Wear of Commutators of DC Machines by Commutation Sparking and Calculation of Commutator Service Life

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

(Presented by the scientific seminar of the Department of Electrical Machines and Apparatus)

One of the units limiting the service life of a DC machine is the commutator-and-brush assembly. For mass-produced general-purpose machines that have a long service life, the durability of the commutator-and-brush assembly is determined by the durability of the commutator. Wear of the brushes down to the minimum size is not a failure, since during the service life the brushes are replaced many times and this is a preventive measure.

The service life of the commutator is limited as a result of frictional and electro-erosive wear of the surface. Calculation of the service life of the commutator is possible only after determining the rate of electro-erosive wear as a function of the degree of sparking. The present article sets forth the results of a study of commutator wear by sparking and proposes a method for calculating the operating time to commutator failure.

First of all, a criterion for failure of the commutator-and-brush assembly must be formulated. According to GOST 183—66, a failure is an event consisting in a loss of operability of a product. In the present case it is difficult to formulate a failure criterion, since the machine remains operable even with substantial damage to the working surface of the commutator. We carried out tests of P22M machines under very intense sparking (three points and higher). The machine operated for more than 1600 hours; the working surface of the commutator was everywhere subjected to erosion to a depth of up to 28–32 microns, the polish was absent, and nevertheless the machines remained operable. Evidently, in many cases the requirements imposed on the permissible level of sparking are unjustifiably too high. This is connected with the absence of methods for calculating the service life of a commutator at different levels of sparking. Determining the relationship between wear and sparking and creating a method for calculating commutator service life would make it possible to establish economically optimal requirements for the permissible level of sparking and the service life of the machine.

To obtain data that would make it possible to formulate a criterion for commutator failure due to electric-spark wear, we tested ten P22M machines at an elevated level of sparking. The machines have an equal-section winding, as do all mass-produced machines with power up to 100 kW. Four sections are located in a slot across the slot width. The sequence of wear is as follows. First, each fourth commutator bar, which in the commutation process opens the last section of the slot, is subjected to electric-spark erosion. On these bars, erosion marks appear with a depth of 4–8 microns, whose shape corresponds to the shape of the working surface of the brush. The width of the erosion mark increases over time in the direction of the bar width. The remaining bars are not subjected to erosion; the surfac-

surface of the commutator, despite intense sparking, remains smooth, and the patina is normal. Brush wear is slight and corresponds to the normal wear rate under weak sparking. Such a condition of the commutator does not lead to a qualitative change in the commutation process and must be regarded as permissible.

A new stage of wear begins after “burn-through” over the entire width of the fourth bars, that is, the bars connected with the last section of the slot. From this moment erosion begins of the penultimate bars in the slot. They “burn through” over their entire width very rapidly, after which erosion begins of the next bars in order, until erosion traces form on the commutator surface around the circumference, whose shape and dimensions correspond exactly to those of the contact surface of the brushes. This process develops very rapidly, since after “burn-through” of the last bars in the slot a qualitative deterioration of commutation occurs: the two last sections commutate as one section with twice the number of turns.

Further wear only increases the depth of the erosion marks, without changing their area. The working tracks of the commutator are subjected to erosion nonuniformly around the circumference, with interruptions. The number of depressions is equal to the number of slots. The commutator profile assumes a sawtooth form with a depth of hollows up to 28–32 microns; the surface is rough, without patina. The rate of brush wear in this case increases severalfold. Although machine failure does not occur, intense wear of the commutator and brushes leads to a sharp reduction in the service life of the machine. Therefore, for machines designed for a long service life, such a condition of the commutator should be considered inoperative, and the failure criterion should be taken as the moment when the bars connected with the last sections of the slot burn through (or at least one of such bars). The time to failure will be determined by the operating time of the machine up to the moment of failure, understood as the moment of burn-through of the “last” bars over their entire width. If the rate of advance of erosion marks with time across the width of the bar as a function of the sparking level is known, then it is possible to calculate the time to failure, the service life of the machine at a given sparking level, taking into account preventive turnings of the commutator after “failure,” and other reliability characteristics.

Let us introduce the notation:

\(b_k\) — width of a bar in millimeters;
\(\beta\) — deviation of the bar width, mm;
\(b\) — width of the erosion mark on the bar, mm;
\(\alpha\) — rate of widening of the erosion mark, mm/hour;
\(\tau_g\) — duration of burning of the commutation arc when the contact between brush and bar opens, microseconds (µs);
\(u_g\) — arc-burning voltage, V;
\(L_p\) — inductance of the break of the last section of the slot, H.

