Yu. P. Galishnikov, A. I. Skorospeleshkin

On the Calculation of Commutation and Commutation Stability of DC Machines on a Computer

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

The main purpose of calculating commutation in a DC machine is, as is known, to determine the reactive emf, on the basis of which the parameters of the commutating poles are determined. However, an equally important problem is the calculation of the commutation stability of the machine at the design stage. The significance of such a calculation can hardly be overestimated, especially in designing highly utilized machines. Nevertheless, the methods of refined calculation proposed up to the present time have not found wide practical application, primarily because the assumptions adopted concerning the behavior of the brush contact during commutation do not have a sufficiently deep physical justification and, as a consequence, lead to considerable discrepancies between calculated and experimental data.

One of the first attempts at the practical use of a computer for calculating commutation in DC machines was presented in the work of Alger and Buly [1]. As the principal criterion for assessing the quality of commutation, the value of the interlamellar voltage is adopted there, which in itself appears correct. However, the calculation of this voltage is carried out under the assumption of constancy of the contact resistance. The latter contradicts the results of experimental investigation of the characteristics of sliding contact and, in particular, the data presented by the authors in a preceding article of the present collection.

There is also the known work [2], devoted to the principles of modeling the commutation process of DC machines on an analog machine. In the modeling, the static current-voltage characteristic of the brush contact was used, although, as noted in [4], such a characteristic describes the sliding contact only under a uniform distribution of current over the face surface of the brush, which occurs only with rectilinear commutation, whereas in practice the overwhelming majority of DC machines operate in an overcommutation mode.

Finally, one of the most recent works in terms of its appearance is that of O. G. Wegner [3]. The calculation of the zone of sparkless operation is given there on the basis of the author’s basic assumption \(\Delta U_{\text{щ}} = \text{const}\). It should be noted that the convenience and clarity of the author’s method of approximating the dynamic characteristics of the sliding contact by no means compensate for the probable error of the calculations, since what is introduced into the calculation is, in essence, a partial current-voltage characteristic.

Common in the practice of calculations and modeling of commutation on a computer is the use of some current-voltage characteristic,

obtained by averaging the characteristics of the anode- and cathode-polarized brushes, which is far from correct. In fact, analysis of the experimental material for EG-43 brushes shows that the discrepancy between the voltages corresponding to the anode and cathode branches reaches relative values of 1.5 and higher.

It is therefore clear that averaging characteristics, as well as using static dynamic volt-ampere characteristics obtained under arbitrary current conditions, sharply reduces the accuracy of the calculations being performed. The potential capabilities of computers in this respect remain unrealized, and the calculations themselves cannot claim to give an exact quantitative solution. On the other hand, a certain complication of the calculation when using the true contact characteristics, caused by the complexity of the process under investigation itself, is not an insurmountable obstacle at the present level of computing technology.

The method proposed in the present article for calculating commutation and commutation stability of DC machines is based on the use of a family of particular natural volt-ampere characteristics of the brush contact, obtained experimentally over a wide range of changes in the conditions of contact formation and processed accordingly. These characteristics are considered in greater detail in the preceding article. The computational version of the expression for the generalized volt-ampere characteristic has the following form:

\[ U=\left(C_1-C_2\frac{j_0-j_{t_k}}{t_k}\right)\cdot j^m, \tag{1} \]

where \(C_1\) and \(C_2\) are constant coefficients,

\(j_0\) is the current density in the contact at the beginning of contact,

\(j_{t_k}\) is the current density \(t_k\) μsec after the beginning of contact,

\(t_k\) is the commutator period.

The corresponding graphs for

\[ U_0=C_1-C_2\frac{j_0-j_{t_k}}{t_k} \tag{2} \]

are given in Fig. 1. The coefficient \(m\) in expression (1) for anode- and cathode-polarized brushes is equal, respectively, to 0.3 and 0.4.

