Relay Control Systems for Electric Motor Currents

A. P. Zaitsev, R. K. Gachik, L. V. Volynets

(Presented by the Scientific and Technical Seminar of the Research Institute of Automation and Electromechanics at TPI).

Relay regulators, possessing such advantages as simplicity of structural design, high speed of response, and the possibility of obtaining large transmission coefficients by comparatively simple means, have opened up a class of discrete automatic systems. They have found the greatest application in automatic control systems (ACS) for the voltage of aircraft and automobile direct-current generators and, until recently, were used almost not at all in ACS for electric-motor currents. This is explained by the fact that the load of direct-current generators, as a rule, has considerable inductance, which ensures a comparatively small level of ripple of the output quantity during the regulation process, whereas the inductance of the power circuit of direct-current electric motors is insignificant; and the use of relay regulators in ACS for electric-motor currents imposes very stringent requirements on the dynamic properties of all elements of the regulator (current sensors, comparison devices, amplifiers, etc.). With the appearance of contactless elements (from sensors to power regulating links), the main drawback of relay regulators—a comparatively short service life—has been overcome, which opens broad prospects for their application for the indicated purposes.

One such application is a system, developed by the Research Institute of Automation and Electromechanics at the Tomsk Polytechnic Institute jointly with an enterprise, for stabilizing the starting and braking currents of series electric motors. Figure 1 shows the circuit of the power loop, and Fig. 2 a simplified block diagram, in which the following are denoted:

BU — control unit;
RE — relay element;
ED — electric motor;
DT — current sensors;
TK — thyristor switch.

Regulation is carried out by shunting, by means of a relay element (for example, a thyristor switch TK), an additional resistance in the circuit of the armature or excitation winding. The feedback voltage $U_i$, proportional to the current instantaneous value $i$, is compared with the reference voltage (setting) $U_0$, proportional to the required value of the armature current, in the comparison device. For the reason indicated above, this device must have a high speed of response and be implemented on electronic devices. All comparison devices actually in existence have nonuni-

a significant (loop) static “input–output” characteristic. Such a characteristic is often attributed to a relay element. If the thyristor switch is connected in parallel to the added resistance in the armature circuit, then the resistance value may be taken as the output coordinate \(F\) of the relay element; it increases abruptly to \(r_d\) if the current, while increasing, reaches a certain value \(i_{\max}\), and drops abruptly to zero if the current, while decreasing,

Fig. 1.

Fig. 2.

reaches a certain value \(i_{\min}\). Since the static characteristic of the relay element is multivalued, the self-oscillation mode is a characteristic operating mode of the system. The thyristor switch is practically an inertia-free element; therefore the order of the system, even with allowance for leakage and the influence of eddy currents, is equal to two, and the system itself is a poor filter of higher harmonics, which makes it difficult to study it by one of the well-developed frequency methods. When the relay element switches, there is a stepwise change in the total resistance of the power circuit and, consequently, in the coefficients of the differential equation of the system. Therefore, to study quasi-steady processes it is expedient to apply the phase-plane method.

For one of the states of the thyristor switch, the system is described by the following differential equations:

\[ \left. \begin{aligned} L_{\Sigma}\frac{di}{dt}+M\frac{di_{\mathrm{k}}}{dt}+r_{\Sigma}i&=U_c-E_{\mathrm{d}};\\ L_{\mathrm{k}}\frac{di_{\mathrm{k}}}{dt}+M\frac{di}{dt}+r_{\mathrm{k}}i_{\mathrm{k}}&=0. \end{aligned} \right\} \tag{1} \]

It is assumed here that the controlled object is connected with large masses and, moreover, that the pulsations of the armature current do not exceed 10%. This makes it possible, when considering dynamic electromagnetic processes, not to take into account the change in speed and, consequently, the counter-e.m.f. of the motor, and not to take into account the nonlinearity of the motor characteristic.

The following notation is adopted (Fig. 1):
\(L_{\Sigma}\) — inductance of the armature winding, taking into account the winding of the additional poles, the field winding, and the leakage of the main poles;
\(L_{\mathrm{k}}\) — inductance of the eddy-current circuit;
\(M\) — coefficient of mutual induction between the field winding and the equivalent eddy-current circuit;
\(r_{\Sigma}\) — total resistance of the power circuit for the given state of the thyristor switch;
\(r_{\mathrm{k}}\) — resistance of the eddy-current circuit;
\(U_c\) — supply-network voltage;
\(E_{\mathrm{d}}\) — counter-e.m.f. of the motor.

