PROCESSES AND CHARACTERISTICS OF MAGNETIC–SEMICONDUCTOR DEVICES WITH FORCED COMMUTATION OF THE SWITCHING ELEMENT

V. P. OBRUSNIK, A. V. KOBZEV, Yu. M. LEBEDEV

(Presented by the scientific and technical seminar of the Research Institute of AEM of the EPA Department)

The use of magnetic–semiconductor devices in voltage regulation–stabilization systems makes it possible to solve the technical problem in the simplest way, to reduce weight and dimensions, and to increase reliability. The operating principle and the physics of the processes in magnetic–semiconductor devices (MSD) are set forth in detail in [1].

Of greatest interest is control of an MSD with stabilization of the phase of the output voltage [2], in which the switch \(K\) (Fig. 1, \(a\)) is closed at the beginning of each period for a time not exceeding a half-period; this is classified as an MSD with forced commutation of the switching element.

The basic properties and characteristics of such devices are determined by the quality of the core material used; therefore, quantitative dependences cannot be obtained with sufficient accuracy using an ideal magnetization curve. Here, the hysteresis loop of the real core was approximated by segments of straight lines (Fig. 1, \(b\)). The equations of sections I–IV are represented by expressions (1)

\[ \begin{aligned} B_{\mathrm{I}} &= a_1 H + b_1, \qquad & B_{\mathrm{III}} &= -a_1 H + b_2,\\ B_{\mathrm{II}} &= a_2 H + b_1, \qquad & B_{\mathrm{IV}} &= a_2 H - b_1 \end{aligned} \tag{1} \]

The coefficients in (1) are determined from the approximated hysteresis loop of the real core.

The analysis of the processes for the MSD in Fig. 1, \(a\), controlled according to [2] under an active load, is carried out below under the following assumptions:

a) the resistance of the semiconductor switching element is taken as constant and equal to its dynamic resistance at the rated current;

b) the active and inductive leakage resistances of the windings are equal to zero;

c) with the switch \(K\) open, the magnetic cores of the MSD are not saturated, which is ensured by the corresponding design calculation of the core and of the numbers of turns of windings \(W_1\) and \(W_2\).

A qualitative picture of the processes in such an MSD can be explained from the curve of instantaneous values of the load current (Fig. 1, \(v\)) and the hysteresis loop (Fig. 1, \(b\)). In this case, at the moment of closing of the switch \((\alpha=0)\), the field strength in the core is not equal to zero, and the operating point moves—

... moves along section I of the hysteresis loop from a certain point 7 (0) to point 1 2($\alpha_1$) (Fig. 1,b), while the current $i_\text{n}$ is determined mainly by the applied voltage and the load resistance. Here and below in the article, the operating point is understood to mean a point in the $B$—$H$ plane that determines the state of the core at the given instant of time.

At the instant $\alpha=\alpha_1$, the operating point passes to section II of the hysteresis loop (point 1 ($\alpha_1$) in Fig. 1,b), while the load current changes practically according to the previous law, since winding $W_2$ of the core is short-circuited through a very small resistance $R_k$ and the induction in it changes insignificantly (from point 1 ($\alpha_1$) to a certain point 2 ($\alpha_2$) in Fig. 1,b). After

Fig. 1.

the opening of switch K, the core is demagnetized; the operating point moves along section II from point 2 ($\alpha_2$) to point 3 ($\varepsilon_1$). The load current in this case is very small and is determined by the magnetizing current of the core.

At the instant $\alpha=\varepsilon_1$, determined by the magnitude of the line voltage and the total resistance of the circuit, the operating point passes to section III of the hysteresis loop and moves from point 3 ($\varepsilon_1$) to point 4 ($\varepsilon_2$). The induction changes insignificantly, and the current $i_\text{n}$ is again determined by the line voltage and the load resistance (Fig. 1,c).

At the instant $\alpha=\varepsilon_2$, the operating point passes to section IV, and the core is magnetized along the trajectory 4 ($\varepsilon_2$) → 5 ($\alpha_s$). The load current in this interval is determined by the magnetizing current of the core (see the curve in Fig. 1,c in the interval $\varepsilon_2 \div \alpha_s$). At $\alpha=\alpha_s$ the core becomes saturated, and the operating point passes to section I of the hysteresis loop, moving along the trajectory 5 ($\alpha_s$)—6—7 ($2\pi$). In this case the induction in the core practically does not change, and the current $i_\text{n}$ is determined by the line voltage and the load resistance.

