Influence of Manufacturing Technology on the Output Parameters of Asynchronous Electric Motors
Abstract Generated abstract
This paper examines how scatter in manufacturing and material parameters affects the performance characteristics of asynchronous electric motors. It formulates a matrix relationship between deviations in input parameters, such as air gap, conductor dimensions, winding resistance, core stack length, and losses, and deviations in output parameters including starting and maximum torque, starting current, efficiency, and power factor. Using a probabilistic tolerance calculation, the authors describe both direct estimation of output tolerances from known technological scatter and an inverse procedure for assigning input tolerances to meet specified output limits. Data from production of 1.0 to 7.5 kW motors are used to identify dominant influences, especially rotor aluminum resistivity and air gap, and to show how acceptable tolerance regions can be determined.
Full Text
INFLUENCE OF MANUFACTURING TECHNOLOGY ON THE OUTPUT PARAMETERS OF ASYNCHRONOUS ELECTRIC MOTORS
O. P. MURAVLEV, E. K. STRELBITSKII
(Recommended by the scientific seminar of the departments of electrical machines and general electrical engineering)
Engineering methods for calculating electrical machines have now been developed quite fully. These methods are based on the assumption that all machine parameters strictly correspond to the calculated ones, and that such quantities as the specific resistance of conducting materials, wire diameter, lengths of the rotor and stator packets, etc., have constant values determined from averaged characteristics. However, under real conditions the indicated quantities always have a greater or lesser scatter, i.e., they are random quantities.
The quality of electrical machines is characterized by the degree to which the parameters of the manufactured machine conform to the specified calculated parameters. In this connection there arises the question of the need to calculate tolerances for the machine parameters. In order to assess the influence of manufacturing technology on the output parameters of the machine, we have analytically determined the relationship between the tolerances on the output parameters and the precision indicators of the technology.
We shall regard as output parameters the technical characteristics (starting and maximum torques, starting current, efficiency, and power factor), and as input parameters—the principal and local dimensions, as well as the characteristics of the materials used (the magnitude of the air gap, wire diameter, specific resistance of the winding material, etc.).
Let us introduce the following notation:
\(\overline{x}\) — vector of input parameters;
\(\overline{\Delta x}\) — vector of deviations of the input parameters;
\(\overline{y}\) — vector of output parameters;
\(\overline{\Delta y}\) — vector of deviations of the output parameters;
\(\overline{z}\) — vector of parameters of the electrical equivalent circuit;
\(\overline{\Delta z}\) — vector of deviations of the parameters of the electrical equivalent circuit.
The components of the vectors are given in Appendix 1.
The correspondence between \(\overline{\Delta y}\) and \(\overline{\Delta x}\) is established by the matrix of influence coefficients \(C\)
\[ \overline{\Delta y} = C \cdot \overline{\Delta x}. \tag{1} \]
For asynchronous motors, the matrix \(C\) is best determined by the formula
\[ C=A\cdot B, \tag{2} \]
where
\(A\) is a matrix establishing the dependence of \(\overline{\Delta y}\) on \(\overline{\Delta z}\);
\(B\) is a matrix establishing the dependence of \(\overline{\Delta z}\) on \(\overline{\Delta x}\).
The method for determining the coefficients of matrices \(A\) and \(B\) is given in [1, 2].
In calculating tolerances, two problems arise that can be solved on the basis of expression (1):
- To carry out a checking calculation of tolerances for the output parameters from known tolerances for the inputs—the direct problem of tolerance calculation.
- From known tolerances for the output parameters, to determine the tolerances for the inputs—the inverse problem.
In calculating tolerances, a probabilistic method should be used, since it agrees best with experiment. The equations for calculating tolerances by this method have the form
\[ \delta_{y_i}^2=c_{i1}^2\delta_{x_1}^2+\ldots+c_{ij}^2\delta_{x_j}^2+\ldots+c_{im}^2\delta_{x_m}^2, \tag{3} \]
\[ i=1,2,\ldots,l,\qquad j=1,2,\ldots,m, \]
where
\(c_{ij}\) is the coefficient of influence of the \(j\)-th input parameter on the \(i\)-th output parameter;
\(\delta_{y_i}\) and \(\delta_{x_i}\) are the half-widths of the tolerance field, in percent, for the output and input parameters;
\(l\) is the number of output parameters;
\(m\) is the number of input parameters.
To solve the direct problem it is sufficient to calculate the influence coefficients \(c_{ij}\), determine the actual values of \(\delta_{x_j}\) for the existing technology, and compute \(\delta_{y_i}\) from expression (3). The method for determining \(\delta_{x_j}\) is described in [2].
For all input parameters, the actual scatter fields \(2\delta_{x_j}\) were determined, corresponding to the manufacturing technology of asynchronous motors with a power of \(1.0\text{–}7.5\) kW at the “Sibelektromotor” plant:
\[ \delta_{w_1}=0\text{–}2.0\%;\quad \text{for } w_1<50\quad \delta_{w_1}=0; \]
\[ \delta_{l_m}=0.80\text{–}1.20\%;\quad \delta_{z_2}=0; \]
\[ \delta_{\rho A1}=45.5-2.05P_n,\quad \text{where } P_n \text{ is in kW}; \tag{4} \]
\[ \delta_\delta=12.0\text{–}15.0\%,\quad \text{for } 2p=2; \]
\[ \delta_\delta=13.5\text{–}18.5\%,\quad \text{for } 2p>2; \]
\[ \delta_{l_1}=5.88-0.0173\,l_1;\quad \text{where } l_1 \text{ is in mm}; \tag{5} \]
\[ \delta_{l_2}=7.07-0.0347\,l_2;\quad \text{where } l_2 \text{ is in mm}; \tag{6} \]
\[ \delta_{P_{\mathrm{st}}+P_{\mathrm{mech}}}=60-0.059(P_{\mathrm{st}}+P_{\mathrm{mech}}),\quad \text{where } P_{\mathrm{st}}+P_{\mathrm{mech}} \text{ is in W}; \tag{7} \]
The deviation of the wire diameter \(d_{\mathrm{pr}}\) can be determined according to GOST 2112-62, “Round Copper Electrical Wire,” and GOST 6132-52, “Aluminum Wire.”
