ANALYTICAL STUDY OF SWITCHING BASED ON DYNAMIC CURRENT–VOLTAGE CHARACTERISTICS

A. I. Skorospeshkin, B. I. Kostylev, R. F. Bekishev

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

Analytical study of switching is carried out, as a rule, with considerable assumptions regarding the accounting for the properties of the sliding contact and the nonidentity of switching cycles. In the practice of switching calculations, static current–voltage characteristics are widely used, often obtained by averaging the characteristics of anode- and cathode-polarized brushes, which cannot but affect the accuracy of the calculations performed [1, 2].

On the other hand, the switching process is, in essence, dynamic. Under these conditions the sliding contact is in fact being formed. Therefore, in analytical studies of switching it is quite natural to strive to take into account real dynamic current–voltage characteristics, both for the purpose of refining the calculations and for basing them on correct physical conceptions of the switching process [3].

The present article gives the results of an analytical study of switching in a transverse-field electric-machine amplifier based on the use of dynamic current–voltage characteristics obtained separately for brushes of both polarities; and, in accounting for the properties of the sliding contact, the influence of the capacitance between the commutator bar and the brush during contact-making was taken into account.

The characteristics were obtained from the curves of current switching and voltage drop between the brush and the commutator bar when using brushes of grade EG-8.

It is known that the structure of an empirical formula must correspond to the physical essence of the process under consideration; therefore the approximation of current–voltage characteristics must be carried out with this requirement in mind.

Analysis of the dynamic current–voltage characteristics obtained for the sliding contact made it possible to establish the dependence of the voltage drop on the properties of the contact and the conditions of its formation in the form

\[ \Delta U = \Delta U_0 + K \cdot j. \tag{1} \]

The magnitude of the voltage drop \(\Delta U_0\) in expression (1), corresponding to zero current density, is due to the presence of reactance in the sliding contact; in this case the capacitive character of the contact is most strongly expressed at the maximum area of overlap of the bar by the brush. For example, with a considerable acceleration of switching, the current of the lamella reaches a zero value at the instant of time corresponding to

maximum area of overlap of the collector plate by the brush. Thus, the quantity \(\Delta U_0\) is a variable depending on the degree of commutation acceleration [4]. Therefore the voltage drop for brushes of both polarities may be represented in the form

\[ \Delta U_0=\Delta U'_0\left[1-\left(\frac{T_{\partial}}{T_{\mathrm{l}}}\right)^4\right], \tag{2} \]

where \(T_{\partial}\) is the time during which the current of the lamella reaches zero;

\(T_{\mathrm{l}}\) is the calculated time of contact of the lamella with the brush;

\(\Delta U'_0\) is the greatest value of the voltage drop in the contact at zero current density.

Applying the method of equal sums in processing the experimental data, the values \(\Delta U'_{0K}=1.5\) V were obtained for the cathode-polarized brush and \(\Delta U'_{0A}=1.1\) V for the anode-polarized brush. If the lamella current reached zero at the moment when the plate leaves the brush, \(\Delta U_0\) would be equal to zero, since in this case \(\dfrac{T_{\partial}}{T_{\mathrm{l}}}=1\); however, such correspondence is practically not observed.

The slope of the current-voltage characteristic is determined by the coefficient \(K\), which depends on the properties of the brushes and on the conditions of formation of the sliding contact. As an indicator of the conditions of contact formation, the magnitude of the rate of change of current density is adopted:

\[ \frac{j_0-j_t}{\Delta t}=\frac{dj}{dt}, \tag{3} \]

where \(j_0\) is the current density in the contact at time \(t\);

\(j_t\) is the current density in the contact at time \(t+\Delta t\);

\(\Delta t\) is the time corresponding to the calculation step in the analytical study.

