**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:

$$\mathbf{Z}\_{\mathbf{2}(1)}^{\cdot} = \frac{\mathbf{R}\_{\mathbf{2}(1)}^{\cdot}}{\mathbf{s}\_{\mathbf{2}(1)}} + \mathbf{j} \mathbf{X}\_{\mathbf{2}(1)}^{\cdot} \tag{58}$$

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

$$
\underline{\mathbf{U}\_{\circ(1)}} = \underline{\mathbf{I}}\_{\circ(1)} \cdot \underline{\mathbf{Z}}\_{\circ(1)} \tag{59}
$$

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

$$\stackrel{\cdot}{\mathbf{L}}\_{\mathbf{z}(1)} = \sum\_{\delta=1}^{c} \stackrel{\cdot}{\mathbf{L}}\_{\alpha\delta(1)} = \frac{\underline{\mathbf{U}}\_{\alpha(1)}}{\underline{\Delta}\_{\text{(1)}}} \sum\_{\delta=1}^{c} \underline{\Delta}\_{\delta(1)} \tag{60}$$

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:

$$\underline{\mathbf{U}}\_{\text{c}(1)} = \sum\_{\mathfrak{c}=1}^{\mathfrak{c}} \underline{\mathbf{R}}\_{\text{\&c}(1)} \cdot \underline{\mathbf{I}}\_{\text{cc}(1)} \cdot \underline{\mathbf{c}} = \mathbf{1} \cdot \mathbf{2} \cdot \dots \mathbf{c} \cdot \tag{61}$$

having the expression:

56 Induction Motors – Modelling and Control

The total reactive power for an uniform current repartition in the bar, in the case of a

 CSF <sup>1</sup>

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

2 2

Similarly, for the reactive power corresponding to the harmonic, in the case of an uniform

2 2

By replacing the relations (54) and (55) in relation (53), the expression for the total reactive

By replacing the relations (52) and (56) in relation (48), the expression for the global

k k L kI kI <sup>I</sup> <sup>q</sup> <sup>k</sup>

<sup>q</sup> LI I <sup>I</sup>

1

**5. Determining the equivalent parameters of the winding rotor,** 

 

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

1 n x1 c1 x c <sup>1</sup> c 1 CSF ~ <sup>1</sup> x CSF 2

CSF 1 n c 1 1 n c 1 n c 1 c

2 22 <sup>2</sup>

2 2 x 1 x

c 1 n c 1 c

1 1 q LI L I L I I (56)

1

1

q q q , (53)

1 11 <sup>n</sup> c 1 1 n c 1 q LI LI . (54)

1 c <sup>n</sup> 1 n <sup>c</sup> q LI LI (55)

1

 

c

I

 

1 c 1

I

(57)

2

(52)

22 2

CSF ~ 1 n x 1 c1 x1 c1 x c

2 2

q kL I kI kI

1

 

motor supplied through a frequency converter, is calculated by the relation:

L kI kI

1 n x1 c1 x c

harmonic in case of a uniform current distribution Ic() in the bar:

current Ic() repartition in the bar, the following relation is obtained:

power for a uniform current distribution in the bar becomes:

equivalent factor of the a.c. modifying inductance is obtained:

CSF 2 2

 

**considering the skin effect** 

$$
\underline{\mathbf{R}}\_{(1)} = \begin{vmatrix}
\underline{\mathbf{R}}\_{11(1)} & \cdots & \underline{\mathbf{R}}\_{1n(1)} \\
\cdot & & \cdot \\
\cdot & & \cdot \\
\underline{\mathbf{R}}\_{n1(1)} & \cdots & \underline{\mathbf{R}}\_{nn(1)} \\
\end{vmatrix} \tag{62}
$$

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

$$
\Delta\_{\mathbb{A}(1)} = \begin{vmatrix}
\underline{\underline{\mathbf{R}}}\_{\text{1}(1)} & \cdots & \underline{\underline{\mathbf{R}}}\_{\text{1},\text{ }\delta-\text{1}(1)} & & & \underline{\underline{\mathbf{R}}}\_{\text{1},\text{ }\delta+1(1)} & \cdots & \underline{\underline{\mathbf{R}}}\_{\text{1n}(1)} \\
& \cdot & & & \cdot \\
& \cdot & & & \cdot \\
\underline{\underline{\mathbf{R}}}\_{\text{n}\text{1}(1)} & \cdots & \underline{\underline{\mathbf{R}}}\_{\text{n},\text{ }\delta-\text{1}(1)} & & & \mathbf{1} & & \underline{\underline{\mathbf{R}}}\_{\text{n},\text{ }\delta+1(1)} & \cdots & \underline{\underline{\mathbf{R}}}\_{\text{nn}(1)}
\end{vmatrix}\_{\text{\textquotedblleft}{\underline{\mathbf{R}}}\_{\text{1}}} \tag{63}
$$

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:

$$\underline{\mathbf{Z}}\_{2^{\{1\}}} = \frac{\underline{\Delta}\_{\{1\}}}{\sum\_{\mathcal{S}=1}^{c} \underline{\Delta}\_{\mathcal{S}(1)}} \tag{64}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 59

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

R , (72)

' '' R k R R , 2 r 2c 2i (73)

kR R (74)

<sup>X</sup> , (76)

, (75)

2 CSF R ' 2

'

R

where R'2 is the rotor winding resistance reported to the stator, when the machine is fed in

where kr is the modification factor of the a.c. rotor resistance, in the case of sinusoidal:

'

kRR

R ' ' r 2c 2i

If both the nominator and the denominator of the second member on the relation (74) are

<sup>k</sup> <sup>R</sup> <sup>1</sup> <sup>1</sup> k r

 kr 2 r r 2c r

r 2c r

2i <sup>2</sup> '

R 1 1 1 1 r R k k

' r CSF 2i '

k k <sup>R</sup> <sup>k</sup> <sup>k</sup>

 '

> ' 2

k

2i 2 ' 2c R r const. <sup>R</sup> ,

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

2 CSF X ' 2

'

X

' ' r CSF 2c 2i

2

k

divided by kr and then by R'2c, the following expression is obtained:

R '

reactance X'2(CSF). As in the case of the rotor resistance, we can write:

'

2

' 2

k

machine is fed in the sinusoidal regime, the kR'2 factor is introduced:

the sinusoidal regime:

krkr(1). It is obtained:

where:

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.

$$\frac{\text{R}\_{2(1)}}{\text{s}\_{(1)}} = \text{Re}\left[\underline{\underline{\mathbf{Z}}\_{2(1)}}\right] \tag{65}$$

$$\mathbf{X}\_{\mathbf{2}(1)}^{\cdot} = \mathfrak{T}\mathbf{m}\left[\underline{\mathbf{Z}}\_{\mathbf{2}(1)}^{\cdot}\right] \tag{66}$$

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

$$\underline{\mathbf{Z}}\_{2^{\{\vee\}}}^{\cdot} = \frac{\underline{\Delta}\_{\left(\mathbf{v}\right)}}{\sum\_{\delta=1}^{c} \underline{\Delta}\_{\delta\left(\mathbf{v}\right)}},\tag{67}$$

$$\frac{\text{R}\_{2\text{(v)}}^{\cdot}}{\text{s}\_{\text{(v)}}} = \mathfrak{Re}\left[\underline{\mathbf{Z}}\_{2\text{(v)}}^{\cdot}\right]\_{\text{-}\text{-}\text{}}\tag{68}$$

$$\mathbf{X}\_{2^{\left(\nu\right)}}^{\cdot} = \mathfrak{Term}\left[\underline{\mathbf{Z}}\_{2^{\left(\nu\right)}}^{\cdot}\right] \tag{69}$$