The form of the original system of equations describing the processes in the MSD is determined by the position of switch K in the circuit in Fig. 1,a. With the switch closed, the state of the circuit is described by the system of equations (2) and by the law of total current (3).

\[ \left. \begin{aligned} &i_k R_k+i_n R_n+W_1S\frac{dB}{dt}=U_m\sin\omega t,\\ &i_k R_k+W_2S\frac{dB}{dt}=0,\qquad i_k=i_n+iw_2 \end{aligned} \right\}, \tag{2} \]

\[ Hl=W_1\left(i_n-\frac{1}{K_t}iw_2\right),\quad \text{where } K_t=\frac{W_1}{W_2}. \tag{3} \]

With the switch open:

\[ i_nR_n+W_1S(K_t+1)\frac{dB}{dt}=U_m\sin\omega t, \tag{4} \]

\[ Hl=i_nW_2(1+K_t). \tag{5} \]

The solution of equations (2)—(5) is carried out over intervals determined by the position of the operating point on the hysteresis loop. The constants of integration are determined by matching the corresponding intervals [3]. The load current will be equal to

\[ i_n=\frac{H\cdot l}{W_2(1+K_t)}. \tag{6} \]

Taking into account that the width of the hysteresis loop is small, i.e., the transition from section II to section IV (Fig. 1,b) occurs over a short interval of time, one may take \(\varepsilon_1=\varepsilon_2=\pi\).

The unknown angles \(\alpha_1\) and \(\alpha_s\) are determined from the condition of steady-state operation \(H_i(\alpha)=H_i(\alpha+2\pi)\). Then \(H_1(0)=H_1(2\pi)\), whence, taking into account the solution of the equation on the first section \((0\le \alpha\le \alpha_1)\),

\[ K_1\sin\alpha_1-\cos\alpha_1=0, \tag{7} \]

\[ \alpha_1=\arc\sin\sqrt{\frac{1}{K_1^2+1}}, \tag{8} \]

where

\[ K_1=\frac{lK_t\cdot R_n(K_t-1)\cdot R_k}{\omega a_1\cdot W_1\cdot W_2\cdot S\left[R_k(K_t^2-1)-R_n\right]}. \tag{9} \]

From expressions (8) and (9) it is evident that the angle \(\alpha_1\) does not depend on the load resistance (taking \(R_k\) and \(K_t\) into account) and depends on the circuit parameters and on the slope coefficient of the saturated sections \(1\) of the magnetization curve. For an ideal magnetization curve [1], when the slope coefficient \(a_1=0\), the angle \(\alpha_1=0\).

In Fig. 2,a, a curve is given for the dependence of the angle \(\alpha_1\) on \(a_1\) for an MSD on a toroidal core.

It is seen that \(\alpha_1\) does not exceed \(90^\circ\). When \(a_1\to0\) or \(K_t=1\), always \(\alpha_1=0\).

The angle \(\alpha_s\) is determined from the condition of periodicity of the operating mode by comparing the field-strength values at \(\alpha=\alpha_s\) and \(\alpha=\alpha_s+2\pi\).

In this case the following expression is obtained:

\[ \frac{K_4}{K_3^2+1}\left[(K_3\sin\alpha_k-\cos\alpha_k) -(K_3\sin\alpha_1-\cos\alpha_1)l^{K_3(\alpha_1-\alpha_k)}\right]l^{K_5(\alpha_k-\alpha_s)}+ \]
\[ +\frac{K_6}{K_5^2+1}\left[(K_5\sin\alpha_s-\cos\alpha_s) -(K_5\sin\alpha_1-\cos\alpha_1)l^{K_5(\alpha_k-\alpha_s)}\right]+ \]
\[ +\frac{K_8}{K_7^2+1}\left\{[1-l^{K_5(\alpha_k-\pi)}]l^{K_5(\pi-\alpha_s)} -(K_7\sin\alpha_s-\cos\alpha_s)\right\}=0. \tag{10} \]

Here \(K_2, K_3, K_4, K_5, K_6, K_7, K_8\) are expressed in terms of the circuit parameters (Fig. 1,a).

Fig. 2

Equation (10) is transcendental and is solved by a grapho-analytical method using a digital computer.

In Fig. 2,b the dependence \(\alpha_s=f(\alpha_k)\) is shown for given circuit parameters. It is seen that the values of \(\alpha_k\) and \(\alpha_s\) are not the same; in reality \(\alpha_k>(\alpha_s-2\pi)\). This is explained by the presence of the slope of section 1 of the hysteresis loop and by the presence of losses in the switch circuit.

Using the expressions obtained for the load current on different intervals, one can construct the curves of the instantaneous values of the load current in Fig. 1,b, which are well confirmed by oscillograms. These curves differ from the analogous curves for an ideal core, presented in [1]. The presence of hysteresis plays a significant role both for the physics of the processes taking place and for the character of the voltage and current curves at the load.

From the instantaneous values of the currents \(i_n\) and \(i_k\), the integral characteristics of the MSU can be calculated and the harmonic composition of the currents and voltages determined.

REFERENCES

1. A. V. Kobzev, V. P. Obrusnik, M. G. Savkov. Discrete control of ferromagnetic devices without biasing by direct current. Izv. TPI, vol. 243, Tomsk State University Publishing House, 1972.

2. V. P. Obrusnik, A. V. Kobzev. Method for controlling a magnetic-semiconductor transformer with a shunt. Author’s certificate No. 276171, cl. 21a² 18/08 of 8/I 1969. Bulletin of Inventions, 1970, No. 23.

3. I. N. Lisitskaya, L. A. Sinitskii, Yu. M. Shumkov. Analysis of electrical circuits with magnetic and semiconductor elements. Kiev, “Naukova Dumka,” 1969.