A practical solution of the inverse problem can be obtained if the two most strongly influencing input parameters are singled out and the system of equalities (3) is replaced by a system of inequalities, while the remaining tolerances for the input parameters are taken for the existing technology. From Table 1, which gives the fractional contributions of a number of input parameters, it follows that one may pass to a system of inequalities
\[ c_{i4}^{2}\delta^{2}x_{4}+c_{i5}^{2}\delta^{2}x_{5}\leqslant \delta^{2}y_i-\sum_{j\ne 4;\,5} c_{ij}^{2}\delta^{2}x_j. \tag{8} \]
In calculating the table, the average values of \(c_{ij}\) and \(\delta x_j\) were used for motors of frame sizes \(1\text{—}4\) of the A02 series and frame sizes 3 and 4 of the A (AO) series.
Table 1.
| Input parameters | \(M_{\mathrm{p}}\) | \(M_{\mathrm{t}}\) | \(I_{\mathrm{k}}\) | \(\eta\) | \(\mathrm{Cos}\varphi\) |
|---|---|---|---|---|---|
| \(x_4=\rho_{\mathrm{Al}}\) | 0.928 | — | 0.597 | 0.296 | — |
| \(x_5=\delta\) | 0.064 | 0.716 | 0.288 | — | 0.818 |
| \(x_9=P_{\mathrm{st}}+P_{\mathrm{mech}}\) | — | — | — | 0.648 | — |
| Total share of accounted parameters | 0.922 | 0.716 | 0.885 | 0.944 | 0.818 |
A graphical interpretation of system (8) gives the region of acceptable tolerance values for the specific resistance of the aluminum of the rotor winding and the air gap. When the number of unknown tolerances for the input parameters is greater than two, the solution of the inverse problem may be carried out by finding the Chebyshev point of the system of inequalities by methods of linear programming.
Thus, on the basis of the relationship we have obtained between the design of asynchronous electric motors and the accuracy characteristics of the manufacturing technology, it is possible to calculate tolerances both for the output and for the input parameters of the machine.
Appendix 1
Conventional notation
\(y_1=M_{\mathrm{p}}\) — starting torque;
\(y_2=M_{\mathrm{m}}\) — maximum torque;
\(y_3=I_{\mathrm{k}}\) — short-circuit current;
\(y_4=\eta\) — efficiency;
\(y_5=\cos\varphi\) — power factor;
\(z_1=r_1\) — active resistance of the stator winding;
\(z_2=r'_2\) — referred active resistance of the rotor winding;
\(z_3=x_{\mathrm{k}}\) — short-circuit inductive resistance;
\(x_1\) — leakage inductive resistance of the stator winding;
\(x'_2\) — referred leakage inductive resistance of the rotor winding;
\(z_4=x_{\mathrm{m}}\) — mutual-induction inductive resistance;
\(x_1=w_1\) — number of turns of a phase of the stator winding;
\(x_2=d_{\mathrm{pr}}\) — diameter of the stator-winding conductor;
\(x_3=l_{\mathrm{m}}\) — mean length of the half-turn of the stator winding;
\(x_4=\rho_{\mathrm{Al}}\) — specific resistance of the material of the rotor winding;
\(x_5=\delta\) — magnitude of the air gap;
\(x_6=l_1\) — length of the stator core stack;
\(x_7=l_2\) — length of the rotor core stack;
\(x_8=z_2\) — number of whole bars of the rotor winding;
\(P_{\mathrm{st}}\) — iron losses;
\(P_{\mathrm{mech}}\) — mechanical losses;
\(x_9=P_{\mathrm{st}}+P_{\mathrm{mech}}\);
\(\lambda_{\mathrm{п}1}\) — leakage permeance of the stator slot;
\(\lambda_{\mathrm{к}1}\) — leakage permeance of the stator tooth tips;
\(\lambda_{s1}\) — leakage permeance of the end portions of the stator winding;
\(\lambda_{\mathrm{п}2}\) — leakage permeance of the rotor slot;
\(\lambda_{\mathrm{к}2}\) — leakage permeance of the rotor tooth tips;
\(\lambda_{s2}\) — leakage permeance of the end portions of the rotor winding;
\[ c_1 = 1 + \frac{x_1}{x_m}; \]
\[ x_k = c_1 x_1 + c_1^2 x_2'; \]
\[ r_k = c_1 r_1 + c_1^2 r_2'; \]
\[ \sum \lambda_1 = \lambda_{\mathrm{п}1} + \lambda_{\mathrm{к}1} + \lambda_{s1}; \]
\[ \sum \lambda_2 = \lambda_{\mathrm{п}2} + \lambda_{\mathrm{к}2} + \lambda_{s2}. \]
REFERENCES
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O. P. Muravlev, E. K. Strelbitskii. Ensuring the required accuracy in the manufacture of asynchronous motors. Elektrotekhnika, No. 7, 1966.
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O. P. Muravlev. Study of the influence of the accuracy characteristics of the technological process on the quality and reliability of asynchronous electric motors. Dissertation, Tomsk, 1966.