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:

$$\mathbf{I}\_{2(\text{CSF})}^{\cdot} = \sqrt{\mathbf{I}\_{2(1)}^{\cdot 2} + \sum\_{\mathbf{v} \neq \mathbf{1}} \mathbf{I}\_{2(\mathbf{v})}^{\cdot 2}} \tag{70}$$

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

' ' ' R k R R , 2 CSF r CSF 2c 2i (71)

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:

$$\mathbf{k}\_{\mathbf{R}\_2^{'}} = \frac{\mathbf{R}\_{2\text{(CSF)}}^{'}}{\mathbf{R}\_2^{'}} \; \prime \tag{72}$$

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

$$\mathbf{R}\_{2}^{\cdot} = \mathbf{k}\_{r} \mathbf{R}\_{2c}^{\cdot} + \mathbf{R}\_{24}^{\cdot} \tag{73}$$

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

$$\mathbf{k}\_{\text{g}\_2^{\cdot}} = \frac{\mathbf{k}\_{\text{r}(\text{CSF})} \mathbf{R}\_{2\text{c}}^{\cdot} + \mathbf{R}\_{21}^{\cdot}}{\mathbf{k}\_{\text{r}} \mathbf{R}\_{2\text{c}}^{\cdot} + \mathbf{R}\_{21}^{\cdot}} \tag{74}$$

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:

$$\mathbf{k}\_{\mathbf{k}\_{\mathbf{x}}} = \frac{\frac{\mathbf{k}\_{\mathbf{r}|\mathrm{CSF}}}{\mathbf{k}\_{\mathbf{r}}} + \frac{\mathbf{R}\_{2\mathbf{r}}^{\cdot}}{\mathbf{R}\_{2\mathbf{c}}^{\cdot}} \cdot \frac{\mathbf{1}}{\mathbf{k}\_{\mathbf{r}}}}{\mathbf{1} + \frac{\mathbf{R}\_{2\mathbf{t}}^{\cdot}}{\mathbf{R}\_{2\mathbf{c}}^{\cdot}} \cdot \frac{\mathbf{1}}{\mathbf{k}\_{\mathbf{r}}}} = \frac{\mathbf{k}\_{\mathbf{k}\mathbf{r}} + \mathbf{r}\_{\mathbf{z}} \cdot \frac{\mathbf{1}}{\mathbf{k}\_{\mathbf{r}}}}{\mathbf{1} + \mathbf{r}\_{\mathbf{z}} \cdot \frac{\mathbf{1}}{\mathbf{k}\_{\mathbf{r}}}}\tag{75}$$

where:

58 Induction Motors – Modelling and Control

in (60), the expression of the equivalent impedance of the rotor phase reduced to the stator,

 

Thus, the expressions for the rotor phase resistance and inductance reduced to the stator,

 '

By considering in the motor power supply the ν harmonic only, similar expressions are

 

1

e Z

 '

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

> 

For the rotor phase equivalent resistance reduced to the stator, corresponding to all

harmonics, defined at the fundamental's frequency, one can write:

1

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

2

 

e Z

2 1

2 1 '

'

2 c

2 '

'

'

 

'

R

s   

1 1

Z (64)

s , (65)

X mZ 2 1 2 1 (66)

Z , (67)

, (68)

X mZ <sup>2</sup> 2 (69)

I II (70)

' ' ' R k R R , 2 CSF r CSF 2c 2i (71)

corresponding to the fundamental valid when considering the skin effect is obtained:

' 1 2 1 c

corresponding to the fundamental, both affected by the skin effect can be written.

 

1

'

R

obtained for the corresponding rotor parameters. Thus:

and it is traversed by the I'2(CSF) current:

$$\mathbf{r}\_2 = \frac{\mathbf{R}\_{24}^\cdot}{\mathbf{R}\_{2c}^\cdot} \equiv \text{const.} \,\mu\text{s}$$

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:

$$\mathbf{k}\_{\chi\_{\mathbf{z}\_{\mathbf{z}}}} = \frac{\mathbf{X}\_{\mathbf{z}|\text{CSF}}^{\cdot}}{\mathbf{X}\_{\mathbf{z}}^{\cdot}} \, \, \, \tag{76}$$

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:

$$\mathbf{X}\_{\text{2(CSF)}}^{\cdot} = \mathbf{k}\_{\text{\chi(CSF)}} \mathbf{X}\_{\text{2c}}^{\cdot} + \mathbf{X}\_{\text{2i}}^{\cdot} \tag{77}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 61

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 nonsinusoidal 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

R

fundamental's magnetization frequency f1(1), we obtain:

shown in Fig. 2, the motor equations are:

1m CSF 2

total magnetization current Iμ(CSF), the principle of the superposition effects is applied:

CSF 1

 

1m CSF 1 CSF 1m CSF CSF

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

U Z I U; 1 CSF 1 CSF 1 CSF e1 CSF

2

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

U X R R

 z1 CSF j1 CSF

p p

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

For the equivalent magnetizing reactance, corresponding to all harmonics, determined at the

CSF

 2 2

1

<sup>2</sup> 1 CSF

3I , (83)

I II (84)

I (85)

1m CSF 1m CSF 1m CSF Z R jX (86)

I01(CSF) Iμ(CSF), for R1m(CSF) it is obtained:

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:

$$\mathbf{X}\_{2}^{\cdot} = \mathbf{k}\_{\chi} \mathbf{X}\_{2c}^{\cdot} + \mathbf{X}\_{21}^{\cdot} \tag{78}$$

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:

$$\mathbf{k}\_{\chi\_{\mathbf{1}}} = \frac{\mathbf{k}\_{\chi(\text{CSF})} \mathbf{X}\_{2c}^{\cdot} + \mathbf{X}\_{21}^{\cdot}}{\mathbf{k}\_{\chi} \mathbf{X}\_{2c}^{\cdot} + \mathbf{X}\_{21}^{\cdot}} = \frac{\frac{\mathbf{k}\_{\chi(\text{CSF})}}{\mathbf{k}\_{\chi}} + \frac{\mathbf{X}\_{21}^{\cdot}}{\mathbf{X}\_{2c}^{\cdot}} \cdot \frac{1}{\mathbf{k}\_{\chi}}}{1 + \frac{\mathbf{X}\_{21}^{\cdot}}{\mathbf{X}\_{2c}^{\cdot}} \cdot \frac{1}{\mathbf{k}\_{\chi}}} = \frac{\mathbf{k}\_{\mathbf{k}\_{\chi}} + \mathbf{x}\_{2} \frac{1}{\mathbf{k}\_{\chi}}}{1 + \mathbf{x}\_{2} \frac{1}{\mathbf{k}\_{\chi}}}\tag{79}$$

where:

 ' 2i 2 ' 2c X x <sup>X</sup> ,

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:

$$\stackrel{\cdot}{\mathbf{Z}\_{2\text{(CSF)}}} = \frac{\stackrel{\cdot}{\mathbf{R}\_{2\text{(CSF)}}}}{\stackrel{\cdot}{\mathbf{s}\_{\text{(CSF)}}}} + \stackrel{\cdot}{\mathbf{j}\mathbf{X}}\_{2\text{(CSF)}\text{ }\prime} \,\tag{80}$$

where:

$$\mathbf{s}\_{\text{(CSF)}} = \frac{\mathbf{R}\_{\text{2(CSF)}}^{\cdot}\mathbf{I}\_{\text{2(CSF)}}^{\cdot}}{\mathbf{U}\_{\text{o1(CSF)}}} \tag{81}$$

and:

$$\mathbf{U}\_{\circ1(\text{CSF})} = \sqrt{\mathbf{U}\_{\circ1(1)}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{U}\_{\circ1(\text{v})}^2} \tag{82}$$

