EFFECT OF TECHNOLOGICAL DEVIATIONS ON BRUSH SPARKING IN DC MACHINES

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

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

The principal cause of sparking of electric brushes is the nonidentity of commutation cycles, caused by technological deviations of the geometrical parameters of the main units of a DC machine. Therefore, in order to increase the reliability of the commutating assembly, there arises the need for a justified system of tolerances for the technological deviations affecting commutation. In the linear theory of accuracy, the variance of the output quantity is related to the variances of the input quantities by the equation

\[ D(y)=\sum_{i=1}^{n}\left[\frac{dy(x_1,x_2,\ldots,x_n)}{dx_i}\right]^2 \cdot D(x_i), \tag{1} \]

where

\[ \begin{aligned} D(y) &\text{ — the variance of the quantity } y,\text{ characterizing the commutation of machines of a series;}\\ D(x_i) &\text{ — the variance of the } i\text{-th input technological parameter } x_i,\\ &\text{\quad affecting the commutation index } y;\\ y(x_1,x_2,\ldots,x_n) &\text{ — an } n\text{-dimensional functional dependence of the commutation index } y\\ &\text{\quad on the input technological parameters } x_i. \end{aligned} \]

It is clear from formula (1) that, knowing the variances of the input parameters for a given level of technology, one can calculate the variance of the commutation index of the machines of a series and the probability of rejection due to sparking. The inverse problem can also be solved: for a given variance of the sparking index and a given permissible probability of rejection, the tolerances for deviations of the input parameters can be calculated. As we see, the problem would be solved simply if the dependence of the sparking index on the input parameters \(y(x_1, x_2, \ldots, x_n)\) were known. Determining this dependence is the aim of the present work.

In article [1] it was shown that the most effective method for finding the indicated dependence is the experimental-statistical method. The selected input parameters are varied in the experiment according to a plan constructed in accordance with the requirements of the theory of mathematical design of experiments. In each experiment the commutation index is measured; as this index we take directly the level of brush sparking, determined from the oscillogram of the voltage at the trailing edge of the brushes with the aid of a previously found dependence between sparking and the oscillogram parameters [2]. The form of the unknown function \(y(x_1, x_2, \ldots, x_n)\) is specified by a polynomial of the required degree. The unknown coefficients of the polynomial are determined by methods

regression analysis of the experimental results, the apparatus of which is set forth in article [1]. Preliminary results of the search for a mathematical model of sparking are also given there. The present article presents the final results of the investigation.

The most noticeable influence on commutation is exerted by technological deviations of the parameters of the geometry of the magnetic system and of the commutator–brush assembly. The parameters of these principal assemblies were therefore taken for consideration.

In order to reduce the dimensionality of the sparking model, the size of the experiment, and the laboriousness of the calculations, it is desirable to represent all technological factors by a minimum number of parameters. Individual assemblies in the general model can be represented by generalized parameters that are outputs for these assemblies.

The geometry of the magnetic system is characterized by six gaps:

$\delta_1$ and $\delta_2$ — the gaps between the main poles and the armature, respectively to the left and to the right of the interpole;

$s_1$ and $s_2$ — the gaps between the pole shoes of the main pole and the interpole, to the left and to the right of the latter;

$\delta_{\mathrm{д}}$ — the gap between the armature and the interpole;

$\delta_s$ — the gap between the frame and the interpole.

Investigations by V. G. Galkin showed that deviations of all these gaps lead to a deviation of the induction under the interpole, and the variance of the induction is determined through the variances of the gaps as follows [3]:

\[ \mathrm{D}(B_{\mathrm{д}})=A_1\cdot \mathrm{D}(\delta_{\mathrm{д}}) + A_2\cdot \mathrm{D}(\delta_s) + A_3\cdot \mathrm{D}(s) + A_4\cdot \mathrm{D}(\delta), \tag{2} \]

where the coefficients $A_i$ are determined through the nominal dimensions of the magnetic-circuit geometry. As can be seen, the deviation of the induction in the commutation zone can serve as the output parameter for the magnetic-circuit assembly and can represent this assembly in the sparking model. Since measurement of the induction on production machines is difficult, we express its deviation through the exciting current of the interpole winding:

\[ \Delta I_{\mathrm{д}}\cdot w_n = I_n\cdot w_{\mathrm{д}} = \frac{1}{\mu_0}\cdot \Delta B_{\mathrm{д}}\cdot \delta_{\mathrm{д}}\cdot k_\delta . \tag{3} \]

The variance of the exciting current is determined by the variance of the induction:

\[ \mathrm{D}(I_n)= \left[ \frac{1}{\mu_0\cdot w_{\mathrm{д}}}\cdot \bar{\delta}_{\mathrm{д}}\cdot \bar{k}_{\delta} \right]^2 \cdot \mathrm{D}(B_{\mathrm{д}}). \tag{4} \]

Thus, the variance of the induction can be replaced by the variance of the exciting current, and the exciting current can be taken as the generalized parameter of the magnetic circuit. In carrying out the experiment, for each machine the optimal exciting current was determined from the oscillogram of the voltage at the trailing edge of the brush. When varying $I_n$, the count was made from this optimal value $I_{n0}$ as from the zero value. This eliminated the influence of the state of the mechanics of the sliding contact, which affects the magnitude of the optimal exciting current.

