PROCESSES AND CHARACTERISTICS OF CONTROL CIRCUITS OF SINGLE-PHASE PULSE-BIASED FERROMAGNETIC DEVICES

V. P. OBRUSNIK, A. V. KOBZEV

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

The control circuit of any pulse-biased ferromagnetic device (FU) may be represented by the circuit in Fig. 1, where \(i_2\sim\) is the second-harmonic current generator, and \(L_y\) is the inductance of the control circuit with unsaturated cores. The validity of such an approach may be justified by the following considerations:

1. The quantitative relations in the equivalent circuit in Fig. 1 are practically independent of the magnitude of the inductance \(L_y\); therefore, within fairly wide limits of its variation, \(L_y\) may be considered constant.

2. In nominal and near-nominal operating conditions, the percentage content of higher harmonics in the bias circuit does not exceed 10% relative to the second harmonic.

Fig. 1

1. Owing to the short duration of the biasing pulses

\[ \left(\frac{t_i}{T} \ll \frac{1}{12}\right) \]

the energy relations in the control-winding circuit depend substantially on the magnitude and phase of the minimum current in this winding and practically do not depend on its waveform.

The qualitative properties and quantitative characteristics of pulse-biased FUs are revealed most fully in the analysis of the power ratio of biasing sources with continuous and pulsed control, as a function of the pulse parameters:

\[ K_{p1}=\frac{P_{y\,\mathrm{cont}}}{P_{y\,\mathrm{pulse}}}=f(\gamma_i,\alpha_i). \tag{1} \]

The indicated relationships can be established by formulating and solving the differential and difference equations of the circuit under study. Thus, for the pulse interval in the \(n\)-th period, when \(K_1\) is closed and \(nT\leq t\leq nT+\gamma T\), we obtain:

\[ U_y=R_y\cdot i_d+L_y\frac{di_d}{dt}+L_y\frac{di_\Gamma}{dt}. \tag{2} \]

Here

\(n\) is the number of the period;

\(T\) is the pulse repetition period;

\[ \gamma=\frac{t_i}{T} \]

is the relative pulse duration.

For the pause interval, when \(K_1\) is open and \(nT+\gamma T\leq t\leq (n+1)T\), we shall have:

\[ 0=R_y\cdot i_d+L_y\frac{di_d}{dt}+L_y\frac{di_\Gamma}{dt}. \tag{3} \]

Dividing (2) and (3) by \(R_y\) and introducing the notation

\[ \frac{U_y}{R_y}=I;\qquad \frac{L_y}{R_y}=T_y;\qquad \frac{T}{T_y}=\beta;\qquad \bar t=\frac{t}{T}, \]

and also taking into account that \(i_\Gamma=I_m\sin(\omega t+\psi)\), \(\omega=\dfrac{2\pi}{T}\), we can obtain:

\[ \beta I=\frac{di_d}{d\bar t}+\beta i_d+ \frac{d\{I_m(\bar t)\sin(2\pi\bar t+\psi)\}}{d\bar t}, \quad \text{when } n\leq \bar t\leq n+\gamma, \tag{4} \]

\[ 0=\frac{di_d}{d\bar t}+\beta i_d+ \frac{d[I_m(\bar t)\sin(2\pi\bar t+\psi)]}{d\bar t}, \quad \text{when } n+\gamma\leq \bar t\leq n+1. \tag{5} \]

The further transformation of these equations depends on the form of the function \(I_m(\bar t)\), which represents the amplitude of the current \(i_\Gamma(\bar t)\). In transient regimes \(I_m(\bar t)\) varies continuously, but because the duration of transient processes in real ferromagnetic biasing devices is large (20 or more periods), the quantity \(I_m(\bar t)\) may, without significant error, be taken within each period as constant and as a function only of the period number, that is,

\[ I_m(\bar t)=I_m(n). \tag{6} \]

Taking (6) into account, the solutions of equations (4), (5) can be found by known methods in the form

\[ i_{d1}(\bar t)=Ae^{-\beta(\bar t-n)}+I- \frac{I_m(n)\cdot 2\pi}{\beta^2+4\pi^2} [2\pi\sin(2\pi\bar t+\psi)+\beta\cos(2\pi\bar t+\psi)] \tag{7} \]

for \(\delta<\bar t-n\leq\gamma\);

\(\delta\to 0\).

\[ i_{d2}(\bar t)=A_1\cdot e^{-\beta(\bar t-n)}- I_m(n)\frac{2\pi}{\beta^2+4\pi^2} [2\pi\sin(2\pi\bar t+\psi)+\beta\cos(2\pi\bar t+\psi)]. \tag{8} \]

for \(\gamma \leq \bar t-n<1-\delta\),
\(\delta \to 0\).

