Study of the Electrical Conductivity of the Short-Circuited Rotor Winding of Induction Motors Using the Statistical Method of Experimental Design

STUDY OF THE SPECIFIC ELECTRICAL CONDUCTIVITY OF THE SHORT-CIRCUITED ROTOR WINDING OF INDUCTION MOTORS USING A STATISTICAL EXPERIMENTAL DESIGN METHOD

S. N. Marukhin, O. P. Muravlev, E. K. Strelbitskii

(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)

In the design and investigation of induction motors with a short-circuited rotor, one of the principal parameters is the active resistance of the rotor winding, which, as is known, depends on the specific electrical conductivity $\gamma$ of the winding material. The windings of the short-circuited rotors of the overwhelming majority of induction motors with ratings from several hundred watts to several hundred kW are made by pressure casting the rotor slots with pure aluminum [1, 2].

Factory calculation methods [3, 4, 5], manuals on the design of electrical machines [6, 7, 8], and reference technical literature [1, 9, 10] give highly varied data on the specific electrical conductivity of aluminum (from 27.5 to 33.3 m/Ohm·mm$^2$). Moreover, as long-term experience in manufacturing short-circuited rotors shows, the specific electrical conductivity of pure aluminum changes during the casting process. This is pointed out by many authors of publications [1, 2, 6]. However, comprehensive studies of the simultaneous influence of various technological and design-technological factors on the specific electrical conductivity of the aluminum rotor winding have not yet been carried out.

The aim of the present work is to obtain a quantitative relationship between various technological and design-technological factors in the manufacture of short-circuited rotors and the value of the specific electrical conductivity of the aluminum rotor winding.

For the study, induction motors of the AO and AO2 series, manufactured for a long time by the Sibelektromotor plant (Tomsk), were selected. The technological process of their manufacture has been developed sufficiently well and is quite established. For the manufacture of the rotors of these motors, electrical steel of grade E12 (or E13) according to GOST 802—58 and pure (primary) aluminum of grade A6 (or A6P) according to GOST 11069—64 are used.

Many technological and design-technological factors affect the value of the specific electrical conductivity of the aluminum rotor winding: the temperature of the aluminum being cast, the content of impurities in the aluminum, the pressing speed during casting, the cross-section and configuration of the rotor slots, the length of the rotor core stack, and others. Since these factors act simultaneously, and the motors are manufactured in series, i.e., they represent a statistical population, the most effective method of investigation is the experimental-statistical one, which is based on the mathematical theory of experimental design [11]. On the basis of a detailed study of the technology of casting rotors at the plant-

at the Sibelektromotor plant, numerous published data on specific electrical conductivity, and also various statistical methods for designing experiments, a rotatable second-order design was selected.

The regression equation that should be obtained as a result of the planned experiment has, in the general case, the following form:

\[ \hat{Y}=b_0+\sum_{i=1}^{k} b_i x_i+\sum_{i=1}^{k} b_{ii}x_i^2+\sum_{\substack{i,j=1\\(i<j)}}^{C_k^2} b_{ij}x_i x_j+\ldots , \tag{1} \]

where \(\hat{Y}\) is the value of the specific electrical conductivity of aluminum predicted by equation (1); \(x_i\) are the factors (varied variables); \(b_0, b_i\ldots\) are regression coefficients; \(k\) is the number of factors; \(C_k^2\) is the number of combinations of \(k\) taken 2 at a time.

A central composite rotatable second-order uniform design was compiled [11]. The following were chosen as independent variables (factors): \(x_1\)—the temperature of the aluminum during casting; \(x_2\)—the iron content in the aluminum; \(x_3\)—the rotor-stack length; \(x_4\)—the total cross-sectional area of the rotor slots. The variables were varied at five levels. On the conditional (coded) scale these levels are equal to: \(-2; -1; 0; 1; 2\), where 0 corresponds to the midpoint of the range of variation, and 1 is the variation step. Table 1 gives the values of the independent variables with tolerances.

Table 1

In assigning the ranges of variation, the following considerations were used:

1. The lower limit of the aluminum temperature during casting, \(670^\circ\mathrm{C}\), is determined by the melting temperature of pure aluminum. The upper limit, \(870^\circ\mathrm{C}\), was chosen according to literature data [1, 10]. When this temperature is exceeded, irreversible physicochemical transformations occur in aluminum, substantially changing its properties.

