Influence of Stator Core Unstacking on the Output Characteristics of an Induction Motor
Abstract Generated abstract
This paper examines how restacking, or radial displacement of stator-core laminations during manufacture, changes the effective air gap and output characteristics of AO2-series induction motors of the third and fourth frame sizes. Using an analytical expression for the air-gap increase and tolerance data for stator sheets, stacking mandrels, and rotors, the authors calculate equivalent air-gap increases and compare them with measurements from sectioned stator cores examined microscopically. The calculations indicate air-gap increases of roughly 8 to 17 percent, while experimental averages show comparable increases for several pole counts. Applying influence coefficients, the study finds that these changes can noticeably affect starting and maximum torque, locked-rotor current, efficiency, and power factor, supporting the need to include restacking effects in motor calculations and tooling tolerance assignments.
Full Text
EFFECT OF RESTACKING OF THE STATOR CORE ON THE OUTPUT CHARACTERISTICS OF AN INDUCTION MOTOR
A. G. Verzesh, A. D. Nemtsev
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
The air gap in induction motors is one of the important parameters affecting the output characteristics \((M_{\text{п}}, M_{\text{м}}, I_{\text{к}}, \cos \varphi\) and efficiency) of an induction motor. In theoretical studies and calculations of induction motors it is usually assumed that the air gap between the stator and the rotor is constant. But already during manufacture of an electric motor, nonuniformity of the gap arises, caused by the toothed structure of the stator, eccentricity of the rotor relative to the stator bore, and corrugation of the stator stack.
Corrugation, or restacking, is due to deviations of the inner diameter of the stator sheet in the die \(D_3\) and of the diameter of the stacking mandrel \(D_1\) (Fig. 1), on which the sheets of the stator are stacked and fastened into a packet. The magnitude of the air gap is determined by the intermediate dimension between \(D_1\) and \(D_3\) and by the outer diameter of the rotor, \(D_2\).
Fig. 1
There are a number of works [1—5] devoted to the study of the effect of nonuniformity of the air gap on the technical characteristics of elec-
motors. However, in most of their work the researchers took into account the change in the air gap only due to the tooth form of the stator and the eccentricity of the rotor relative to the stator bore.
The purpose of the present work is to determine the effect of restacking on the output characteristics of motors of the 3rd and 4th frame sizes of the AO2 series.
In [4] the authors obtained an analytical expression for determining the change in the air gap due to restacking
\[ \frac{\delta_{\mathrm{e}}}{\delta_{\mathrm{n}}} = \frac{\lambda}{\ln(1+\lambda)}, \tag{1} \]
where
\[ \lambda = 2\frac{D_3-D_1}{D_1-D_2}. \tag{2} \]
Expression (1) is valid when the following conditions are satisfied [4]:
\[ \frac{\delta_1}{b}=0.1 \div 2 \quad \text{and} \quad \frac{\delta_2}{b}=0.2 \div 1.6, \tag{3} \]
where
\[
\delta_1 \text{ and } \delta_2
\]
are the dimensions of the air gap between the rotor surface and the possible displacements of the stator laminations caused by technological tolerances in the stacking fixture and in the stator laminations;
\[
b=0.5\ \text{mm}
\]
is the thickness of the electrical steel.
For the motors under consideration, conditions (3) were checked; the results of the calculations are presented in Table 1.
As can be seen from Table 1, conditions (3) are satisfied for all motors. Therefore expression (1) may be used. An example calculation for determining the effect of restacking on the air gap for the motor AO2-32-2.
Scheme a.
From Table 1
\[ D_1 = 106_{-0.027},\quad D_2 = 105\ \text{mm},\quad D_3 = 106^{+0.060}\ \text{mm} \]
\[ \lambda = 2\frac{D_3-D_1}{D_1-D_2} = 2\frac{106.060-105.973}{105.973-105.0} = 0.178. \]
The relative change in the air gap due to restacking is
\[ \frac{\delta_{\mathrm{e}}}{\delta_{\mathrm{n}}} = \frac{\lambda}{\ln(1+\lambda)} = \frac{0.178}{\ln 1.178} = 1.09, \]
i.e., an increase of 9%.
Scheme b.
From Table 1
\[ D_1 = 106_{-0.040},\quad D_2 = 105\ \text{mm},\quad D_3 = 106^{+0.060}\ \text{mm} \]
\[ \lambda = 2\frac{106.060-105.960}{105.960-105.0} = 0.208, \]
\[ \frac{\delta_{\mathrm{e}}}{\delta_{\mathrm{n}}} = \frac{0.208}{\ln 1.208} = 1.10, \]
i.e., an increase of 10.0%.