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 nonsinusoidal 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:

60 Induction Motors – Modelling and Control

machine when it is fed in the sinusoidal regime:

X ' ' '

'

2

where:

where:

and:

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:

and X'2 is the reactance of the rotor phase reduced to the stator which characterizes the

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:

<sup>k</sup> <sup>X</sup> <sup>1</sup> <sup>1</sup> k x k XX k k <sup>X</sup> <sup>k</sup> <sup>k</sup>

 '

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

> 

2 CSF 2 CSF CSF

> 

e1 CSF

' ' 2 CSF 2 CSF

R I

 2 2 e1 CSF e1 1 e1

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,

1

'

R Z jX

2 CSF '

x

stator in the case of a machine supplied by a power converter, receives the form:

CSF

s

'

2i 2 ' 2c X

<sup>X</sup> ,

k 2 X CSF 2c 2i X X 2c <sup>X</sup>

X 2c 2i 2i <sup>2</sup> '

kX X X 1 1

X CSF 2i ' ' '

1 1 x X k k

2c X X

'

' ' ' X k XX 2 CSF X CSF 2c 2i , (77)

' '' X k X X (78) 2 X 2c 2i

X

s , (80)

U (81)

U UU (82)

, (79)

$$\mathbf{R}\_{1\text{m(CSF)}} = \frac{\mathbf{P}\_{\text{z1(CSF)}} + \mathbf{P}\_{\text{jl(CSF)}}}{\mathbf{3}\mathbf{I}\_{\mu(\text{CSF})}^2},\tag{83}$$

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:

$$\mathbf{I}\_{\mu(\text{CSF})} = \sqrt{\mathbf{I}\_{\mu(1)}^2 + \sum\_{\mathbf{v} \bullet \mathbf{1}} \mathbf{I}\_{\mu(\mathbf{v})}^2} \tag{84}$$

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

$$\mathcal{X}\_{\text{1m(CSF)}} \cong \sqrt{\left(\frac{\mathcal{U}\_{\text{1(CSF)}}}{\mathcal{I}\_{\mu(\text{CSF})}}\right)^2 - \left(\mathcal{R}\_{\text{1(CSF)}} + \mathcal{R}\_{\text{1m(CSF)}}\right)^2} \tag{85}$$

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

$$\underline{\mathbf{Z}}\_{\rm 1m(CSF)} = \mathbf{R}\_{\rm 1m(CSF)} + \mathbf{j} \cdot \mathbf{X}\_{\rm 1m(CSF)} \tag{86}$$

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:

$$\underline{\mathbf{U}}\_{\mathbf{1}(\text{CSF})} = \underline{\mathbf{Z}}\_{\mathbf{1}(\text{CSF})} \cdot \underline{\mathbf{I}}\_{\mathbf{1}(\text{CSF})} - \underline{\mathbf{U}}\_{\mathbf{1}(\text{CSF})} \tilde{\mathbf{Y}}$$

$$
\underline{\mathbf{U}}\_{\circ2(\rm{CSF})} = \underline{\mathbf{Z}}\_{\circ2(\rm{CSF})} \cdot \underline{\mathbf{I}}\_{\circ2(\rm{CSF})} = \underline{\mathbf{U}}\_{\circ1(\rm{CSF})}.
$$

$$
\underline{\mathbf{U}}\_{\circ1(\rm{CSF})} = -\underline{\mathbf{Z}}\_{\circ1(\rm{CSF})} \cdot \underline{\mathbf{I}}\_{\circ0(\rm{CSF})}.\tag{87}
$$

$$
\underline{\mathbf{I}}\_{\circ01(\rm{CSF})} = \underline{\mathbf{I}}\_{\circ1(\rm{CSF})} + \stackrel{\cdot}{\underline{\mathbf{I}}}\_{\circ2(\rm{CSF})}
$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 63

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

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

22 2

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

2

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

f , (89)

1

z1 zh h 1 zw w 1 z1m z1 p k f k f B G , (88)

theoretical study carried out in the paper.

**7.1. Statoric iron losses** 

*7.1.1. The main stator iron losses* 

**A. The main stator teeth losses** 

**7. Theoretical analysis of the magnetic losses** 

for the magnetization frequency f1 is the following:

where ξ is the refulation factor.

the magnetic hysteresis losses, pzlh and the eddy currents losses, pzlw:

## **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**

62 Induction Motors – Modelling and Control

**6. Experimental validation** 

2(CSF)

R' (calculated)

R'

2

calculation, corresponding to 0.37 [kW] x 1500 [rpm] MAS.

2(CSF)

R' (calculated)

R'

2

calculation, corresponding to 1.1 [kW] x 1500 [rpm] MAS.

R'2

k

kR'2 (measured)

kR'2 (measured)

R'2

k

motors tested.

Nr. f1(1) [Hz]

Nr. f1(1) [Hz]

 ' '' U Z I U; e2 CSF 2 CSF 2 CSF e1 CSF

> ' 01 CSF 1 CSF 2 CSF I I I

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

> εkR'2 [%]

1. 25 1,048 1,11 5,58 0,863 0,894 3,6 2. 30 1,026 1,077 4,97 0,912 0,857 -6,03 3. 40 1,021 1,061 3,77 0,944 0,884 -6,35 4. 50 1,014 1,075 6,01 0,967 0,897 -7,23 5. 60 1,011 1,079 6,82 0,975 0,914 -6,25

**Table 1.** The theoretical and experimental values of factors kR'2 and kX'2, respectively the errors of

εkR'2 [%]

1. 20 1,098 1,185 7,92 0,812 0,821 1,108 2. 30 1,041 1,120 7,58 0,886 0,916 3,386 3. 40 1,034 1,106 6,96 0,926 0,891 -3,77 4. 50 1,023 1,089 6,45 0,956 0,863 -9,72 5. 60 1,018 1,082 6,28 0,966 0,871 -9,83

**Table 2.** The theoretical and experimental values of factors kR'2 şi kX'2, respectively the errors of

U Z I; e1 CSF m CSF 01 CSF (87)

2(CSF)

2(CSF)

X' (calculated)

X'

2

X'2

k

X' (calculated)

X'

2

kX'2 (measured)

kX'2 (measured) εkX'2 [%]

εkX'2 [%]

X'2

k

### *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:

$$\mathbf{p}\_{\rm z1} = \left(\mathbf{k}\_{\rm zh} \cdot \boldsymbol{\sigma}\_{\rm h} \cdot \mathbf{f}\_{\rm l} + \mathbf{k}\_{\rm zw} \cdot \boldsymbol{\sigma}\_{\rm w} \cdot \mathbf{f}\_{\rm l}^2 \cdot \boldsymbol{\Delta}^2\right) \cdot \mathbf{B}\_{\rm x1m}^2 \cdot \mathbf{G}\_{\rm x1} \,\tag{88}$$

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:

 2 1 f , (89)

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:

$$\mathbf{p}\_{\rm x1(1)} = \left(\mathbf{k}\_{\rm xh} \cdot \boldsymbol{\sigma}\_h \cdot \mathbf{f}\_1 + \mathbf{k}\_{\rm xw} \cdot \boldsymbol{\sigma}\_w \cdot \mathbf{f}\_1^2 \cdot \boldsymbol{\Delta}^2\right)^2 \cdot \mathbf{B}\_{\rm x1n(1)}^2 \cdot \mathbf{G}\_{\rm x1} \,\,\,\,\,\tag{90}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 65

z1 zh <sup>h</sup> h 1 zw <sup>w</sup> w 1 z1m z1 p kk fk k f B G (95)

 

> 

p k , (100)

1 z1m 1

fk fk , (97)

 

B

2

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

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:

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

> 

 h h z z

\* 22 2 2

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:

z1 1 z1e 1 p k k 1 kk

z1 CSF zw w 1 z1m 1 z1 z1e 1 z1e w

which the relations (93), (94) and (99) are taken into account we obtain:

consist of the hysteresis losses, pj1h and eddy currents losses, pj1w:

K K k k <sup>1</sup> k 1 <sup>1</sup>

h w

sh sin sh sin <sup>3</sup> k ; k ; 2 ch cos ch cos (96)

1 1 w w

222 2 z1m

p k f B Gk k k B (99)

 z1 CSF z1e 2 2 pz1 w Bz1 ,1

In the case of the direct – mode supplying system of the machine, the principal yoke losses

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

z1 z1w z1e z1w z1e zw <sup>w</sup> w 1 z1m z1 p p k p k kk f B G (98)

22 2 2

of the eddy currents:

the case of the fundamental, we obtain:

z1e 2

where kBz(ν,1) = Bz1m(ν) / Bz1m(1).

