ACCOUNTING FOR THE INFLUENCE OF TECHNOLOGY ON COMMUTATION IN THE DESIGN OF DC MACHINES

E. K. Strelbitskii, V. S. Stukach, A. Ya. Tsirulik

(Presented by the scientific seminar of the Departments of Electrical Machines and Apparatus and General Electrical Engineering)

The requirement of satisfactory commutation is one of the constraints in design, since commutational stability decreases as the level of machine utilization increases. Technological deviations of the electromagnetic, geometric, and mechanical parameters of the machine and of the properties of brush materials have a substantial effect on commutation. Ensuring reliable commutation is to a large extent connected with the development of scientifically substantiated methods for calculating tolerances for technological deviations. The most promising method is one that provides for calculating tolerances at the design stage, which makes it possible to balance scientifically the expenditures on materials and on ensuring tolerances and thereby to minimize the cost of the machine. The traditional study of the influence of individual factors does not make it possible to take into account the effects of interaction between different factors. The assignment of tolerances by the maximum–minimum method is unjustified from the physical point of view. Technological deviations are random; therefore, methods for calculating tolerances for deviations must be probabilistic-statistical. A scientific approach to tolerance calculation requires recourse to the general theory of errors, using the mathematical apparatus of analysis of variance and making it possible to determine the multidimensional dependence of the variance of the output quantity \(y\), characterizing commutation, on the variances of the input technological parameters \(x_i\) according to the formula

\[ D(y)=\sum_{1}^{n}\left[\frac{\partial y(X)}{\partial_i X}\right]^2 \cdot D(x_i), \tag{1} \]

where \(y(X)=y(x_1, x_2, \ldots x_i, \ldots x_n)\) is the multidimensional dependence of the parameter \(y\) on the input parameters. The transition from variances to tolerances is carried out by the formulas:

\[ \delta_i=3\cdot\sqrt{D(x_i)}, \qquad \delta_y=3\cdot\sqrt{D(y)}, \tag{2} \]

where \(\delta\) is half the tolerance field. The parameter \(y\) must characterize the state of commutation directly, in terms of the degree of destruction of the commutator and brushes; only in this case is it possible to calculate the reliability of the machine. To calculate tolerances by formula (1), it is necessary to know the multidimensional dependence \(y(X)\). In the present case it is unknown. Na-

...its occurrence can be carried out either by an experimental-statistical method or analytically. The experimental method leads most quickly to the goal when assigning tolerances for designed machines. Such a method is best implemented on the basis of the mathematical theory of experimental design, as shown in [1, 2]. However, the experimental method does not give the general solution needed for calculating tolerances at the design stage for series of machines of any type. Therefore we have attempted to establish analytical relationships between the nominal parameters of the machine, manufacturing errors, and the parameters of random influences in operation, on the one hand, and the probability of sparking, on the other hand, as follows.

Let us assume that the flux linkage \(\psi_p\) of the coil varies linearly during the commutation cycle of the coil, and that the commutating properties of the brush appear only at the final stage of commutation, when the current density under the trailing edge of the brush becomes greater than the critical value \(j_{\mathrm{cr}}\). The change of flux linkage in any coil with time, with exact adjustment of the interpoles and in the absence of technological errors, is described by the equation (line 1, Fig. 1):

\[ \psi_p(t) = L_p \cdot i_a - \alpha_p \cdot t, \tag{3} \]

where

\[ \alpha_p = \frac{2L_p \cdot i_a \cdot v_k}{t_k(\beta + u_p - 1)} . \]

\(L_p\) — total inductance of the coil, H;
\(i_a\) — current of the parallel branch, A;
\(\beta\) — brush coverage;
\(U_p\) — number of sections across the width of the coil;
\(t_k\) — commutator pitch, cm;
\(v_k\) — peripheral speed of the commutator, cm/s.

Let us consider the influence of individual errors.

1. A delay in the start of commutation of the first section due to instability of contact, caused by imperfection of the commutator geometry, vibrations, etc., by a time \(x_1\), shifts the line to the right (line 2, Fig. 1) and creates a residual flux linkage \(a_1 x_1\), where \(a_1 = \alpha_p\). It can be shown that a delay of commutation in the remaining sections of the coil does not have a significant effect on the law of variation of the flux linkage.

