ON THE CALCULATION OF THE DYNAMIC STABILITY OF ENGINE SPEED CONTROL SYSTEMS BY VARYING THE EXCITATION FLUX

V. A. Beinarovich, A. I. Sapozhnikov,
Yu. P. Serdyukov, V. K. Smirnov, V. M. Stepanov

(Presented by the Scientific and Technical Seminar of the NII AEM)

When regulating the speed of separately excited motors by varying the excitation flux, certain special features are encountered in the study of the dynamic properties of such systems. These features are due to the nonlinearity of the motor magnetization characteristic, the presence in the system of nonlinear multiplication links, and the inverse dependence between the control action (the excitation flux) and the controlled quantity (the rotational speed).

Fig. 1

Figure 1 presents the block diagram of a motor speed control (stabilization) system developed at NII AEM for excitation control, when the excitation winding is supplied through a thyristor PWM regulator from the same source as the motor armature circuit. The block diagram shown corresponds to the generally accepted assumptions concerning linearization of the parameters and characteristics of the system, except for allowance for the nonlinearities of the motor magnetization characteristic, ...

...the engine and the multiplication links. It is assumed that, of the external actions, only the quantities of the specified speed \(\omega_z\), the supply voltage \(U_c\), and the load torque \(M_c\) are significant (these actions are indicated by circles in the block diagram). For proper functioning of the control system, the measuring device must perform the function of signal inversion, which is taken into account by the negative sign of its transfer coefficient \((-K_i)\). The remaining elements of the system, presented in the block diagram of Fig. 1, evidently require no explanation. It may only be said that the measuring device \((-K_i)\), the intermediate amplifier \((K_y)\), the pulse-duration modulator \((K_{\text{pdm}})\), and the thyristor voltage regulator \((U_v=\gamma U_c)\) are regarded as inertia-free elements, while the motor parameters—the resistance of the field winding \(R_v\), the resistance of the armature circuit \(R_a\), the time constants of the field circuit \(T_v\) and of the armature \(T_a\), the moment of inertia \(I\), the coefficient \(C\), and the magnetization characteristic—are regarded as unchanged.

From consideration of the block diagram in Fig. 1 it follows that the investigation of the dynamic properties of such a nonlinear system in general form, with simultaneous allowance for all actions, is not possible by analytical methods, and is extremely difficult by graphical and graphical-analytical methods. Such an investigation can be carried out with the aid of an electronic computer, but only for conditions in which the input actions are specified functions of time or of the system coordinates. At the same time, in designing control systems it is first necessary to evaluate their operability by investigating dynamic stability. Usually a system may be regarded as operable if, under the most unfavorable conditions, when investigated “in the small,” the system has a sufficient stability margin. In this case the investigation is performed from simplified equations in deviations for free motions. This procedure can also be used for the excitation-control system under consideration.

After the corresponding transformations [1], the transfer function of the open-loop control system, obtained from the equation in deviations for free motions in the absence of external actions, takes the form

\[ W_p=-K \frac{ \left(1-2\frac{E_0}{U_{c0}}\right)+\left(1-\frac{E_0}{U_{c0}}\right)pT_a }{ (1+pT_v)(1+pT_M+p^2T_aT_M) }, \tag{1} \]

where

\[ \begin{aligned} K &\text{— the overall gain coefficient of the regulator;}\\ E_0 &\text{— the initial value of the motor emf;}\\ U_{c0} &\text{— the initial value of the supply-line voltage;}\\ T_M &\text{— the electromechanical time constant.} \end{aligned} \]

Analysis of expression (1) shows that the control system contains an aperiodic and an oscillatory element, as well as differentiating elements connected in parallel; moreover, the coefficients of these last two elements may have different signs depending on the ratio of the initial values of the emf \(E_0\) and the supply voltage \(U_{c0}\). The block diagram for this case is shown in Fig. 2. Investigation of the stability of the resulting block diagram for given parameters presents no difficulty and can be carried out by known methods.

