On the Issue of Commutation in Direct-Current Machines

F. A. Serdyuk, A. I. Skorospelkin

(Presented by the scientific seminar of the Electromechanical Faculty)

The complexity of the commutation process makes experimental and theoretical investigations difficult.

At present there is no theory of commutation that would, with sufficient completeness, encompass the entire commutation process and be valid for any types of machines and any operating conditions.

In the analytical study of the commutation process, the chief difficulty and complexity lies in the correct determination of the resistances of the entering and leaving edges of the brush, \(r_1\) and \(r_2\), which enter into the equation of the commutated circuit shown in Fig. 1.

Fig. 1. Diagram of the commutated circuit.

The equation is written in the following form:

\[ L_c \frac{di}{dt} + i r_c + i_2 r_2 - i_1 r_1 = \pm e_k . \tag{1} \]

In equation (1) and in Fig. 1 the following notation is used:
\(L_c\)—inductance of the section;
\(r_c\)—active resistance;
\(i_a\)—current of the parallel branch;

\[ \begin{aligned} i&\text{—current of the section undergoing commutation;}\\ i_1&=i_a-i\text{—current corresponding to the entering part of the brush;}\\ i_2&=i_a+i\text{—current corresponding to the leaving part of the brush;}\\ t&\text{—time from the beginning of commutation;}\\ T_k&\text{—period of commutation;}\\ r_1 \text{ and } r_2&\text{—resistances of the entering and leaving parts of the brush;}\\ e_k&\text{—commutating e.m.f.} \end{aligned} \]

Different authors, in studying the process of commutation, adopted certain assumptions on the basis of which they carried out the calculation of commutation.

The founder of the classical theory, Arnold [1], proceeds from the assumption that the current in the brush contact is distributed uniformly over the entire surface of the brush. In this case the resistance \(r_{\mathrm{sh}}\), corresponding to the full width of the brush, remains unchanged, while the resistances of the entering and leaving parts of the brush vary inversely in proportion to the contact surface.

Proceeding from this assumption, Arnold obtained expressions for the current curves of the section undergoing commutation and carried out extensive studies of the commutation process.

The classical theory found the greatest application in calculating commutation both in large and in small machines. It made it possible to carry out calculation and adjustment of commutation, explained rather fully the role of interpoles and, in part, the causes of sparking at the brushes.

Nevertheless, the propositions of the classical theory began to be disputed by many investigators, which led to the emergence of new views on commutation.

O. G. Wegner [2,3], on the basis of investigations carried out, comes to the conclusion that the propositions of the classical theory do not correspond to reality and, instead, proposes a new one based on the assumption that the transient voltage drop in the brush contact remains constant, i.e. \(\Delta u=\mathrm{const}\).

Proceeding from this assumption, he obtains an equation for the current of the section undergoing commutation and proposes, as the most expedient for commutation, a curve with a small-current step (Fig. 2).

Fig. 2. Current curve.

The absence of break current at the completion of the commutation process is a positive factor. The operating conditions of the leaving edge of the brush are thereby considerably facilitated.

But the main difficulty consists in obtaining the degree of small current. For this, an appropriate distribution of the commutating field and of the winding is necessary, and a definite grade of brushes is required.

In some cases, apparently, it is simpler to provide a certain degree of acceleration of the commutation process and thereby to facilitate the operation of the trailing edge of the brush. In doing so one may proceed from the propositions of the classical theory.

The studies carried out by M. F. Karasev [4] are distinctive. The investigations were conducted on artificial apparatus reproducing the processes that take place in a real machine.

On the basis of the investigations, the author [4] comes to the conclusion that the real commutation process may be represented as consisting of the short-circuiting of a section, to which there corresponds a drop of the current to zero, and then the opening of the section, associated with the establishment in the section of a current of the opposite direction (Fig. 3).

Fig. 3. Current curve.

Section 1 in the figure corresponds to the short-circuiting of the section, section 2 to its opening.

The horizontal section 3 determines the operation of the leading and trailing edges of the brush contact. Here, moving it upward improves the operation of the leading edge of the brush, while moving it downward improves that of the trailing edge. Its most rational position can be obtained by adjusting commutation.

