DETERMINATION OF THE SWITCHING FREQUENCY OF A SYSTEM UNDER PULSE EXCITATION OF ELECTRICAL MACHINES

A. I. ZAITSEV, S. A. KITAEVA

Under pulse regulation of excitation, the excitation current has a pulsating character. The presence of current pulsations leads to iron losses due to eddy currents and to additional losses in the winding resistance, caused by the difference between the effective value of the current under pulsating conditions and in the absence of pulsations.

In addition, the presence of current pulsations in the excitation winding causes the generator emf also to have an oscillatory character, thereby producing amplitude modulation, which leads to a deterioration of the dynamic qualities of the system.

In this connection, when designing pulse systems, special attention must be paid to the magnitude of the modulation of the output voltage. From this point of view, the form of the current and voltage curves in the steady-state regime should approach the smooth component as closely as possible.

In the present article we consider questions of pulsations of the excitation current and of the voltage at the armature of a dc generator in a regulation system in which rectangular voltage pulses with frequency $T_1$ and duty factor $\gamma_1$ are applied to the excitation winding during the time interval $T\gamma$, and pulses with period $T_2$ and duty factor $\gamma_2$ during the interval $T(1-\gamma)$ [1], Fig. 1.

Fig. 1.

In such a system the current pulsation coefficient, on the basis of the condition that during the time interval $T\gamma$ the voltage $U_{\gamma_1}$ is applied to the excitation winding, and during the time interval $T(1-\gamma)$ the voltage $-U_{\gamma_2}$ is applied, can be determined as

\[ \chi=\frac{I_{\max}}{I_{\min}}= \frac{\gamma_1(1-e^{-\beta\gamma})+\gamma_2[1-e^{-\beta(1-\gamma)}]e^{-\beta\gamma}} {\gamma_1 e^{-\beta(1-\gamma)}(1-e^{-\beta\gamma})+\gamma_2[1-e^{-\beta(1-\gamma)}]} \tag{1} \]

where

\[ \beta=\frac{T}{T_{\mathrm{B}}}; \qquad T_{\mathrm{B}}=\frac{L_{\mathrm{B}}}{R_{\mathrm{B}}}, \]

\(R_v, L_v\) are the active resistance and inductance of the excitation winding.

A reduction in the ripple factor can be obtained by increasing the switching frequency. For this purpose the switching frequency should be as high as possible. However, the frequency can be increased only up to a certain level, at which switching losses lead to failure of the switching semiconductor devices. This frequency is specified for each type of semiconductor device.

Therefore, from the point of view of heating of the semiconductor devices and the reliability of system operation, it is advisable to choose the switching frequency according to the ripple conditions at a specified relative on-time duration [2].

The switching frequency for the system is determined from (1). Since the exact expression for determining the period of oscillations is complicated, we assume that for \(T_v \ll T\)

\[ 1-e^{-\beta\gamma}\approx \beta\gamma;\qquad 1-e^{-\beta(1-\gamma)}\approx \beta(1-\gamma), \]

then

\[ T=\frac{(x-1)\,[\gamma_2(1-\gamma)+\gamma_1\gamma]} {\gamma(1-\gamma)(x\gamma-\gamma_2)}\,T_v \tag{2} \]

On the basis of (2) one can construct the law of variation of the repetition period \(T=f(\gamma)\) for a constant specified ripple factor (Fig. 2).

When the dc generator operates on the linear portion of the magnetization curve, the dependence of the emf on the excitation current \(E=f(i_v)\) is linear and has the form

\[ E=i\tan\sigma, \]

where \(\sigma=\mathrm{const}\) and is the angle of inclination of the no-load characteristic to the abscissa axis. In this case the total ripple swing of the generator emf can be determined as [3]

Fig. 2.

\[ \Delta E=k\Delta i=E_{\max}-E_{\min} = \frac{ E_{\mathrm{st}}(\gamma_1-\gamma_2)(1-e^{-\beta\gamma})[1-e^{-\beta(1-\gamma)}] }{ 1-e^{-\beta} }, \tag{3} \]

where

\[ E_{\mathrm{st}}=\frac{U\tan\sigma}{R_v}, \]

and expression (1) for the current ripple factor will also be valid for the emf ripple factor.

