ANALYTICAL STUDY OF COMMUTATION IN COMMUTATOR ELECTRICAL MACHINES USING AN ELECTRONIC DIGITAL COMPUTER

A. I. Skorospelkin, B. I. Kostylev, G. G. Konstantinov

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

Analytical studies of commutation, as a rule, are carried out with significant assumptions with respect to taking into account the mutual inductive couplings between the sections being commutated and the nonlinearity of the brush contact. This is explained by the complexity of the commutation process, which does not allow a solution to be obtained in general form. Numerical methods for solving the commutation equations without the use of calculating and computing machines require large expenditures of time in order to obtain satisfactory accuracy.

With the development of analog and digital computing machines, the possibility has arisen for the most complete and in-depth analytical study of commutation not only for finished machines, but also in the process of designing them.

Unfortunately, electronic computing machines have not yet found sufficiently wide application for the study of commutation. There are only a few works [1, 2, 3] concerning the study of commutation with the aid of computing machines. This situation, apparently, is explained by the absence of a unified calculation procedure that takes into account the maximum number of factors influencing the commutation process.

Nevertheless, only analytical studies make it possible to solve the commutation problem of commutator electrical machines before their manufacture, to prescribe the necessary accuracy of the technological process, and to specify the corresponding electromagnetic loadings, section parameters, and brush grade.

The commutation process is characterized to a significant degree by the law of variation of the current in the sections being commutated, which is a consequence of the action of the entire complex of factors. To determine this law it is necessary to solve a system of differential equations that take into account all the interrelations of the sections being commutated. The difficulty in solving these equations lies in the discontinuity of the mutual-inductive influence and, chiefly, in determining the voltage drops in the brush contact. It is precisely in the brush contact that the principal influences on the commutation process are concentrated (mechanics, temperature, humidity, pressure, etc.). Therefore, depending on how correctly the brush contact is represented, the accuracy of the solution will also be determined.

It is very difficult to take into account all the factors acting on the brush contact and, apparently, this is possible only with the use of prob—

probability methods. But for a specific machine during its prolonged operation it may be assumed that only the current density remains variable, while the other factors may be regarded as established. And if the volt–ampere characteristic of the brush contact is available for these operating conditions, then it already takes all these factors into account sufficiently completely. At the same time, many years of investigations have shown that the dependence of the specific resistance of the brush contact on current density is nonlinear and differs from the assumptions both of the classical theory, \(\rho_{\text{sh}}=\text{const}\), and of O. G. Venner, \(\Delta I=\text{const}\).

The indicated assumptions are only special cases and turn out to be valid only for certain grades of brushes and within a definite range of current densities. Therefore, the use in the calculation of the real nonlinear dependence of the brush contact, \(\Delta U=f(j)\), which is quite possible when automatic computing machines are available, should give a more accurate solution over the entire range of current densities.

The compilation of the system of differential equations of commutation is the same for all machines. For this it is necessary to have the order of commutation of the sections under the various brushes and the winding data of the machine.

The solution time, or the computational period, is determined by the time from the instant the commutation of the first section of the slot begins to the instant when data can be obtained for all sections of one slot. In this case it is quite natural to assume that the commutation conditions for sections identically located in different slots are identical, provided the commutator has been made sufficiently accurately. The computational period is divided into a number of intervals, the beginning and end of which are characterized either by the entry of some section into commutation or by its exit from commutation, as a result of which the energy content in the circuits changes.

The computational period and its intervals in each particular case will be their own and depend on the number of sections in a slot and on the brush overlap. In the general case, if it is assumed that the volt–ampere characteristic of brushes of both polarities is the same, and if the direction of the current and of the winding of the sections under one of the brushes [1] is changed, then for the lap winding of an elementary machine the computational period will be equal, when \(K\) is an odd number, to

\[ U_{\text{p}}\frac{T_{\text{k}}}{2}, \]

and when \(K\) is an even number—to

\[ U_{\text{p}}\cdot T_{\text{k}}, \]

where \(U_{\text{p}}\) is the number of section sides in a slot, and \(T_{\text{k}}\) is the time corresponding to the displacement of the armature by one commutator division.