Table 1
| Motor type | \(D_1\), mm: nominal size | \(D_1\), mm: limiting deviation | \(D_2\), mm: nominal size | \(D_2\), mm: limiting deviation | \(D_3\), mm: nominal size | \(D_3\), mm: limiting deviation | For scheme \(a\): \(\delta_1\), mm | For scheme \(a\): \(\dfrac{\delta_1}{\varepsilon}\) | For scheme \(a\): \(\delta_2\), mm | For scheme \(a\): \(\dfrac{\delta_2}{\varepsilon}\) | For scheme \(b\): \(\delta_1\), mm | For scheme \(b\): \(\dfrac{\delta_1}{\varepsilon}\) | For scheme \(b\): \(\delta_2\), mm | For scheme \(b\): \(\dfrac{\delta_2}{\varepsilon}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A02-31 (2)-2 | 106,0 | −0,027 −0,040 |
105,0 | ±0,035 | 106,0 | +0,08 +0,06 |
0,486 | 0,937 | 0,573 | 1,148 | 0,480 | 0,960 | 0,580 | 1,160 |
| A02-31 (2)-4 | 112,0 | −0,027 −0,040 |
111,4 | ±0,035 | 112,0 | +0,08 +0,06 |
0,286 | 0,573 | 0,373 | 0,746 | 0,280 | 0,560 | 0,380 | 0,761 |
| A02-31 (2)-6 | 122,0 | −0,027 −0,040 |
121,4 | ±0,035 | 122,0 | +0,08 +0,06 |
0,286 | 0,573 | 0,373 | 0,746 | 0,280 | 0,560 | 0,380 | 0,761 |
| A02-41 (2)-2 | 123,0 | −0,027 −0,045 |
121,8 | ±0,04 | 123,0 | +0,08 +0,06 |
0,586 | 1,170 | 0,673 | 1,345 | 0,580 | 1,160 | 0,685 | 1,370 |
| A02-41 (2)-4 | 133,0 | −0,027 −0,045 |
132,3 | ±0,04 | 133,0 | +0,08 +0,06 |
0,336 | 0,672 | 0,423 | 0,847 | 0,327 | 0,627 | 0,432 | 0,864 |
| A02-41 (2)-6,8 | 144,0 | −0,027 −0,045 |
143,3 | ±0,04 | 144,0 | +0,08 +0,06 |
0,336 | 0,672 | 0,423 | 0,847 | 0,327 | 0,627 | 0,432 | 0,864 |
The results of calculating the effect of restacking on the change in the nominal air gap for the 3rd and 4th frame sizes of the AO2 series are given in Table 2.
Table 2
| Motor type | Increase in the equivalent air gap, % for circuit a |
Increase in the equivalent air gap, % for circuit b |
|---|---|---|
| AO2-31 (2)-2 | 9.0 | 10.0 |
| AO2-31 (2)-4 (6) | 15.0 | 17.0 |
| AO2-41 (2)-2 | 8.0 | 9.0 |
| AO2-41 (2)-4 (6, 8) | 13.5 | 15.5 |
To confirm the reliability of the results obtained, an experiment was carried out. The purpose of the experiment was to determine the actual displacement of the stator-core laminations in the radial direction. For this purpose the stator cores were cut in half along the longitudinal axis, and the cut surface was ground. The measurements were made with a microscope. The measurement results are presented in Table 3.
Table 3
| No. | Magnitude of relative displacement, mm | Frequency 3rd frame size |
Frequency 4th frame size |
|---|---|---|---|
| 1 | 0.025 | 123 | 142 |
| 2 | 0.050 | 63 | 194 |
| 3 | 0.075 | 12 | 42 |
| 4 | 0.100 | 1 | 20 |
| 5 | 0.125 | 1 | 4 |
| 6 | 0.150 | — | 2 |
For these data, using the expression
\[ \bar{x}=\frac{\sum n_i \cdot x_i}{\sum n_i} \tag{4} \]
the average values of lamination displacement were calculated for the third \(\bar{x}_3\) and fourth \(\bar{x}_4\) frame sizes: \(\bar{x}_3=0.0368\) mm; \(\bar{x}_4=0.0476\) mm.
If one takes into account that the air gap is determined by the difference between the diameters of the laminated core bore and the rotor, then the obtained average values of displacement of the stator-core laminations indicate that the actual size of the air gap increases on average by 0.0368 mm in motors of the third frame size and by 0.0476 mm in motors of the fourth frame size, which amounts respectively to 12.2% and 13.6% for 4-, 6-, and 8-pole machines. The data obtained experimentally differ only slightly from the calculated data given in Table 2.
Since the size of the air gap increases significantly as a result of restacking, quantitative determination of the effect of restacking on the output characteristics of induction motors is of practical interest. This can be done using coefficients of the influence of input parameters on the output parameters [5]. The results of the calculation to determine the effect of the change in the air gap due to restacking on the output characteristics are given in Table 4.
