# Power indexes of induction motors and electromagnetic efficiency their windings/Asinchroniniu varikliu energiniai rodikliai ir ju apviju elektromagnetinis efektyvumas.

Introduction

Large part of electrical energy used is transformed into mechanical energy in the electric drives of various machines and devices. Three-phase cage rotor induction motors are used to transform the energy in these drives since their construction is simple, they are most reliable during exploitation, have the least relative weight and are least expensive. Three-phase stator winding is one of the most important construction parts of these engines. Main energy interchange processes take place in this winding therefore it essentially determined the operation of the motor.

When electric currents forming the symmetric three-phase system of the currents flow through the three-phase winding of induction motor they create non-sinusoidal magnetic fields which move in space and periodically change their shape in the course of time. Usually only odd space harmonics except for the multiples of three exist in the harmonic spectrum of these non-sinusoidal magnetic fields.

There are many different constructions of the three-phase windings of induction motors and each of them have distinctive parameters [1, 2]. Therefore harmonic spectrum of the magnetic fields created by these windings and at the same time the electromagnetic properties differ and they in turn determine the power indexes and operation quality of induction motors . Electromagnetic efficiency factor is used to evaluate electromagnetic properties of three-phase windings .

The aim of this paper is to perform a theoretical analysis of electromagnetic efficiencies of two types of three-phase windings and to relate them theoretically and experimentally to the power indexes of particular induction motors.

Object of research

Standard dimensioned 1,5 kW three-phase induction motor with single-layer former winding and the same motor with stator winding replaced with sinusoidal winding is investigated in this work. Common parameters of stator for both motors are the following: number of phases m = 3; number of stator magnetic circuit slots Z = = 24; number of poles 2p = 2; number of pole and phase slots q = Z/(2p m) = 24 /(2 x 3) = 4; pole pitch [tau] = Z/2p = = 24 / 2 = 12; slot span expressed in electrical degrees [alpha] = 360[degrees]p /Z = = 360[degrees] x 1/24 = 15[degrees]. The relative magnitude of number of turns of any coil for the single-layer former winding sections with four coils is [N.sup.*.sub.1] = 1/q = 1/4 = 0,25. Relative magnitudes of number of turns of any section in sinusoidal winding calculated according corresponding formulas  are obtained: [N.sup.*.sub.21] = 0,114; [N.sup.*.sub.22] = 0,1862; [N.sup.*.sub.23] = 0,13165; [N.sup.*.sub.24] = 0,06815.

Distribution of elements of the analyzed windings is given in Tables 1 and 2.

Relative magnitudes of the instantaneous values of electric currents in both windings in time moment t = 0 are [i.sup.*.sub.U] = sin 0[degrees] = 0; [i.sup.*.sub.V] = sin 120[degrees] = 0,866; [i.sup.*.sub.W] = sin 240[degrees] = = -0,866. Conditional magnetomotive force changes [DELTA]F = [i.sup.*] [N.sup.*] in the slots of magnetic circuit of the stator in time moment t = 0 (Tables 3 and 4) are calculated according to the determined number of coil turns and relative magnitudes of electric currents by using the layouts of electric circuits of the analyzed windings.

[FIGURE 1 OMITTED]

Electric circuit layout of the sinusoidal three-phase winding is formed according to the data presented in Table 2 (Fig. 2, a).

[FIGURE 2 OMITTED]

Space distributions of rotating magnetomotive force in the defined moment of time are determined according to the results from Tables 3 and 4 (Fig. 1., b and Fig. 2, b).

Research method

On the base of Fig. 1, b and Fig. 2, b, the amplitude value of rotating magnetomotive force of the v-th harmonic [F.sub.mv] is calculated using this analytical expression :

[F.sub.mv] = 4/[pi] v [k.summation over (i=1)] [F.sub.is] sin(v [[beta].sub.i]/2); (1)

here k--number of rectangles forming the half-period of rotating magnetomotive force; [F.sub.is]--height of the i-th rectangle of the stair-shaped magnetomotive force; [[beta].sub.i] width of the i-th rectangle of the stair-shaped magnetomotive force expressed in electrical degrees of the fundamental space harmonic; [upsilon]--number of harmonic.

Then relative magnitudes of harmonics of rotating magnetomotive force are calculated on the base of results of harmonic analysis of rotating magnetomotive force functions (Fig. 1, b and Fig. 2, b) :

[f.sub.v] = [F.sub.mv]/[F.sub.m1]; (2)

here [F.sub.m1]--amplitude value of the first (fundamental) harmonic of rotating magnetomotive force.

Electromagnetic efficiency factors of the considered three-phase windings are calculated according to this expression :

[k.sub.ef] = 1 - [square root of [[infinity].summation over (v=1)] [f.sup.2.sub.v] - 1]. (3)

All power indexes of the standard dimensioned induction motor with single-layer former winding and motor with stator winding replaced with sinusoidal three-phase winding are calculated after completing their no-load and load tests of the motor by using the segregated-losses method. Respective power indexes of asynchronous motors are compared under the indicated load.