The integration constant \(A\) is found from the initial conditions \(\bar t=n\), \(i_d(t)=i_d(n)\):

\[ A=i_d(n)-I+K_1 I_m(n), \tag{9} \]

where

\[ K_1=\frac{2\pi}{\beta^2+4\pi^2}\cdot(2\pi\sin\psi+\beta\cos\psi). \]

The value \(A_1\) is determined after matching (7) and (8) at \(\bar t=n+\gamma\):

\[ A_1=i_d(n)-I+K_1 I_m(n)+I e^{\beta\gamma}. \tag{10} \]

If we further assume that

\[ I_m(n)=K_{Mi}\cdot I_{d\mathrm{cp}\,n}, \tag{11} \]

where

\[ K_{Mi}=\frac{I_{d\mathrm{cp}\,n}-I_{d\,n\min}}{I_{d\mathrm{cp}\,n}}\leq 1 \]

(Fig. 2) is the modulation coefficient of the biasing current by even harmonics, then, by integrating equations (7) and (8) over a period, we find

Fig. 2

\[ I_m(n)=\frac{1}{K_3\cdot\beta}\,i_d(n)(1-e^{-\beta})+\frac{1}{K_3}I\cdot\gamma+\frac{1}{K_3\beta}I\cdot e^{-\beta}(1-e^{\beta\gamma}), \tag{12} \]

where

\[ K_3=\left[\frac{1}{K_{Mi}}-\frac{K_1}{\beta}(1-e^{-\beta})\right], \]

so that one unknown function remains—\(i_d(n)\), the sample of the current at the beginning of each pulse.

This function can be found by solving a difference equation, composed by the method of [2], from the condition of continuity of the current in the inductance \(L_y\). It has the form

\[ i_d[n+1]-D\cdot i_d[n]=Q, \tag{13} \]

where

\[ Q=\frac{ I\left[ \frac{K_1}{K_3}\gamma(e^{-\beta}-1)+\mathrm{Б}(e^{-\beta}-1)-e^{-\beta}+e^{\beta(\gamma+1)} \right] }{1+\Gamma}, \]

\[ D=\frac{e^{-\beta}-A(1-e^{-\beta})+\Gamma}{1+\Gamma}; \qquad A=\frac{K_1}{K_3\cdot\beta}\cdot(1-e^{-\beta}); \]

\[ \mathrm{Б}=-\frac{K_1}{K_3\cdot \beta}\cdot e^{-\beta}(1-e^{\beta\gamma}); \qquad \Gamma=\frac{\sin\gamma}{K_3\cdot \beta}\cdot (1-e^{-\beta}). \]

Solving equation (13) by the classical method according to [1], we obtain:

\[ i_d[n]= \frac{I\left[K_{M_i}\cdot K_1\cdot \gamma(e^{-\beta}-1)+e^{-\beta}(e^{\beta\gamma}-1)\right]} {1-e^{-\beta}}\cdot (1-e^{-\beta_\mathrm{e} n})- \frac{K_{M_i}\cdot I\sin\psi\left[\beta\gamma-e^{-\beta}(1-e^{\beta\gamma})\right]} {\beta-K_1\cdot K_{M_i}(1-e^{-\beta})+K_{M_i}\cdot \sin\psi(1-e^{-\beta})}\cdot e^{-\beta_\mathrm{e} n}, \tag{14} \]

where \(\beta_\mathrm{e}=\ln\left(\frac{1}{D}\right)\) is the equivalent time constant.

If (14) is substituted into (12), (10), (9), and then into (7) and (8), the law of variation of the current \(i_d(t)\) is found at any instant of time, including the transient process.

Analysis of (14) makes it possible to conclude that the dynamic properties of pulse- and continuously controlled FDs are identical. First, the equivalent constant \(\beta_\mathrm{e}\) depends only on the parameters of the bias circuit and does not depend on the pulse parameters; second, if in (14) the exponential \(e^{\beta\gamma}\) is replaced by its Taylor-series expansion and quantities of fourth order of smallness with respect to \(\beta\) and \(\gamma\) are neglected, then (14) reduces to the form

\[ i_d[n]=I\cdot \gamma\left[ \frac{K_{M_i}\cdot K_1(e^{-\beta}-1)+e^{-\beta}\cdot \beta} {1-e^{-\beta}}(1-e^{-\beta_\mathrm{e} n})- \frac{K_{M_i}\cdot \sin\psi\cdot (1+e^{-\beta})} {\beta-K_{M_i}\cdot K_1(1-e^{-\beta})+K_{M_i}\cdot \sin\psi(1-e^{-\beta})}\cdot e^{-\beta_\mathrm{e} n} \right]. \tag{15} \]

It turns out that if the step of the continuous control voltage is equal to the step of the average value of the pulsed voltage, then \(i_d[n]\) changes qualitatively and quantitatively in the same way both under continuous \((I\gamma=I_{\text{uninterr}})\) and under pulse control.