In this connection, the role of additional poles is explained in a new way: it is reduced to distributing the current between the trailing and leading parts of the brushes, and not to compensating the reactive e.m.f.

However, it should be borne in mind that division of the current curve of the commutated section into separate portions is not always possible. Such division is certainly easier to carry out for curves obtained on an artificial apparatus.

I. S. Elokhin [5] obtains an equation for the current curve of the commutated section on the assumption that the resistances of the leading and trailing edges of the brush contact are equal and remain constant, i.e. \(r_1 = r_2 = \mathrm{const}\).

Equation [5] is recommended for machines of low power, where comparatively large inductances and total resistances occur. It should be noted that this equation is applicable only in the special case of a definite combination of parameters.

Equation [5] differs from [2] by the introduction of the resistance of the brush contact. Proceeding from this, the author [5] comes to the conclusion that equation [2] does not take into account the properties of the brush contact.

Such a conclusion is not entirely correct.

The assumption \(\Delta u=\mathrm{const}\) corresponds to a hyperbolic dependence of the transient resistance of the brush contact on current density. With such a dependence, the resistances of the entering and leaving edges of the brush contact, being approximately identical, mutually compensate one another, and the process of current variation in the section is determined only by its parameters. Therefore the resistances of the entering and leaving edges of the brush contact are not included in equation [2].

In addition, the author of [5], on the basis of his investigations, arrives at the conclusion that the classical theory is unrealistic.

In reality, under certain conditions, when good contact is present (meaning the seating-in of the brushes and minimal vibrations), equations [1] are applicable and provide greater accuracy than [2 and 5].

Attention is drawn to the sequence in the application of equations [1, 2, 5] established by V. A. Faleev [6].

It should be noted that it is difficult to be guided by it, since equations [1, 2, 5] cannot be placed in any sequential order by taking as a guide only the increase of \(r_c dI_c\). These equations are obtained from very different assumptions, and each of them can be valid only under certain conditions, which are determined by the type of machine, its operating mode, rotational speed, the parameters and grade of the brushes.

Therefore, knowing only \(r_c\) and \(L_c\), it is difficult to say anything about the application of equations [2 and 5].

Nor should the influence of temperature on the character of the section-current curve in isolation from the current load be considered.

Indeed, by changing the temperature it is possible to modify the character of the current curve as well. But it must be taken into account that temperature is a factor accompanying the presence of a certain current density and also depends on cooling conditions.

In this respect the work of Günter Hilgarth [7] is characteristic. In it formulas are derived that establish the relationship between resistance, temperature, and current load at the contacts under various conditions.

As an example, below is given one of the formulas obtained by the author of [7], in which temperature and current load are related. The formula is written in the following form:

\[ \Theta_r=\frac{1}{\alpha}\left[\exp\left(\frac{I^2\rho_0\alpha}{4\pi^2 r^4 C_0}\,t\right)-1\right]. \tag{1} \]

Here \(\alpha\) — temperature coefficient;

\[ \rho_0 \]
— specific contact resistance at ambient temperature;

\[ C_0 \]
— specific heat capacity;

\[ t \]
— time;

\[ r \]
— distance to various places of the contact;

\[ \Theta_r \]
— temperature.

Accordingly, for the resistance the formula has the form:

\[ r_k=\frac{\rho_0}{\pi}\int_b^\infty \frac{\exp\left(\dfrac{m}{r^4}\right)}{r^2}\,dr, \tag{2} \]

where

\[ m=\frac{I^2\rho_0zt}{4\pi^2 C_0}. \]

Formula (2) establishes the relationship between the contact resistance and the current load.

V. A. Faleev [6] investigated the final stage of commutation. The latter is the result of the change in current over the entire preceding period.

Proceeding from the condition that, during its existence, the electric arc forms with the section a parallel circuit and turns out to be shunted by the section, the author [6] constructs an equivalent circuit for the final stage of commutation and obtains expressions for the current of the breaking arc, the arc-burning time, and the energy released.