Thus, a reduction in the modulation of the output voltage can be achieved either by increasing the time constant of the excitation winding, or by increasing the switching frequency. Increasing the time constant of the excitation winding is associated with a decrease in the speed of response of the regulating system. Therefore, the basic method of reducing modulation is to increase the frequency of self-oscillations.

The switching frequency in the given system is determined by the switching frequency of the relay element. In such a nonlinear system, the frequency of self-oscillations and the amplitude are determined by known methods.

nonlinear theory and depend on the properties of the system. For a second-order system with the characteristic of the relay element shown in Fig. 3, to determine the switching frequency and the amplitude in analytical form it is convenient to use the approximate method of harmonic balance.

Fig. 3.

In view of the fact that the relay element has an asymmetric characteristic, asymmetric self-oscillations with a constant component arise in the automatic control system:

\[ U = U_0 + A \sin \omega t, \]

where \(U_0\) is the constant component;
\(A\) and \(\omega\) are the amplitude and frequency of the self-oscillations.

Analyzing the automatic control system, one can obtain expressions for determining the frequency and amplitude of the self-oscillations, expressed through the system parameters.

The switching frequency of the system is determined as

\[ \omega = \sqrt{ \frac{T_1 + T_{\mathrm{в}} + (1+k\beta)\tau} {T_1 T_{\mathrm{в}}\tau} }, \tag{4} \]

\[ A = \frac{ 2ck \cos \left[ \pi - \left( \frac{C}{2} + U_0(1+k\beta) + b I_{\mathrm{я}} R_{\mathrm{я}} \right)\frac{\pi}{c} \right] }{ \pi \omega \sqrt{ T_1^2 T_{\mathrm{в}}^2 \omega^2 + (T_1 + T_{\mathrm{в}})^2 } }, \tag{5} \]

where: \(k\) is the gain coefficient of the system;
\(T_1; T_{\mathrm{в}}\) are the time constants of the system;
\(\tau\) is the delay in the system introduced by the pulse element;
\(I_{\mathrm{я}}R_{\mathrm{я}}\) is the voltage drop across the armature, varying as a function of the load variation;
\(\beta\) is the feedback coefficient.

The displacement of the center of the self-oscillations is determined by the parameters of the system and by the magnitude of the load.

\[ U_0 = \frac{ k\left( \frac{c}{2} - \frac{c}{\pi}\arcsin \frac{U_0}{A} \right) + b I_{\mathrm{я}} R_{\mathrm{я}} }{ 1+k\beta }. \tag{6} \]

An increase in the load on the generator leads to an increase in the amplitude modulation of the generator output voltage. This is explained by the fact that, with an increase in load, the displacement of the center of the self-oscillations occurs in the direction of increase, and this leads to a decrease in the value of the duty factor \(\gamma\); as a result, the magnitude of the voltage ripple increases.

Analysis of expressions (4), (5), (6) makes it possible to give practical recommendations for selecting the parameters of the system from the condition of obtaining the greatest switching frequency and the smallest amplitude of self-oscillations. The law of variation of the oscillation amplitude with frequency is shown in Fig. 4.

An experimental study was carried out on a pulse system for regulating the voltage of a direct-current generator. The studies were conducted in the established thermal regime; at the same time

the magnitude of current ripple and the amplitude of voltage oscillations at various loads were determined.

The greatest magnitude of ripple in the generator excitation current was observed at the rated load and was 1.22; in this case the amplitude of self-oscillations was equal to 0.15 V, which amounts to 0.55%.

Comparison of the experimental data with the values calculated from the derived equations gives a discrepancy of 10–15%.

Fig. 4.

REFERENCES

1. A. I. Zaitsev, S. A. Kitaeva. Relay-pulse voltage regulator for direct-current generators, Izvestiya TPI, vol. 139, 1965.

2. A. P. Zaitsev. Development and investigation of certain pulse devices for controlling electrical machines. Dissertation, TPI, 1964.

3. E. N. Popov, I. P. Pal’tov. Approximate methods for investigating nonlinear automatic systems, Fizmatgiz, 1960.