NONLINEARITY OF THE ELECTRICAL PARAMETERS OF SMALL DIRECT-CURRENT MACHINES

V. M. ARSHINSKII, A. A. LUKONIN, V. I. KUZ’MIN, V. L. KIRSHNER

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

In automatic-control systems, small direct-current machines are widely used, performing the functions of active elements of various kinds of actuating mechanisms. As an example one may cite the actuating mechanism of a flat controller, driven by a direct-current machine with an independent field winding of type SL-621, with a power of 172 kW.

In designing automatic-control circuits in which small direct-current machines are used, knowledge of their dynamic properties is required. If the supply of the field winding is maintained during the transient process, then one usually confines oneself only to taking into account the electromechanical time constant of the actuating mechanism, i.e., describes its behavior by a linear differential equation of the first order. The complexity of the problem increases when an attempt is made to take into account the inductance of the armature winding. In the most complete version of the problem, with variable excitation, the entire transient process is considered as the aggregate of the electromechanical transient process, the transient process in the armature winding, and the transient process in the field winding.

If the armature reaction is neglected, then it may be assumed that the transient processes in the field winding and in the armature winding proceed independently. But such an assumption is permissible only for large machines. In small motors, however, there are no additional poles and their brushes are shifted from the neutral. As a result, any changes in the current of one of the windings will cause the appearance of an emf of mutual induction in the other winding.

Thus, for the armature winding and the field winding we have, respectively:

\[ U_{\mathrm{a}} = R_{\mathrm{a}} i_{\mathrm{a}} + L_{\mathrm{a}} \frac{d i_{\mathrm{a}}}{dt} + M \frac{d i_{\mathrm{f}}}{dt} + e_q \tag{1} \]

\[ U_{\mathrm{f}} = R_{\mathrm{f}} i_{\mathrm{f}} + L_{\mathrm{f}} \frac{d i_{\mathrm{f}}}{dt} + M \frac{d i_{\mathrm{a}}}{dt}. \tag{2} \]

Here

\(U_{\mathrm{a}}; U_{\mathrm{f}}\) — voltages applied to the armature winding and to the field winding;

\(i_{\mathrm{a}}; i_{\mathrm{f}}\) — winding currents;

\(R_{\mathrm{a}}; R_{\mathrm{f}}\) — winding resistances;

\(L_{\ya};\ L_{\v}\) are the differential inductances of the windings;
\(M\) is the differential mutual inductance;
\(e_{\d}\) is the counter-emf of the rotating machine.

Equations (1) and (2) are typically nonlinear differential equations, since both the inductances and the mutual inductance depend primarily on the degree of saturation of the magnetic circuit.

A change in magnetic flux over time is accompanied by the appearance of eddy currents in the massive parts of the magnetic circuit. The influence of eddy currents on the machine winding can formally be taken into account within expressions (1) and (2), if the electrical parameters of the windings are assumed to depend on the rate of change of the magnetic flux, or, in other words, on the rate at which the transient process proceeds.

Thus, in the general case

\[ \begin{aligned} R_{\ya} &= R_{\ya}\left(i_{\ya}, i_{\v}, \frac{di_{\ya}}{dt}, \frac{di_{\v}}{dt}\right)\\ L_{\ya} &= L_{\ya}\left(i_{\ya}, i_{\v}, \frac{di_{\ya}}{dt}, \frac{di_{\v}}{dt}\right)\\ R_{\v} &= R_{\v}\left(i_{\ya}, i_{\v}, \frac{di_{\ya}}{dt}, \frac{di_{\v}}{dt}\right)\\ L_{\v} &= L_{\v}\left(i_{\ya}, i_{\v}, \frac{di_{\ya}}{dt}, \frac{di_{\v}}{dt}\right)\\ M &= M\left(i_{\ya}, i_{\v}, \frac{di_{\ya}}{dt}, \frac{di_{\v}}{dt}\right). \end{aligned} \tag{3} \]

In order to determine the dependences (3), we carried out two series of experiments.