**B. The principal losses in the stator yoke** 

where Bz1m(1) represents the magnetic induction from the middle of the tooth, B B . z1m 1 z1m 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:

$$\mathbf{p}\_{\mathbf{z}1\le(1)}^{\bullet} = \mathbf{p}\_{\mathbf{z}1(1)}\tag{91}$$

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:

$$\mathbf{p}\_{\mathbf{z}1\le(1)}^{\bullet} = \mathbf{k}\_{\mathbf{z}1\alpha(1)} \cdot \mathbf{p}\_{\mathbf{z}1\le(1)}\tag{92}$$

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

$$\left(\mathbf{k}\_{\rm zh} \cdot \boldsymbol{\sigma}\_{\rm h} \cdot \mathbf{f}\_{\rm z} + \mathbf{k}\_{\rm zw} \cdot \boldsymbol{\sigma}\_{\rm w} \cdot \mathbf{f}\_{\rm z}^{2} \cdot \boldsymbol{\Delta}^{2}\right) \cdot \mathbf{B}\_{\rm z1m(1)}^{2} \cdot \mathbf{G}\_{\rm z1} = \mathbf{k}\_{\rm z1o(1)} \cdot \mathbf{k}\_{\rm zw} \cdot \boldsymbol{\sigma}\_{\rm w} \cdot \mathbf{f}\_{\rm z}^{2} \cdot \boldsymbol{\Delta}^{2} \cdot \mathbf{B}\_{\rm z1m(1)}^{2} \cdot \mathbf{G}\_{\rm z1} \,\, . \tag{93}$$

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

$$\mathbf{k}\_{x1s(1)} = \mathbf{1} + \frac{\mathbf{K}\_{x\Delta}}{\mathbf{f}\_1} \tag{94}$$

where we have

$$\mathbf{K}\_{\mathbf{z}\boldsymbol{\Lambda}} = \mathbf{K}\_{\mathbf{z}} / \boldsymbol{\Delta}^2 \text{ with } \mathbf{K}\_{\mathbf{z}} = \frac{\boldsymbol{\sigma}\_{\mathbf{h}} \cdot \mathbf{k}\_{\mathbf{z}\boldsymbol{\Lambda}}}{\boldsymbol{\sigma}\_{\mathbf{w}} \cdot \mathbf{k}\_{\mathbf{z}\boldsymbol{\text{w}}}}.$$

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:

$$\mathbf{P}\_{x1(\text{v})} = \left(\mathbf{k}\_{\text{zh}} \cdot \mathbf{k}\_{\text{h}(\text{v})} \cdot \boldsymbol{\sigma}\_{\text{h}} \cdot \mathbf{v} \cdot \mathbf{f}\_{1} + \mathbf{k}\_{\text{xw}} \cdot \mathbf{k}\_{\text{w}(\text{v})} \cdot \boldsymbol{\sigma}\_{\text{w}} \cdot \mathbf{v}^{2} \cdot \mathbf{f}\_{1}^{2} \cdot \boldsymbol{\Delta}^{2}\right) \cdot \mathbf{B}\_{x1\text{m}(\text{v})}^{2} \cdot \mathbf{G}\_{x1} \tag{95}$$

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:

$$\mathbf{k}\_{\mathsf{h}(\boldsymbol{\nu})} = \frac{\mathsf{\tilde{\xi}}\_{\mathsf{(\boldsymbol{\nu})}}}{\mathsf{\mathcal{D}}} \cdot \frac{\operatorname{sh}\mathsf{\tilde{\xi}}\_{\mathsf{(\boldsymbol{\nu})}} + \sin\mathsf{\tilde{\xi}}\_{\mathsf{(\boldsymbol{\nu})}}}{\operatorname{ch}\mathsf{\tilde{\xi}}\_{\mathsf{(\boldsymbol{\nu})}} - \cos\mathsf{\tilde{\xi}}\_{\mathsf{(\boldsymbol{\nu})}}}; \ \mathsf{k}\_{\mathsf{w}(\boldsymbol{\nu})} = \frac{\mathsf{\tilde{\xi}}}{\mathsf{\tilde{\xi}}\_{\mathsf{(\boldsymbol{\nu})}}} \cdot \frac{\operatorname{sh}\mathsf{\tilde{\xi}}\_{\mathsf{(\boldsymbol{\nu})}} - \sin\mathsf{\tilde{\xi}}\_{\mathsf{(\boldsymbol{\nu})}}}{\operatorname{ch}\mathsf{\tilde{\xi}}\_{\mathsf{(\boldsymbol{\nu})}} - \cos\mathsf{\tilde{\xi}}\_{\mathsf{(\boldsymbol{\nu})}}}; \tag{96}$$

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:

$$\mathbf{k}\_{\rm x1o(v)} = 1 + \frac{\mathbf{K}\_{\rm x}}{\Delta^2} \cdot \frac{1}{\mathbf{v} \cdot \mathbf{f}\_1} \cdot \frac{\mathbf{k}\_{\rm h(v)}}{\mathbf{k}\_{\rm w(v)}} = 1 + \frac{\mathbf{K}\_{\rm x1}}{\mathbf{v} \cdot \mathbf{f}\_1} \cdot \frac{\mathbf{k}\_{\rm h(v)}}{\mathbf{k}\_{\rm w(v)}},\tag{97}$$

$$\mathbf{p}\_{\mathbf{x}\mathbf{1}(\mathbf{v})} = \mathbf{p}\_{\mathbf{x}\mathbf{1}\mathbf{w}(\mathbf{v})}^{\ast} = \mathbf{k}\_{\mathbf{x}\mathbf{1}\mathbf{e}(\mathbf{v})} \cdot \mathbf{p}\_{\mathbf{x}\mathbf{1}\mathbf{w}(\mathbf{v})} = \mathbf{k}\_{\mathbf{x}\mathbf{1}\mathbf{e}(\mathbf{v})} \cdot \mathbf{k}\_{\mathbf{x}\mathbf{w}} \cdot \mathbf{k}\_{\mathbf{w}\mathbf{(v}} \cdot \sigma\_{\mathbf{w}} \cdot \mathbf{v}^{2} \cdot \mathbf{f}\_{\mathbf{1}}^{2} \cdot \Delta^{2} \cdot \mathbf{B}\_{\mathbf{x}\mathbf{1}\mathbf{m}(\mathbf{v})}^{2} \cdot \mathbf{G}\_{\mathbf{x}\mathbf{1}} \tag{98}$$

