Study of the Influence of Manufacturing Errors in the Main Components of Induction Micromachines on Output Parameters
Abstract Generated abstract
This paper examines how manufacturing tolerances in the principal components of two-phase induction micromachines with hollow rotors affect output parameters relevant to interchangeability, including starting current, starting torque, and theoretical no-load speed. It develops a first-order influence-coefficient method, expressed in matrix form, linking deviations in design dimensions and equivalent-circuit parameters to deviations in machine performance. The authors derive coefficients for key circuit quantities and for dimensions such as air gap, stator-stack length, rotor diameter, and rotor wall thickness, then apply the method to DG-series motor-generators using maximum-minimum and probabilistic calculations. Comparison with reported experimental deviations indicates that the obtained expressions can estimate the influence of dimensional tolerances on output parameters with satisfactory accuracy.
Full Text
INVESTIGATION OF THE EFFECT OF MANUFACTURING ERRORS IN THE MAIN COMPONENTS OF INDUCTION MICROMACHINES ON OUTPUT PARAMETERS
A. N. GAVRILOV, A. A. KAMENSKY
The problem of the accuracy of output parameters is one of the principal questions in ensuring physical interchangeability.
In many instruments and elements of control systems, the output parameters are physical quantities, and their accuracy depends both on geometrical errors and on the physical properties of the constituent units and parts.
The deviation of geometrical characteristics from the nominal value causes an error in the physical parameters of the equivalent circuit, which determine the scatter of the output parameters of the product. The magnitude of the error of the output parameters is determined not only by the magnitude of the error of the input parameter, but also by the degree of influence of the input parameter under consideration on the output parameter [1, 2].
The output parameter \(Y_i\) may be regarded as a function of the input parameters \(x_j\).
\[ y_i = f_i(x_1, \ldots x_j, \ldots x_n). \]
Because of manufacturing errors, the output parameter differs from the nominal one.
\[ y_i + \Delta y_i = f_i(x_1 + \Delta x_1, \ldots x_j + \Delta x_j, \ldots x_n + \Delta x_n), \]
where \(y_i, x_j\) are the nominal values of the parameters;
\(\Delta y_i, \Delta x_j\) are deviations of the parameters from the nominal value.
Passing to finite increments and neglecting terms of the second order of smallness, the expression for the error in dimensionless form may be written as follows:
\[ \frac{\Delta y_i}{y_i} = \sum_{1}^{j=n} \frac{\partial f_i(x_1, \ldots x_j, \ldots x_n)}{\partial x_j} \, \frac{x_j}{f_i(x_1, \ldots x_j, \ldots x_n)} \, \frac{\Delta x_j}{x_j}. \]
The expression standing before the error of the input parameter and determining the degree of influence of this error on the error of the output parameter is the influence coefficient.
\[ a_{ij} = \frac{\partial f_i(x_1, \ldots x, \ldots x_n)}{\partial x_j} \, \frac{x_j}{f_i(x_i, \ldots x_j, \ldots x_n)}. \]
In the case where the output parameter \(y_i\) depends on the parameters \(x_j\), and the latter, in turn, are functions of the parameters \(z_k\)
and the analytical dependences between them are sufficiently complex, then, in order to determine the coefficients of influence of the parameters \(z_k\) on the parameters \(y_i\), it is more convenient to use the matrix form of notation. The coefficients of influence \(a_{ij}\) of the parameters \(x_j\) on the parameters \(y_i\) constitute the matrix \(A\), and the coefficients of influence \(b_{jk}\) of the parameters \(z_k\) on the parameters \(x_j\) constitute the matrix \(B\). The coefficients of influence \(c_{ik}\) of the parameters \(z_k\) on the output parameters \(y_i\) are determined by multiplying the matrices \(A\) and \(B\).
In automatic-control systems, induction micromachines have found wide application as actuating elements, especially two-phase machines with a hollow rotor, for which the coefficients of influence have been determined by the above method and the deviations of the output parameters have been calculated.
The coefficients of influence \(a_{ij}\) of the parameters of the equivalent circuit (\(r_c\)—active resistance of the winding, \(x_c\)—inductive reactance, \(x_m\)—inductive reactance of the magnetizing branch, \(r_p\)—active resistance of the hollow rotor) on the output parameters have been determined: \(I_n\)—starting current, \(M_n\)—starting torque, \(\nu_{xx}\)—relative theoretical no-load speed. The coefficients of influence were determined on the basis of the equations given in [3], taking into account the adopted assumptions and under the condition that the parameters of the equivalent circuit of one phase can be expressed through the square of the transformation coefficient and the corresponding parameters of the other phase.