The number of intervals and their duration are determined by the brush overlap.

If, for \(K\) odd, the brush overlap is expressed as

\[ \beta = 0.5n+\varepsilon, \]

where \(n\) is an integer and \(\varepsilon<0.5\), then the computational cycle is divided into intervals of duration \(\varepsilon T_{\text{k}}\) and \((0.5-\varepsilon)T_{\text{k}}\).

If, for \(K\) even, the brush overlap is expressed as

\[ \beta = n+\gamma, \]

where \(n\) is an integer and \(\gamma<1\), then the computational cycle is divided into intervals of duration \(\gamma T_{\text{k}}\) and \((1-\gamma)T_{\text{k}}\).

Moreover, the first of these intervals are characterized at their end by the exit of one of the sections from commutation, and the second by the entry of a new section into commutation. The number of intervals of either duration will be the same and equal to the number of section sides in a slot.

Thus, the computational cycle does not depend on the brush overlap. It only determines the number and duration of the intervals of the computational cycle and the number of equations for each of them.

Thus, for odd \(K\), on the interval \(\varepsilon T_k\) the number of equations is \(n+1\), and on the interval \((0.5-\varepsilon)T_k\) it will be equal to \(n\). For even \(K\), respectively, we shall have \(2(n+1)\) and \(2n\).

The equation for each section in general form is written as:

\[ L_x \frac{di_x}{dt} + \sum_{k=1}^{n} M_{xk}\frac{di_k}{dt} = \sum e - i_x R - \Delta U_x + \Delta U_{x+1}, \]

where \(L_x\) is the coefficient of self-inductance of the section;
\(M_{xk}\) is the coefficient of mutual inductance of the section with the remaining simultaneously commutating sections;
\(\sum e\) is the total emf acting in the section at the instant of commutation;
\(R\) is the resistance of the section;
\(\Delta U\) is the voltage drop on the commutator bar to which the beginning of the commutating section is connected;
\(\Delta U_x\) is the voltage drop on the commutator bar to which the end of the commutating section is connected.

The terms of the equation that take mutual-inductive couplings into account will be constant only on individual intervals of the computational cycle. Therefore it is expedient to carry out the calculation precisely by intervals. The voltage drop in the brush contact will be determined by the current through the commutator bar and by the area of the brush contact with the aid of the volt-ampere characteristic \(\Delta U=f(j)\). The current through the bar, as a current node, is determined by

\[ i_n = i_x - i_{x+1}, \]

where \(i_x\) is the current of the section connected to the bar at its beginning;
\(i_{x+1}\) is the current of the section connected to the bar at its end.

The area of contact of the lamella with the brush is a function of time and, for brush overlap \(\beta>1\), will vary according to a trapezoidal law, which is very difficult to represent by a single expression. But if the calculation is carried out by intervals and the designations of the unknowns in each of them are made identically from the standpoint of the instant of commutation, and not according to their belonging to a section, then the system of equations and the law of variation of the contact areas will be identical. The inconsistency of the designations can easily be taken into account when processing the calculation results.

The total emf \(\sum e\) acting in the commutating section includes extraneous emfs arising in the section at the instant of commutation either due to rotation of the armature in the magnetic field or by transformer action.

Such a calculation method makes it possible to carry out calculations both for direct-current and alternating-current machines without introducing substantial changes.

We checked the method for an EMU of a transverse-field type EMU-12A, having the following basic data: \(P_n=1.2\) kW; \(U_n=115\) V; \(I_n=10.4\) A; \(I_{dn}=3.4\) A; \(W_c=4\); \(U_n=3\); \(K=87\); \(2p=2\); \(\beta=1.5\); slot pitch \(1\text{—}15\); \(\eta n=2900\) rpm. In this case the initial circuit and the time diagram of commutation have the form shown in Fig. 1. The system of equations is written as:

\[ L\frac{di_1}{dt} + M_{12}\frac{di_2}{dt} + M_{13}\frac{di_3}{dt} = \sum e - i_1R - \Delta U_1 + \Delta U_2, \]

\[ L\frac{di_2}{dt} + M_{21}\frac{di_1}{dt} + M_{23}\frac{di_3}{dt} = \sum e - i_2R - \Delta U_3 + \Delta U_4, \]

\[ L\frac{di_3}{dt} + M_{31}\frac{di_1}{dt} + M_{32}\frac{di_2}{dt} = \sum e - i_3R - \Delta U_5 + \Delta U_1, \]