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:

$$\mathbf{p}\_{\rm x1(CSF)} = \mathbf{k}\_{\rm xw} \cdot \boldsymbol{\sigma}\_{\rm w} \cdot \mathbf{f}\_{1}^{2} \cdot \boldsymbol{\Delta}^{2} \cdot \mathbf{B}\_{\rm x1m(1)}^{2} \cdot \mathbf{G}\_{\rm x1} \bigg| \mathbf{k}\_{\rm x1s(1)} + \sum\_{\mathbf{v} \neq 1} \mathbf{k}\_{\rm x1s(v)} \cdot \mathbf{k}\_{\rm w(v)} \cdot \mathbf{v}^{2} \bigg| \frac{\mathbf{B}\_{\rm x1m(v)}}{\mathbf{B}\_{\rm x1m(1)}} \bigg|^{2} \Bigg] \tag{99}$$

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:

$$\mathbf{k}\_{\rm px1} = \frac{\mathbf{P}\_{\rm x1(CSF)}}{\mathbf{P}\_{\rm x1}} = \mathbf{1} + \sum\_{\rm v} \left( \frac{\mathbf{k}\_{\rm x1o(v)}}{\mathbf{k}\_{\rm x1o(1)}} \cdot \mathbf{k}\_{\rm w(v)} \cdot \mathbf{v}^2 \cdot \mathbf{k}\_{\rm Rx1(v,1)}^2 \right) \tag{100}$$

where kBz(ν,1) = Bz1m(ν) / Bz1m(1).

64 Induction Motors – Modelling and Control

teeth, can be written as following:

"pz1w(1)" – fundamental-mode supplying state:

can consider that:

where we have

teeth losses is obtained. The relation (91) made explicit becomes:

<sup>z</sup>

supplying mode, at which usually we have f1≤120 [Hz], the principal losses from the stators

where Bz1m(1) represents the magnetic induction from the middle of the tooth, B B . z1m 1 z1m 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:

\*

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

\*

We consider that through this equalization factor a covering value of the principal stator

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

<sup>K</sup> k 1

<sup>2</sup> K K/ z z with

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

zh h 1 zw w 1 z1m 1 z1 z1e 1 zw w 1 z1m 1 z1 k fk f B G k k f B G . (93)

1

z

<sup>k</sup> <sup>K</sup>

h zh

w zw

k

22 2 <sup>222</sup>

z1e 1

z1 1 zh h 1 zw w 1 z1m 1 z1 p k f k f B G , (90)

z1w 1 z1 1 p p (91)

z1w 1 z1e 1 z1w 1 p k p (92)

f (94)

<sup>2</sup> 22 2

### **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:

$$\mathbf{p}\_{\rm j1} = \left(\boldsymbol{\sigma}\_{\rm h} \cdot \mathbf{f}\_{\rm l} \cdot \mathbf{k}\_{\rm j1h} + \boldsymbol{\sigma}\_{\rm w} \cdot \boldsymbol{\Delta}^2 \cdot \mathbf{f}\_{\rm l}^2 \cdot \mathbf{k}\_{\rm j1w}\right) \cdot \mathbf{B}\_{\rm j1}^2 \cdot \mathbf{G}\_{\rm j1} \tag{101}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 67

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,

> 

 h h w w

In conclusion, the principal losses in the stator yoke, corresponding to the order time

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

j1 1 j1e 1 p k k 1 kk

p f k B Gk k k <sup>B</sup>

j1 CSF w 1 j1w j1 1 j1 j1e 1 j1e w

divide the relation (110) at (101). After finishing the computations we have:

c1

K K k k <sup>1</sup> k 1 <sup>1</sup>

 

j1 j1w j1e j1w pp kp , (108)

1 j1 1

1,5 <sup>2</sup> c1 41

p k , (111)

 j1 CSF j1e 2 2 pj1 w Bj1 ,1

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

<sup>1</sup> <sup>b</sup> P l D kNn kB

1 o c2 c2 2 2

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

j1 22 2 2

j1w w <sup>w</sup> 1 j1w j1 j1 p k f k B G (109)

fk fk (107)

 

B

2

(110)

(112)

1 1 w w

with the "pj1w()" type losses:

where:

harmonic can be written by equalizing as:

\*

2 22 2

principle on the theoretical linear model we can write:

*7.1.2. The supplementary stator iron losses* 

2

**A. Surface supplementary losses** 

where: kBj(ν,1) = Bj1(ν) / Bj1(1).

expression:

j1e 2

where: Bjl is the magnetic induction in the stator yoke, Gjl represents the weight of the stator yoke, j1w j1w1 j1w2 k kk , 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:

$$\mathbf{p}\_{\rm jt(1)} = \left(\boldsymbol{\sigma}\_{\rm h} \cdot \mathbf{f}\_{\rm l} \cdot \mathbf{k}\_{\rm j1h} + \boldsymbol{\sigma}\_{\rm w} \cdot \boldsymbol{\Delta}^2 \cdot \mathbf{f}\_{\rm l}^2 \cdot \mathbf{k}\_{\rm j1w}\right) \cdot \mathbf{B}\_{\rm jt(1)}^2 \cdot \mathbf{G}\_{\rm l1} \tag{102}$$

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):

$$\mathbf{p}^{\mathbf{"}}\_{\text{l}^{\text{l}\text{w}}(1)} = \mathbf{p}\_{\text{l}^{\text{l}(1)}} \tag{103}$$

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):

$$\mathbf{p}^\*\_{\parallel \simeq (1)} = \mathbf{k}\_{\parallel \simeq (1)} \cdot \mathbf{p}\_{\parallel \simeq (1)} \tag{104}$$

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

$$\mathbf{k}\_{\rm jlat} = \mathbf{1} + \frac{\mathbf{K}\_w}{\boldsymbol{\Delta}^2 \cdot \mathbf{f}\_1} = \mathbf{1} + \frac{\mathbf{K}\_{\rm w\Lambda}}{\mathbf{f}\_1} \tag{105}$$

where we have:

$$\mathbf{K}\_{\rm w} = \frac{\boldsymbol{\sigma}\_{\rm h} \cdot \mathbf{k}\_{\rm jlh}}{\boldsymbol{\sigma}\_{\rm w} \cdot \mathbf{k}\_{\rm j1w}} \text{ and } \mathbf{K}\_{\rm wA} = \frac{\mathbf{K}\_{\rm w}}{\boldsymbol{\Delta}^2}$$

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.

$$\mathbf{p}\_{\rm jt(v)} = \left(\mathbf{k}\_{\rm h(v)} \cdot \boldsymbol{\sigma}\_{\rm h} \cdot \mathbf{v} \cdot \mathbf{f}\_{\rm l} \cdot \mathbf{k}\_{\rm jlh} + \mathbf{k}\_{\rm w(v)} \cdot \boldsymbol{\sigma}\_{\rm w} \cdot \boldsymbol{\Delta}^2 \cdot \mathbf{v}^2 \cdot \mathbf{f}\_1^2 \cdot \mathbf{k}\_{\rm jlh}\right) \cdot \mathbf{B}\_{\rm |il(v)}^2 \cdot \mathbf{G}\_{\rm |il} \tag{106}$$

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:

$$\mathbf{k}\_{\rm j\_{h^{\rm (v)}}} = 1 + \frac{\mathbf{K}\_{\rm w}}{\Delta^2} \cdot \frac{1}{\mathbf{v} \cdot \mathbf{f}\_1} \cdot \frac{\mathbf{k}\_{\rm h^{\rm (v)}}}{\mathbf{k}\_{\rm w^{\rm (v)}}} = 1 + \frac{\mathbf{K}\_{\rm w \rm h}}{\mathbf{v} \cdot \mathbf{f}\_1} \cdot \frac{\mathbf{k}\_{\rm h^{\rm (v)}}}{\mathbf{k}\_{\rm w^{\rm (v)}}} \tag{107}$$

