STUDY OF THE INFLUENCE OF MECHANICAL FACTORS ON THE COMMUTATION OF COMMUTATOR ELECTRICAL MACHINES

L. Ya. Zinner, A. I. Skorospeshkin, E. K. Damm

(Recommended by the seminar of the Department of Electrical Machines
and General Electrical Engineering)

Radial and tangential vibrations of brushes, caused by the specific operation of the sliding contact, are in many cases the cause of unsatisfactory commutation of commutator electrical machines. This is due to the fact that brush vibrations lead to modulation of the transient resistance of the brush—commutator and brush—segment contacts and, consequently, to modulation of the contact voltage drop.

The present work gives the results of a study of the influence of mechanics on the commutation of commutator machines.

In view of the fact that tangential and radial vibrations can be considered independently of one another, the investigations were carried out in two directions:

a) study of the influence of tangential vibrations on commutation;

b) study of the influence of radial vibrations of brushes on commutation.

In [1] it was established that tangential vibrations are always accompanied by a redistribution of currents over the parts of the brushes, which modulates the contact voltage drop and the transient resistance.

The measurements carried out showed that the depth of modulation of these quantities reaches values of 0.2—0.5 and more, and the transient resistance of the contact can be described by an expression of the form

\[ R = R_0(1 + m\sin 2\pi ft), \tag{1} \]

where

\(R_0\) — transient resistance in the absence of tangential vibrations;

\(f\) — frequency of tangential vibrations;

\(m\) — modulation depth, depending on the amplitude and frequency of the tangential vibrations.

On the basis of (1) it can be shown that the transient resistance of the brush—segment contact also does not remain constant during the short-circuiting of the commutated section. Indeed, taking into account that the frequency of the tangential vibrations is several times greater than the segment frequency (\(f_t = 3—8\) mgc), we have for the resistance of the entering and leaving layers of the brush:

\[ r_1 = \frac{R_0}{\tau}\left[1 + m\sin(2\pi fT_c + \varphi_0)\right], \tag{2} \]

\[ r_2=\frac{R_0}{1-\tau}\left[1+m\sin(2\pi fT\tau+\varphi_0)\right], \tag{3} \]

where

\[ \tau=\frac{t}{T} \]

\(T\) is the commutation period,
\(\Phi_0\) is the angle taking into account the phase of the brush position at the instant commutation begins.

Expressions (2) and (3) indicate a qualitative relationship between the tangential vibrations of the brushes and the behavior of the brush–commutator-bar contact. In this case, modulation of the transient brush–bar resistance leads to modulation of the contact voltage drop and, consequently, directly affects the form of the section-current curve.

Taking (2, 3) into account, we write the equation for the current of the commutated section (Fig. 1) in the form

\[ L\frac{di}{dt}+i_2r_1(t)-r_2(t)i_1+e_k=0. \tag{4} \]

Considering that, during rotation of the commutator, the brush takes up the clearance in the brush holder by tipping onto the leading edge and stabilizes the operation of the leading edge, it may be assumed with sufficient accuracy that

\[ r_1=R_0\frac{T}{t}. \]

Then, after substituting the values \(di=di_2\), \(i_1=2i_a-i_2\), we have

\[ \frac{di_2}{d\tau}+i_2\frac{R_0T}{L\tau(1-\tau)} \left[1+m\sin(\omega\tau+\varphi_0)\right] = \frac{2i_aR_0T-e_kT\tau}{L\tau}, \tag{5} \]

where

\[ \omega=2\pi fT. \]

Solving (5) with respect to \(i_2\) leads to an extremely cumbersome expression containing the sine and cosine integrals as terms.

In order to check the degree of influence of tangential brush vibrations on commutation, we carried out calculations of equation (5) on the “Promin” computer for the following data (Table 1).

Table 1

The results of calculations by (5) for the initial data of Table 1 are presented as graphical dependences in Figs. 2, 3.

