COMMUTATION STABILITY OF DC MOTORS AND ITS RELATIONSHIP WITH THE SPARKLESS OPERATING ZONE

V. V. IVASHIN, I. A. MILORADOV

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

The concept of the sparkless operating zone of a machine, first introduced into the practice of adjusting DC machines by V. T. Kas’yanov [1], in the general case does not unambiguously determine the state of a machine with respect to commutation. The use of the concept of the sparkless operating zone for evaluating the commutation state is associated with such difficulties as comparing machines of different designs with one another—for example, machines with and without a compensating winding, with a full and a reduced number of interpoles. If all these machines have, for example, identical sparkless operating zones, taken by the same method and plotted in the same coordinates, this still does not mean that the machines operate identically in terms of commutation, or, more precisely, that the commutation stability of these machines is the same. In this case it is assumed that the commutation stability of a machine is determined by the ratio of the difference of the commutating voltages at the upper and lower boundaries of the sparkless operating zone to the mean value of these commutating voltages.

Between the sparkless operating zones of machines of different designs and their commutation stability there exist definite relationships, which can be calculated in designing the machine. The aim of the present article is to find these relationships and to examine their physical meaning.

Based on the concept of the sparkless operating zone, to determine its magnitude in the coordinates $\dfrac{i_{\text{p}}}{I_{\text{a}}}, I_{\text{a}}$, one may write the expression

\[ \theta = \frac{i_{\text{p}1} - (-i_{\text{p}2})}{I_{\text{a}}}, \tag{1} \]

where

\[ \begin{aligned} i_{\text{p}1}\ \text{and}\ i_{\text{p}2} &\text{— the excitation currents of the interpole winding at the upper}\\ &\text{and lower boundaries of the zone,}\\ I_{\text{a}} &\text{— the armature current of the machine.} \end{aligned} \]

Taking into account that the excitation currents are directly proportional to the commutating emfs (in the case where the pole systems are unsaturated), it is not difficult to obtain an expression for the zone of a machine with a full number of interpoles and without a compensating winding in the form [2]

\[ \theta = \frac{F_{\delta}}{F_{\delta} + F_{\text{ra}}}\cdot \frac{e_1 - e_2}{e_0}. \tag{2} \]

In (2) we have:

\(e_1, e_2, e_0\) are the commutating EMFs at the upper and lower boundaries of the zone and at its middle:

\[ e_0=\frac{e_1+e_2}{2}, \tag{3} \]

\(F_{\mathrm{ra}}\) is the magnetomotive force of the armature reaction,

\(F_{\delta}\) is the magnetomotive force required to create, in the gaps under the interpoles, a magnetic field corresponding to the rotational EMF \(e_0\).

For a machine with a compensating winding

\[ \theta_{\mathrm{ko}}= \frac{F_{\delta}}{F_{\delta}+\left(F_{\mathrm{ra}}-F_{\mathrm{ko}}\right)} \cdot \frac{e_1-e_2}{e_0}. \tag{4} \]

\(F_{\mathrm{ko}}\) is the magnetomotive force of the compensating winding.

For a machine with the number of interpoles reduced by a factor of 2,

\[ \theta_{\mathrm{p}}= \frac{2F_{\delta}\cdot k} {2F_{\delta}\cdot k+F_{\mathrm{ra}}+F'_{\mathrm{ra}}} \cdot \frac{e_1-e_2}{e_0}. \tag{5} \]

In this formula,

\(k\) is a coefficient taking into account the increase in the magnetomotive force of the interpole due to the passage of its flux through the gap of the machine’s main pole;

\(F'_{\mathrm{ra}}\) is the magnetomotive force of the interpole that creates in the gap a field equal to the field in the commutation zone without an interpole.

Comparison of formulas (2), (4), and (5) for the sparkless operating zones of machines of different designs shows that they differ only in the first coefficient; formulas (2), (4), and (5), therefore, may be written in the form

\[ \theta = k_{\mathrm{d}}\cdot k_{\mathrm{st}}, \tag{6} \]

\[ \theta_{\mathrm{ko}} = k_{\mathrm{dko}}\cdot k_{\mathrm{st}}, \tag{7} \]

\[ \theta_{\mathrm{p}} = k_{\mathrm{dp}}\cdot k_{\mathrm{st}}. \tag{8} \]

The coefficient \(k_{\mathrm{st}}\) expresses the static commutation stability of the machine, which does not depend on the design of the excitation system.

The coefficients \(k_{\mathrm{d}}, k_{\mathrm{dko}}, k_{\mathrm{dp}}\) depend on the design and affect the operation of the machine only under sharply varying operating conditions and when any sections of the machine’s magnetic circuits are saturated. These coefficients may be called dynamic, since their influence is manifested chiefly in transient conditions.