The calculation period will be equal to

\[ T_{\mathrm{c}}=U_{\mathrm{p}}\frac{T_{\mathrm{k}}}{2}=1.5T_{\mathrm{k}}. \]

Since the brush overlap is \(\beta=1+0.5\), the calculation cycle is divided into three intervals of duration \(0.5T_{\mathrm{k}}\). If the calculation is begun from the instant at which commutation starts in the section that is the first to commutate in the slot, then in the first interval we shall have:

\(M_{12}=M_{21}=M_{23}=M_{32}=M_{\mathrm{vn}}\) — the mutual inductance between the upper and lower section sides in the slot, or the mutual inductance between sections commutating under brushes of different polarity.

\(M_{13}=M_{31}=M\approx O\) — the mutual inductance between sections commutating under one brush.

In the second and third intervals:

\[ M_{12}=M_{21}=M_{23}=M_{32}=M_{\mathrm{vn}},\quad M_{13}=M_{31}=M\ne 0. \]

In the first interval \(M\approx O\), since the sections commutating under one brush are located in adjacent slots and the inductive coupling between them is very weak; whereas in the second and third intervals they are located in the same slot, and the mutual-inductance coefficient will be close to the value of the self-inductance coefficient for one section, provided the sections are equally wound.

Thus, we can rewrite the system in the form:

\[ L\frac{di_1}{dt}+M_{\mathrm{vn}}\frac{di_2}{dt}+M\frac{di_3}{dt} =\sum e-i_1R-\Delta U_1+\Delta U_2 . \]

\[ L\frac{di_2}{dt}+M_{\mathrm{vn}}\frac{di_1}{dt}+M_{\mathrm{vn}}\frac{di_3}{dt} =\sum e-i_1R-\Delta U_3+\Delta U_4, \]

\[ L\frac{di_3}{dt}+M\frac{di_1}{dt}+M_{\mathrm{vn}}\frac{di_2}{dt} =\sum e-i_3R-\Delta U_5+\Delta U_1, \]

where \(M\) is variable over the intervals.

The total emf over the calculation cycle may be taken as constant and identical for all sections, since the field in the commutation zone is practically constant, and, in the presence of transformer emf, the change of the latter over the commutation period may also be neglected.

For the example under consideration, when the volt-ampere characteristic is approximated by an expression of the form

\[ J=\frac{a}{S}\Delta U^k \]

the voltage drops are determined as:

\[ \Delta U_1=[C(i_1-i_3)]^{\frac{1}{k}}; \qquad \Delta U_2=\left[C\frac{i_{\mathrm{an}}-i_1}{t/T_{\mathrm{k}}}\right]^{\frac{1}{k}}; \]

\[ \Delta U_3=\left[C\frac{i_2-i_{\mathrm{ak}}}{1-t/T_{\mathrm{k}}}\right]^{\frac{1}{k}}; \qquad \Delta U_4=\left[C\frac{i_{\mathrm{an}}-i_2}{0.5+t/T_{\mathrm{k}}}\right]^{\frac{1}{k}}, \]

\[ \Delta U_5=\left[C\frac{i_3-i_{\mathrm{ak}}}{0.5-t/T_{\mathrm{k}}}\right]^{\frac{1}{k}}, \]

where

\[ C=\frac{\beta}{a} \]

\(i_{a n}\) is the current of the parallel branch or section before the beginning of commutation,
\(i_{a k}\) is the current of the parallel branch or section after commutation.