Figure 2 shows the dependences of the section current on the depth of modulation at \(\varphi_0=0\). As can be seen, larger values of \(m\) correspond to more accelerated commutation and a higher rate of current change \(\frac{di}{dt}\) in the concluding phase of the commutation process, which leads to a sharp

Fig. 1. Circuit of the commutated section

Fig. 2. Calculated curves of the current in the commutated section, taking into account tangential oscillations of the brush at different values of \(m\)

to an increase in the current density at the trailing edge of the brushes and, consequently, to the appearance of break arcs. Here we considered the case when \(\varphi_0 = 0\). In fact, since the lamellar frequency and the frequency of tangential oscillations are not synchronous, the initial phase from one revolution to the next revo-

Fig. 3. Calculated current curves in the commutated section, taking into account tangential oscillations of the brush at different initial phases of the brush position \(\varphi\)

[[unclear: beginning of sentence]] revolution for one and the same lamella can assume the most diverse values and lead to nonidentity of the commutation cycles. Fig. 3 presents calculated curves of the section current for \(m=0.6\) and different initial phases \(\varphi_0\). In this case, positive initial phases correspond to the case in which, at the moment the section is short-circuited, the brush is tilted toward the leading edge, while negative \(\varphi_0\) corresponds to the case in which the brush is tilted toward the trailing edge.

As is seen from Fig. 3, the initial phase has a greater influence on commutation than does the depth of modulation. This is due to the fact that the initial phase directly determines the position of the trailing edge of the brush at the moment commutation ends. In this case two limiting cases occur: the brush is pressed against the trailing lamella, which corresponds to positive initial phases, and is more uniformly loaded with current; the brush is tilted toward the leading edge, the case of a negative initial phase, and a break in the section current occurs. In the first case, commutation is completely determined by the parameters of the section and by the value \(e_k\), whereas in the second it is determined by the magnitude of the section current and by the instant at which the section opens. In addition, as is seen from Figs. 2 and 3, tangential oscillations have a certain influence on the course of the current curve throughout the entire commutation period, consisting in the fact that a harmonic component, whose period is equal to the period of the tangential oscillations, is superposed on the steady value of the section current. This is further confirmed by the fact that, when the section current is recorded on an oscillograph, the current curve is clearly unstable from revolution to revolution, which can be explained only by the presence of tangential oscillations of the brushes. The presence of tangential oscillations of the brushes also explains the fact that, in experimentally obtained curves of section currents, steps and dips occur, indicating modulation of the section current.

Thus, tangential oscillations of the brushes may be the cause of considerable nonidentity of the commutation cycles and of unsatisfactory commutation. This applies primarily to machi-

for machines of low and medium power operating with one brush on the brush stud. In machines operating with several brushes connected in parallel, the influence of tangential vibrations of the brushes is considerably smaller, because the positions of the brushes are out of phase relative to the trailing commutator bar. Nevertheless, here too tangential oscillations lead to excessive current loading of some brushes and to de-energizing of others, which narrows the zone of sparkless operation at high loads and increased speed.

In addition to tangential oscillations of the brushes, radial oscillations, caused by irregularities in the commutator profile, also have a significant influence on commutation. At the same time, the nature of the influence of radial oscillations is somewhat different. The point is that radial oscillations of the brushes lead to simultaneous weakening or strengthening of the brush contact both with the entering and with the leaving commutator bars. In other words, they do not exert a noticeable influence on the commutation process of a completely definite section (the case of protrusion of an individual commutator bar at a fixed rotational speed is not considered here). Considering the commutation process of the machine as a whole, it is evident that the presence of harmonic disturbances of the profile can lead to a considerable nonidentity of the commutation cycles. This is due to modulation of the transient resistance of the brush–commutator contact by the irregularities of the commutator profile.

The connection of the transient contact resistance with the state of the commutator surface is indicated by the fact that, when operating under current on a short-circuited commutator having considerable runout, modulation of the contact voltage drop is always observed. This is due to the fact that the presence of runout leads to modulation of the contact pressure and, consequently, to modulation of the transient resistance.

In order to be able to estimate the degree of influence of radial vibrations of the brushes, it is necessary to establish the dependence of the transient resistance on the contact pressure or loading. Figure 4 gives the dependence of the contact voltage drop \(\Delta U\) on the loading \(P\). As can be seen, \(\Delta U\) depends strongly on the loading. This is explained by the fact that, with an increase in \(P\), the effective contact area increases, leading to a decrease in the transient resistance—

\[ \Delta U = f(P). \]

Fig. 4. Voltage curve \(\Delta U\) as a function of the specific pressure on the brush.

…and to an increase in the contact voltage drop at a fixed current value.