Thus, the coefficient \(K\) for brushes of both polarities is determined as

\[ K=\left[C-C_1\left(\frac{j_0-j_t}{\Delta t}\right)\right]. \tag{4} \]

In the approximation, the coefficients \(C\) and \(C_1\), reflecting the properties of the brushes, were adopted; their values after the corresponding processing were determined as follows: \(C_K=0.042\), \(C_{K1}=0.013\cdot10^{-6}\) for the cathode-polarized brush and \(C_A=0.033\), \(C_{A1}=0.01\cdot10^{-6}\) for the anode-polarized brush. Then the dynamic current-voltage characteristic of the sliding contact, respectively for cathode- and anode-polarized brushes, may be written as follows:

\[ \Delta U_K=\Delta U'_{0K}\left[1-\left(\frac{T_{\partial}}{T_{\mathrm{l}}}\right)^4_K\right] +\left[C_K-C_{K1}\left(\frac{j_0-j_t}{\Delta t}\right)\right]\cdot j, \]

\[ \Delta U_A=\Delta U'_{0A}\left[1-\left(\frac{T_{\partial}}{T_{\mathrm{l}}}\right)^4_A\right] +\left[C_A-C_{A1}\left(\frac{j_0-j_t}{\Delta t}\right)\right]\cdot j. \tag{5} \]

These expressions characterize a family of dynamic current-voltage characteristics under a wide variation in the conditions of formation of the sliding contact and in the nature of commutation. This family indicates that the commutation of sections lying side by side in a slot proceeds nonuniformly, even under ideal mechanical operating conditions of the contact. The last sections in the slot have retarded commutation, whereas the preceding sections of the slot commutate at an accelerated rate. Taking this circumstance into account, it may be concluded that the conditions of formation

boundaries of contact conductivity vary over wide limits during the commutation of sections of one slot.

Figure 1 presents the dynamic current-voltage characteristics of the sliding contact for the first and last sections of one slot (solid lines), which indicate differences in the conditions of formation of the sliding contact of different commutator laminations, caused by electromagnetic factors. The same figure also presents current-voltage characteristics calculated according to the approximating expressions (5) (dashed lines); comparison of these with the experimental ones confirms the correctness of the adopted approximation of the voltage drop.

Fig. 1. Dynamic current-voltage characteristics of the sliding contact during commutation of different sections of one slot of a cathode-polarized brush: experimental — solid lines, calculated — dashed lines

In the analytical study of commutation, the armature winding is conventionally divided into a number of groups similar to one another with respect to the commutation conditions. Such a group is represented by sections located in one armature slot. According to the method for setting up differential equations set forth in [2], and in the time diagram of commutation of the EMU (Fig. 2), the calculated period, or cycle, consists of six identical intervals, the duration of each interval being equal to \(0.5T_k\), that is,

\[ T_c = 3T_k. \]

Table 1 gives the currents of the slot sections commutated simultaneously by brushes of cathode and anode polarity during the calculated period by intervals,

where \(i_1, i_2, i_3\) are the currents of sections commutated by brushes of cathode polarity,

\(i'_1, i'_2, i'_3\) are the currents of sections commutated by brushes of anode polarity.

The commutation calculation was carried out for an EMU-12A having the following data: \(P = 1.2\) kW, \(U_n = 115\) V, \(I_q = 3.4\) A, \(I_d = 10.4\) A, armature wind—

Fig. 2. Time diagram of EMU commutation

Table 1

... — simple lap, number of sections per slot $n_s = 3$, number of collector plates $K = 87$, brushes of grade EG-8, size $2 \times 5 \times 15 \times 20$ mm, brush overlap $\beta = 1.5$. The parameters of the commutated sections are as follows:

\[ \begin{aligned} L_q &= 18.4 \cdot 10^{-6}\ \text{H}, & L_d &= 27 \cdot 10^{-6}\ \text{H},\\ M_{xyq} &= M_{yzq} = 6.9 \cdot 10^{-9}\ \text{H}, & M_{xyd} &= M_{yzd} = 11.2 \cdot 10^{-6}\ \text{H},\\ M_{xzq} &= 15.64 \cdot 10^{-6}\ \text{H}, & M_{xzd} &= 23 \cdot 10^{-6}\ \text{H},\\ T_k &= 2.4 \cdot 10^{-4}\ \text{s}, & r_c &= 0.032\ \Omega . \end{aligned} \]