When the sparkless zone of a machine is taken in the static mode of its operation, the coefficients \(k_{\mathrm{d}}, k_{\mathrm{dko}}, k_{\mathrm{dp}}\) are scale coefficients. The coefficient \(k_{\mathrm{st}}\) unambiguously determines the static commutation stability and can be found experimentally from the value of the actual sparkless operating zone and the dynamic coefficient \(k_{\mathrm{d}}\), calculated from the values of the magnetomotive forces obtained in the electromagnetic calculation of the machine.

In order to make it possible to compare the zones of machines of different designs, their experimentally determined zones may be reduced to a single scale; that is, the zone may be recalculated to the scale of some machine chosen as the base or comparative machine.

As the comparative machine one may choose a machine of any design, but apparently it is most expedient to choose one ...

choose a machine without a compensating winding and with the full number of interpoles. Such a machine has found the widest application. The cited zone of sparkless operation for a machine with a compensating winding is

\[ \Theta_{\mathrm{PKO}}=\Theta_{\mathrm{KO}}\cdot \frac{k_{\mathrm{d}}}{k_{\mathrm{DKO}}}. \tag{9} \]

For a machine with the number of poles reduced by a factor of 2,

\[ \Theta_{\mathrm{PP}}=\Theta_{\mathrm{P}}\cdot \frac{k_{\mathrm{d}}}{k_{\mathrm{DPP}}}. \tag{10} \]

Let us consider the operating features of identical machines with different coefficients \(k_{\mathrm{d}}\), that is, machines made, for example, with different gaps in the magnetic circuit of the interpole. A large value of \(k_{\mathrm{d}}\) corresponds to a large value of the equivalent air gap. During abrupt increases or decreases of the armature current, the magnetomotive force of the interpole winding is superposed by the magnetomotive force of eddy currents induced in the magnetic circuit along the flux paths of the interpole. The magnetomotive force of the eddy currents depends only on the rate of change of the magnetic flux (for the same material of the magnetic circuit) and, consequently, will introduce the smaller disturbances into the magnetic field under the interpole, the larger the value of the machine’s dynamic coefficient.

It is known from electrical-machine engineering practice that in machines operating under transient conditions, for example in traction machines, it is expedient, in order to improve commutation, to make a second air gap between the interpole and the frame. The magnetomotive force \(F_{\delta}\) then increases correspondingly. The coefficient \(k_{\mathrm{d}}\) also increases with increasing \(F_{\delta}\), which corresponds to the increased dynamic stability of machines with enlarged gaps in the path of the interpole flux.

In machines with a compensating winding, as well as in machines with a reduced number of interpoles, the values of \(k_{\mathrm{d}}\) are greater than in machines without a compensating winding and with the full number of interpoles, which indicates their increased dynamic commutation stability.

During prolonged current overloads, when eddy currents may be neglected, machines with increased \(k_{\mathrm{d}}\) should also operate better, since saturation of the magnetic circuit in the path of the interpole flux affects the induction under the interpole the less, the larger this coefficient is.

Static commutation stability is determined by the relation

\[ k_{\mathrm{st}}=\frac{e_{1}-e_{2}}{e_{0}}. \tag{11} \]

The quantity \(e_{1}-e_{2}\) depends on many causes, including the type of machine winding, the number of armature teeth, and the commutating properties of the brushes, which in turn depend on the maximum permissible voltages across the contact resistance of the brush contact under the entering and leaving brush edges. The commutating emf \(e_{0}\) depends on the type and power of the machine, the armature current, the type of winding, the shape of the slots of the armature iron, and other factors.

It is of certain interest to analyze the nature of the change in the commutation stability of machines with various syste-

...with excitation currents, as well as their sparkless operating zones as functions of the armature current.

In the general case, for machines with either shunt or series excitation, the emf \(e_0\) is expressed by the equation

\[ e_0 = k_1 \omega I_a, \tag{12} \]

where \(\omega\) is the angular velocity of the machine, const;

\(k_1\) is the proportionality coefficient.

The quantity \(e_1 - e_2\) for each winding can be expressed in terms of the commutating ability of the brushes \(\psi_{\mathrm{br}}\), the inductance of the sections, and the mutual inductance between simultaneously commutating sections,

\[ e_1 - e_2 = 2\Delta e = f(\psi_{\mathrm{br}},\, L,\, M). \tag{13} \]

It is known that the commutating ability of brushes increases somewhat as the armature current of the machine increases. It also depends on the rotational speed of the machine. First we shall determine the change in the operating zone of the machines, assuming that the commutating ability of the brushes is constant and that it corresponds to the value \(e_1 - e_2\), equal to \(2\Delta e_0\); then, qualitatively taking into account the increase in the commutating properties of the brushes with increasing load, we shall correct the obtained curve \(k_{\mathrm{st}}\).