The approximating expression for the curves for all brush grades has the form

\[ \Delta U = aP^b . \tag{6} \]

The only difference is that the values of the coefficients \(a\) and \(b\) are different for different brush grades and brushes of different cross section. Thus, for an EG-49 brush (Fig. 4) with a cross section of \(1\ \text{cm}^2\), \(a=9.584\) and \(b=-0.46\).

From (6), assuming that the current did not change during the experiment, we have

\[ R = 0.96P^{-0.46}. \tag{7} \]

When a machine operates with a deformed commutator, modulation of the brush pressure and of the transition resistance occurs continuously. In the general case, taking into account tangential and radial vibrations of the brushes, the transition resistance may be written as

\[ R = R_0(1 + m\sin 2\pi ft)\cdot(1 + m''\sin \xi\omega t), \tag{8} \]

where:

\[ \begin{aligned} m'' &\text{— depth of modulation of the resistance by commutator irregularities,}\\ \xi &\text{— number of the harmonic of the disturbing force,}\\ \omega &\text{— angular velocity of the disturbing force corresponding to the eccentricity.} \end{aligned} \]

Taking into account that the pressure is modulated by the commutator irregularities, we have

\[ P = P_0(1 + m\sin \xi\omega t), \tag{9} \]

and for the total contact resistance

\[ R = a[P_0(1 + m'\sin \xi\omega t)]^b, \tag{10} \]

where

\[ m'=\frac{A m_{\text{br}}(\xi\omega)^2}{P_0} \]

is the depth of pressure modulation,

\[ \begin{aligned} A &\text{— amplitude of the }\xi\text{-th harmonic,}\\ m_{\text{br}} &\text{— mass of the brush.} \end{aligned} \]

With continuous motion of the brush over the commutator, it may be assumed that \(m''=m'\). Then, for the resistances of the leading and trailing edges of the brush short-circuiting the section connected to the \(i\)-th and \((i+1)\)-th commutator bars, we have:

\[ r_1 = R_0\left(1 + m''\sin 2\pi\xi\frac{i}{K}\right)\cdot\frac{T}{t}, \]

\[ r_2 = R_0\left(1 + m''\sin 2\pi\xi\frac{i}{K}\right)\cdot\frac{T}{T-t}, \tag{11} \]

where

\[ \begin{aligned} i &\text{— ordinal number of the bars,}\\ K &\text{— number of commutator bars.} \end{aligned} \]

Taking (11) into account, the equation for the current of the commutated section (Fig. 1) is written in the form

\[ L\frac{di}{dt} + i_2\frac{R_0T}{T-t}\left(1 + m''\sin 2\pi\xi\frac{i}{K}\right) - i_1\frac{R_0T}{t}\left(1 + m''\sin 2\pi\xi\frac{i}{K}\right) = \]

\[ = -e_k . \tag{12} \]

For calculating the section current on the “Promin” digital computer, equation (12), after substituting into it the values \(i_1=2i_a-i_2\) and \(di=di_2\), was reduced to the form

\[ \frac{di_2}{d\tau}+ \frac{0.46 i_2\left(1+m''\sin 2\pi \xi \frac{i}{K}\right)}{1-\tau} = \frac{2i_a\cdot 0.46\left(1+m''\sin 2\pi \xi \frac{i}{K}\right)+15.5\tau}{\tau}. \tag{13} \]

The calculation was carried out for the following parameters of the commutated circuit (Table 2).

Table 2

In Fig. 5 are given the curves of the currents of sections short-circuited, respectively, by commutator bars 0 and 1, 8 and 9, 18 and 19, 54 and 55, 63 and 64, for \(m''=0.2\).

The value \(m''=0.2\) corresponds to quite real operating conditions of the commutator-and-brush unit at \(e=50\,\mu\), \(P_0=200\ \text{g}/\text{cm}^2\), \(n=3000\ \text{rpm}\), \(\xi=1\), \(m_{\mathrm{sh}}=3\ \text{g sec}^2/\text{m}\).

As can be seen from Fig. 5, all sections are considerably overcommutated in comparison with curve 1, corresponding to rectilinear commutation. Curve \(i'_{0,1}\) corresponds to the case when there are no mechanical effects on the brush, since at this point \(\sin 2\pi \xi \dfrac{i}{K}=0\). The dif-

Fig. 5. Calculated curves of the currents of sections connected to different pairs of commutator bars, taking account of radial brush vibrations at \(m''=0.2\).