Fig. 1. Dependence \(U_0=U_0\left(\dfrac{j_0-j_{t_k}}{t_k}\right)\) for EG-43 brushes

As an example, let us consider the construction of the differential equations of the commutated sections for a low-power 2-pole DC motor. The armature winding is a simple lap winding with a diametral pitch. In one real slot there are 4 elementary slots. The number of commutator bars simultaneously covered by a brush is 3.

Table 1 gives the currents of the sections commutated simultaneously by the anode and cathode brushes.

Table 1

The indices 1, 2, 3, 4 refer to the sections commutated by the anode brush.

The indices \(1'\), \(2'\), \(3'\), \(4'\) refer to the sections commutated by the cathode brush.

The system of differential equations of the circuits commutated by brushes of both polarities over the period from 0 to \(6t_k\) has the form:

\[ L\frac{di_1}{dt} + M_{nn}\left(\frac{di_2}{dt}+\frac{di_3}{dt}\right) + M_{vn}\left(\frac{di_1'}{dt}+\frac{di_2'}{dt}+\frac{di_3'}{dt}\right) = \]

\[ = e_{k1}(t)-R_c i_1-\Delta U_2+\Delta U_1. \]

\[ L\frac{di_2}{dt} + M_{nn}\left(\frac{di_1}{dt}+\frac{di_3}{dt}+\frac{di_4}{dt}\right) + M_{vn}\frac{di_1'}{dt} +\frac{di_2'}{dt} +\frac{di_3'}{dt} +\frac{di_4'}{dt} = \]

\[ = e_{k2}(t)-R_c i_2-\Delta U_3+\Delta U_2. \]

\[ L\frac{di_3}{dt} + M_{nn}\left(\frac{di_1}{dt}+\frac{di_2}{dt}+\frac{di_4}{dt}\right) + M_{vn}\left(\frac{di_1'}{dt}+\frac{di_2'}{dt}+\frac{di_3'}{dt}+\frac{di_4'}{dt}\right) = \]

\[ = e_{k3}(t)-R_c i_3-\Delta U_4+\Delta U_3. \]

\[ L\frac{di_4}{dt} + M_{nn}\left(\frac{di_2}{dt}+\frac{di_3}{dt}\right) + M_{vn}\left(\frac{di_2'}{dt}+\frac{di_3'}{dt}+\frac{di_4'}{dt}\right) = \]

\[ = e_{k4}(t)-R_c i_4-\Delta U_5+\Delta U_4. \tag{3} \]

\[ L\frac{di_1'}{dt} + M_{nn}\left(\frac{di_2'}{dt}+\frac{di_3'}{dt}\right) + M_{vn}\left(\frac{di_1}{dt}+\frac{di_2}{dt}+\frac{di_3}{dt}\right) = \]

\[ e_{k1}'(t)-R_c i_1'-\Delta U_2' + \Delta U_1' \]

\[ L\frac{di_2'}{dt} + M_{vn}\left(\frac{di_1'}{dt}+\frac{di_3'}{dt}+\frac{di_4'}{dt}\right) + M_{vn}\left(\frac{di_1}{dt}+\frac{di_2}{dt} +\frac{di_3}{dt}+\frac{di_4}{dt}\right) = e_{k1}'(t)-R_c i_2'-\Delta U_3' + \Delta U_2' \]

\[ L\frac{di_3'}{dt} + M_{nn}\left(\frac{di_1'}{dt}+\frac{di_2'}{dt}+\frac{di_4'}{dt}\right) + M_{vn}\left(\frac{di_1}{dt}+\frac{di_2}{dt}+\right. \]

\[ + \frac{di_3}{dt} + \frac{di_4}{dt} \biggr) = e_{k3}'(t) - R_c \cdot i_3' - \Delta U_4' + \Delta U_3' \]

\[ L \frac{di_4'}{dt} + M_{\mathrm{nn}}\left(\frac{di_1'}{dt} + \frac{di_3'}{dt}\right) + M_{\mathrm{vn}}\left(\frac{di_2}{dt} + \frac{di_3}{dt} + \frac{di_4}{dt}\right) = \]

\[ = e_{k4}'(t) - R_c \cdot i_4' - \Delta U_5' + \Delta U_4' \]