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

$$\mathbf{p}\_{\rm int(v)} = \mathbf{p}\_{\rm int(v)}^{"\prime} = \mathbf{k}\_{\rm int(v)} \cdot \mathbf{p}\_{\rm 1:n(v)} \, \prime \, \tag{108}$$

where:

66 Induction Motors – Modelling and Control

22 2

where: Bjl is the magnetic induction in the stator yoke, Gjl represents the weight of the stator yoke, j1w j1w1 j1w2 k kk , 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:

losses from the stator yoke described with the relation (102):

\*

\*

where we have:

22 2

If we have p\*j1w(1) as losses in eddy currents, than these must be equalized with the principal

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):

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

2 22 2

K K k1 1

w w

1 1

and

j1 h h 1 j1h <sup>w</sup> <sup>w</sup> 1 j1w j1 j1 p k fk k f k B G (106)

w w 2 <sup>K</sup> <sup>K</sup>

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

 h j1h

take into account respectively the skin effect and the eddy currents reaction.

w j1w k

k

w

K

j1e 1 2

j1 h 1 j1h w 1 j1w j1 j1 p fk f k B G (101)

j1 1 h 1 j1h w 1 j1w j1 1 j1 p fk f k B G (102)

j1w 1 j1 1 p p (103)

j1w 1 j1e 1 j1w 1 p k p (104)

<sup>f</sup> <sup>f</sup> , (105)

$$\mathbf{p}\_{\text{jiv}(\text{v})} = \mathbf{k}\_{w(\text{v})} \cdot \boldsymbol{\sigma}\_{w} \cdot \boldsymbol{\Lambda}^2 \cdot \mathbf{v}^2 \cdot \mathbf{f}\_{\text{l}}^2 \cdot \mathbf{k}\_{\text{jiv}} \cdot \mathbf{B}\_{\text{jl}(\text{v})}^2 \cdot \mathbf{G}\_{\text{ll}} \tag{109}$$

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:

$$\mathbf{p}\_{\text{jl(CSF)}} = \boldsymbol{\sigma}\_{\text{w}} \cdot \mathbf{f}\_{1}^{2} \cdot \mathbf{A}^{2} \cdot \mathbf{k}\_{\text{j1w}} \cdot \mathbf{B}\_{\text{jl(1)}}^{2} \cdot \mathbf{G}\_{\text{jl}} \left[ \mathbf{k}\_{\text{j1e}(1)} + \sum\_{\text{v} \neq 1} \mathbf{k}\_{\text{j1e}(\text{v})} \cdot \mathbf{k}\_{\text{v}\text{(v)}} \cdot \mathbf{v}^{2} \left( \frac{\mathbf{B}\_{\text{jl(v)}}}{\mathbf{B}\_{\text{jl(1)}}} \right)^{2} \right] \tag{110}$$

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:

$$\mathbf{k}\_{\rm p|1} = \frac{\mathbf{P}\_{\rm \rm lt(CSF)}}{\mathbf{P}\_{\rm \rm lt}} = 1 + \sum\_{\rm v} \left( \frac{\mathbf{k}\_{\rm lto(v)}}{\mathbf{k}\_{\rm lto(v)}} \cdot \mathbf{k}\_{\rm w(v)} \cdot \mathbf{v}^2 \cdot \mathbf{k}\_{\rm bpl(v, 1)}^2 \right) \tag{111}$$

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:

$$\mathbf{P}\_{o1} = \frac{1}{2} \cdot \mathbf{1} \cdot \boldsymbol{\pi} \cdot \mathbf{D} \cdot \frac{\boldsymbol{\tau}\_{c1} - \mathbf{b}\_{41}}{\boldsymbol{\tau}\_{c1}} \cdot \mathbf{k}\_o \cdot \left(\mathbf{N}\_{c2} \cdot \mathbf{n}\right)^{1.5} \cdot \left(\boldsymbol{\tau}\_{c2} \cdot \boldsymbol{\beta}\_2 \cdot \mathbf{k}\_{\delta 2} \cdot \mathbf{B}\_{\delta}\right)^2 \tag{112}$$

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:

$$P\_{\sigma1(\text{CSF})} = \frac{1}{2} \cdot 1 \cdot \pi \cdot \text{D} \cdot \frac{\tau\_{c1} - \text{b}\_{41}}{\tau\_{c1}} \cdot \text{k}\_o \cdot \left(\text{N}\_{c2} \cdot \text{n}\right)^{1.5} \cdot \left(\tau\_{c2} \cdot \text{p}\_2 \cdot \text{k}\_{o2} \cdot \text{B}\_{\delta(1)}\right)^2 \left|1 + \sum\_{\forall \ast 1} \left(\frac{\text{B}\_{\delta(\text{v})}}{\text{B}\_{\delta(1)}}\right)^2\right| \tag{113}$$

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:

$$\mathbf{k}\_{\rm pe1} = \frac{\mathbf{P}\_{\rm o1(CSF)}}{\mathbf{P}\_{\rm o1}} = \mathbf{1} + \sum\_{\mathbf{v} \neq 1} \left( \frac{\mathbf{B}\_{\delta(\mathbf{v})}}{\mathbf{B}\_{\delta(\mathbf{1})}} \right)^2 = \mathbf{1} + \sum\_{\mathbf{v} \neq 1} \mathbf{k}\_{\rm u(\mathbf{v}, \mathbf{1})}^2 > \mathbf{1} \tag{114}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 69

c1 1 z1m 1

<sup>2</sup> <sup>2</sup> <sup>2</sup> z1m

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

P B (117)

  (116)

machine is supplied by inverters, by applying the over position effect principle, the following

22 B

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

 2

P1 CSF z1m 2 Pp1 Bz1 ,1 P1 1 1 z1m 1

k 1 1k 1

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

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

2 22 2 2

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

\*

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)

 h h z z

K K k k <sup>1</sup> k 1 <sup>1</sup>

1 1 w w

by using the relation (118), after finishing the intermediary computations we obtain:

given by the relation (118) are produced only by eddy currents:

z2e 2

z2 zh <sup>h</sup> <sup>h</sup> 1 zw <sup>w</sup> <sup>w</sup> <sup>1</sup> z2m z2 p kk s fk k s f B G (118)

z2 z2w z2e z2w p p k p , (119)

 

s fk s fk (120)

expression for the supplementary pulsation losses in the stator PP1(CSF) is obtained:

P B

<sup>k</sup> <sup>B</sup> <sup>1</sup> <sup>P</sup> k Nn GB 1

P1 CSF w wP1 c2 z1 z1m 1

supplying system, kPp1:

small value of the

**7.2. Rotor iron losses** 

teeth have the expression:

2 Bz1 ,1 k .

*7.2.1. Principal losses in the rotor iron* 

**A. The principal losses in the rotor's teeth** 

where kBδ(ν,1) = Bδ(ν) / Bδ(1). By analyzing the relation (114) one can notice the fact that the kP<sup>1</sup> 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:

$$\mathbf{P}\_{\rm p1} = \frac{1}{2} \cdot \boldsymbol{\sigma}\_{\rm w} \cdot \mathbf{k}\_{\rm w1} \cdot \left(\boldsymbol{\Lambda} \mathbf{N}\_{\rm c2} \mathbf{n}\right)^{2} \cdot \left(\frac{\gamma\_{\rm z} \delta \mathbf{k}\_{\rm s}}{2\tau\_{\rm c1}}\right)^{2} \cdot \mathbf{G}\_{\rm z1} \cdot \mathbf{B}\_{\rm z1\rm m}^{2} \tag{115}$$