Fig. 6. Calculated curves of the currents of sections connected to different pairs of commutator bars, taking into account radial vibrations of the brush at \(m''=0.6\)

The scatter of the current curves of different armature sections indicates the nonidentity of the commutation cycles caused by the eccentricity of the commutator. Indeed, for sections \(i'_{54,55}\) and \(i'_{63,64}\) we have the most accelerated commutation in comparison with \(i'_{01}\). This is due to the fact that the sections are short-circuited on bars lying on the approaching portion, where strengthening of the contact takes place and, consequently, a decrease in the transition resistance, leading to an increase in the additional current.

The situation is different in the case of commutation of sections \(i'_{9,10}\) and \(i'_{18,19}\), where weakening of the contact, an increase in transition resistance, and a decrease in the additional current occur.

Figure 6 gives the current curves of the same sections as in Fig. 5, calculated for \(m''=0.6\). As can be seen, an increase in the oscillation frequency or, what is the same, an increase in the depth of modulation leads to an increase in nonidentity. The difference in the additional currents of two sections lying on the protruding and recessed portions of the commutator is \(1.6\) A.

Thus, it is quite clear that brush vibrations lead to deterioration of commutation and to nonidentity of the commutation cycles. Therefore it is important to develop a criterion that would make it possible to estimate the expected nonidentity of the commutation cycles, caused by the mechanics, for a known commutator profile.

In this respect, the most convenient criterion, in our opinion, should be taken to be the rate of change of the current in the final phase of commutation. From Figs. 5 and 6 it is seen that the rate of change of the current depends both on the depth of modulation and on which pair of bars the section is connected to. When

\[ \left.\frac{di}{dt}\right|_{t\to T} \to 0 \]

there is complete unloading by the current of the trailing edge of the brush, which corresponds to sparkless commutation. When

\[ \left.\frac{di}{dt}\right|_{t\to T} \to \infty \]

a break in the current of the section occurs, accompanied by arc processes.

Consequently, to determine the nonidentity it is sufficient to determine the derivative of the section current at the final stage of commutation. For this purpose, from (12) it is necessary to determine the limit to which

\[ -\left.\frac{di}{dt}\right|_{\tau \to 1} \]

tends. After transforming (12), we have

\[ -\left.\frac{di}{dt}\right|_{t \to 1} = \frac{2i_a R_0(1 \pm m'') - e_k T} {L\left[1-\frac{R_0}{L}(1 \pm m'')T\right]} . \tag{14} \]

Dwelling on (14) and taking into account that when

\[ \left.\frac{di}{dt}\right|_{\tau \to T} \to 0 \]

sparkless commutation occurs, we note that the nonidentity of commutation cycles caused by harmonic disturbances of the commutator surface can be estimated with sufficient accuracy by the following criterion:

\[ \chi = \frac{e_{k(1+m'')} - e_{k(1-m'')}}{e_{k0}} . \tag{15} \]

Thus, expression (15) makes it possible, in advance, for known parameters of the commutator-brush assembly \((m_{\text{sh}}, e, E,\ \text{and } P_0)\), to determine the expected nonidentity of commutation cycles.

With somewhat greater accuracy, the nonidentity criterion can be calculated by the formula

\[ \chi = \frac{ \displaystyle \frac{di}{d\tau_{(1+m')}} - \displaystyle \frac{di}{d\tau_{(1-m'')}} }{ \displaystyle \frac{di}{d\tau_0} }. \tag{16} \]

The nonidentity of commutation cycles caused by radial vibrations of the brushes, calculated from (16) for the curves in Figs. 4 and 5, amounts to 25–40 percent.

Thus, on the basis of the investigations carried out, the following conclusions may be drawn:

1. Tangential and radial vibrations of the brushes lead to modulation of the transient contact resistance and, consequently, exert a large influence on commutation.

2. The depth of modulation of the transient resistance reaches a considerable magnitude and takes values from 0.2 to 0.8.

3. A method has been developed for taking into account the influence of brush vibrations on commutation, as well as a criterion for the nonidentity of commutation cycles.

REFERENCES

1. L. Ya. Zinner, A. I. Skorospeshkin. Tangential oscillations of brushes. Izvestiya TPI, vol. 190, 1968.