In setting up the equations for the equivalent circuit, the capacitance between the commutator segments and the mutual-inductive coupling between the individual sections and the section with the parallel branch of the armature winding, which in fact exist, are not taken into account.

In Holm’s work [8], which likewise deals with the study of the final stage of commutation, the mutual inductance between the section, the parallel branch of the winding, and the winding of the interpoles is taken into account.

Figure 4 presents the circuit adopted in [8].

Fig. 4. Equivalent circuit for the final stage of commutation.

From Fig. 4 it is clear that, with respect to the arc current arising between the commutator segment and the brush that has left it, both the section and the rest of the winding are in the same relation. In them there will be induced e.m.f.’s associated with the change in current, determined by the parameters of these parts of the windings.

The author [8] determines, in the same way as in [6], the breaking-arc current, the arc-burning time, and the energy released in the process. But the resulta-

[8], obviously, will be more accurate, since they take into account a larger number of parameters.

Thus, having expressions for the arc current at the final stage of commutation, the current of the commutated section can be represented more completely in the form of two components \(i_c=i_{pr}+i_d\), where \(i_{pr}\) is the current in the section up to the final stage of commutation, and \(i_d\) is the arc current.

At the same time, for a more correct determination of the breaking arc current, it is certainly necessary to determine the current at the preceding stage of commutation with sufficient accuracy.

The analysis carried out of the equations [1, 2, 5] for the current of the commutated section reveals their essential shortcomings.

Consequently, there is a need to seek new methods of analytical investigation of the commutation process, which could take into account, with sufficient accuracy and more completely, the physical aspect of the commutation process.

One such possible method is an investigation that takes into account the dependence of the transient resistance of the brush contact on the current density.

Indeed, if there is a steady-state operating mode of the machine, which is determined by the presence of a constant rotational speed, a steady collector temperature, and a constant pressure on the brush, assuming good contact, then the variation of the contact resistance as a function of current density is almost the only factor that substantially influences the character of the current curve of the commutated section.

In this connection, in the work of V. A. Faleev [6] it is noted that “the use in the equations of volt–ampere characteristics and experimentally obtained dependences \(r_{sh}=f(j)\) is the most reliable way to obtain commutation equations free from all assumptions and suppositions. This direction must be developed further.”

The first attempt to carry out an investigation of the commutation process on the basis of static volt–ampere characteristics is the work of B. K. Tur [9].

In the differential equation compiled for the scheme adopted by him, the dependence of the transient voltage drop on the current density and the capacitance between the collector plates are taken into account.

In the calculation, the influence of the transient voltage drop is determined through the capacitance current. Consequently, the determining influence on the character of the change of the current in the commutated section will be exerted not by the transient voltage drop, but by the value of the capacitance adopted in the calculation. This introduces an error into the calculations and conclusions made by B. K. Tur.

It is known that the transient resistance of the brush contact (and, correspondingly, the transient voltage drop) exerts a more substantial influence on the character of the change of the current in the section than the capacitance, especially for the volt–ampere characteristic adopted by B. K. Tur.

Moreover, approximation of the volt–ampere characteristic in the form of a straight line, beginning from \(8\) to \(8080\ \mathrm{a/cm^2}\), is unrealistic [10].

Furthermore, experimental confirmation of the calculation is absent.

All these shortcomings, despite the positive fact of taking into account the dependence \(\Delta u=f(j)\), called into question the results obtained by the author [9].

It must be borne in mind that, when using the dependences \(\Delta u=f(j)\) or \(r_{\mathrm{sh}}=f(j)\), one must know the conditions under which these dependences were obtained. The principal one of these conditions is temperature. The closer the dependence \(r_{\mathrm{sh}}=f(j)\) corresponds to the actual conditions, the more accurate and correct will be the result of solving the differential equation drawn up for the commutated circuit.

The question of which of the dependences \(r_{\mathrm{sh}}=f(j)\) should be used—static or dynamic—is a subject for research.

One of the unsuccessful attempts to use dynamic volt-ampere characteristics for obtaining the current curve of the commutated section is the work of Z. B. Vartanov [11].

