ANALYTICAL EXPRESSIONS FOR THE CURRENT AND E.M.F. CURVES OF THE SHORT-CIRCUITED SECTION OF UNIVERSAL MOTORS OPERATING ON ALTERNATING CURRENT

F. A. Serdyuk

(Presented by the scientific-methodological seminar of the Electromechanical Faculty)

The nature of the commutation process is determined by the close interrelation between the physical processes occurring in the brush contact and in the commutated section during the commutation period.

To explain the processes taking place in a stationary contact, a number of theories were proposed [11, 12, 13]. The propositions set forth in them, with certain corrections, may also be extended to a sliding contact, since it differs from a stationary one in that, within it, the points of direct contact of the two surfaces are in relative motion.

There exist several hypotheses [2, 3, 10] and views [1] explaining the behavior of a sliding contact. Analysis and experimental verification by various investigators of the theories, hypotheses, and views show that, in order to explain the behavior of a sliding contact, the second and third hypotheses, taken together with certain additions, should be used.

As for the physical processes occurring in the commutated sections, a number of works [1–6] have also been devoted to elucidating this process, and in some of them a general theory of commutation is given for all types of machines.

However, experiments carried out comparatively recently by a number of investigators (M. F. Karasev and others) show that a unified theory of commutation for all types of machines is unacceptable.

Taking the foregoing into account, it may be considered that the analytical expressions for the current and e.m.f. curves of the short-circuited section of one type of machine will differ from those for another type of machine. The grade of brushes used will also influence the character of these expressions.

Universal motors possess a whole series of distinctive features. They are intended to operate on both direct and alternating current, have supersynchronous rotational speeds, are fitted with hard grades of brushes, and their individual sections possess considerable active and inductive resistances.

Undoubtedly, these features will exert a definite influence on the course of the commutation process in these motors,

and, consequently, the nature of the analytical expressions for the quantities reflecting the course of the commutation process. Usually, in deriving equations for the current and the self-induction e.m.f. of a short-circuited section, the resistance of the coils and the active resistance of the sections are neglected. Since the individual sections of universal motors have a rather large active resistance, it undoubtedly cannot be neglected; as for the resistances of the coils, in the total resistance they constitute only a small part, and therefore, for simplicity, we shall neglect them.

The nature of the variation of the current and e.m.f. curves of the short-circuited section is substantially affected by the resistances of the entering \(r_1\) and leaving \(r_2\) edges of the brush; therefore, before obtaining these equations it is necessary to determine the law of variation of these resistances. It may be considered proven [1, 7] that the smooth variation of these resistances, adopted in the classical theory of commutation [2], does not occur in reality.

E. M. Sinelnikov [9], in deriving the criterion for sparkless operation of direct-current machines, assumed that “the contact resistances do not depend on the area of contact between the brush and the commutator.”

I. S. Elokhin [1], using the results of his own experiments and the experimental results of O. G. Wegner [10] and Stine [3], came to the conclusion that the contact resistances remain constant and do not depend on the brush surface.

Yu. E. Nebolyubov [8], on the basis of experiments carried out on models reproducing alternating-current machines, also comes to the conclusion that the working surface of the brush has no substantial influence on the processes of making and breaking contact by the brush with the commutator segments.

Individual sections of motors of the universal type have a high value of active resistance, exceeding several times the value of the transient resistances of the brushes at operating current densities; therefore, for the greater part of the commutation period, they will play the principal role, and only at the very beginning and at the end of the short-circuiting of the sections will the transient resistances of the brushes exert some influence.

Taking this fact into account and using the experimental conclusions of certain authors [1], as a first approximation let us assume that the transient resistances of the brush contact are constant and equal to one another \((r_1 = r_2 = \mathrm{const})\).

The assumption of constancy and equality of the transient resistances of the entering \(r_1\) and leaving \(r_2\) edges of the brush is to some extent physically unjustified [7], but, taking into account that the values of these resistances are small in comparison with the active resistances of the sections and, in calculation, are taken from curves of their dependence on load current, they will not exert a substantial influence on the commutation equation of universal motors.