The system of differential equations of the circuits of sections commutated by brushes of both polarities over the calculation period will consist of 18 equations. However, at any instant during the calculation period there are no more than three simultaneously commutated sections under the brushes of both polarities; therefore, applying the cyclic method of calculation by intervals and assuming identical commutation conditions in the individual winding groups, this system can be reduced to a system of three equations with three unknowns.

This system is a system of nonhomogeneous differential equations with variable coefficients, whose solution in general form is impossible; therefore it is necessary to use approximate methods with the application of an electronic digital computer.

\[ L \frac{di_x}{dt} + M_{xy}\frac{di_y}{dt} + M_{xz}\frac{di_z}{dt} = \sum e - i_x r_c + \Delta U_2 - \Delta U_1, \]

\[ L \frac{di_y}{dt} + M_{yx}\frac{di_x}{dt} + M_{yz}\frac{di_z}{dt} = \sum e - i_y r_c + \Delta U_3 - \Delta W_4, \tag{6} \]

\[ L \frac{di_z}{dt} + M_{zy}\frac{di_y}{dt} + M_{zx}\frac{di_x}{dt} = \sum e - i_z r_c + \Delta U_1 - \Delta U_5. \]

Substitution of variables into the system of equations by intervals is performed automatically by the program in accordance with Table 1.

The alternation of sections commutated successively by brushes of cathode and anode polarity proceeds by intervals in accordance with the time diagram of commutation (Fig. 2).

The voltage drops in the contact are determined by an expression of the form:

\[ \Delta U_{Ki} = \Delta U'_{OK} \left[ 1 - \left( \frac{T_\partial}{T_\ell} \right)^4_K \right] + \left[ C_K - C_{K_i} \left( \frac{dj_i}{dt} \right) \right] j_i . \]

The expressions for the voltage drops for an anode-polarized brush are compiled analogously, using the constants \(\Delta U'_{OA}\), \(C_A\), \(C_{A1}\).

The current densities are determined from the current through the contact and the overlap area of the lamella by the brush at the given instant of time.

\[ j_1 = \frac{i_x - i_z}{S_{\text{sh}}/\beta}; \qquad j_2 = \frac{i_n - i_x}{\dfrac{S_{\text{sh}}}{\beta}\cdot \dfrac{t}{T_K}}; \qquad j_3 = \frac{i_y - i_k}{\dfrac{S_{\text{sh}}}{\beta}\left(1-\dfrac{t}{T_K}\right)}; \]

\[ j_4 = \frac{i_n - i_y}{\dfrac{S_{\text{sh}}}{\beta}\left(0.5+\dfrac{t}{T_K}\right)}; \qquad j_5 = \frac{i_z - i_k}{\dfrac{S_{\text{sh}}}{\beta}\left(0.5-\dfrac{t}{T_K}\right)}; \]

The initial values \(i_x\), \(i_y\), \(i_z\) for the first interval are specified from the condition of straight-line commutation. The solution is carried out by intervals, with the time counted from \(t=0\) in each interval. For each subsequent interval, the final conditions of the preceding interval serve as the initial conditions. The solution is performed by the method of successive iterations to refine the input data specified approximately. An M-20 electronic digital computer was used for the solution.

For the first iteration the ratio \(\dfrac{T_\partial}{T_\ell}\) is taken equal to unity. For subsequent iterations, the quantity \(T_\partial\) is determined from the calculation in accordance with Table 2 for each of the six lamellae.

As a result of the calculation, the currents of the commutated sections and the voltage drops between the brush and the collector lamella were determined in each interval.

According to Table 2, on the basis of the calculation results, curves are plotted for the voltage drops between different collector lamellae and brushes of both polarities.