2. Premature completion of commutation of the last section of the coil, for analogous reasons, by a time \(x_2\), increases the residual flux linkage by the amount \(a_2 x_2\), where \(a_2 = \alpha_p\).

3. A deviation of the induction \(B_d\) under the interpole by an amount \(x_3\) changes the slope of the straight line \(\psi_p(t)\) (line 3, Fig. 1) and changes the residual flux linkage by the amount \(a_3 x_3\), where

\[ a_3 = \frac{l_a \cdot v_a \cdot w_p \cdot \beta}{(\beta + u_p - 1)} , \tag{4} \]

\(l_a\) — length of the armature, cm;
\(v_a\) — peripheral speed of the armature, cm/s;
\(w_p\) — sum of the turns of the sections of one layer of the coil.

The residual flux linkage \(\psi_0\) is equal to

\[ \psi_0 = a_1 \cdot x_1 + a_2 \cdot x_2 - a_3 x_3 . \tag{5} \]

The arc burning time under these conditions is

\[ t_d = \frac{\psi_0}{U_d}, \tag{6} \]

Fig. 1

Fig. 2

where \(U_{\mathrm{a}}\) is the arc voltage in V. The arc-burning time, according to experimental data, is closely correlated with the level of sparking and the wear of the commutator and brushes, and can serve as a quantitative measure of the degree of sparking and wear.

The commutating properties of the brush at the final stage are taken into account as follows (Fig. 2). The current density \(j_{\mathrm{n}}\) under the entering part of the brush changes only slightly at the final stage and, ...

is considered constant. Therefore the voltage drop under the entering part of the brush, \(U_{\mathrm{sh\,n}}\), is also considered constant. The current density at the trailing edge of the brush increases in a short time to very large values; therefore we take the voltage at the trailing edge, \(U_{\mathrm{sh\,s}}\), to be equal to \(U_{\mathrm{sh\,max}}\). Thus, at the final stage of commutation a voltage is introduced into the section

\[ U_{\mathrm{sh}}=U_{\mathrm{sh\,max}}(j_{\mathrm{kr}})-U_{\mathrm{sh\,n}}(j_{\mathrm{n}}), \tag{7} \]

where \(U_{\mathrm{sh}}\) is not the same for accelerated and retarded commutation, since for retarded commutation \(U_{\mathrm{sh\,max}}(j_{\mathrm{kr}})\) is positive, and for accelerated commutation it is negative. The time \(\tau\) during which \(U_{\mathrm{sh}}\) acts is determined as follows:

\[ \tau=\frac{\psi_0-a_2x_2}{L_c l_{\mathrm{sh}} v_k j_{\mathrm{kr}}-\alpha}, \tag{8} \]

where \(L_c\) is the inductance of the section, H;

\(l_{\mathrm{sh}}\) is the length of the brush along the commutator axis, cm;

\[ \alpha=\alpha_p+a_3x_3. \]

During the time in which the commutating properties are manifested, the brush will reduce the residual flux linkage by the amount

\[ \Delta\psi=U_{\mathrm{sh}}(\tau-x_2), \tag{9} \]

and the actual residual flux linkage will be equal to

\[ \psi=\psi_0-\Delta\psi =a_1 A x_1+(a_2+U_{\mathrm{sh}})x_2-a_3 A x_3, \tag{10} \]

where

\[ A=1-\frac{U_{\mathrm{sh}}}{L_c l_{\mathrm{sh}} v_k i_{\mathrm{kr}}-\alpha} =1-\frac{U_{\mathrm{sh}}(\beta+U_{\mathrm{p}}-1)t_k} {\left[L_c l_{\mathrm{sh}} j_{\mathrm{kr}}(\beta+U_{\mathrm{p}}-1)-2L_{\mathrm{p}} i_a\right]v_k}. \tag{11} \]

