=kw(ν)σwΔ2ν2f12kj1wBj1(ν)2Gj1E122

As a following we have considered the situation of the machine supplied by the fundamental and the superior time harmonics as well. Taking pj1(CSF) as the global losses occurring in the stator yoke due to the converter supplying mode, by applying the over position effect principle on the theoretical linear model we can write:

pj1(CSF)=σwf12Δ2kj1wBj1(1)2Gj1[kj1e(1)+ν1kj1e(ν)kw(ν)ν2(Bj1(ν)Bj1(1))2]E123

In order to analyze the changes that the principal losses from the stator yoke suffer when the machine is being supplied through an inverter versus the sine-mode supplying case, we divide the relation (110) at (101). After finishing the computations we have:

kpj1=pj1(CSF)pj1=1+ν1(kj1e(ν)kj1e(1)kw(ν)ν2kBj1(ν,1)2)E124

where: kBj(ν,1) = Bj1(ν) / Bj1(1).

#### 7.1.2. The supplementary stator iron losses

A. Surface supplementary losses

In the case of a network supplying mode, the magnetic induction distribution curve over the polar step is not very different from a sine-curve. The surface stator losses are given by the expression:

Pσ1=12lπDτc1b41τc1ko(Nc2n)1,5(τc2β2kδ2Bδ)2E125

In the relation (112) the significance of the sizes is the following: D is the inner diameter of the stator, c1 is the step of the stator slot and c2 is the step of the rotor slot, b41 is the opening of the stator slot, Nc2 is the number of stator slots, n is the rotation speed, 2 is a factor dependent on the ratio b42/ (b42 is the opening of the rotor slot), k2 is an air gap factor, ko is an adjustment factor which depends on the materials resistivity and its magnetic permeability. In the case of the inverter supplying method, due to the deforming state at the supplementary losses produced by the fundamental, the surface losses produced by the superior time harmonics must be considered. Because of the fact that the surface losses in the polar pieces are treated as the eddy current losses developed in the inductor sheets, we can apply the over position effect principle without any further parallelism. Therefore, the surface supplementary losses in the stator in the case of a machine supplied by inverters can be computed with the relation:

Pσ1(CSF)=12lπDτc1b41τc1ko(Nc2n)1,5(τc2β2kδ2Bδ(1))2[1+ν1(Bδ(ν)Bδ(1))2]E126

Dividing the supplementary losses in the stator surface when having an inverter supplying system for the machine, P1(CSF), by the supplementary losses in the stator surface when we have the sine-mode supplying system for the machine, P1, and making the intermediary computations we obtain the increment factor of the supplementary stator surface losses in the inverter versus the sine-mode supplying case, kP1, as following:

kPσ1=Pσ1(CSF)Pσ1=1+ν1(Bδ(ν)Bδ(1))2=1+ν1kBδ(ν,1)2>1E127

where kBδ(ν,1) = Bδ(ν) / Bδ(1). By analyzing the relation (114) one can notice the fact that the kP1 factor tends to 1 because of the fact that the value is practically very low. Consequently, the surface supplementary losses increase due to the inverter supplying system to an extent that is not to be taken into consideration.

## B. The pulsation supplementary losses

In the case of the sine-mode supplying system, the pulsation supplementary losses in the stator, provided that the magnetic field along the polar step is not much different from a sine-wave, has the following expression:

PP1=12σwkwP1(ΔNc2n)2(γ2δkδ2τc1)2Gz1Bz1m2E128

where kwP1 is an increment coefficient of the stator losses by eddy currents due to processing, k is the total air gap factor and 2 is constant for the one and the same machine, depended on the opening of the stator slot and the air gap dimension. In the situation in which the machine is supplied by inverters, by applying the over position effect principle, the following expression for the supplementary pulsation losses in the stator PP1(CSF) is obtained:

PP1(CSF)=12σwkwP1(ΔNc2n)2(γ2δkδ2τc1)2Gz1Bz1m(1)2[1+ν1(Bz1m(ν)Bz1m(1))2]E129

Dividing the pulsation stator losses in the case of the inverter supplying system PP1(CSF), by the pulsation stator losses in the case of sine-mode supplying system PP1, we obtain the increment factor of the supplementary pulsation losses in the inverter versus sine-wave supplying system, kPp1:

kPp1=PP1(CSF)PP1=1+ν1(Bz1m(ν)Bz1m(1))2=1+ν1kBz1(ν,1)2>1E130

By analyzing the relation (117) we can state that in the case of an inverter supplied machine we have not obtained a significant increment of the pulsation losses in the stator due to the small value of thekBz1(ν,1)2.