2. The lower limit of the iron content, 0.2%, was chosen on the basis of the results of statistical processing of the chemical composition of primary aluminum as delivered (according to certificates from supplier plants). The upper limit, 1.4%, is determined by the technical capabilities of the equipment of the laboratory of the Sibelektromotor plant, where measurements of the iron content in aluminum were to be carried out.

3. The stack lengths and rotor-slot cross sections were chosen so as to cover the greatest number of motor standard sizes and to make maximum use of existing technological tooling. The total cross-sectional area of the slots was calculated from the nominal slot dimensions of the base motors, i.e., those series-produced motors whose die stamps were used in manufacturing the experimental rotors.

Table 2

Table 2 gives the experimental design in coded units, the number of parallel tests (the number of identical rotors), and the types of base motors. The experimental rotors were manufactured at the Sibelektromotor plant under conditions as close as possible to serial production. The rotors were cast on POLAK pressure die-casting machines.

The aluminum for casting the rotors, grade A6P from the Irkutsk aluminum plant, was prepared in a melting-and-holding furnace with a coated cast-iron crucible of 200 kg capacity. Iron was gradually added to the aluminum (from 0.2 to 1.4%) in the form of fine gray-cast-iron turnings. The iron content was determined by the spectral method in the plant laboratory on samples specially cast in a permanent mold. The temperature of the aluminum in the crucible was monitored with a chromel-alumel thermocouple with an accuracy of \(\pm 5^\circ\mathrm{C}\).

In all, 116 experimental rotors were manufactured. Since casting involved rearranging the casting molds, prolonged dissolution of the cast-iron turnings in the aluminum, setting the corresponding temperature in the crucible, etc., the process of casting all the rotors took about 12 hours. After casting, the lengths of the rotor stacks were measured at normal ambient temperature. They lay within the limits indicated in Table 1. On the short-circuiting rings of the rotor winding (on the end pads between the ventilation vanes), the specific electrical conductivity \(\gamma\) was measured in three different places for each ring. Then eight ventilation vanes were cut off from each rotor (four from each side of the rotor), and the specific electrical conductivity was also measured on these vanes on both sides. After this, the short-circuiting rings were cut off, and from each rotor, by milling, three winding bars were extracted. The extraction was carried out carefully, in order to exclude

deformation of the bars. Each bar was milled down to approximately half its thickness, and on the resulting flat surface the specific electrical conductivity was measured at five locations (along the axial length of the bar). All measurements* of the specific electrical conductivity $\gamma$ were made in the laboratory of the Sibélektromotor plant with an IE-1 type instrument [12]. On the ventilation blades a total of 1856 measurements were obtained, on the bars—1740, and on the short-circuiting rings—696. The total number of measurements of specific electrical conductivity with the IE-1 instrument on all experimental rotors was 4292.

Table 3

The mean values of $\gamma$ were calculated and reduced to a temperature of $+20^\circ$C. The results are placed in Table 3, in which the values of specific electrical conductivity are expressed in m/Ohm·mm$^2$. From Table 3 it is seen that the highest values of $\gamma$ are found in the short-circuiting rings, the lowest in the bars, while the ventilation blades occupy an intermediate position. Moreover, the values of $\gamma$ on the side where the aluminum enters the rotor during casting (on the runner side), as a rule, exceed the corresponding values of $\gamma$ on the opposite side of the rotor.

During operation of an induction motor, the ventilation blades practically do not participate in conducting the electric current of the rotor. As can be seen from Table 3, the values of $\gamma$ for the short-circuiting rings on the two sides of each rotor type differ only slightly. The values of $\gamma$ for the winding bars, however, differ substantially from the corresponding values of $\gamma$ for the short-circuiting rings. Therefore, regression equations of form (1) were obtained separately for the short-circuiting rings (for

* In carrying out the measurements, great assistance was provided by the senior engineer of the laboratory of the Sibélektromotor plant, L. N. Selivyanchova.