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:

$$\mathbf{P}\_{\rm Pl\text{(c3F)}} = \frac{1}{2} \cdot \boldsymbol{\sigma}\_{\rm w} \cdot \mathbf{k}\_{\rm w\rm p1} \cdot \left(\Delta \mathbf{N}\_{\rm c2} \mathbf{n}\right)^{2} \cdot \left(\frac{\gamma\_{2} \delta \mathbf{k}\_{\rm s}}{2\pi\_{\rm c1}}\right)^{2} \cdot \mathbf{G}\_{\rm x1} \cdot \mathbf{B}\_{\rm x1m(1)}^{2} \left|1 + \sum\_{\mathbf{v} \neq 1} \left(\frac{\mathbf{B}\_{\rm x1m(v)}}{\mathbf{B}\_{\rm x1m(1)}}\right)^{2}\right| \tag{116}$$

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:

$$\mathbf{k}\_{\rm p\_{\rm p1}} = \frac{\mathbf{P}\_{\rm p1(CSF)}}{\mathbf{P}\_{\rm p1}} = 1 + \sum\_{\mathbf{v} \neq 1} \left( \frac{\mathbf{B}\_{x1\mathbf{m}(\mathbf{v})}}{\mathbf{B}\_{x1\mathbf{m}(\mathbf{1})}} \right)^2 = 1 + \sum\_{\mathbf{v} \neq 1} \mathbf{k}\_{\rm p1}^2 \mathbf{r}\_{\rm (v, 1)} > 1 \tag{117}$$

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 the 2 Bz1 ,1 k .

### **7.2. Rotor iron losses**

68 Induction Motors – Modelling and Control

be computed with the relation:

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

P B

<sup>b</sup> <sup>B</sup> <sup>1</sup> P l D kNn kB 1

c1 1 1

c1

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

2 2 , (115)

P B , (114)

 B ,1 2

2 B (113)

<sup>2</sup> 1,5 c1 41

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

k 1 1k 1

1 1 1 1

where kBδ(ν,1) = Bδ(ν) / Bδ(1). By analyzing the relation (114) one can notice the fact that the kP<sup>1</sup> 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

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

P1 w wP1 c2 z1 z1m

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

<sup>1</sup> <sup>k</sup> P k Nn G B

 

1 CSF 2

the inverter versus the sine-mode supplying case, kP1, as following:

is not to be taken into consideration.

**B. The pulsation supplementary losses** 

sine-wave, has the following expression:

P 1

1 CSF o c2 c2 2 2 1

 

2

### *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:

$$\mathbf{p}\_{\mathbf{x}2\text{(v)}} = \left(\mathbf{k}\_{\text{zh}} \cdot \mathbf{k}\_{\text{h}\text{(v)}} \cdot \boldsymbol{\sigma}\_{\text{h}} \cdot \mathbf{s}\_{\text{(v)}} \cdot \mathbf{v} \cdot \mathbf{f}\_{\text{1}} + \mathbf{k}\_{\text{zw}} \cdot \mathbf{k}\_{\text{w}\text{(v)}} \cdot \boldsymbol{\sigma}\_{\text{w}} \cdot \mathbf{s}\_{\text{(v)}}^{2} \cdot \mathbf{v}^{2} \cdot \mathbf{f}\_{\text{1}}^{2} \cdot \boldsymbol{\Delta}^{2}\right) \mathbf{B}\_{\text{z2m}(v)}^{2} \cdot \mathbf{G}\_{\text{zz}} \tag{118}$$

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:

$$\mathbf{p}\_{\mathbf{x}2\text{(v)}} = \mathbf{p}\_{\mathbf{x}2\text{w}\text{(v)}}^{\text{"}} = \mathbf{k}\_{\mathbf{x}2\text{w}\text{(v)}} \cdot \mathbf{p}\_{\mathbf{x}2\text{w}\text{(v)}} \text{ \textquotedbl{}}\tag{119}$$

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:

$$\mathbf{k}\_{\mathbf{z}\mathbf{z}\circ(\mathbf{v})} = \mathbf{1} + \frac{\mathbf{K}\_{\mathbf{z}}}{\mathbf{A}^2} \cdot \frac{\mathbf{1}}{\mathbf{s}\_{\left(\mathbf{v}\right)} \cdot \mathbf{v} \cdot \mathbf{f}\_1} \cdot \frac{\mathbf{k}\_{\mathbf{h}\left(\mathbf{v}\right)}}{\mathbf{k}\_{\mathbf{w}\left(\mathbf{v}\right)}} = \mathbf{1} + \frac{\mathbf{K}\_{\mathbf{z}\mathbf{A}}}{\mathbf{s}\_{\left(\mathbf{v}\right)} \cdot \mathbf{v} \cdot \mathbf{f}\_1} \cdot \frac{\mathbf{k}\_{\mathbf{h}\left(\mathbf{v}\right)}}{\mathbf{k}\_{\mathbf{w}\left(\mathbf{v}\right)}}\tag{120}$$

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

$$\mathbf{p}\_{x2(v)} = \mathbf{k}\_{x2a(v)} \cdot \mathbf{k}\_{xw} \cdot \mathbf{k}\_{w(v)} \cdot \boldsymbol{\sigma}\_w \cdot \mathbf{s}\_{(v)}^2 \cdot \mathbf{v}^2 \cdot \mathbf{f}\_1^2 \cdot \boldsymbol{\Delta}^2 \cdot \mathbf{B}\_{x2m(v)}^2 \cdot \mathbf{G}\_{x2} \tag{121}$$

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:

$$\mathbf{p}\_{x2(\text{CSF})} = \sum\_{\mathbf{v} \neq 1} \mathbf{p}\_{x2(\mathbf{v})} \tag{122}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 71

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

> 

k 1 1k k1

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

P2 w wP2 c1 P2 z2 <sup>1</sup> P k N nB G

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

 P2 CSF <sup>2</sup> Pp2 B ,1 P2 1

k 1k 1

This paper aims to study the theoretical behavior of asynchronous three-phase motor in the case of supplying through a power frequency converter. This study has aimed to develop the theory of the asynchronous three-phase motor in non-sinusoidal periodic regime to serve as a starting point in optimizing the design methodology. Given that the asynchronous threephase motor is fed through a static frequency converter, the machine operation in the presence of higher time harmonics in the supply voltage can be described by a single mathematical model. The model consists of a single equivalent scheme corresponding to all

 z2m

k

B ,1

B B , (132)

P

P

B B

z2m 1 1

 2

2 CSF 2 P 2 B ,1 P 1 2 1 1 1

 c2 42

c2 <sup>1</sup> <sup>b</sup> P pl 2 2 , (128)

2 o c1 c1 1 1 p kN n k B (129)

P B (130)

(131)

(133)

where the specific rotor surface losses p2 have the expression:

**B. The supplementary pulsation losses** 

harmonics and it is defined at the fundamental frequency.

supplying system, kP2:

following expression:

the fact that:

we obtain:

**8. Conclusions** 

1,5 <sup>2</sup>

2 2

P B

<sup>2</sup>

2

### **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:

$$\mathbf{P}\_{|\mathbf{2}(\mathbf{v})} = \left(\mathbf{k}\_{\mathrm{h}(\mathbf{v})} \cdot \boldsymbol{\sigma}\_{\mathrm{h}} \cdot \mathbf{s}\_{\mathrm{(v)}} \cdot \mathbf{v} \cdot \mathbf{f}\_{\mathrm{i}} \cdot \mathbf{k}\_{\mathrm{jlh}} + \mathbf{k}\_{\mathrm{w}(\mathbf{v})} \cdot \boldsymbol{\sigma}\_{\mathrm{w}} \cdot \mathbf{s}\_{\mathrm{(v)}}^{2} \cdot \mathbf{v}^{2} \cdot \mathbf{f}\_{\mathrm{i}}^{2} \cdot \boldsymbol{\Delta}^{2} \cdot \mathbf{k}\_{\mathrm{jzw}}\right) \cdot \mathbf{B}\_{|2\langle \mathbf{v} \rangle}^{2} \cdot \mathbf{G}\_{|2} \tag{123}$$

