Quasi-Steady-State Processes in Relay Current Control in Compound-Excited Motors

L. I. Sklyar, R. K. Gachik, V. M. Zhmurovskii

(Presented by the scientific-technical seminar of the Scientific Research Institute of AEM)

There are a number of mechanisms equipped with compound-excited DC motors in which it is necessary to regulate or limit the motor armature current (torque). The use of relay control systems for these purposes is due to such well-known advantages of relay systems as high speed of response, a wide control range, simplicity, etc. A feature of relay automatic control systems for the current of compound-excited motors is the presence of two magnetically coupled motor excitation circuits and a relatively small time constant of the armature circuit. The article gives a description of a compound-excited motor as an object of discrete control and substantiates admissible simplifications that make it possible, in a first approximation, to determine the parameters of quasi-steady-state processes by known methods.

Figure 1 presents the block diagram of the motor-current control system and the static characteristic of the relay element, where

RE — relay element, including a thyristor converter;

$W_{\mathrm{e}}$ — transfer function of the electric motor;

DE — delay element;

PE — pulse element;

$i_0$ — setpoint current;

$U_{\mathrm{e}}$ — motor voltage.

Fig. 1.

If the current $i < i_0$, then the RE is switched on; if the current $i > i_0$, then the RE is switched off. The inclusion in the feedback circuit of the pulse element PE, which closes briefly at a high frequency, eliminates the influence of hysteresis of the static characteristic of the real relay element RE. Technically this is implemented by using armature-current sensors [1], wh...

which convert a continuous input signal into rectangular high-frequency voltage pulses, or by using, as a comparison device, comparators switched at high frequency [2].

The circuit delay \(\tau\), caused by the devices of the control channel, is taken into account by a conventional delay element.

To investigate the electromagnetic processes in the motor, we compose the equivalent circuit (Fig. 2), where

Fig. 2

\[ L_a \]
is the inductance of the motor armature winding, taking the additional poles into account;

\[ L_{fs},\ L_{fsh} \]
are the magnetizing inductances of the series and shunt field windings, respectively;

\[ L_{sc},\ L_{ssh} \]
are the leakage inductances of the series and shunt field windings, respectively;

\[ r_{ac} \]
is the active resistance of the armature circuit;

\[ r_{sh} \]
is the active resistance of the shunt field winding;

\[ U_c,\ U_f \]
are the supply voltages of the armature and the field winding, respectively.

In pulse current control of motors it is necessary to take into account the influence of eddy currents. Therefore, allowing for the influence of eddy currents by means of a short-circuited loop without leakage fluxes and using the theory of the reduced transformer, the circuit of Fig. 2 can be transformed into the circuit of Fig. 3, where

\[ L_{ac}=L_a+L_{fs}+L_{sc},\quad r_{ac} \]
are, respectively, the inductance and active resistance of the armature circuit;

\[ L_\mu \]
is the inductance of the magnetizing loop;

\[ L'_{ssh},\ r'_{sh} \]
are, respectively, the reduced leakage inductance and active resistance of the shunt field winding;

\[ r'_k \]
is the reduced active resistance of the eddy-current loop;

\[ U_c,\ U'_f \]
are, respectively, the armature supply voltage and the reduced supply voltage of the field winding;

\[ i_a,\ i_\mu,\ i_k,\ i_f \]
are, respectively, the currents of the armature, the magnetizing loop, the short-circuited eddy-current loop, and the shunt field winding.

Fig. 3

The reduced circuit (Fig. 3) for the case of a stalled motor can be described by the following system of equations:

\[ \left. \begin{aligned} L_{yats}\frac{di_{ya}}{dt}+L_{\mu}\frac{di_{\mu}}{dt}+r_{yats}i_{ya}&=U_c,\\ L'_{sh}\frac{di_v}{dt}-L_{\mu}\frac{di_{\mu}}{dt}+r'_{vsh}i_v&=U'_v,\\ L_{\mu}\frac{di_{\mu}}{dt}&=i_k r'_k,\\ i_{ya}&=i_v+i_{\mu}+i_k \end{aligned} \right\} \tag{1} \]