Figure 3 presents the calculated (dashed lines) and experimental current curves of three commutated sections of the longitudinal EMF circuit, located in one slot, during their commutation by a cathode-polarized brush.

Comparing the calculated and experimental curves, it is necessary to note the sufficient agreement of the curves for all sections of the slot. Both the experimental and the calculated curves reveal a substantial nonidentity of the commutation cycles, caused by electromagnetic reasons.

Table 2

Fig. 3. Commutation-current curves of three phase sections: experimental — solid lines, calculated — dashed lines.

of this nature; therefore, a sufficient convergence of the results can serve as evidence of correct allowance for the conditions of formation of the contact conductance and the properties of the contact pair consisting of the brush and the commutator segment.

The root-mean-square error is

\[ m=\pm \sqrt{\frac{\sum x^2}{n}}=\pm 0.51\ \text{A}, \]

which amounts to

\[ \gamma\%=\pm \frac{m}{I_{\text{n}}}\,100=10.2\%. \]

This accuracy is quite sufficient for engineering calculations.

Figure 4 presents the curves of the voltage drop between a cathode-polarized brush and three commutator plates connected to the sections of one slot. In other words, each of the presented voltage-drop curves is directly associated with the current of the corresponding segment, which determines the form and outline of the voltage-drop curves.

Fig. 4. Curves of the contact voltage drop of three commutator plates arranged in a row; experimental—solid lines, calculated—dashed lines.

Analyzing the experimentally obtained (solid lines) and calculated voltage-drop curves, one may note their fairly accurate agreement. Comparing, in time, the current curves of the commutated sections and the voltage drop between the brush and the commutator segment connected with these sections, one may note that, when the current of the segment passes through zero, there is a definite value of the voltage drop which, both according to the experimental data and according to the calculation, is \((1.0\text{–}1.5)\) V.

The latter circumstance confirms the validity of the adopted physical model of sliding contact, according to which one of the important properties of a sliding contact is the capacitance of the contact layer of the brush—commutator segment [4].

Similar results were obtained in the commutation of sections by an anode-polarized brush. The current curves of the sections in this case have

somewhat greater acceleration, since the contact voltage drop for an anode-polarized brush is somewhat lower than for a cathode-polarized one.

Calculations carried out by the indicated method for the transverse circuit of the EMU-12A amplifier reveal sufficient agreement with the experimental curves of section currents and voltage drops for brushes of both polarities.

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

1. The proposed approximation of the contact voltage drop makes it possible to take into account the properties of the brush–commutator-bar contact pair and the conditions under which the sliding contact is formed.

2. The sufficient agreement of the calculated curves of currents and voltage drops with those obtained experimentally, both along the longitudinal and the transverse axes of the EMU-12A, confirms the validity of the adopted physical model of the sliding contact, which takes into account the capacitance of the contact layer of the brush–commutator-bar pair.

3. The indicated method of calculating commutation by individual sections, with a specified alternation of unknowns, makes it possible to halve the number of differential equations of the section circuits subject to simultaneous solution.

REFERENCES

1. E. M. Sinelnikov, A. G. Nazinyan, V. V. Kleimenov, F. I. Chernyavskii. Application of continuous-action calculating machines for the study of commutation in direct-current machines. IVUZ, Electromechanics, No. 10, 1960.

2. A. I. Skorospeshkin, B. I. Kostylev, G. G. Konstantinov. Analytical investigation of commutation in commutator electrical machines using an electronic digital computer. Izvestiya TPI, vol. 160, 1966.

3. Yu. P. Galishnikov, A. I. Skorospeshkin. Natural volt-ampere characteristics of the sliding contact of a carbon S brush—commutator. Izvestiya TPI, vol. 172, 1967.

4. A. I. Skorospeshkin, B. I. Kostylev, R. F. Bekishev. Electrical properties of a sliding contact. Izvestiya TPI, vol. 190, 1968.