To determine the rate of spark wear of the commutator with time as a function of the sparking level, tests were carried out on P22M machines under increased sparking. The sparking level was evaluated by the parameters of the voltage of the sparking arcs, which was measured and oscillographed by means of an additional brush (the method of M. F. Karasev). Studies by L. L. Lavrinovich established (and our studies confirm this) that the arc voltage when the bar–brush contact opens remains practically unchanged during the arc-burning time, while the current in the arc decreases to zero according to a linear law. Therefore, the quantity of electricity \(Q\) transferred during the time \(\tau_g\), and the energy \(A\) released in the arc (and, consequently, the wear of the bars), are uniquely related to \(\tau_g\) for a specific machine frame size, when \(L_p = \text{const}\).

\[ Q = \frac{U_g}{2L_p}\cdot \tau_g^{2}, \qquad A = \frac{U_g^{2}}{2L_p}\cdot \tau_g^{2}. \]

For constant \(U_g\) and \(L_p\), sparking can be unambiguously assessed by the duration of burning of the sparking arcs. In carrying out the experiment, \(\tau_g\) was measured on each lamella of the commutators of all machines, and the corresponding dimensions of the erosion traces were measured after specified intervals of time. The numerous obtained realizations of wear at different values of \(\tau_g\) show that the wear rate is practically constant for each value of \(\tau_g\). Figure 1 shows, from the experimental results, the dependence of the width of the erosion trace on \(\tau_g^2\), from which the mean wear rate in time \(\overline{\alpha}\) and the variance of the wear rate \(D(\alpha)\) are determined as functions of \(\tau_g\):

\[ \overline{\alpha}=7.27\cdot 10^{-6}\cdot \overline{\tau_g^2}, \tag{1} \]

\[ D(\alpha)=2.62\cdot 10^{-12}\cdot \overline{\tau_g^4}+52.8\cdot 10^{-12}\cdot D(\tau^2) \tag{2} \]

In expression [2] the first term characterizes the scatter of the wear rate caused by the nonuniformity of the composition and structure of the copper of individual commutator lamellae and by measurement errors. The second term explains the scatter of the wear rate of commutators of different machines of one type size, caused by the variance \(D(\tau^2)\) of arc durations. This variance is due to the technological scatter of the parameters of the sections, the commutator-brush assembly, and the magnetic circuit of machines of one series.

Fig. 1. Dependence of the width of the erosion trace on the lamellae on the square of the duration of sparking arcs for the time instant 103 hours from the start of tests.

Since the wear rate is proportional to the number of spark discharges per unit time (the armature rotational speed), inversely proportional to the total length of the trailing edge of the brushes \(l_{\text{щ}}\) from one brush set along the commutator axis, and the reserve of energy in the arc at identical \(\tau_g\) is inversely proportional to the inductance of the break, the wear rate in the general case must be determined by the formula

\[ \alpha=7.27\cdot 10^{-6}\cdot \frac{l_{\text{щ}1}L_{p1}}{n_1}\cdot \frac{n}{l_{\text{щ}}L_p}\cdot \tau_g^2, \tag{3} \]

where \(n_1\), \(l_{\text{щ}1}\), \(L_{p1}\) are the parameters of the machines investigated, and \(n\), \(l_{\text{щ}}\), and \(L_p\) are the parameters of the machine for which the calculation is to be made. The dimensions of these parameters are, respectively, rpm, cm, H. Substituting the values of the parameters of the machines investigated, we obtain:

\[ \alpha=8.73\cdot 10^{-13}\cdot \frac{n}{l_{\text{щ}}\cdot L_p}\cdot \tau_g^2, \tag{4} \]

\[ D(\alpha)=76.6\cdot 10^{-26}\cdot \frac{\overline{n^2}}{\overline{l}_{\text{щ}}^{\,2}\cdot \overline{L}_p^{\,2}}\cdot \left[0.0497+v^2(\tau_g^2)+v^2(n)+v^2(l_{\text{щ}})+v^2(L_p)\right]\cdot \overline{\tau_{\text{д}}^4}(5), \]

where \(v(z)\) are the coefficients of variation of the parameter \(z\). The scatter of a large number of machine parameters affects the scatter of the wear rate \(\alpha\); the scatter of these parameters, in turn, is caused by the random influence of many technological factors. Therefore the distribution of \(\alpha\) may be regarded as normal.