The motor field winding may have different designs, because the required ampere-turns of excitation are determined by the product of the excitation current and the number of turns of the field winding. However, in the production of electric motors with a thyristor speed stabilizer, it is expedient to make the field winding so that its time constant corresponds to the best combination of parameters under the conditions of dynamic stability of the stabilization system. Such a problem was posed by electrical-machine designers to the developers of the speed-control system as applied to electric motors of types P-81, P-82, P-112.

The study of the influence of \(T_{\mathrm v}\) on the dynamic stability of the stabilization system, taking into account the approach indicated above, is expediently performed by the frequency method of \(D\)-partition in the domain of two parameters—the gain coefficient \(K\) and the time constant of the field-winding circuit \(T_{\mathrm v}\). The \(D\)-partition curves are constructed from the characteristic equation of the closed-loop system obtained from (1). The special straight lines [2] are expressed by equations (2) and (3):

Fig. 2

\[ 1+K\left(2\frac{E_0}{U_{co}}-1\right)=0 \qquad \text{for } \omega=0, \tag{2} \]

\[ T_{\mathrm m}\cdot T_{\mathrm a}\cdot T_{\mathrm v}=0 \qquad \text{for } \omega=\infty . \tag{3} \]

The equation of the \(D\)-partition curve (stability boundary) with respect to the parameters \(K\) and \(T_{\mathrm v}\) has the form

\[ K\left[ T_{\mathrm m}\cdot T_{\mathrm a}\cdot T_{\mathrm v}\frac{E_0}{U_{co}} + T_{\mathrm m}\cdot T_{\mathrm a}^{2}\left(1-\frac{E_0}{U_{co}}\right) \right] = T_{\mathrm m}^{2}\cdot T_{\mathrm v} + T_{\mathrm v}^{2}\cdot T_{\mathrm m} + T_{\mathrm m}\cdot T_{\mathrm a}. \tag{4} \]

Substituting into (4) the parameters of the motors of the greatest \((T_{\mathrm m1}=0.037\ \mathrm{s},\ T_{\mathrm a1}=0.02\ \mathrm{s})\) and the least \((T_{\mathrm m2}=0.027\ \mathrm{s},\ T_{\mathrm a}=0.011\ \mathrm{s})\) power of the series under consideration, we obtain two stability boundaries in the plane of the parameters \(K\) and \(T_{\mathrm v}\), shown in Fig. 3. From them it is seen that with the higher-power motor the admissible gain coefficient of the system is much lower than with the lower-power motor. The overall gain coefficient in the open-loop system, for a specified stabilization accuracy \(x=0.01\), is determined [2] as

\[ K_{\Sigma}=K\left(2\frac{E_0}{U_{co}}-1\right)\ge \frac{1-x}{x}\simeq 100, \tag{5} \]

and since usually \(E_0\simeq U_{co}\), then \(K_{\Sigma}=K\), i.e., the gain coefficient of the entire system is determined by the value of the gain coefficient of the regulator.

\(K = 100\). It is evident from Fig. 3 that, with the required gain coefficient, it is practically impossible to ensure the stability of the system solely by selecting the time constant of the excitation circuit \(T_{\mathrm{v}}\).

When designing speed-stabilization systems, it is recommended to make the excitation windings with a circuit time constant on the order of \(0.3\) sec (taking into account the full value of the additional resistance for field weakening) and to use a series integral-differentiating circuit [2, 3], whose parameters are selected by the method of logarithmic frequency characteristics [2] for motors of the P-112 type of the greatest power. As the equivalent additional resistance is reduced during operation of the PWM regulator of the excitation current, the time constant of the excitation circuit increases, which leads to an increase in the stability margin. The use of a regulator with parameters selected in this way will ensure stability when operating with motors P-81, P-82, since systems with these motors have an increased stability margin (Fig. 3).

Fig. 3

References

1. A. S. Shatalov. Structural Methods in Control Theory and Electrical Automation. Moscow—Leningrad, 1962.

2. A. A. Voronov. Fundamentals of the Theory of Automatic Control. Part I, Moscow—Leningrad, “Energiya,” 1965.

3. V. A. Besekerskii. Dynamic Synthesis of Automatic Control Systems. Moscow, “Nauka,” 1970.