For unperturbed motion \((dU_c/dt=0)\), system (1) can be reduced to the equation

\[ \frac{d^2 i}{dt^2}=f-a\frac{di}{dt}-bi, \tag{2} \]

where \(a\), \(b\), \(f\) are constant quantities.

The equation of the integral curves is

\[ \frac{dy}{dx}=\frac{f-ay-bx}{y}, \tag{3} \]

from which the phase portrait of the system can be constructed for one of the states of the relay element. By supplementing it with the phase portrait for the other state of the relay element, it is not difficult to obtain a clearly expressed limit cycle. The parameters of the self-oscillations may be found from the limit cycle, or determined by a more accurate method—the method of point transformations.

The magnitude of the amplitudes of the self-oscillations is determined by this method from the expression

\[ (X_i)^{\frac{p_1}{p_2}} = \frac{(p_2)^{p_2}(y_0+p_1x_0)^{p_1}} {(p_1)^{p_1}(y_0+p_2x_0)^{p_2}}, \tag{3} \]

where

\(p_1, p_2\) — roots of the characteristic equation of the system;
\(x, y\) — initial values of the armature current and of its first derivative on the switching line of the relay element.

It is assumed here that the origin of coordinates has been shifted to the “singular” point.

The duration of the half-periods \(t_i\) of the self-oscillations is determined by the expression

\[ t_i=\frac{1}{p_2-p_1}\ln\frac{p_2(y_0+p_1x_0)}{p_1(y_0+p_2x_0)}, \tag{4} \]

and here it is also assumed that the origin of coordinates has been shifted to the “singular” point.

The self-oscillations have a strongly pronounced relaxation character, and the limit cycle is asymmetric, which is due, on the one hand, to the current value of the motor counter-e.m.f. and, on the other, to the stepwise change in the coefficients of the differential equation of the system when the relay element is switched. Therefore, within the limit cycle, the values of the amplitudes and the durations of the half-periods of the self-oscillations should be found separately for the open and closed states of the thyristor swit-

...and the pulsation swing and the period of self-oscillations are found as the sum of the amplitudes and half-periods, respectively.

The switching element may be connected in series in the power circuit. After the motor has been accelerated to its natural characteristic, the thyristor switch is connected in parallel with the field winding, and further acceleration is carried out by smoothly weakening the field. In the dynamic-braking mode with self-excitation, the thyristor switch is connected in series in the braking circuit. A feature of the automatic control system is that the logic of operation of the relay element remains unchanged regardless of the operating mode of the electric motor: to increase the armature current it is necessary to open the switch, and to decrease it—to close it.

The mode of dynamic braking with self-excitation of series motors is the simplest braking mode. The creation of relay systems for regenerative braking with self-excitation of series-excitation traction motors presents considerable difficulties, since special measures must be taken to ensure a satisfactory transient process under step-like changes of the contact-line voltage. It is much easier to ensure satisfactory quality of transients under regenerative braking with independent excitation of traction motors. In addition, at high speeds it is necessary to limit the braking current according to the potential conditions at the motor commutator. Such a regenerative-braking system was developed at the Research Institute of Automated Electric Drives under the Tomsk Polytechnic Institute jointly with an enterprise and is described in [3]. Analysis and synthesis of relay systems for regulating the armature current of motors by changing the field current is difficult, since even when inertia-free control elements are used the system is described by a fourth-order differential equation and contains several nonlinearities (the relay element, saturation, variation of time constants, product of system coordinates, etc.). Since the controlled object is associated with large rotating masses, the electromechanical time constant exceeds the electromagnetic ones by several orders of magnitude. This circumstance makes it possible to investigate the electromagnetic and electromechanical processes separately; i.e., the electromagnetic processes are investigated at unchanged speeds of motion, and the electromechanical processes are investigated with respect to the smooth component of the current.

When controlling the motor from the side of the field winding, the system is a good high-frequency filter; therefore, for investigation of the electromagnetic processes, in a first approximation (especially of self-oscillatory modes) the method of harmonic linearization may be used. Since the self-oscillation mode for relay systems with an ambiguous characteristic of the relay element is the only possible one, determining the amplitude and frequency of the self-oscillations of the output coordinate of the system solves the problem of investigating the electromagnetic quasi-steady processes. Let us analyze the quasi-steady processes in the above-mentioned system, whose block diagram after harmonic linearization is shown in Fig. 3. The behavior of the system is described by the following system of differential equations:

\[ \left. \begin{aligned} \frac{\Delta U_{\mathrm{v}}}{r_{\mathrm{v}}}\,(T_{\mathrm{k}}p+1) &= \Delta I_{\mathrm{v}}\left[T_{\mathrm{s}}T_{\mathrm{k}}p^{2}+(T_{\mathrm{s}}+T_{\mathrm{k}}+T_{\mu})p+1\right];\\ \Delta I_{\mathrm{v}}\,K_{1} &= \Delta E\,(T_{\mathrm{k}}p+1);\\ \frac{\Delta E}{r_{\mathrm{a}}} &= \Delta I\,(T_{\mathrm{a}}p+1);\\ \Delta U_{\mathrm{v}} &= \Delta U_{\mathrm{y}}\frac{4B}{\pi A}e^{-j\varphi};\\ \varphi &= \arcsin\frac{b}{A};\\ \Delta U_{\mathrm{y}} &= \Delta I\cdot K_{2}\cdot K_{3}; \end{aligned} \right\} \tag{1} \]

where

\[ \begin{aligned} U_{\mathrm{v}}&\text{— voltage on the field winding;}\\ I_{\mathrm{v}}&\text{— motor field current;}\\ E&\text{— rotational emf of the motor;}\\ I&\text{— motor armature current;}\\ U_{\mathrm{y}}&\text{— output voltage of the computing device;}\\ T_{\mathrm{k}}&\text{— time constant of the eddy-current circuit;}\\ T_s&\text{— time constant due to the leakage inductance of the main poles.} \end{aligned} \]

Fig. 3

\[ \begin{aligned} T_{\mu}&\text{— magnetization time constant of the main poles;}\\ r_{\mathrm{v}}&\text{— resistance of the field circuit;}\\ r_{\mathrm{a}}&\text{— resistance of the armature circuit;}\\ B&\text{— amplitude of the output voltage of the relay element;}\\ b&\text{— half-width of the ambiguity zone of the relay element;}\\ A&\text{— amplitude of the first harmonic of self-oscillations at the input of the relay element;}\\ K_2,\ K_3&\text{— transfer coefficients of the armature-current sensor and of the computing device, respectively.} \end{aligned} \]

Taking the amplitude \(A_0\) and the frequency \(\omega_0\) of the self-oscillations of the voltage at the input of the system relay element, in the first approximation, to be equal to the amplitude and frequency of the self-oscillations of the same quantity in a harmonically linearized system lying on the stability boundary, the root-locus method may be used to determine \(A\) and \(\omega_0\). The transfer function of the open-loop system is written as

\[ W(p)= \frac{K_{\mathrm{l}}\cdot K(A)} {\left[T_sT_{\mathrm{k}}p^2+\left(T_s+T_{\mathrm{k}}+T_{\mu}\right)p+1\right]\left(T_{\mathrm{a}}p+1\right)}, \tag{2} \]

where

\[ K_{\mathrm{l}}\text{— gain coefficient of the linear part of the system;} \]

\[ K(A)=\frac{4B}{\pi A_0}e^{-j\varphi} \quad \text{and} \quad \varphi=\arcsin\frac{b}{A_0} \text{— harmonic transfer coefficient of the relay element.} \]

The root-locus equation is written as

\[ \varphi+\sum_{i=1}^{3}\operatorname{arc\,tg}\frac{\omega_0}{p_i}=\pm\pi, \tag{3} \]

where

\[ p_i\text{— poles of transfer function (2).} \]

The value of the limiting gain coefficient is determined by

\[ K_{\mathrm{гp}}=K_{\mathrm{l}}\cdot K(A) = C\prod_{i=1}^{3}\sqrt{\omega_0^2+p_i^2}. \tag{4} \]

Assigning several values of \(\omega_0\) from (3), (4) and the expression for \(K(A)\), we determine the dependences \(\omega_0 = f_1(K_{\text{л}})\) and \(A_0 = f_2(K_{\text{л}})\), Fig. 4. From the dependences obtained, one can determine the quantities \(A_0\) and \(\omega_0\) for the developed system, or, in synthesizing the system, determine the required \(K(\text{Л})\) from the desired \(A_0\) and \(\omega_0\). When the numerical parameters of the system are substituted, the result of the calculations agrees with experiment to within 10–15%.

[Figure: curves \(\omega_0\) and \(A_0\) plotted against \(K_{\text{л}}\).]

Fig. 4

REFERENCES

1. A. A. Andronov, A. A. Vitt, S. E. Khaikin. Theory of Oscillations. Fizmatgiz, 1959.

2. E. G. Uderman. Approximate Study of Self-Oscillations by the Method of the Root Hodograph. “Energiya,” 1967.

3. A. P. Zaitsev, R. K. Gachik. Optimal control system for electric braking of an electric train. Proceedings of the Fifth Scientific and Technical Conference on Problems of Production Automation, vol. 1, 1967, Tomsk.