The quantities \(b_{\mathrm{k}}\) and \(\beta\) are independent and random. The mean value \(\overline{b}_{\mathrm{k}}\) is determined by the expression:

\[ \overline{b}_{\mathrm{k}}=\frac{\pi \cdot \overline{D}_{\mathrm{k}}}{K}-\overline{\Delta}_{\mathrm{i}}, \tag{6} \]

where \(\overline{D}\) is the nominal commutator diameter, mm;
\(K\) is the number of commutator segments;
\(\overline{\Delta}_{\mathrm{i}}\) is the mean width of the intersegment gap, mm.

Random deviations \(\beta\) of the segment width \(b_{\mathrm{k}}\) are explained by technological factors: the scatter of commutator diameters and of the width of the intersegment gap, and the inaccuracy of manufacture and operation of dividing devices that rotate the commutator during slotting.

Fig. 2. Relation of the sparking level in points according to GOST 183–66 to the duration of burning of sparking arcs.

Since deviations are equally probable in both directions, the mean deviation \(\overline{\beta}\) may be taken as equal to zero. As statistical studies show, the distribution of deviations \(\beta\) can, with sufficient accuracy, be regarded as normal. Then the wear \(b(t)\) also has a normal distribution with parameters:

\[ \overline{b}(t)=\overline{\alpha}\cdot t+\overline{\beta}=\overline{\alpha}\cdot t, \tag{7} \]

\[ D\{b(t)\}=t^{2}\cdot D(\alpha)+D(\beta) \tag{8} \]

If \(b(T)=b_{\mathrm{k}}\), failure occurs, and \(T\) is the operating time to failure, h. The mean operating time to failure is

\[ \overline{T}=\frac{\overline{b}_{\mathrm{k}}}{\overline{\alpha}}. \tag{9} \]

From the normality of \(b(t)\) it follows that the distribution of the time \(T\) is described by the Bernstein dispersion distribution [1]. According to this law, the probability of failure-free operation is

\[ P\{T>t\}=1-\Phi\left[\frac{t-C}{\sqrt{A\cdot t^{2}+B}}\right], \tag{10} \]

where \(\Phi|z\) is the Laplace integral function,

\[ A=\frac{D(\alpha)}{\overline{\alpha}^{2}},\quad B=\frac{D(\beta)}{\overline{\alpha}^{2}},\quad C=\frac{\overline{b}_{\mathrm{k}}-\overline{\beta}}{\overline{\alpha}}. \tag{11} \]

Formulas [4, 5, 6, 10, 11] make it possible, for any instant of time, to calculate \(P\{t\}\) and to construct the probability curve of failure-free operation of the commutator of a ma

DC machines, for a known sparking level and a known relationship between $\tau_d$ and the sparking level. At present, studies are being conducted to determine this relationship. For P22M machines this relationship is shown in Fig. 2. Fig. 3 gives the dependences $P\{T > t\}$ for P22M machines at different values of $\tau$, calculated from the formulas derived. As can be seen from Fig. 2, with sparking in the range of 1.25–1.5 points according to GOST 183–66, the maximum possible arc duration does not exceed 3 $\mu$sec. With such sparking, the machine can operate for more than 30 thousand hours with a probability of failure-free commutator operation greater than 0.95 (Fig. 3). This indicates the possibility of increasing the guaranteed service life of these machines (at least with respect to the commutator-brush assembly) by several times. At present this period is set at 12.5 thousand hours. The total service life of the commutator, taking into account possible turning or grinding of the commutator after “failure,” may reach 100 thousand hours.

Fig. 3. Dependence of the probability of failure-free operation of the commutator on time and on the sparking level, determined by arc duration.

Conclusions

1. The failure of the commutator of DC machines with an equal-section armature winding, designed for a long service life, should be taken to be the moment when the width of the trace of electrospark erosion on the segments becomes equal to the width of a segment (the moment of “burn-through” of the segment over its entire width).

2. A method has been developed for calculating the commutator time to failure and the probability of failure-free operation.

References

1. I. B. Gertsbakh, Kh. B. Kordonskii. Failure Models, “Sovetskoe radio,” Moscow, 1966.