The very fact of taking dynamic volt-ampere characteristics into account in order to determine the current curve of a section is positive. But the approach to solving this question and the results cannot fail to raise objections.

First, if in static volt-ampere characteristics a certain instability is observed in connection with the influence of various factors, then for dynamic characteristics this instability has no bounds.

Therefore, adopting any mathematical approximation for the most diverse dynamic volt-ampere characteristics seems almost impossible.

Above all, it is necessary to find methods for obtaining stable dynamic characteristics corresponding to the real physical process taking place during commutation.

Second, the method by which these characteristics were obtained by the author of [11] is too far removed from the real conditions of the machine.

The imitation of the operation of the trailing edge of the brush and of the edge running onto the collector, and the choice of a triangular impulse determining the dynamic characteristic, correspond poorly to real conditions.

Finally, the method of obtaining the equation for the current of the commutated section contains rather crude approximations.

Thus, the results of [11] cannot be applied in calculating commutation.

Under present conditions it appears possible to conduct an investigation of commutation taking into account the static dependences of the transition resistance of the brush contact on current density.

The basis for this may be considered to be that

1) they most fully take into account the properties of the brush contact;

2) they have a clearly expressed regular character, which can be mathematically approximated;

3) the static characteristics are stable under definite conditions (pressure, temperature).

Such investigations, as applied to electrical-machine amplifiers with a transverse field, have been carried out in works [12 and 13]. In this connection it is found that the dependence of the transition resistance of the brush contact on current density has a substantial influence on the character of the current curve of the commutated section.

This dependence should be taken into account when investigating the commutation process in other direct-current machines as well.

Comparison of the current curves obtained according to [1, 2, 5] and with allowance for \(r_{\mathrm{sh}}=f(j)\) [12, 13] with experimental ones shows that the curves [12, 13] have the greatest similarity to the experimental curves.

On the basis of the foregoing, the following conclusions may be drawn.

1. One of the possible paths for investigating the commutation process in direct-current machines is investigation with allowance for the dependence of the transition resistance of the brush contact on dens-

current density, since this dependence most fully takes into account the properties of the brush contact.

The use of the dependence \(r_{\mathrm{sh}} = f(j)\) makes it possible to conduct more in-depth studies of commutation (for example, taking mutual inductance into account). In doing so, it is necessary that the dependence \(r_{\mathrm{sh}} = f(j)\) also correspond to real conditions.

1. Equations [1, 2, 5] are applicable only under definite conditions.

REFERENCES

1. Arnold la Cour. Direct-Current Machines, vol. I, 1931.

2. O. G. Vetner. On certain questions of current commutation with the aid of the commutator and brushes. Proceedings of LMI, No. 7, 1938.

3. O. G. Vetner. Questions of the modern theory of current commutation in commutator machines. Electricity, No. 7, 1956.

4. M. F. Karasev. Commutation of Direct-Current Machines. GEI, 1955.

5. I. S. Elokhin. Investigation of the curves of current and self-induction emf during commutation. Dissertation, Tomsk, 1953.

6. V. A. Faleev. Investigation of the sparking process during commutation in electrical machines. Dissertation, Tomsk, 1957.

7. Gunter Hilgarth. The relation between contact voltage and temperature at contacts under briefly loaded high-current contacts. ETZ, No. 13, 1958.

8. On the theory of commutation of direct-current machines. Express Information, issue 25 (No. 143–147), 1959.

9. B. K. Tur. On the question of commutation in direct-current machines. Electricity, No. 11, 1956.

10. M. F. Karasev. Equations of the current and emf curves during commutation in commutator machines. Electricity, No. 12, 1957.

11. E. B. Vartanov. Impulse volt-ampere characteristics of brush contact. VESh, No. 2, 1957.

12. D. I. Korolevsky. On the commutation of low-power direct-current machines. Proceedings of Higher Educational Institutions: Electromechanics, No. 2, 1959.

13. A. I. Skorospeshkii. On the commutation of electromechanical amplifiers with a transverse field. Proceedings of Higher Educational Institutions: Electromechanics, No. 6, 1959.