For a motor with shunt excitation,

\[ \omega = \frac{u - I_a \cdot R_a}{k_2 \Phi} = a_1 - a_2 I_a \tag{14} \]

and for \(k_{\mathrm{st}}\) we obtain

\[ k_{\mathrm{st}} = \frac{2\Delta e_0}{a_1 I_a - a_2 I_a^2}. \tag{15} \]

Suppose that \(R_a = 0\). Then \(a_2 = 0\), and

\[ k_{\mathrm{st}} = \frac{2\Delta e_0}{a_1 I_a}, \tag{16} \]

that is, in a motor with an absolutely rigid characteristic and with constant commutating ability of the brushes, the static commutation stability as a function of load current changes according to a hyperbola (Fig. 1, curve 1). Taking into account the active resistance \(R_a\) in the armature circuit and its increase transfers the operation of the motor to the natural and rheostatic characteristics. The static commutation stability of the machine then increases (curves 2 in Fig. 1) and can be determined from (15). When the increase in the commutating ability of the brushes with increasing load current is taken into account, the commutation-stability curve passes somewhat higher; its ordinates increase in proportion to the increase in the commutating properties of the brushes. In Fig. 1, curve 3 corresponds to the commutation stability at \(R = R_1\), taking into account \(2\Delta e = f(I_a)\). The sparkless operating zone, calculated from the ordinates of curve 3 and a dynamic coefficient of 0.3, is shown by curves 4. The real sparkless operating zones of machines with shunt excitation have outlines approximately the same as the zone bounded by curves 4 [3]. For a more accurate construction of the sparkless operating zone or of the static stability of a machine, it is necessary to know the dependence of the increase in the commutating ability of the brushes on the load current and speed.

In a series-excitation motor the excitation magnetic flux depends linearly (if saturation is not taken into account) on the armature current; therefore expression (14) for the angular velocity takes the form

\[ \omega = \frac{\beta_1}{I_a} - \beta_2, \tag{17} \]

Fig. 1. Sparkless operating zone of a shunt-wound motor and its static stability for various armature-circuit resistances.
\(1\) — \(R_a=0\),
\(2\) — \(R_a>0,\ R_3>R_2>R_1\),
\(3\) — \(R_a=R_1,\ 2\Delta e \ne \mathrm{const}\),
\(4\) — sparkless operating zone for \(\kappa_d=0.3,\ R_a=R_1\)

and the commutation stability

\[ \kappa_{\mathrm{st}}=-\frac{2\Delta e_0}{\beta_1-\beta_2 I_a}. \tag{18} \]

Study of this expression shows that, for \(R_a=0,\ \Delta e_0=\mathrm{const}\), as the armature current increases the commutation stability remains constant (Fig. 2, curve 1). Taking \(R_a\) into account and with an increase in the commutating ability of the brushes, it increases and assumes the form of a flare (Fig. 2, curves 2 and 3).

In real machines, when \(I_a \gg I_{an}\), individual sections of the magnetic circuit along the path of the excitation flux of the main poles begin to saturate; the angular velocity of the machine, with increasing armature current, decreases to a lesser degree than according to (17), and the commutation stability of the machine gradually becomes smaller. The character of its variation approaches the character of the variation in commutation stability of a shunt-wound machine (curve 3 for \(I_a>2I_{an}\)). It should be noted that at large armature currents the speed of a series-wound machine is low, which leads to a substantial influence on commutation stability of the active resistance of the sections. Owing to the influence of the active resistances of the commutating sections, the sparkless operating zone should rise upward.

At small currents the speed of the machine increases, and its commutation stability may in fact decrease to zero because of the influence on commutation of factors of a mechanical nature.

Fig. 2. Sparkless operating zone of a series-excited motor and its static stability.

1 — \(R_a=0\),
2 — \(R_a\ne 0,\ 2\Delta e=\mathrm{const}\),
3 — \(R_a\ne 0,\ 2\Delta e\ne \mathrm{const}\),
4 — sparkless operating zone for \(k_d=0.3,\ R_a\ne 0\).

The influence of the active resistance of the sections and of factors of a mechanical nature is not taken into account in the case considered.

Conclusions

1. The sparkless zone of a machine does not unambiguously determine its operation with respect to commutation, and by its magnitude one cannot compare the commutation stability of machines with different designs of excitation systems.

2. The static commutation stability of a machine can be evaluated by the ratio of the difference between the commutating voltages at the upper and lower boundaries of the sparkless operating zone to their mean value.

3. The nature of the change in the sparkless operating zone of series- and shunt-excited motors is well explained by introducing the concept of static commutation stability.

References

1. V. T. Kasyanov. Regulation of commutating poles of direct-current machines. Elektrichestvo, 1934, No. 20; 1935, No. 1.

2. V. V. Ivashin, I. A. Miloradov. Investigations of sparkless operating zones of a direct-current motor with brush-valve commutation. Izvestiya TPI, vol. 172, 1967.

3. M. F. Karasev, V. P. Suvorov et al. Features of adjusting the commutation of traction machines. Scientific Works of OMIIT, issue 54, 1965.