Research results

According to the expression (1) and determined parameters of rotating magnetomotive force half-period (k = 4; [F.sub.1s] = -0,2165; [F.sub.2s] = -0,2165; [F.sub.3s] = -0,2165; [F.sub.4s] = -0,2165; [[beta].sub.1] = 165[degrees]; [[beta].sub.2] = 135[degrees]; [[beta].sub.3] = 105[degrees]; [[beta].sub.4] = 75[degrees]) the harmonic analysis of instantaneous rotating magnetomotive force function (Fig. 1, b) of single-layer former three-phase winding (Fig. 1, a) was completed and relative magnitudes of its space harmonics were calculated (Table 5).

According to expression (1) and determined parameters of rotating magnetomotive force half-period (k = 6; [F.sub.1s] = = -0,1140; [F.sub.2s] = -0,2203; [F.sub.3s] = -0,1975; [F.sub.4s] = -0,1613; [F.sub.5s] = -0,1140; [F.sub.6s] = -0,0590; [[beta].sub.1] = 180[degrees]; [[beta].sub.2] = 150[degrees]; [[beta].sub.3] = 120[degrees]; [[beta].sub.4] = 90[degrees]; [[beta].sub.5] = 60[degrees]; [[beta].sub.6] = 30[degrees]) the harmonic analysis of the instantaneous rotating magnetomotive force function (Fig. 2, b) of the sinusoidal three-phase winding (Fig. 2, a) was performed and relative magnitudes of its space harmonics were calculated (Table 6).

According to expression (3) the respective electromagnetic efficiency factors [k.sub.ef] of the single layer former and sinusoidal three-phase windings with q = 4 are calculated: [k.sub.ef1] = 0,9139; [k.sub.ef2] = = 0,9335. Electromagnetic efficiency factor of the sinusoidal three-phase winding is obtained by 2,14 % higher than in case of single-layer former winding.

Experimental tests of the standard dimensioned asynchronous motor with the researched single-layer former winding and motor with stator winding replaced with sinusoidal three-phase winding (under no-load and load conditions) were performed and power indexes of analyzed motors were calculated according to received results using the segregated-losses method  (Tables 7 and 8).

In Tables 7 and 8 [I.sub.1]--phase current of stator winding; [P.sub.1]--consumed power; n--rotational speed of rotor; M--electromagnetic torque; [summation]P--total power losses of motor; [P.sub.2]--useful power; [eta]--efficiency; cos [phi]--power factor.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

After comparing experimental and calculation results under indicated load from Tables 7 and 8 it is received that in case of asynchronous motor with stator winding replaced with sinusoidal three-phase winding the phase current of the stator winding decreased by 6,9 %, power taken from electric grid decreased by 5,0 %, power losses decreased by 11,7 %, efficiency factor increased by 2,4 % and power factor increased by 3,4 %.

Conclusions

* Electromagnetic properties of the three-phase windings can be evaluated by performing harmonic analysis of the rotating magnetomotive force created by them and by calculating electromagnetic efficiency factors based on the results of this analysis.

* It was determined theoretically that electromagnetic efficiency factor of the single-layer former three-phase winding [k.sub.ef1] = 0,9139 and of sinusoidal three-phase winding--[k.sub.ef2] = 0,9335, i.e. by 2,14 % higher than respective factor of the first winding.

* In case of induction motor with sinusoidal three-phase winding under the indicated load the phase current of the stator winding decreased by 6,9 %, power taken from electric grid decreased by 5,0 %, power losses decreased by 11,7 %, efficiency factor increased by 2,4 % and power factor increased by 3,4 % compare to the respective power indexes of the same motor with single-layer former winding obtained under the same load.

* Induction motors with the stator winding electromagnetic efficiency factors closer to one have better power indexes.

References

[1.] Timar P. L. Induction motor fiel analysis. In: Engelmann R. H., Middendorf W. H. (eds). Handbook of Electric Motors. Marcel Dekker, New York, 1995.--P. 228-273.

[2.] [TEXT NOT REPRODUCIBLE IN ASCII], 1989.--399 c.

[3.] Gecys S., Smolskas P. Copper-squirrel-cage Solid Rotor Teeth Zone Parameter Rational Choice for Induction Motor Operating under Geophysical Conditions // Electronics and Electrical Engineering.--Kaunas: Technologija, 2009.--No. 1(89).--P. 91-94.

[4.] Buksnaitis J. Kintamosios sroves trifaziu elektros masinu apviju elektromagnetinis efektyvumas: monografija.--Kaunas: Technologija, 2007.--196 p.

[5.] Kostrauskas P. Asinchronines elektros masinos.--Kaunas: Technologija, 2000.--116 p.

