Statistical Processing of Profilograms of the Commutator of DC Machines
Abstract Generated abstract
This paper addresses errors in evaluating DC machine commutator profilograms when differences between adjacent minimeter readings are treated directly as differences in segment height. It argues that eccentricity and ellipticity introduce first and second harmonic components that can significantly distort apparent bar steps, even when their direct influence on commutation may be small in low speed machines. The authors propose a statistical processing method based on trigonometric regression to determine eccentricity, ellipticity, and corrected bar heights, followed by calculation of true inter-bar height differences and their variance. They also estimate how many uniformly distributed profilogram points are needed for acceptable accuracy, showing that the method can be implemented efficiently with digital computers or, at reduced accuracy, by limited manual computation.
Full Text
STATISTICAL PROCESSING OF PROFILOGRAMS OF THE COMMUTATOR OF DC MACHINES
E. K. STRELBITSKII, V. S. STUKACH, A. Ya. TSIRULNIK
The practice of operating DC machines and numerous studies show that factors of a mechanical nature exert a very substantial influence on the commutation process. Among these factors, the principal importance belongs to the characteristics of the working surface of the commutator—eccentricity, ellipticity, and differences in height between adjacent segments. The geometry of the commutator surface is usually evaluated by analyzing profilograms taken with minimeters relative to a stationary base. In this process the total runout of the commutator, caused by the ellipticity and eccentricity of the surface, is determined, as are the differences in height of adjacent segments as the difference between minimeter readings on adjacent segments.
This method of determining the differences between segments is fundamentally incorrect. The difference in minimeter readings on different segments is caused not only by the unequal heights of the segments, but also by the eccentric seating of the commutator on the shaft and by the ellipticity of the commutator surface. When differences are determined in this way, the eccentricity of the commutator surface introduces an error that varies sinusoidally in magnitude with the rotation frequency. Ellipticity introduces an error that varies sinusoidally with double frequency.
The maximum errors \(\Delta h_{\text{ecc}}\) and \(\Delta h_{\text{ell}}\), introduced respectively by eccentricity and ellipticity, are equal to:
\[ \Delta h_{\text{ecc}}=e\cdot \sin \frac{2\pi}{K}, \qquad \Delta h_{\text{ell}}=E\cdot \sin \frac{4\pi}{K}, \tag{1} \]
where \(e\) is the eccentricity of the commutator, \(\mu\);
\(E\) is the ellipticity, determined by the half-difference of the lengths of the semi-axes of the ellipse, \(\mu\);
\(K\) is the number of commutator segments.
For series P machines studied at the Department of Electrical Machines, with commutator runout \(2e=2E=30\ \mu\) permitted by the technical specifications and with \(K=56\), these errors take the values:
\[ \Delta h_{\text{ecc}}=1.67\ \mu,\qquad \Delta h_{\text{ell}}=3.33\ \mu, \]
i.e., they may exceed the actual differences by a factor of 2–3. The root-mean-square deviation of the differences \(\sigma\) for these machines often does not exceed one micron.
The nature of the influence of the eccentricity and ellipticity of the commutator on brush sparking differs from the nature of the influence of unequal segment heights.
The influence of eccentricity and ellipticity can be appreciable only in the case where the accelerations of the brush produced by these factors exceed the acceleration of the brush produced by the spring, and the brush will lift off from the commutator surface. To prevent this phenomenon, it is necessary that the eccentricity and ellipticity not exceed the values determined by the expressions:
\[ e < \frac{P \cdot g}{4\pi^{2}G \cdot n^{2}} \quad \text{and} \quad E < \frac{P \cdot g}{16\pi^{2}G \cdot n^{2}}, \tag{2} \]
where \(P\) is the spring pressure;
\(g\) is the acceleration due to gravity;
\(G\) is the weight of the brush;
\(n\) is the rotational speed of the commutator.
For machines of the П series of the second frame size, at \(P = 0.3\) kg, \(G = 0.03\) kg, \(n = 2000\) rpm, the permissible values of eccentricity and ellipticity could be \(e = 2250\ \mu\), \(E = 563\ \mu\), which is hundreds of times greater than the values specified by the technical conditions (\(30\ \mu\)). This indicates that, at least in machines with low rotational speeds, ellipticity and eccentricity do not reach magnitudes at which they could have an appreciable effect on commutation. At the Department of Electrical Machines, the influence on commutation of various factors was investigated. The components of the sparking variance caused by the influence of eccentricity and ellipticity are respectively 0.0058 and 0.137, while the component caused by differences in bar levels is 0.102. The nonidentity of the commutation cycles caused by mechanical factors is associated primarily with the nonuniform heights of the commutator bars, which leads to instability of the brush contact.
What has been said explains the need for correct determination of the differences in height of the commutator bars.