For a wave winding, without great error one may take
\(L_1 = L_2 = L_3 = L_4 = L_1' = L_2' = L_3' = L_4' = L\). The coefficient \(M_{\mathrm{nn}}\) characterizes the mutual-inductive coupling between sections whose sides are located in the same layers of the slot. In this case the empirical value is \(M_{\mathrm{nn}} = 0.7\text{--}0.8L\). For sections whose sides lie in different layers of the slot (sections commutated by brushes of different polarity), \(M_{\mathrm{vn}} = 0.3\text{--}0.4L\). With the cyclic method of calculation, the system of equations (3) can easily be reduced to 6 equations with the corresponding number of unknown quantities.

As is known, the commutation of the machine as a whole is determined by the behavior of the independently commutated armature sections. According to the definition [6], sections are called independent if they finish commutating in the absence of short-circuited circuits that have a close magnetic coupling with the former. For a simple lap winding with diametral pitch, each last section of the slot is independent, since its magnetic coupling with the sections of the adjacent slot that are short-circuited during this period of time is very weak. The independently commutated armature sections are highly critical with respect to the magnitude and distribution of the interpole field. Thus, to determine the commutation stability of the machine, it is necessary and sufficient to evaluate the behavior of each fourth section of the slot.

In accordance with this, as a criterion for evaluating the quality of commutation, the authors adopted the voltage drop between the brush and the commutator bar leaving the state of contact with the brush when the last section of the slot is broken. In equations (3), this quantity for the anode and cathode brushes is denoted as \(\Delta U_4\) and \(\Delta U_4'\).

Analysis of oscillograms of the contact voltage drop shows that sparking of \(1\frac{1}{4}\text{--}1\frac{1}{2}\) points corresponds to a voltage under the trailing part of the brush of approximately 4 V. A similar value of the sparking voltage, as a weak form of an arc, is given in [5], a work devoted to the experimental study of sparking and arc formation in commutator machines. Thus, a contact voltage of 4 V can, with sufficient justification, characterize the boundaries of the sparkless operating zone of the machine. The condition of optimum commutation in this case is equality to zero of the potential (relative to the brush) of the commutator bar connected with the last section of the slot and leaving the state of contact with the brush at the moment when commutation of this section is completed.

In summary, it should be noted that in the present work a new method is proposed for the refined calculation of commutation and commutation stability of DC machines, based on the use of the natural volt-ampere characteristics of the brush contact, obtained experimentally for a wide range of variation of commutation conditions and represented in the form of analytical expressions that separately describe families of particular curves for anode- and cathode-polarized brushes.

References

1. I. Alger, D. Bewley. Analysis of commutation in DC machines, Power Appar. and Syst., 31, 1957.

2. V. M. Sinelnikov, A. G. Nazikyan, V. V. Kleimenov, V. I. Chernyavskii. Application of continuous-operation calculating machines for the study of commutation in DC machines. Izvestiya vuzov, “Elektromekhanika,” 1960, No. 10.

3. O. G. Wegner. Calculation of the commutation process and of the width of the sparkless operating zone of DC machines with the aid of an electronic digital computer. Izvestiya vuzov, “Elektromekhanika,” 1966, 4.

4. Yu. P. Galushnikov, A. I. Skorospeshkin. On the question of instability of the dynamic volt–ampere characteristics of the sliding contact carbon brush—collector. Izvestiya TPI, vol. 160, 1967.

5. B. Swinnerton, M. Turner, I. Thompson. Arc formation in the transition layer between graphite and copper, Conf. Commut. Rotat. Mach., London, 1964.

6. L. Dreyfus. Commutation of large DC machines. Theory of commutation disturbances, Acta Polytechnica, 1954.