In the first of them, the task was set: under the assumption that \(R_{\ya}\) and \(R_{\v}\) are constant, to establish the functional relationships

\[ \begin{aligned} L_{\ya} &= f_1(i_{\ya}, i_{\v})\\ L_{\v} &= f_2(i_{\ya}, i_{\v})\\ M &= f_3(i_{\ya}, i_{\v}). \end{aligned} \tag{4} \]

To solve the problem, with the machine stationary, oscillograms were recorded of transient processes caused by: 1) short-circuiting the field winding while maintaining power to the armature winding; 2) short-circuiting the armature winding while maintaining excitation.

In the initial modes, various \(I_{\ya 0}\) and \(I_{\v 0}\) were specified. The nature of the oscillograms is shown in Fig. 1. Let us consider the oscillogram of Fig. 1, a.

At the times \(t_1\) and \(t_2\),

\[ \frac{di_{\ya}}{dt}=0. \]

Moreover, for \(t>t_3\), practically

\[ \frac{di_{\ya}}{dt}\simeq 0. \]

Then from (1), taking \(e_{\d}=0\), and (2), for the moments \(t=t_1\), \(t=t_2\), and \(t \gg t_3\), we may write:

\[ M=\frac{U_{\ya}-R_{\ya}i_{\ya}}{-\dfrac{di_{\v}}{dt}} \tag{5} \]

\[ L_{\mathrm{в}}=-\frac{R_{\mathrm{в}} i_{\mathrm{в}}}{\dfrac{di_{\mathrm{в}}}{dt}} \tag{6} \]

Fig. 1. Oscillograms of transient processes:
a) when short-circuiting the field winding;
b) when short-circuiting the armature winding.

Similarly, from the oscillogram in Fig. 1,b it follows that at the instants \(t=t_4;\ t=t_5\) and \(t>t_6\), \(\dfrac{di_{\mathrm{в}}}{dt}\) is either equal to zero or close to it. Therefore, from (1) and (2) we shall have

\[ L_{\mathrm{я}}=-\frac{R_{\mathrm{я}} i_{\mathrm{я}}}{\dfrac{di_{\mathrm{я}}}{dt}} \tag{7} \]

\[ M=\frac{U_{\mathrm{в}}-R_{\mathrm{в}} i_{\mathrm{в}}}{\dfrac{di_{\mathrm{я}}}{dt}} . \tag{8} \]

An illustration of the results obtained in the course of the first series of experiments (for the motor SL-621) is given in Fig. 2. Considering (Fig. 2,a) the family of curves \(L_{\mathrm{я}}=f_1(i_{\mathrm{я}}, i_{\mathrm{в}})\), we find that both with an increase in the armature currents and with an increase in the field current, the equivalent magnetic resistance to the armature flux

in flux increases. This is easily explained by the increase in saturation of the magnetic circuit.

Fig. 2. Curves of the dependences of the winding inductances on the armature current
and the excitation current:
a) for the excitation winding;
b) for the armature winding.

Let us now turn to the family of curves \(L_{\mathrm{в}} = f_2(i_{\mathrm{я}}, i_{\mathrm{в}})\). Here (Fig. 2,b), as the armature current increases in the region of small excitation currents, \(L_{\mathrm{в}}\) decreases, while in the region of large excitation currents it increases. This character of the curves is due to the fact that the main part of the magnetic resistance to the excitation flux falls on the frame.

In some sections of the frame, the components of the armature and excitation magnetic fluxes are directed in the same direction, while in others they are opposed. As the excitation current increases, the saturation of the sections with oppositely directed fluxes decreases and increases in the sections with the same direction. As a result, the differential magnetic permeability of the first sections increases (and more intensely), while in the other sections it decreases. The equivalent magnetic resistance falls, and the inductance \(L_{\mathrm{в}}\) increases.

This process is observed, apparently, until the intensity of the increase in the differential magnetic permeability of sections with opposing fluxes becomes equal, in terms of the results of its action, to the intensity of the decrease in the differential magnetic permeability of sections with aiding fluxes. The inductance of the excitation winding reaches a maximum and then, with a further increase in the excitation current, decreases.

Large values of armature currents also require large values of excitation currents at which an extremum of the curve \(L_{\mathrm{v}} = f(i_{\mathrm{v}})\) is observed. This means that, at the moment the extremum is reached, the sections with the aiding direction of the fluxes will be saturated to a greater degree; the equivalent magnetic resistance proves to be greater, and the extremal value of the inductance \(L_{\mathrm{v}}\) smaller.