From formulas (10) and (11) the influence of the nominal parameters on the magnitude of the residual flux linkage is clear. Substituting into expression (10) the mean values of the random parameters \(x_i\), we obtain the mean residual flux linkage \(\overline{\psi}\). It is natural to require that \(\overline{\psi}=0\); then \(\overline{x}_3\), which is necessary in order to ensure the equality \(\overline{\psi}=0\), is equal to

\[ \overline{x}_3=\frac{a_1}{a_3}\,\overline{x}_1+ \frac{a_2+U_{\mathrm{sh}}}{a_3 A}\,\overline{x}_2. \tag{12} \]

Equation (12) shows by how much the induction under the additional pole must be increased in order to compensate for the retarding influence of the systematic deviations \(\overline{x}_1\) and \(\overline{x}_2\), and makes it possible at the design stage to take this influence into account if \(\overline{x}_1\) and \(\overline{x}_2\) are known from statistical data. By increasing the angle of inclination of the curve \(\psi_{\mathrm{p}}(t)\) to the value \(\alpha_p+a_3\overline{x}_3\), we obtain, on average, \(\overline{\psi}=0\). Now the residual flux linkage will be determined by the random deviations of all parameters from their mean values:

\[ \psi=a_1' A' \Delta x_1+(a_2'+U_{\mathrm{sh}})\Delta x_2-a_3' A'\Delta x_3, \tag{13} \]

where

\[ A'=1-\frac{U_{\mathrm{sh}}}{L_c' l_{\mathrm{sh}} v_k i_{\mathrm{kr}}-\alpha_p'-a_3'\Delta x_3}, \]

\[ a_3'=\frac{l_a v_a w_{\mathrm{p}}\beta t_k} {(\beta+U_{\mathrm{p}}-1)t_k-(x_1+x_2)v_k}. \]

The variance of the residual flux linkage is equal to:

\[ D(\psi)=\sum\left(\frac{\partial \psi}{\partial y_i}\right)^2 \cdot D(y_i). \tag{14} \]

Owing to the inequality of \(U_{\mathrm{sh}}\) under delayed and accelerated commutation, the variances of the flux linkage in these cases are not equal in magnitude, that is, \(D(\psi)_{\mathrm{d}} \ne D(\psi)_{\mathrm{a}}\). Therefore an additional displacement \(\Delta x_{\mathrm{d}}=\Delta B_{\mathrm{d}}\) of such magnitude should be provided that the probabilities of the occurrence of limiting flux linkages \(\psi_{\mathrm{lim\,d}}\) and \(\psi_{\mathrm{lim\,a}}\) are the same. This displacement can be determined from the relation

\[ F\left( -\frac{\psi_{\mathrm{lim\,d}}+a'_{\mathrm{d}}\cdot \Delta \bar{x}_{\mathrm{d}}} {\sqrt{D(\psi)_{\mathrm{d}}}} \right) = F\left( -\frac{\psi_{\mathrm{lim\,a}}-a'_{\mathrm{d}}\cdot \Delta \bar{x}_{\mathrm{d}}} {\sqrt{D(\psi)_{\mathrm{a}}}} \right), \tag{15} \]

where \(F(z)\) is the Laplace function.

In equation (14), \(y_i\) should be understood as all the parameters of equation (13) that may have technological deviations. Such parameters are \(\Delta x_1\), \(\Delta x_2\), \(\Delta x_3\), \(t_k\), \(j_{\mathrm{cr}}\), \(U_{\mathrm{sh}}\). Thus, the nominal electromagnetic parameters of the machine have been related by equation (13) to all technological errors and random effects of operating conditions. Equation (14) makes it possible to decompose the variance of the residual flux linkage into components caused by different factors, to determine the degree of influence of each factor, to calculate tolerances for all deviations at the design stage, and to analyze the dependence of the magnitude of the tolerances on the power and utilization level of the machine.

LITERATURE

1. E. K. Strelbitskii, V. S. Stukach, A. Ya. Tsirulik. Application of the method of mathematical statistics to the study of commutation. Izvestiya TPI, vol. 160, 1966.

2. A. Ya. Tsirulik. Mathematical models of commutation in direct-current machines and their application to the calculation of tolerances and commutation reliability. Dissertation, Tomsk, 1967.