Processes in a Magnetic-Semiconductor AC Voltage Regulator When Supplied from a Rectangular-EMF Source

V. P. Obrusnik, A. V. Kobzev, G. A. Shadrin, Yu. M. Lebedev

(Presented by the scientific and technical seminar of the Research Institute of Automation of Electromechanics)

Magnetic-semiconductor voltage regulators (MSR), described in [1] with forced commutation of the switching element by the method of [2], make it possible in a number of cases to solve the technical problem of regulation—voltage stabilization—in the simplest way. From the standpoint of reducing weight and dimensions and improving energy-performance indices, the use of an MSR is expedient at an increased frequency [5]. The latter is very often generated with a rectangular voltage waveform.

Below we consider the processes in an MSR supplied from a source of rectangular EMF under the following assumptions:

1) the load is purely active,
2) the active and inductive leakage resistances of the windings are equal to zero,
3) the resistance of the semiconductor switching element is taken to be constant and equal to its dynamic resistance at the rated current.

Fig. 1.

The law of variation in time of the supply voltage is described by the system of equations (1)

\[ U(t)= \begin{cases} +U_m, & \text{for } 0+2\pi n < \alpha \leq \pi+2\pi n,\\ -U_m, & \text{for } \pi+2\pi n \leq \alpha \leq 2\pi(n+1). \end{cases} \tag{1} \]

The choke core (Fig. 1, a) is remagnetized along the hysteresis loop of Fig. 1, b, whose approximation is made by segments of straight lines. The equations of segments I—IV are represented by the expressions

\[ \begin{aligned} B_{\mathrm{I}}&=a_1H+b_1\\ B_{\mathrm{II}}&=a_2H+b_1\\ B_{\mathrm{III}}&=a_1H+b_2\\ B_{\mathrm{IV}}&=a_2H-b_1 \end{aligned} \tag{2} \]

The qualitative picture in such an MGR can be explained from the curve of instantaneous values of the current and load (Fig. 2) and the hysteresis loop (Fig. 1, b).

A characteristic feature of supplying an MGR with a rectangular voltage is the circumstance that, at the moment when the polarity of the supply voltage changes at \(\alpha=0\) (shown without parentheses in Fig. 1, a), the core, as a result of the action of the preceding half-period (the polarity in parentheses), is in a state of deep saturation at point \(6(0)\) in Fig. 1, b. Since at this same instant the switch K closes, the processes in the core and in the load proceed practically independently. The current in the load abruptly assumes the opposite direction, while the current of the previous direction, determined by the reserve of electromagnetic energy in the residual inductance of the core, closes through the circuit of the switch with a very small resistance and slowly decays. From the moment the switch opens at \(\alpha=\alpha_k\), the core winding is connected in series with the load, and the current determined by the reserve of electromagnetic energy of the core closes through the network and the load resistance, producing in the latter a negative voltage overshoot from the moment \(\alpha=\alpha_k\) to \(\alpha=\alpha_1\) (Fig. 2). At \(\alpha=\alpha_k\) the field strength in the core becomes equal to zero (point \(2(\alpha_1)\) in Fig. 1, b) and the load current also becomes equal to zero.

Fig. 2.

Then it changes sign and becomes equal to the demagnetizing current of the core. The operating point thereby moves along section II of the hysteresis loop from point \(2(\alpha_1)\) to point \(3(\varepsilon_1)\).

In view of the fact that the width of the hysteresis loop is small, that is, the transition of the operating point from section II to section IV occurs in a short interval of time, the angle \(\varepsilon_1\), determined by point 3, may be taken equal to the angle \(\varepsilon_2\), determined by point 4, that is, \(\varepsilon_1=\varepsilon_2=\pi\). Further, beginning from the moment \(\varepsilon_2=\pi\) (point 4), the core begins to be magnetized and the operating point moves along section IV from point 4 to point \(5(\alpha_s)\). The load current \(i_n\) is then small and equal to the magnetizing current.

(see Fig. 2 in the interval \(\alpha_1 \div \alpha_s\)). At \(\alpha=\alpha_s\) the core becomes saturated and the operating point moves to section 1, following the trajectory \(5(\alpha_s)—6(0)\). The load current is then determined by the supply voltage and the load resistance.

The states of the circuit (Fig. 1, \(a\)) are described by the following systems of equations:

a) with the switch closed

\[ \left. \begin{aligned} i_n R_n + i_k R_k + W_1 S \frac{dB}{dt} &= U(t),\\ i_k R_k + W_2 S \frac{dB}{dt} &= 0,\\ i_k &= i_n + i_{w_2},\\ Hl &= W_1\left(i_n-\frac{1}{k_t}i_{w_2}\right),\quad \text{where } k_t=\frac{W_1}{W_2} \end{aligned} \right\} \tag{3} \]

b) with the switch open

\[ \left. \begin{aligned} i_n R_n + W_2(k_t+1)S\frac{dB}{dt} &= U(t),\\ Hl &= i_n W_2(1+k_t) \end{aligned} \right\} \tag{4} \]

Table

where

\(S\) is the active cross-section of the magnetic core,

\(l\) is the length of the mean line of force.