Let us now consider the family of curves \(M = f_3(i_{\mathrm{ya}}, i_{\mathrm{v}})\) (Fig. 2, c). The nature of the change in mutual inductance under the action of the increasing excitation current is explained, as above, by a decrease in the differential magnetic permeability of the equivalent circuit. The increase of \(M\) with increasing armature current is caused by the following circumstances.

The magnetic circuit of the machine is a bridge. In this case the stator forms the arms of the bridge, and the windings are located along the diagonals. If the bridge is balanced (\(i_{\mathrm{ya}} = 0\)), then the inductive coupling between the windings is minimal. As the armature current increases, saturation of the bridge arms with opposing fluxes decreases, while that of the bridge arms with aiding fluxes increases. The equilibrium of the bridge circuit is increasingly disturbed, which leads to the appearance and further increase of diagonal fluxes that increase the magnetic coupling of the windings.

In the second series of experiments, the aim was to determine the sensitivity of the electrical parameters of the machine to the rate of the transient process. The experimental method was reduced to determining the dependence of the corresponding parameters on the frequency of the supply alternating voltage. For this purpose a set of low-frequency equipment for studying the frequency characteristics of elements of automatic-control systems was used.

The electrical circuit of the experimental setup (Fig. 3) included:

DPV — a double peak voltmeter measuring the amplitudes of voltage drops on the winding under test \(U_2\) and on a certain known resistance \(U_1\);

LF — a low-frequency phase meter making it possible to calculate the phase-shift angle \(\varphi\) between the voltages \(U_1\) and \(U_2\);

LFGPO — a low-frequency generator of periodic oscillations, whose signals were amplified by an electric-machine amplifier EMU. The latter, in addition, performs the function of decoupling the instrument circuits.

Determining in the course of the experiment, at different frequencies, \(U_2\), \(\varphi\), and also \(U_1 = I \cdot R_1\), we then, by simple calculation, find the total resistance of the winding under test

\[ Z(\omega)=\frac{U_2}{I}=\frac{U_2}{U_1}R_1 \tag{9} \]

its inductance

\[ L(\omega)=\frac{Z_w}{\omega}\sin\varphi \tag{10} \]

and the active resistance

\[ R(\omega)=\sqrt{[Z(\omega)]^2-[L(\omega)]^2} \tag{11} \]

![Schematic diagram with labels: EMU, \(U_1\), \(U_2\), \(R\), \(R_{\mathrm{в}}\), \(L_{\mathrm{в}}\), DPV, LF, LF EMU, and NGPK.]

Fig. 3. Diagram of the setup for determining the dependence of the inductance of the excitation winding on the frequency of the excitation current.

The results of processing the experimental data showed a weak sensitivity of \(R_{\mathrm{я}}\), \(R_{\mathrm{в}}\), \(L_{\mathrm{я}}\), and \(M\) to frequencies in the range of low values. The inductance of the excitation winding depends most significantly on the rate of the transient process (Fig. 4). Such a phenomenon is easily explained, since the magnetic circuit of the excitation flux

![Curve of the dependence of the excitation-winding inductance on excitation-current frequency. Vertical axis: \(L_{\mathrm{в}}, \text{H}\), with marks 100, 200, 300, 400. Horizontal axis: \(f, \text{Hz}\), with marks 0, 0.1, 1, 10.]

Fig. 4. Curve of the dependence of the inductance of the excitation winding on the frequency of the excitation current.

...of excitation consists, to a considerable extent, of massive steel structural elements, which create favorable conditions for the occurrence of eddy currents.

Conclusions

1. The electrical parameters of small direct-current machines are highly nonlinear quantities.

2. A qualitative description of transient processes in automatic-control systems using small machines is possible only provided that these nonlinearities are taken into account.

3. Special attention must be paid to the sensitivity of the electrical parameters to the rate at which the transient process proceeds.

4. The method for experimentally determining the dependence of the electrical parameters on frequency develops and substantially refines the method known from [1].

LITERATURE

1. V. P. Andreev, Yu. A. Sabinin. Fundamentals of Electric Drive, Gosenergoizdat, 1963.