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

$$\mathbf{p}\_{\vert 2^{\langle \mathbf{v} \rangle}} = \mathbf{p}\_{\vert 2^{\mathbf{w}} \langle \mathbf{v} \rangle}^{\cdot} = \mathbf{k}\_{\vert 2^{\langle \mathbf{v} \rangle} \langle \mathbf{v} \rangle} \cdot \mathbf{p}\_{\vert 2^{\langle \mathbf{w} \rangle} \langle \mathbf{v} \rangle} \tag{124}$$

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:

$$\mathbf{k}\_{\text{j2a}(\text{v})} = \mathbf{1} + \frac{\mathbf{K}\_{\text{w}}}{\Delta^2} \cdot \frac{\mathbf{1}}{\mathbf{s}\_{\text{(v)}} \cdot \mathbf{v} \cdot \mathbf{f}\_1} \cdot \frac{\mathbf{k}\_{\text{h}(\text{v})}}{\mathbf{k}\_{\text{w}(\text{v})}} = \mathbf{1} + \frac{\mathbf{K}\_{\text{w}\text{A}}}{\mathbf{s}\_{\text{(v)}} \cdot \mathbf{v} \cdot \mathbf{f}\_1} \cdot \frac{\mathbf{k}\_{\text{h}(\text{v})}}{\mathbf{k}\_{\text{w}(\text{v})}} \tag{125}$$

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

$$\mathbf{p}\_{\text{/2(v)}} = \mathbf{k}\_{\text{/2o(v)}} \cdot \mathbf{k}\_{\text{/2w}} \cdot \mathbf{k}\_{\text{w(v)}} \cdot \boldsymbol{\sigma}\_{\text{w}} \cdot \mathbf{s}\_{\text{(v)}}^2 \cdot \mathbf{v}^2 \cdot \mathbf{f}\_1^2 \cdot \boldsymbol{\Delta}^2 \cdot \mathbf{B}\_{\text{/2(v)}}^2 \cdot \mathbf{G}\_{\text{/2}} \tag{126}$$

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:

$$\mathbf{P}\_{|\mathbb{2}^{\{\text{CSF}\}}} = \sum\_{\mathbf{v} \neq \mathbf{1}} \mathbf{P}\_{|\mathbb{2}^{\{\text{v}\}}} \tag{127}$$

### *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:

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 71

$$\mathbf{P}\_{a2} = \frac{1}{2} \cdot \mathbf{p}\_{a2} \cdot \mathbf{l} \cdot \pi \cdot \left(\Lambda - 2\delta\right) \cdot \frac{\mathbf{r}\_{c2} - \mathbf{b}\_{a2}}{\mathbf{r}\_{c2}} \; , \tag{128}$$

where the specific rotor surface losses p2 have the expression:

$$\mathbf{p}\_{a2} = \mathbf{k}\_o \left(\mathbf{N}\_{c1} \cdot \mathbf{n}\right)^{1.5} \cdot \left(\mathbf{r}\_{c1} \cdot \mathbf{\beta}\_1 \cdot \mathbf{k}\_{\delta1} \cdot \mathbf{B}\_{\delta}\right)^2 \tag{129}$$

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:

$$\mathbf{k}\_{\rm pe2} = \frac{\mathbf{P}\_{\rm o2(CSF)}}{\mathbf{P}\_{\rm o2}} = \mathbf{1} + \sum\_{\mathbf{v} \neq 1} \left( \frac{\mathbf{B}\_{\vartheta(\mathbf{v})}}{\mathbf{B}\_{\vartheta(\mathbf{1})}} \right)^2 = \mathbf{1} + \sum\_{\mathbf{v} \neq 1} \mathbf{k}\_{\rm Bi(\vartheta, 1)}^2 = \mathbf{k}\_{\rm pe1} > \mathbf{1} \tag{130}$$

### **B. The supplementary pulsation losses**

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

$$\mathbf{P}\_{\rm p\_2} = \frac{1}{2} \boldsymbol{\sigma}\_{\rm w} \cdot \mathbf{k}\_{w\rm p\_2} \left(\boldsymbol{\Delta} \cdot \mathbf{N}\_{\rm c1} \cdot \mathbf{n} \cdot \mathbf{B}\_{\rm p\_2}\right)^2 \cdot \mathbf{G}\_{\rm r2} \tag{131}$$

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

$$\frac{\mathbf{B}\_{\times 2m(\mathbf{v})}}{\mathbf{B}\_{\times 2m(\mathbf{1})}} = \frac{\mathbf{B}\_{\delta(\mathbf{v})}}{\mathbf{B}\_{\delta(\mathbf{1})}} = \mathbf{k}\_{\mathbb{R}^{\delta(\mathbf{v}, 1)}} \, \, \, \tag{132}$$

we obtain:

70 Induction Motors – Modelling and Control

harmonic can be written by equalization as it follows:

**B. The principal losses from the rotor's yoke** 

theoretical linear model we can obtain the equality:

written by equalization in the form:

j2e 2

in the rotor yoke, pj2(CSF), are computed with the relation:

*7.2.2. The supplementary losses in the rotor iron* 

**A. The surface supplementary losses** 

losses are calculated with the relation:

2 22 2 2

real principal losses induced by it in the rotor yoke have the expression:

in the rotor yoke, only with losses of the type "pj2w()" type as it follows:

2 22 2 <sup>2</sup>

harmonics are present, the principal losses in the rotor teeth can be written as:

Therefore, the principal losses from the rotor teeth, corresponding to the order time

In the conditions in which in the supplying system of the machine all the superior time

 z2 CSF z2 1

In the hypotheses in which in the supplying system only the order harmonic is present, the

Through the energetic equalization, due to the replacement of the real machine by a

\*

Reasoning as in the previous cases, we can determine the equalizing factor of the real losses

 h h w w

Consequently, the principal rotor yoke losses corresponding to the order harmonic can be

2 22 2 2

Disregarding all these, in the case of the inverter supplying system the total principal losses

 j2 CSF j2 1

If the machine is directly supplied from the power supply, the surface supplementary rotor

K K k k <sup>1</sup> k 1 <sup>1</sup>

1 1 w w

j2 j2e j2w <sup>w</sup> <sup>w</sup> <sup>1</sup> j2 j2 p k kk s f B G (126)

j2 h h 1j1h w 1j <sup>w</sup> 2w j j2 <sup>2</sup> p k s f k k s f k B G (123)

z2 z2e zw <sup>w</sup> <sup>w</sup> <sup>1</sup> z2m z2 p k kk s f B G (121)

p p (122)

j2 j2w j2e j2w p p k p (124)

 

s fk s fk (125)

p p (127)

$$\mathbf{k}\_{\rm Pr2} = \frac{\mathbf{P}\_{\rm p2(CSF)}}{\mathbf{P}\_{\rm p2}} = 1 + \sum\_{\mathbf{v} \neq 1} \mathbf{k}\_{\rm R4(\mathbf{v}, 1)}^2 > 1 \tag{133}$$

### **8. Conclusions**

This paper aims to study the theoretical behavior of asynchronous three-phase motor in the case of supplying through a power frequency converter. This study has aimed to develop the theory of the asynchronous three-phase motor in non-sinusoidal periodic regime to serve as a starting point in optimizing the design methodology. Given that the asynchronous threephase motor is fed through a static frequency converter, the machine operation in the presence of higher time harmonics in the supply voltage can be described by a single mathematical model. The model consists of a single equivalent scheme corresponding to all harmonics and it is defined at the fundamental frequency.