The systems (2), (3), (4) are solved by the matching method [4] with respect to the field strength \(H\) on the corresponding intervals of the curve of instantaneous values of the load current \(i_{\mathrm{н}}\) (Fig. 2). The results of the calculations are summarized in Table 1.

Here the coefficients \(k_i\) \((i=1,2,\ldots,8)\), which depend on the circuit parameters and the load resistance, are determined by the expressions:

\[ k_1=\frac{l k_{\mathrm{т}} R_k R_{\mathrm{н}}} {\omega a_1 W_1 W_2\left[R_k\left(k_{\mathrm{т}}^2-1\right)-R_{\mathrm{н}}\right]}; \qquad k_2=\frac{U_m R_k\left(k_{\mathrm{т}}+1\right)} {\omega a_1 W_2 S\left[R_k\left(k_{\mathrm{т}}^2-1\right)-R_{\mathrm{н}}\right]}; \]
\[ k_3=\frac{lR_{\mathrm{н}}}{a_1\omega S W_2^2(1+k_{\mathrm{т}})^2}; \qquad k_4=\frac{U_m}{a_1\omega S W_2(1+k_{\mathrm{т}})}; \]
\[ k_5=k_3\cdot\frac{a_1}{a_2}; \qquad k_6=k_4\cdot\frac{a_1}{a_2}; \qquad k_7=k_3; \qquad k_8=-k_4, \]

where

\(\omega=2\pi f\) is the angular frequency of the supply voltage.

The unknown angle \(\alpha_1\) is determined by substituting equation (10) into the condition of periodic operation,

\[ H_5(0)=H_5(0+2\pi). \tag{11} \]

As a result of this substitution, the expression for \(\alpha_1\) has the form

\[ \alpha_1= \frac{ \ln k_3+\ln\left(\frac{k_8}{k_7}-\frac{k_2}{k_3}\right)-\ln k_4-k_1\alpha_k }{k_3} +\alpha_k, \tag{12} \]

i.e., the value of the angle \(\alpha_1\) differs little from the value of the commutation angle \(\alpha_k\), and the duration of the negative voltage overshoot over the interval \(\alpha_k\leq \alpha\leq\alpha_1\) is small. When the regulator is supplied from a sinusoidal a.c. source [3], the angle \(\alpha_1\) is determined by the formula

\[ \alpha_1=\arcsin \sqrt{\frac{1}{k_1^2-1}}, \]

i.e., it lies within the interval \(0\leq\alpha\leq\alpha_k\) and does not depend on the commutation angle. In this case the negative voltage overshoot is absent.

By specifying the circuit parameters in Fig. 1, \(a\), one can use formula (12) to construct the dependence \(\alpha_1=f(\alpha_k)\), as well as dependences of the form \(\alpha_1=f(a_1,a_2,k_{\mathrm{т}})\) for \(\alpha_k=\mathrm{const}\), and from them determine the influence of particular parameters on the magnitude of \(\alpha_1\).

The values of the angle \(\alpha_s\) are determined from the equality of formulas (10) and (9) when the current value of \(\alpha\) is replaced by \(\alpha_s\). The final expression has the form

\[ \frac{k_6}{k_5} \left[\left(2-e^{k_5(\alpha_1-\pi)}\right)e^{k_5(\pi-\alpha_s)}+1\right] - \left\{ \frac{k_2}{k_1} + \left[ \frac{k_4}{k_3}\left(1-e^{k_3(\alpha_1-\alpha_k)}\right) - \frac{k_2}{k_1} \right] e^{k_1\alpha_k-k_7(2\pi n+\alpha_s)} \right\} + \frac{k_8}{k_7} \left[ e^{-k_7(2\pi n+\alpha_s)}-1 \right] =0. \tag{13} \]

The transcendental equation (13) is solved by a graphical-analytical method using a digital computer [3]. From the results obtained, the dependences \(\alpha_s=f(\alpha_k)\) are constructed for the specified circuit parameters. On the basis of expressions for the instantaneous values of the currents \(i_{\mathrm{н}}\) and \(i_k\), one can determine the harmonic composition of the currents and voltages and calculate the integral characteristics.

References

1. A. V. Kobzev, V. P. Obrusnik. Theory of processes in discretely controlled ferromagnetic devices not biased by direct current. Elektromekhanika, 1972, No. 5.

2. V. P. Obrusnik, A. V. Kobzev. Method for controlling a magnetic-semiconductor transformer with a shunt. Author’s certificate No. 276171. Bulletin of Inventions No. 23, 1970.

3. V. P. Obrusnik, A. V. Kobzev, Yu. M. Lebedev. Processes and characteristics of magnetic-semiconductor devices with forced commutation of the switching element. Izvestiya TPI, vol. 261, Publishing House of TSU, 1972.

4. I. N. Lisitskaya, L. A. Sinitskii, Yu. M. Shumkov. Analysis of electric circuits with magnetic and semiconductor elements. Kiev, Naukova Dumka, 1969.

5. G. A. Shadrin, B. I. Konovalov. Magnetic-valve voltage regulator of increased frequency. Proceedings of the II Scientific and Technical Conference. Electromechanical Devices of Automation Systems. Tomsk, 1972.