of the mean values \(\gamma\)) and for the rotor winding bars. In the calculation the following quantities were used:

for the short-circuiting rings—

\[ \gamma_{\mathrm{kk}}=\gamma_{\mathrm{Ae}}-\gamma_{\mathrm{kk}}; \tag{2} \]

for the winding bars—

\[ \gamma_{\mathrm{st}}=\gamma_{\mathrm{Ae}}-\gamma_{\mathrm{st}}, \tag{3} \]

where \(\gamma_{\mathrm{kk}}\) and \(\gamma_{\mathrm{st}}\) are the mean values of the specific electrical conductivity for the short-circuiting rings and bars, respectively (Table 3), and \(\gamma_{\mathrm{Ae}}=35.5\ \mathrm{m}/\Omega\cdot\mathrm{mm}^2\) is the base quantity. The specific electrical conductivity of technically pure (refined) aluminum was adopted as the base quantity [6, 9].

As a result of statistical processing of the experimental data according to the method of [11, 13], regression equations were obtained in the form of a second-order polynomial (on the coded scale):

for the short-circuiting rings—

\[ \hat{y}_{\mathrm{kk}}= 3.9714+0.07083x_1+1.6125x_2-0.05417x_3 -0.2292x_4-0.1564x_1^2-0.1314x_2^2+0.0686x_3^2 +0.0686x_4^2-0.1938x_1x_2-0.0438x_1x_3-0.0312x_1x_4 -0.0812x_2x_3-0.0438x_2x_4+0.1062x_3x_4; \tag{4} \]

for the winding bars—

\[ \hat{y}_{\mathrm{st}}= 8.7571-0.1292x_1+1.3625x_2+0.0625x_3 -0.1542x_4+0.0118x_1^2-0.2007x_2^2-0.0882x_3^2 -0.6382x_4^2-0.3812x_1x_2-0.0438x_1x_3-0.7188x_1x_4 -0.2688x_2x_3-0.5188x_2x_4-0.2812x_3x_4. \tag{5} \]

Testing for adequacy by Fisher’s \(F\)-criterion [11, 13] showed that equations (4) and (5) are adequate at the 5% significance level, i.e., these equations are mathematical models of the process under consideration.

When the regression coefficients were tested for significance by Student’s criterion [11, 13], it turned out that, at the 5% level, most of the coefficients of equations (4) and (5) are insignificant, i.e., they may be replaced by zeros. Then the regression equations take the following simple form, convenient for practical calculations:

for the short-circuiting rings—

\[ \hat{y}_{\mathrm{kk}}=3.9714+1.6125x_2-0.2292x_4-0.1564x_1^2; \tag{6} \]

for the winding bars—

\[ \hat{y}_{\mathrm{st}}=8.7571+1.3625x_2-0.6382x_4^2-0.7188x_1x_4. \tag{7} \]

To obtain the dependences of the specific electrical conductivities \(\hat{\gamma}_{\mathrm{kk}}\) and \(\hat{\gamma}_{\mathrm{st}}\) on the factors, the following substitution must be made in equations (4)–(7):

\[ \hat{\gamma}_{\mathrm{kk}}=35.5-\hat{y}_{\mathrm{kk}};\qquad \hat{\gamma}_{\mathrm{st}}=35.5-\hat{y}_{\mathrm{st}}. \tag{8} \]

To convert equations (4)–(7) to the natural scale, it is necessary to use the formulas:

\[ x_1=\frac{X_1-770}{50};\qquad x_2=\frac{X_2-0.8}{0.3}; \]

\[ x_3=\frac{X_3-126.5}{12.25};\qquad x_4=\frac{X_4-2024}{630}, \tag{9} \]

where \(X_1, X_2, X_3\), and \(X_4\) are the factors in physical units of measurement;

Main Conclusions

1. Quantitative relationships (4)–(7) have been obtained between deviations from the base value of the specific electrical conductivity of the aluminum of the rotor winding and the following factors: the temperature and purity of the aluminum, the stack length, and the cross section of the rotor slots. These relationships are nonlinear functions.

2. The specific electrical conductivities of the aluminum of the short-circuiting rings and the bars of the winding differ substantially.

3. Equations (4)–(7) make it possible to take into account technological and design-technological factors when designing induction motors similar in design to those investigated, and thereby to increase the accuracy of calculating their characteristics.

REFERENCES

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