### 7.2. Rotor iron losses

#### 7.2.1. Principal losses in the rotor iron

A. The principal losses in the rotor’s teeth

Firstly, only one superior time harmonic is considered present in the supplying system of the machine, of an average order . The real losses that this harmonic produces in the rotor teeth have the expression:

pz2(ν)=(kzhkh(ν)σhs(ν)νf1+kzwkw(ν)σws(ν)2ν2f12Δ2)Bz2m(ν)2Gz2E131

In the relation (118), Bz2m() represents the magnetic induction corresponding to the order harmonic from the middle of the rotor tooth. In the theoretical model adopted, these losses given by the relation (118) are produced only by eddy currents:

pz2(ν)=pz2w(ν)*=kz2e(ν)pz2w(ν)E132

where kz2e() is an equalizing factor of the real losses from the rotor teeth, only with the losses of “pz2w()” type, corresponding to the order time harmonic. Developing the relation (119) by using the relation (118), after finishing the intermediary computations we obtain:

kz2e(ν)=1+KzΔ21s(ν)νf1kh(ν)kw(ν)=1+KzΔs(ν)νf1kh(ν)kw(ν)E133

Therefore, the principal losses from the rotor teeth, corresponding to the order time harmonic can be written by equalization as it follows:

pz2(ν)=kz2e(ν)kzwkw(ν)σws(ν)2ν2f12Δ2Bz2m(ν)2Gz2E134

In the conditions in which in the supplying system of the machine all the superior time harmonics are present, the principal losses in the rotor teeth can be written as:

pz2(CSF)=ν1pz2(ν)E135

B. The principal losses from the rotor’s yoke

In the hypotheses in which in the supplying system only the order harmonic is present, the real principal losses induced by it in the rotor yoke have the expression:

pj2(ν)=(kh(ν)σhs(ν)νf1kj1h+kw(ν)σws(ν)2ν2f12Δ2kj2w)Bj2(ν)2Gj2E136

Through the energetic equalization, due to the replacement of the real machine by a theoretical linear model we can obtain the equality:

pj2(ν)=pj2w(ν)*=kj2e(ν)pj2w(ν)E137

Reasoning as in the previous cases, we can determine the equalizing factor of the real losses in the rotor yoke, only with losses of the type “pj2w()” type as it follows:

kj2e(ν)=1+KwΔ21s(ν)νf1kh(ν)kw(ν)=1+KwΔs(ν)νf1kh(ν)kw(ν)E138

Consequently, the principal rotor yoke losses corresponding to the order harmonic can be written by equalization in the form:

pj2(ν)=kj2e(ν)kj2wkw(ν)σws(ν)2ν2f12Δ2Bj2(ν)2Gj2E139

Disregarding all these, in the case of the inverter supplying system the total principal losses in the rotor yoke, pj2(CSF), are computed with the relation:

pj2(CSF)=ν1pj2(ν)E140
(127)

#### 7.2.2. The supplementary losses in the rotor iron

A. The surface supplementary losses

If the machine is directly supplied from the power supply, the surface supplementary rotor losses are calculated with the relation:

Pσ2=12pσ2lπ(Δ2δ)τc2b42τc2E141

where the specific rotor surface losses p2 have the expression:

pσ2=ko(Nc1n)1,5(τc1β1kδ1Bδ)2E142

In the relations (128) and (129) we noted by b42 the opening of the rotor slot, Nc1 the number of rotor slots, 1 a factor dependent on the b41/ ratio and k1 the air gap factor. Proceeding similarly we can obtain the expression of the increment factor of the supplementary losses in the rotor surface while the machine is being supplied by inverters versus the sine-mode supplying system, kP2:

kPσ2=Pσ2(CSF)Pσ2=1+ν1(Bδ(ν)Bδ(1))2=1+ν1kBδ(ν,1)2=kPσ1>1E143

B. The supplementary pulsation losses

The supplementary pulsation rotor losses, in the sine-mode supplying system have the following expression:

PP2=12σwkwP2(ΔNc1nBP2)2Gz2E144

BP2 represents the pulsation induction in the rotor teeth. Consequently, taking into account the fact that:

Bz2m(ν)Bz2m(1)=Bδ(ν)Bδ(1)=kBδ(ν,1)E145

we obtain:

kPp2=PP2(CSF)PP2=1+ν1kBδ(ν,1)2>1E146