As the parameters of the commutator–brush assembly, the contact pressure $P$, the eccentricity of the working surface of the commutator $e$, the ellipticity of the commutator surface $\rho$, and the root-mean-square deviation of the level differences of the bars $\sigma_{\Delta}$ were adopted. The commutator parameters were determined from profilograms, as described in [4]. Statistical analysis of profilograms of commutators of machines of the P1-6 series of frame sizes showed that the distribution law of bar level differences is normal with a mean equal to zero; therefore the parameter $\sigma_{\Delta}$ completely determines the distribution law of the level differences and unambiguously determines the probability of the appearance of one, two,

etc., jumps exceeding the critical level at which sparkless commutation is impossible.

Preliminary screening experiments made it possible to identify and discard factors that were insignificant for the machines under study. At an armature rotational speed of 1500 rpm, the eccentricity and ellipticity of the commutator proved to have little influence when varied within the tolerance limits. Thus, the input parameters of the sparking model were the excitation current \(I_{\mathrm{n}}=x_1\), representing six parameters of the magnetic circuit, the contact pressure \(P=x_2\), and the commutator parameter \(\sigma_\Delta=x_3\). The output quantity is the sparking level \(\mathrm{И}\), in points.

The form of the sparking model is specified by a second-degree polynomial:

\[ \hat{\mathrm{И}}=b_0+\sum_1^3 b_i x_i+\sum_1^3 b_{ii}x_i^2+\sum_{i<j}^3 b_{ij}x_i x_j . \tag{5} \]

The intervals of variation of the variables in the experiment (Table 1) were assigned on the basis of an analysis of the actual fields of their scatter as applied to the technology of the Elektromashina plant (Prokop’evsk). The parameter \(\sigma_\Delta\) was varied by selecting machines with the corresponding commutators.

Table 1

The input parameters were varied according to a central composite rotatable uniform design of the second order, presented in Table 2 [5]. This table also gives the sparking values according to visual assessment and according to objective assessment on the basis of signal parameters.

Table 2

After processing the experiment, we obtain the following dependence of brush sparking on the selected system of parameters:

\[ I = 2.523 - 0.114 \cdot I_{\mathrm{p}} - 1.1 \cdot P - 0.761 \cdot \sigma_{\Delta} + 0.0077 \cdot I_{\mathrm{p}}^{2} + 0.156 \cdot P^{2}+ \]
\[ +0.894 \cdot \sigma_{\Delta}^{2} +0.0336 \cdot I_{\mathrm{p}} \cdot P +0.0108 \cdot I_{\mathrm{p}} \cdot \sigma_{\Delta} -0.033 \cdot P \cdot \sigma_{\Delta}. \]

Here \(I_{\mathrm{p}}\) is in percent of \(I_{\mathrm{n}}\), \(P\) is in hundreds of grams per \(1\ \mathrm{cm}^{2}\), and \(\sigma_{\Delta}\) is in microns.

To check the adequacy of the obtained sparking model, we compute the residual sum of squares of deviations of the theoretical values \(\hat I\) from the experimental values \(I\):

\[ S_R = \sum_{k=1}^{20} (I_k - \hat I_k)^2 = 0.135. \tag{7} \]

The multiple correlation coefficient characterizing the adequacy of the model is

\[ R = \sqrt{1 - \frac{S_R}{\sum (I_k - \bar I_k)^2}} = 0.989 \]

with a lower confidence limit \(R_{\mathrm{n}} = 0.96\) at a 99 percent confidence probability. The high correlation coefficient shows that, for machines of the P1-6 frame series with an armature rotation speed of 1500 rpm, all technological factors that substantially affect commutation have been taken into account in the sparking model. Among these factors are six parameters of the geometry of the magnetic system and two parameters of the commutator–brush assembly.

The high adequacy of the obtained dependence makes it possible to use it for constructing a system of tolerances for the parameters considered.

REFERENCES

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. E. K. Strelbitskii, V. S. Stukach, A. Ya. Tsirulik. Evaluation of sparking at the commutator in DC machines. Izvestiya TPI, present collection.

3. V. G. Galkin. Investigation of the influence of technological deviations in the geometry of the magnetic system of traction motors on the state of commutation. Dissertation, Omsk, 1965.

4. E. K. Strelbitskii, V. S. Stukach, A. Ya. Tsirulik. Statistical processing of commutator profilograms of DC machines. Izvestiya TPI, vol. 160, 1966.

5. V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extremal Experiments. Nauka, Moscow, 1965.