J. Buksnaitis

Department of Agroenergetics, Lithuanian University of Agriculture, Akademija, LT- 53361 Kaunas distr., Lithuania, phone: +370 7 397529, e-mail: juozas.buksnaitis@lzuu.lt
```Table 1. Distribution of elements of single-layer
former three-phase winding

Phase         U1      W2       V1       U2        W1        V2
alteration

Number         4       4       4         4         4        4
of coils
in a
section

Slot No.     1; 2;   5; 6;   9; 10;   13; 14;   17; 18;   21; 22
3; 4    7; 8    11; 12   15; 16    19; 20    23; 24

Table 2. Distribution of elements
of sinusoidal three-phase winding

Phase alteration    U1       W2       V1
Number of coils     4        4        4
in a section

Slot No.      Z    1; 2;    5; 6;    9; 10;
3; 4     7; 8    11; 12

Z   10; 11   14; 15   18; 19
12; 13   16; 17   20; 21

Phase alteration    U2       W1       V2
Number of coils     4        4        4
in a section

Slot No.      Z   13; 14   17; 18   21; 22
15; 16   19; 20   23; 24

Z   22; 23   2; 3;    6; 7;
24; 1     4; 5     8; 9

Table 3. Conditional changes of magnetomotive force in slots of
single-layer former three-phase winding in time moment t = 0

Slot No.     1        2        3        4        5        6
[DELTA]F     0        0        0        0      -0,216   -0,216

Slot No.     7        8        9        10       11       12
[DELTA]F   -0,216   -0,216   -0,216   -0,216   -0,216   -0,216

Table 4. Conditional changes of magnetomotive force in slots of
sinusoidal three-phase winding in time moment t = 0(b)

Slot No.     1        2         3          4          5         6
AF        0      -0,059   -0,114     -0,1613     -0,1975   -0,220

Slot No.     7        8         9         10         11        12
AF      -0,228   -0,220   -0,1975    -0,1613    -0,1140   -0,0590

Table 5. Results of harmonic analysis of the instantaneous
rotating magnetomotive force function of the single-layer
former three-phase winding with q = 4 and relative
magnitudes of its space harmonics

v               1         5         7        11        13
[F.sub.mv]   -0,914    0,0390    0,0210    -0,0110   -0,0090
[F.sub.v]       1      0,0429    0,0235    0,01197   0,01013

v              17        19        23        25
[F.sub.mv]   0,0090    0,0100    -0,0400   0,0370
[F.sub.v]    0,00968   0,01129   0,0435    0,0400

Table 6. Results of harmonic analysis of the
Instantaneous rotating magnetomotive force
function of the sinusoidal three-phase winding
with q = 4 and relative magnitudes of its space
harmonics

v          1      5      7        11     13

[F.sub.mv]   -0,871   0      0        0      0
[F.sub.v]      1      0      0        0      0

v          17     19     23       25

[F.sub.mv]     0      0    0,0380   -0,035
[F.sub.v]      0      0    0,0436   0,0402

Table 7. Experimental and calculation results of
the standard dimensioned asynchronous motor with
single-layer former winding

No.   [I.sub.1], A   [P.sub.1], A        n ,         M , Nm
[min.sup.-1]

1        1,75           405            2983          0,586
2        2,03           840            2961          1,93
3        2,30           1110           2945          2,71
4        2,70           1410           2924          3,61
5        3,13           1725           2899          4,50
6        3,65           2100           2870          5,55
7        4,13           2370           2851          6,23
8        4,98           2805           2810          7,38

No.    [SIGMA]P W    [P.sub.2], A      [eta] %      cos [phi]

1        315             90            22,2          0,351
2        333            507            60,4          0,627
3        361            749            67,5          0,731
4        402            1008           71,5          0,791
5        457            1268           73,5          0,835
6        535            1565           74,5          0,872
7        610            1760           74,3          0,869
8        741            2064           73,6          0,853

Table 8. Experimental and calculation results of the
asynchronous motor with stator winding replaced with
sinusoidal three-phase winding

No.   [I.sub.1], A   [P.sub.1], A   n, [min.   M, nm
sup.-1]

1         1,80           385         2948      0,83
2         2,20           1215        2923      3,31
3         2,50           1440        2902      3,95
4         2,87           1695        2882      4,65
5         3,03           1800        2868      4,92
6         3,35           1995        2847      5,45
7         3,60           2145        2825      5,83
8         3,95           2345        2798      6,33
9         4,50           2655        2756      7,08

[SIGMA]                      [eta]     cos
No.       P,W        [P.sub.2], W      %      [phi]

1         232            153         39,7     0,324
2         298            917         75,5     0,837
3         337            1103        76,6     0,873
4         387            1308        77,2     0,895
5         419            1381        76,7     0,900
6         472            1523        76,3     0,902
7         522            1623        75,7     0,903
8         596            1749        74,6     0,899
9         718            1937        73,0     0,894
```
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