The authors have developed and used, in analyzing the commutation of machines of the П series, a procedure for processing profilograms of commutators recorded with the aid of a minimeter, making it possible to determine eccentricity, ellipticity, and the true differences between bars.
The essence of the procedure is as follows. From the profilogram the following are determined:
\[ a_{1} = \frac{ 2 \cdot \sum_{i=1}^{k} y_{i} \cdot \sin \frac{2\pi}{K} \cdot i }{K}; \]
\[ a_{2} = \frac{ 2 \cdot \sum_{i=1}^{k} y_{i} \cdot \cos \frac{2\pi}{K} \cdot i }{K}; \]
\[ \tag{3} \]
\[ a_{3} = \frac{ 2 \cdot \sum_{i=1}^{k} y_{i} \cdot \sin \frac{4\pi}{K} \cdot i }{K}; \]
\[ a_{4} = \frac{ 2 \cdot \sum_{i=1}^{k} y_{i} \cdot \cos \frac{4\pi}{K} \cdot i }{K}. \]
Here \(a_1\) and \(a_2\) are the amplitudes, respectively, of the sine and cosine components of the first harmonic due to eccentricity, \(\mu\);
\(a_3\) and \(a_4\) are the amplitudes, respectively, of the sine and cosine components of the second harmonic due to ellipticity, \(\mu\);
\(y_i\) is the \(i\)-th value of the profilogram, corresponding to the \(i\)-th commutator bar, \(\mu\);
\(K\) is the number of commutator bars.
The eccentricity and ellipticity are determined from the formulas:
\[ e=\sqrt{a_1^2+a_2^2}, \qquad E=\sqrt{a_3^2+a_4^2}. \tag{4} \]
Next, from all values \(y_i\), the values of the indicated harmonics corresponding to the \(i\)-th commutator bar are subtracted; as a result, the “pure” heights of the bars \(h_i\), not distorted by eccentricity and ellipticity, are obtained. The steps between bars are determined by the expression:
\[ \Delta h_i=h_i-h_{i+1}, \qquad (i=1,2,\ldots,K-1), \]
\[ \Delta h_k=h_k-h_1. \tag{5} \]
The variance of the steps is
\[ \sigma^2=\frac{\displaystyle\sum_{i=1}^{k}(\Delta h_i)^2}{K}. \tag{6} \]
For \(K=56\) and \(K=72\), the calculations prove laborious; therefore the processing of profilograms was carried out on the “Promin” and “Minsk-1” electronic digital computers.
Processing profilograms in the form presented above is laborious when the number of bars is large, but it can be simplified if, in order to find the trigonometric regression line, only a portion of the profilogram points, uniformly distributed around the circumference of the commutator, is used. It is possible to determine the minimum number of points that ensures the required accuracy. Since ellipticity introduces a larger error than an eccentricity of the same magnitude, we shall carry out the calculation for ellipticity.
The magnitude of ellipticity is
\[ E=\hat{E}\pm \sqrt{\frac{D}{N-m-1}}, \tag{7} \]
where
\(\hat{E}\) is an estimate of \(E\), found as a result of regression analysis,
\(N\) is the number of profilogram points required for the calculation,
\(m=4\) is the number of parameters of the regression line.
\(D\) is the residual variance, in the present case—the variance of the bar steps.
On the basis of expression (1),
\[ \Delta h_{\text{ell}}=\hat{E}\cdot \sin\frac{4\pi}{K} \pm \sqrt{\frac{D}{N-m-1}}\cdot \sin\frac{4\pi}{K} =\Delta \hat{h}_{\text{ell}}\pm \Delta, \tag{8} \]
the unaccounted error in determining the step is
\[ \Delta=\sqrt{\frac{D}{N-m-1}}\cdot \sin\frac{4\pi}{K}. \tag{9} \]
Since the steps are measured with a minimeter to an accuracy of up to one micron, it is expedient to take \(\Delta\) equal to \(0.1\)–\(0.15\ \mu\). Then, for machines of series P, the minimum required number of profilogram points
for determining \(e\) and \(E\), for \(K = 56, 72,\) and \(84\), will be equal, respectively, to 32, 22, and 17 for \(\Delta = 0.1\ \mu\text{m}\), and 17, 12, and 10 for \(\Delta = 0.15\ \mu\text{m}\).
Conclusions
- The proposed method for processing commutator profilograms makes it possible:
a) to determine, from the profilogram, the eccentricity and ellipticity of the commutator;
b) to determine the true differences in lamella heights, not distorted by ellipticity and eccentricity.
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The processing method is very simple when an electronic digital computer is used.
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If an error of 10–15 percent is allowed, the amount of computation is small, which makes it possible to process profilograms without mechanized means of computation.