In deriving the equations of the current and e.m.f. of the short-circuited section, we shall also assume that the width of the brush is equal to the width of the commutator segment; the insulating spacer between the commutator segments is infinitely small; the commutator and brush apparatus are mechanically perfect; the capacitance between commutator segments and individual sections is small; the actual commutation process consists of two stages—initial and final, the initial stage beginning when the section is short-circuited by the brush and continuing until the beginning of the breaking of the section, while the final stage occurs through-

an extremely short time, when the short-circuited section is opened by the brush [1, 5, 8].

In this case, the equivalent circuit of a universal motor during commutation may be represented by Fig. 1.

Fig. 1.

In the figure: \(r_1\) and \(r_2\) are the transition resistances of the leading and trailing edges of the brush;

\(i_a \sin \omega (t_0 + t)\) is the current of the parallel branch of the armature winding;

\(R_c\) and \(L_c\) are the resistance and inductance of the section;

\(M\) is the mutual inductance between the field winding (F.W.) and the short-circuited section during the commutation period;

\(i\) is the current of the short-circuited section.

Contact 1 replaces the leading edge of the brush. Commutation begins at the moment when this contact closes (with contact 2 closed).

On the basis of Kirchhoff’s first law, for the nodal points 3 and 4 one may write that

\[ i_1 = i_a \sin \omega (t_0 + t) + i, \]

\[ i_2 = i_a \sin \omega (t_0 + t) - i. \]

These currents cause the following voltage drops in the contact:

\[ i_1 r_1 = [i_a \sin \omega (t_0 + t) + i] r_1, \]

\[ i_2 r_2 = [i_a \sin \omega (t_0 + t) - i] r_2. \]

In addition to these voltage drops in the short-circuited section, there will be voltage drops in the resistance of the section and in the resistance of the connecting leads,

\[ R_c i,\quad R_v i_1,\quad R_v i_2. \]

Neglecting the resistance of the connecting leads and taking into account that, in addition to the voltage drops indicated above, an emf of self-induction acts in the short-circuit circuit and inductive coupling takes place between the main motor current and the short-circuited section, we shall have

\[ [i_a \sin \omega (t_0 + t) + i] r_1 + i R_c - [i_a \sin \omega (t_0 + t) - i] r_2 = \]

\[ = - L_c \frac{di}{dt} + M \frac{d}{dt} \left\{ 2 i_a [\sin \omega (t_0 + t)] \right\}. \]

Since the current of the main circuit varies sinusoidally, then

\[ \frac{d}{dt}\{2 i_a[\sin \omega(t_0+t)]\}=2\omega i_a\cos\omega(t_0+t). \]

Denoting \(r_1+r_2+R_c=\sum r\), \(\dfrac{\sum r}{L_c}=K\), and \(\dfrac{2Mi_a}{L_c}=K_1\), and taking into account the assumptions adopted, after some transformations we obtain

\[ \frac{di}{dt}+iK=K_1\omega\cos\omega(t_0+t). \]

The equation obtained is a first-order differential equation.

As a result of solving this equation, we obtain the expression for the current of the short-circuited section

\[ \begin{aligned} i={}&-i_a\sin\omega t_0\, e^{-\frac{\sum r}{L_c}t} +\frac{2Mi_a}{L_c\left[1+\dfrac{\sum r^2}{(\omega L_c)^2}\right]} \Bigg[ \sin\omega(t_0+t) \\ &\quad +\frac{\sum r}{\omega L_c}\cos\omega(t_0+t) -e^{-\frac{\sum r}{L_c}t} \left( \sin\omega t_0+\frac{\sum r}{\omega L_c}\cos\omega t_0 \right) \Bigg]. \end{aligned} \tag{1} \]

Since the self-induction e.m.f. is \(e_s=-L_c\dfrac{di}{dt}\), the expression for the self-induction e.m.f. is

\[ \begin{aligned} e_s={}&-\sum r\, i_a\sin\omega t_0\, e^{-\frac{\sum r}{L_c}t} -\frac{2Mi_a}{\left[1+\dfrac{\sum r^2}{(\omega L_c)^2}\right]} \Bigg[ \omega\cos\omega(t_0+t) \\ &\quad -\frac{\sum r}{L_c}\sin\omega(t_0+t) +\frac{\sum r}{L_c}e^{-\frac{\sum r}{L_c}t} \left( \sin\omega t_0+\frac{\sum r}{\omega L_c}\cos\omega t_0 \right) \Bigg]. \end{aligned} \tag{2} \]

Having general expressions for the current and the self-induction e.m.f. of the short-circuited section, it is easy to find expressions for the current and the self-induction e.m.f. for various commutation phases.