For the steady-state mode \(n=\infty\), the current \(i_d\) at any instant of time will be found from the expressions:

\[ i_{d_1}(\varepsilon)= [C-I+K_{M_i}\cdot \gamma\cdot I\cdot K_1]e^{-\beta\gamma} -I-K\cdot K_{M_i}\cdot I\gamma \left[2\pi\sin(2\pi\varepsilon+\psi)+\beta\cos(2\pi\varepsilon+\psi)\right], \tag{16a} \]

\[ 0\leqslant \varepsilon\leqslant \gamma. \]

\[ i_{d_2}(\varepsilon)= [C-I+K_1\cdot K_{M_i}\cdot I\gamma+I\cdot e^{\beta\gamma}]e^{-\beta\varepsilon} -K\cdot K_{M_i}I\cdot \gamma \left[2\pi\sin(2\pi\varepsilon+\psi)+\beta\cos(2\pi\varepsilon+\psi)\right], \tag{16} \]

\[ \gamma\leqslant \varepsilon\leqslant 1. \]

Using (16a), (16), one can find the average value of the magnetizing current

\[ I_{d\,\mathrm{av}}=\int_0^1 i_d(\varepsilon)d\varepsilon =\int_0^\gamma i_{d_1}(\varepsilon)d\varepsilon+ \int_\gamma^1 i_{d_2}(\varepsilon)d\varepsilon =I\cdot \gamma \tag{17} \]

and of the control current

\[ I_{u\,\mathrm{av}}=\int_0^1 i_d(\varepsilon)d\varepsilon =\int_0^\gamma i_{d_1}(\varepsilon)d\varepsilon = \frac{I[1-e^{-\beta(1-\gamma)}](e^{-\beta\gamma}-1)} {\beta(1-e^{-\beta})} +I\cdot \gamma +K\cdot K_{M_i}\cdot I\cdot \gamma \left[ \cos(2\pi\gamma+\psi)-\cos\psi -\frac{\beta}{2\pi}\sin(2\pi\gamma+\psi) +\frac{\beta}{2\pi}\sin\psi \right]. \tag{18} \]

Substituting (17) and (18) into (1), we obtain the relation of the coefficient \(K_{p1}\) with the parameters \(I;\ \gamma;\ K_{mi};\ \beta;\ \psi\).

Using (1), taking account of (17) and (18), on the Mir-1 digital computer, for a number of FDs the curves \(K_{p1}=f(\psi,\gamma);\ K_{p1\max}=f(K_{mi},\gamma);\ K_{p1\max}=f(\beta,\gamma)\) were calculated. Their character is shown in Fig. 3, a, b, c, and the analysis makes it possible to obtain a number of interesting conclusions and recommendations.

Fig. 3

1. The power transfer coefficient in the biasing circuit \(K_{p1}\) depends very strongly on the phase of the current of the second harmonic relative to the pulse. For example (Fig. 3, a), if \(\psi>150^\circ\), then \(K_{p1}<1\), and with pulse control even more power will be required than with continuous control. At the same time, for \(\psi=75^\circ \div 85^\circ\), the value of \(K_{p1}\) increases by several tens of times. This circumstance must be borne in mind when designing intermediate semiconductor amplifiers for pulse-biased FDs. In particular, taking into account that the phase of the second-harmonic current depends on a number of factors (degree of biasing, nature and magnitude of the load, etc.), the amplifier must be built so that the pulse phase changes in accordance with the change in the phase of the second harmonic.

2. The maximum value \(K_{p1\max}\) depends on the relative duration of the pulses \(\gamma\), and a smaller \(\gamma\) corresponds to a larger value of \(K_{p1\max}\) (Fig. 3, b). It follows from this that, in order to maintain the power gain at a sufficiently high level, it is necessary to limit the maximum pulse width and to choose the parameters of the semiconductor amplifier so that the conditions of its matching with the FD are satisfied at the smallest possible duration.

3. \(K_{p1\max}\) depends only weakly on the quantity \(\beta=\dfrac{T}{T_y}\) (Fig. 3, c), as a result of...

outside of which the curves in Figs. 3a and 3b can be used for ferromagnetic amplifiers of various powers.

1. \(K_{p1\max}\) depends very strongly on the value of the modulation coefficient of the bias current \(K_{m_i}\). Other parameters being equal, a larger \(K_{m_i}\) corresponds to a larger value of \(K_{p1\max}\). Consequently, in order to obtain a large power gain coefficient, it is necessary to have a sufficiently large value of \(K_{m_i}\).

REFERENCES

1. A. O. Gelfond. Calculus of Finite Differences. “Nauka,” 1967.

2. Ya. Z. Tsypkin. Theory of Linear Pulse Systems. GIFML, 1963.

3. V. P. Obrusnik, A. V. Kobzev. Method of pulse control, Inventor’s Certificate No. 245842, cl. 21a² 18/08. Bulletin of Inventions, No. 20, 1969.