\[ a_{I,x_c}=-\frac{Xx_c}{R^2+X^2}, \]
\[ a_{I,r_c}=-\frac{Rr_c}{R^2+X^2}. \]
\[ a_{I,x_m}=-\frac{x_m}{R^2+X^2} \left[ R\frac{2r_p^3x_m}{(r_p^2+x_m^2)^2} + X\frac{(r_p^2-x_m^2)r_p^2}{(r_p^2+x_m^2)^2} \right], \]
\[ a_{I,r_p}=-\frac{r_p}{R^2+X^2} \left[ R\frac{(x_m^2-r_p^2)x_m^2}{(r_p^2+x_m^2)^2} + X\frac{2x_m^3r_p}{(r_p^2+x_m^2)^2} \right], \]
where
\[ R=r_c+\frac{x_m^2r_p}{r_p^2+x_m^2}; \qquad X=x_c+\frac{x_mr_p^2}{r_p^2+x_m^2}; \]
\[ a_{M,r_c}=a_{I,r_c}; \qquad a_{M,x_c}=a_{I,x_c}; \]
\[ a_{M,x_m}=2\left(\frac{r_p^2}{r_p^2+x_m^2}+a_{I,x_m}\right); \]
\[ a_{M,r_p}=\frac{x_m^2-r_p^2}{r_p^2+x_m^2}+2a_{I,r_p}; \]
\[ a_{\nu,r_c}= \frac{2(1-\alpha^2)^2r_cx_m(2x_c+x_m)} {(1+\alpha^2)^2r_p(2r_c+2x_c+x_m)^2} \frac{1}{\nu_{xx}}; \]
\[ a_{\nu,x_c}=- \frac{4(1-\alpha^2)^2r_cx_cx_m} {(1+\alpha^2)^2r_p(2r_c+2x_c+x_m)^2} \frac{1}{\nu_{xx}}; \]
\[ a_{\nu,x_m}= \frac{2(1-\alpha^2)^2r_cx_m(2r_c+2x_c)} {(1+\alpha^2)^2r_p(2r_c+2x_c+x_m)^2} \frac{1}{\nu_{xx}}; \]
\[ a_{\nu,r_p}=- \frac{2(1-\alpha^2)^2r_cx_m} {(1+\alpha^2)^2r_p(2r_c+2x_c+x_m)^2} \frac{1}{\nu_{xx}}, \]
where the theoretical no-load speed is expressed by the empirical approximate formula
\[ V_{xx}=\frac{2a}{1+a^2}+\frac{2(1-a^2)^2 r_c x_m}{(1+a^2)^2 r_p(2r_c+2x_c+x_m)}. \]
The accuracy of the formula increases as the signal coefficient \(a\) increases, and for \(a=0.6 \div 1.0\) and real parameters of the equivalent circuit the error is less than 1% of the values calculated by the formula given in [3].
The coefficients of influence of the design dimensions—namely, the air-gap value \(\delta\), the stator-stack length \(l\), the hollow-rotor diameter \(D_p\), and its wall thickness \(\Delta\)—on the parameters of the equivalent circuit have the following form:
\[ b_{x_m,l}=1;\qquad b_{x_m,\delta}=-1; \]
\[ b_{r_p,l}=\frac{2lp^2S}{D_p^2+2lp^2S};\qquad b_{r_p,\Delta}=-1; \]
\[ b_{r_p,D_p}=\frac{D_p^2-2lp^2S}{D_p^2+2lp^2S};\qquad b_{x_c,l}=1, \]
where \(p\) is the number of pole pairs;
\(S\) is the overhang of the rotor cup.
For motor-generators of the DG series, deviations of the output parameters were calculated according to the tolerances for the design dimensions \((\delta, l, D_p, \Delta)\). For the active resistance of the winding, the tolerance was also taken as a design tolerance, since the electrical resistance of the winding is determined mainly by variation in the resistivity of the conductor and in its diameter.
Table 1 presents the calculated deviations, in percent, of the output parameters of the DG, and the actual deviations agree fairly well with the calculated ones. Thus, for example, for the DG-1TA the actual deviations of the starting torque \(M_{\text{п}}\) are \(\pm 6\%\), of the starting currents \(I_{\text{п}}\) \(\pm 3.5\%\), and of the no-load speed \(\pm 2\%\).
Table 1
| Calculation method | Output parameters | DG-1TA | DG-2TA | DG-3TA | DG-5TA |
|---|---|---|---|---|---|
| Max–min method | \(M_{\text{п}}\) | 7.456 | 6.722 | 6.712 | 5.912 |
| Max–min method | \(I_{\text{п}}\) | 6.988 | 5.718 | 4.935 | 4.859 |
| Max–min method | \(v_{xx}\) | 3.533 | 3.174 | 2.354 | 2.682 |
| Theoretical-probabilistic method | \(M_{\text{п}}\) | 4.864 | 4.546 | 4.654 | 3.931 |
| Theoretical-probabilistic method | \(I_{\text{п}}\) | 3.75 | 2.861 | 2.416 | 2.399 |
| Theoretical-probabilistic method | \(v_{xx}\) | 2.493 | 2.228 | 1.673 | 1.897 |
The results of experiment and calculation showed that, on the basis of the expressions obtained, it is possible to estimate with sufficient accuracy the influence of tolerances in the main design dimensions of a machine on its output parameters.
LITERATURE
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V. P. Gusev, A. V. Fomin et al. Calculation of Electrical Tolerances of Radio-Electronic Equipment, Sovetskoe radio, 1963.
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Yu. S. Chechet. Electrical Micromachines of Automatic Devices, Energia, Moscow, 1964.