By the commutation phase we shall here mean the instant of time when the commutation process begins. For example, if the commutation process begins when the instantaneous value of the motor current is equal to zero, then the expression for the current will be

\[ i= \frac{2Mi_a}{L_c\left[1+\dfrac{\sum r^2}{(\omega L_c)^2}\right]} \left( \sin\omega t+\frac{\sum r}{\omega L_c}\cos\omega t -\frac{\sum r}{\omega L_c}e^{-\frac{\sum r}{L_c}t} \right), \tag{3} \]

and the expression for the self-induction e.m.f.

\[ e_s=-\frac{2Mi_a}{\left[1+\frac{\Sigma r^2}{(\omega L_c)^2}\right]} \left(\omega\cos\omega t-\frac{\Sigma r}{L_c}\sin\omega t+ \frac{\Sigma r^2}{\omega L_c^2}e^{-\frac{\Sigma r}{L_c}t}\right), \tag{4} \]

if the commutation process begins when the instantaneous value of the motor current is equal to the amplitude, \(t_0=\frac{1}{4}T\), where \(T\) is the commutation period, then

\[ i=-i_a e^{-\frac{\Sigma r}{L_c}t} +\frac{2Mi_a}{L_c\left[1+\frac{\Sigma r^2}{(\omega L_c)^2}\right]} \left(\cos\omega t- \frac{\Sigma r}{\omega L_c}\sin\omega t-e^{-\frac{\Sigma r}{L_c}t}\right), \tag{5} \]

\[ e_s=-\Sigma r\, i_a e^{-\frac{\Sigma r}{L_c}t} +\frac{2Mi_a}{\left[1+\frac{\Sigma r^2}{(\omega L_c)^2}\right]} \left(\omega\sin\omega t+ \frac{\Sigma r}{L_c}\cos\omega t- \frac{\Sigma r}{L_c}e^{-\frac{\Sigma r}{L_c}t}\right) \quad \text{etc.} \tag{6} \]

In the case where there is no transformer coupling between the field winding and the short-circuited section, the analytical expressions for them are simplified:

\[ i=-i_a\sin\omega t_0 e^{-\frac{\Sigma r}{L_c}t}, \tag{7} \]

\[ e_s=-\Sigma r\, i_a\sin\omega t_0 e^{-\frac{\Sigma r}{L_c}t}. \tag{8} \]

Comparing the expressions for the current and the self-induction emf in the presence of transformer coupling and without it (7, 8), one can see that the first term in expression (1) is the current caused by the self-induction emf, \(i_s\), while the second is the current caused by the transformer emf, \(i_t\), that is,

\[ i_s=-i_a\sin\omega t_0 e^{-\frac{\Sigma r}{L_c}t}, \tag{9} \]

\[ i_t= -\frac{2Mi_a}{L_c\left[1+\frac{\Sigma r^2}{(\omega L_c)^2}\right]} \left[ \sin\omega(t_0+t)+\frac{\Sigma r}{\omega L_c}\cos\omega(t_0+t) - e^{-\frac{\Sigma r}{L_c}t} \left(\sin\omega t_0+\frac{\Sigma r}{\omega L_c}\cos\omega t_0\right) \right]. \tag{10} \]

Consequently, the expression for the current of the short-circuited section can be represented as the sum of two currents,

\[ i=i_s+i_t. \tag{11} \]

As a result of investigating the equations obtained, it has been established:

1. For the first phase, at $\omega t_0=0^\circ$, the current of the short-circuited section $i$ reaches its maximum value at the end of the initial stage of commutation, and then drops sharply to the value of the current of that parallel branch into which the section is switched at the end of the final stage of commutation. The maximum magnitude of this current exceeds the amplitude value of the armature current; the switching current $i_n$ is large, and the component of the current caused by the self-induction emf, $i_s$, is absent (Fig. 2).

Fig. 2.

The curve of the self-induction emf obeys the law

\[ e_s=-L_c\,\frac{di}{dt}. \]

1. For the second phase, at $\omega t_0=45^\circ$, the character of the change in the current curve and in the self-induction emf is the same, but the switching current has an insignificant magnitude—its maximum value is less than the amplitude value (Fig. 3).