Transforming the system of equations (1), we obtain the following differential equation:

\[ \begin{aligned} &\bigl[T_{ya}T_kT_sD^3+ (T_{ya}T_s+T_{ya}T_k+T_{\mu s}T_{ya}+T_sT_k+T_{\mu}T_s)D^2\\ &\quad +(T_{ya}+T_s+T_k+T_{\mu s}+T_{\mu})D+1\bigr]i_{ya}\\ &=K_cU_c\bigl[T_sT_kD^2+(T_s+T_k+T_{\mu s})D+1\bigr] +K_vT_{\mu}DU'_v, \end{aligned} \tag{2} \]

where

\[ T_{ya}=\frac{L_{yats}}{r_{yats}};\quad T_{\mu}=\frac{L_{\mu}}{r_{yats}};\quad T_s=\frac{L'_{sh}}{r'_{vsh}};\quad T_{\mu s}=\frac{L_{\mu}}{r'_{vsh}};\quad T_k=\frac{L_{\mu}}{r'_k};\quad K_c=\frac{1}{r_{yats}}; \]

\[ K_v=\frac{1}{r'_{vsh}};\quad D=\frac{d}{dt}. \]

With a pulsed armature supply voltage and an unchanged supply voltage of the shunt excitation winding, equation (2) takes the form:

\[ \begin{aligned} &\bigl[T_{ya}T_sT_kD^3+ (T_{ya}T_s+T_{ya}T_k+T_{\mu s}T_{ya}+T_sT_k+T_{\mu}T_s)D^2\\ &\quad +(T_{ya}+T_s+T_k+T_{\mu s}+T_{\mu})D+1\bigr]i_{ya}\\ &=K_cU_c\bigl[T_sT_kD^2+(T_s+T_k+T_{\mu s})D+1\bigr]. \end{aligned} \tag{3} \]

At present, the literature contains no methods for studying relay systems whose linear part is described by a third-order differential equation with a derivative of the input action. Therefore, in the present work, an MN-7 electronic analog modeling installation was used to study quasi-steady-state modes. The transfer function of the motor and the control system corresponding to the block diagram of the system in Fig. 1 were assembled.

To determine the possibility of using computers to study such systems by existing methods, investigations were carried out on an electronic model of the same system; however, in the motor equivalent circuit, either the leakage inductance of the shunt excitation winding or the influence of the eddy-current circuit was neglected. In this case equation (3) is transformed into:

a) for \(L'_{sh}=0\)

\[ \bigl[(T_{ya}T_k+T_{\mu s}T_{ya})D^2+(T_{ya}+T_k+T_{\mu s}+T_{\mu})D+1\bigr]i_{ya} =K_cU_c\bigl[(T_k+T_{\mu s})D+1\bigr], \tag{4} \]

b) for \(r'_k=\infty\)

\[ \bigl[(T_{ya}T_s+T_{\mu s}T_{ya})D^2+(T_{ya}+T_s+T_{\mu s}+T_{\mu})D+1\bigr]i_{ya} =K_cU_c\bigl[(T_s+T_{\mu s})D+1\bigr]. \tag{5} \]

Equations (4) and (5) are second-order differential equations with a derivative of the input action. Systems described by such differential equations can be investigated by methods existing in the literature, for example, by the phase-plane method.

The results of the study on the MN-7 of quasi-steady-state conditions in the described system are presented in Table 1, where

\(i_{\min}, i_{\max}\) are, respectively, the minimum and maximum currents of the electric motor in amperes;

\(f\) is the frequency of pulsations of the electric-motor current in hertz;

\(\tau\) is the lag in the system in microseconds.

An electric motor of type P-52 was selected as the electric motor under study.

Table 1

It is evident from Table 1 that, for motors of the P series, when studying in the first approximation one may use the differential equation without accounting for the leakage inductance of the shunt field winding. In this case the error in determining the amplitude of pulsations of the regulated current is practically equal to zero, while the error in determining the frequency of pulsa-

current does not exceed 20%. The influence of eddy currents may be neglected only when it is necessary to determine only the magnitude of the current pulsations. In this case the error does not exceed 10%. The frequency of the current pulsations in this case exceeds the actual one by several times.

REFERENCES

1. V. V. Bolotov, A. P. Zaitsev. Converter of direct current into alternating current. Inventor’s certificate No. 311354. Discoveries, Inventions, Industrial Designs, Trademarks. Official Bulletin, 1971, No. 24.

2. A. P. Zaitsev, Yu. F. Mikheev, L. V. Volyntsev. Voltage comparison device. Inventor’s certificate No. 271574. Discoveries, Inventions, Industrial Designs, Trademarks. Official Bulletin, 1971, No. 18.