The system of differential equations of commutation is easily transformed to the normal form

\[ -\frac{d i_k}{d t}=F(i_1; i_2 \ldots; i_k; t), \]

which is convenient for solving the system on an electronic computer. The initial conditions are determined exactly on the first interval for the section entering commutation, and approximately for the remaining sections. For the other intervals, the final conditions of the preceding interval serve as the initial conditions. The solution is carried out several times until the initially approximate input data have been refined. The solution is obtained sufficiently rapidly and makes it possible to obtain the necessary data easily: the current, the voltage drop between the brush and the commutator bar, and the derivative of the current (on a certain scale, the reactive e.m.f.). For this it is only necessary to enter into the program a command to print the corresponding quantity.

The amplifier taken as an example had the following initial data for the calculations:

\[ S_{\mathrm{sh}}=1.5\ \mathrm{cm}^2;\quad R=0.032\ \Omega;\quad T_k=2.4\cdot 10^{-4}\ \mathrm{s}. \]

The parameters of the commutated sections, obtained by calculation, were:

for the transverse circuit:

\[ \begin{aligned} L&=18.4\cdot 10^{-6}\ \mathrm{H},\\ M_{\mathrm{vn}}&=6.9\cdot 10^{-6}\ \mathrm{H},\\ M&=15.64\cdot 10^{-6}\ \mathrm{H}, \end{aligned} \]

for the longitudinal circuit:

\[ \begin{aligned} L&=27\cdot 10^{-6}\ \mathrm{H},\\ M_{\mathrm{vn}}&=11.2\cdot 10^{-6}\ \mathrm{H},\\ M&=23\cdot 10^{-6}\ \mathrm{H}. \end{aligned} \]

The difference in the parameters of the section for the longitudinal and transverse circuits arises because the additional poles are located only along the longitudinal axis.

The amplifier is equipped with brushes of the EG-8 grade; the volt-ampere characteristic of these brushes, taken on a short-circuited commutator, is approximated by the expression

\[ J=2\Delta U^2 \]

Numerous calculations and experiments for the EMU-12A, carried out for both direct and alternating current, confirmed that the proposed calculation method gives satisfactory accuracy. On alternating current, in contrast to direct current, the instantaneous values of the currents of the parallel branches and the total e.m.f. of the section were determined with allowance for the phase shift.

Figures 2 and 3 present a number of commutation-current curves for sections of one slot pitch, obtained by calculation and experiment. Figure 2 gives the current curves of sections commutated in the transverse circuit, which is supplied from an independent source of direct (2) and alternating (3) current. Curve 1 was obtained by calculation.

Figure 3 gives the current curves of sections commutated in the longitudinal circuit when the amplifier operates on alternating current of frequency 50 Hz, for the instant of time corresponding to the maximum value of the current in the longitudinal circuit and to a small transformer e.m.f.

The curves obtained show that the proposed method gives satisfactory accuracy. Some deviations of the experimental curves from the calculated ones, expressed as sharp peaks, can be explained by the fact that

Fig. 1. Initial circuit and time diagram of commutation of an EMU.

Fig. 2. Current curves of the commutated sections of the transverse circuit of an EMU.
1 — calculation;
2, 3 — experiment for alternating and direct current.

Fig. 3. Current curves of the commutated sections of the longitudinal circuit of an EMU.

mechanical contact disturbances and failure to take into account the capacitance of the circuit in the final stage of commutation.

Thus, the proposed method for calculating the commutation process gives satisfactory accuracy for both direct and alternating current and, with further improvement, may find wide application in analytical studies of commutation in commutator electrical machines.

References

1. E. M. Sinelnikov, A. G. Nazikin, V. V. Kleimenov, F. I. Chernyavsky. Application of continuous-action computing machines to the study of commutation in direct-current machines, Proceedings of Higher Educational Institutions. Electromechanics, No. 10, 1960.

2. Alger R. M., Bewley D. T. An Analysis of D.—C. Machine Commutation. Power Apparatus and Systems. 1957, August.

3. V. V. Kleimenov. Analysis of commutation in direct-current machines using electronic computers, Collection of Works of the All-Union Scientific-Research and Design-Engineering Institute of Electric Locomotive Engineering, No. 4, 1964.