2. For the third phase, at $\omega t_0=90^\circ$, the switching current has a larger magnitude, and its value is the greater, the smaller the commutation period. The current caused by the transformer emf has a negative value. The curve of the self-induction emf crosses the abscissa axis and acquires a positive value (Fig. 4).

Fig. 3.

In order to be able to judge to what extent the proposed equations for the current curve of the short-circuited section are confirmed

experimentally, a comparison was made of the calculated and experimental data obtained on the artificial apparatus for several commutation phases.

Fig. 4.

The schematic circuit diagram of the apparatus used for recording the curves of current and self-induction emf is shown in Fig. 5, where

\(B\)—a commutator having 28 bars insulated from one another by insulating spacers;

\(L_a, R_a\)—the inductance and active resistance simulating the parameters of the parallel branches;

Fig. 5.

\(OV\)—an unsaturated choke with an air gap, by varying which one can obtain different current strengths. Several test sections \(A\), with different numbers of turns and active resistances, are wound on the choke;

Fig. 6.

Fig. 7.

Fig. 8.

\(r\) is a shunt, the voltage drop across which is applied to one of the beams of the cathode oscillograph, and the current curve is recorded;

\(K_1, K_2, K_3\) are three additional contact rings with brushes \(a, b,\) and \(c\) applied to them. The commutator plates of the commutator are connected in such a way that two commutation cycles are completed during one revolution.

The inductance and active resistance of the parallel branches are taken to be many times greater than the inductance and active resistance of the section.

The commutation apparatus made it possible to oscillograph the curves of the current and the self-induction emf of the short-circuited section.

Figures 6–8 show oscillograms of the current of the short-circuited section for a UG-2 brush for various commutation phases, obtained on the apparatus under the following conditions: \(w_k = 40\) turns, \(L_c = 0.36\) mH, \(j = 6.8\ \mathrm{A/cm^2}\), \(e_l = 5.38\ \mathrm{V}\), \(T_d = 0.953 \cdot 10^{-3}\ \mathrm{s}\), \(M = 2.63\) mH, \(\Sigma r = 1.394\ \Omega\), and also the current curves obtained by calculation from the proposed formulas (3, 5) and according to the classical theory of commutation.

Comparison of the current curves obtained from the proposed equations (1, 3, 5) with the current oscillograms, and with the curves obtained by calculation according to the classical theory of commutation, shows that these equations are in good agreement with the experimental data, whereas the equations obtained according to the classical theory are not confirmed experimentally.

Conclusions

1. The classical theory of commutation is not applicable in calculating the commutation of universal motors.

2. In calculating the commutation of universal motors, one should proceed from the assumption of constancy and equality of the resistances of the brush contact.

Literature

1. Karasev M. F. Commutation of Direct-Current Machines. Gosenergoizdat, 1955.

2. Arnold E. and La-Cour. Direct-Current Machines. Gostekhizdat, 1931.

3. Neikirchen I. Carbon Brushes and the Causes of Nonconstancy of the Conditions of Commutation of Direct-Current Machines. ONTI, NKTP USSR, 1935.

4. Sosnovskaya M. S. Phenomena in the brush contact. Electricity, No. 5, 1934.

5. Karasev M. F. The nature of the brush contact of direct-current electrical machines. Electricity, No. 10, 1948.

6. Neikirch V. The Problem of Sliding Contact. 1935.

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

8. Nebolyubov Yu. E. Modeling the commutation process in single-phase alternating-current commutator motors. Electricity, No. 3, 1953.

9. Sinelnikov E. M. On the question of commutation of direct-current machines. Electricity, No. 5, 1952.

10. Vegner O. G. On certain questions of commutation by means of the commutator and brushes. Transactions of the Leningrad Industrial Institute, No. 7, 1938.

11. R. Holm. Die Technische Physik der Electrischen Contacte I, Springer, 1941.

12. P. F. Sopper, B. Sc, Ph. D. Carbon-brush Contact Phenomena in Electrical Machinery. The Proceedings of the Institution of El. Eng., vol. 96, Part II, No. 52, August, 1949.

13. W. Shalfchlin. Contact Resistance of Electric Switching Apparature, Electric Journal, 8, 1928.