Table 4
Change in the output characteristics of the induction motor, %
| Motor type | For circuit \(a\): \(M_{\mathrm{p}}\) | For circuit \(a\): \(M_{\mathrm{m}}\) | For circuit \(a\): \(I_{\mathrm{k}}\) | For circuit \(a\): efficiency | For circuit \(a\): \(\cos \varphi\) | For circuit \(b\): \(M_{\mathrm{p}}\) | For circuit \(b\): \(M_{\mathrm{m}}\) | For circuit \(b\): \(I_{\mathrm{k}}\) | For circuit \(b\): efficiency | For circuit \(b\): \(\cos \varphi\) |
|---|---|---|---|---|---|---|---|---|---|---|
| A02-31-2 | 5,22 | 2,32 | 2,61 | −0,081 | −0,520 | 5,81 | 2,58 | 2,90 | −0,090 | −0,578 |
| A02-31-4 | 8,48 | 3,88 | 4,23 | −0,282 | −1,750 | 9,60 | 4,40 | 4,80 | −0,320 | −1,980 |
| A02-31-6 | 6,62 | 3,03 | 3,30 | −0,593 | −3,200 | 7,50 | 3,44 | 3,74 | −0,672 | −3,620 |
| A02-32-2 | 5,87 | 2,63 | 2,94 | −0,055 | −0,370 | 6,52 | 2,92 | 3,26 | −0,061 | −0,412 |
| A02-32-4 | 9,17 | 4,26 | 4,58 | −0,206 | −1,420 | 10,40 | 5,00 | 5,20 | −0,233 | −1,615 |
| A02-32-6 | 7,30 | 3,40 | 3,64 | −0,447 | −2,790 | 8,27 | 3,86 | 4,13 | −0,507 | −3,160 |
| A02-41-2 | 5,23 | 2,38 | 2,61 | −0,046 | −0,398 | 5,88 | 2,68 | 2,94 | −0,052 | −0,448 |
| A02-41-4 | 6,30 | 3,73 | 4,15 | −0,225 | −1,010 | 9,50 | 4,28 | 4,75 | −0,144 | −1,155 |
| A02-41-6 | 7,08 | 3,24 | 3,54 | −0,408 | −3,290 | 8,12 | 3,72 | 4,07 | −0,468 | −3,780 |
| A02-41-8 | 8,07 | 3,74 | 4,03 | −0,857 | −5,860 | 9,28 | 4,29 | 4,63 | −0,982 | −6,730 |
| A02-42-2 | 5,91 | 3,02 | 2,95 | −0,029 | −0,252 | 6,65 | 3,40 | 3,32 | −0,032 | −0,283 |
| A02-42-4 | 8,92 | 4,08 | 4,47 | −0,129 | −1,570 | 10,20 | 4,68 | 5,13 | −0,149 | −1,805 |
| A02-42-6 | 7,51 | 3,46 | 3,76 | −0,360 | −3,140 | 8,63 | 3,97 | 4,33 | −0,414 | −3,620 |
| A02-42-8 | 8,32 | 3,88 | 4,16 | −0,676 | −5,130 | 9,55 | 4,45 | 4,77 | −0,776 | −5,900 |
Analyzing the data presented in Table 4, we see that increasing the air gap as a result of restacking leads to significant changes in the output characteristics and amounts to 15–90% of the tolerances for \(M_{\mathrm{p}}\), \(M_{\mathrm{m}}\), \(I_{\mathrm{k}}\), \(\cos \varphi\), and efficiency established by GOST 183-66. Therefore, the increase in the air gap due to restacking must be taken into account when carrying out the electrical calculation of induction motors. This will make it possible to improve the accuracy of the calculation of technical characteristics. In addition, knowledge of the amount by which the air gap increases as a result of restacking will make it possible to take a more substantiated approach to assigning tolerances for tooling (the stacking mandrel and dies for punching the iron).
REFERENCES
-
A. N. Kostylev et al. Methods for measuring air-gap nonuniformity in an induction generator. Elektrotekhnika, 1971, No. 10.
-
A. P. Voskresenskii. Influence of air-gap nonuniformity on the characteristics of an induction motor. Vestnik elektropromyshlennosti, 1957, No. 5.
-
N. A. Kiklevich. Influence of air-gap nonuniformity on the characteristics and operating reliability of induction electric motors. Elektrichestvo, 1949, No. 12.
-
F. P. Davidyan, A. A. Terzyan. The magnetic field of the gap in the longitudinal section of electrical machines, taking technological deviations into account. Elektrichestvo, 1971, No. 10.
-
O. P. Muravlev, E. K. Strelbitskii. Ensuring the required precision in the manufacture of induction motors. Elektrotekhnika, 1966, No. 7.