Proceedings
of the Tomsk Polytechnic Institute named after S. M. Kirov,
Order of the October Revolution and Order of the Red Banner of Labor

Volume 243
1972

PHYSICAL PROCESSES IN FERROMAGNETIC BIASED DEVICES UNDER PULSED CONTROL

V. P. OBRUSNIK, A. V. KOBZEV

(Submitted by the Scientific and Technical Seminar of the Department of Static Converters, Research Institute of AE&M at TPI)

When constructing systems for regulation—stabilization of voltage, current, and power with power ferromagnetic biased devices (FBD) (for example, magnetic amplifiers (MA) or biased transformers (BT)), considerable attention is paid to the choice of the intermediate amplifier. In this case, owing to the interaction of these two elements [3, 4, 5], the choice of the latter largely determines such characteristics of the regulator as the power-circuit scheme, efficiency, overload and short-circuit stability, speed of response, and power gain. The authors of the present article have established that the greatest positive effect from using this interaction can be obtained if, for controlling the simplest FBD (Fig. 1, a) with one control winding, an intermediate class-D semiconductor amplifier is used, implementing the method of pulsed biasing described in detail in [1, 2]. Briefly, it consists in applying narrow, unipolar voltage pulses with twice the mains frequency to the control winding, shunted by a reverse valve, at the instants when the current in this winding is minimal. In this case the required power of the control unit is sharply reduced, the power-circuit scheme is simplified, and its efficiency is increased. Characteristics providing natural protection against overloads and short circuits are readily formed. This method of pulsed biasing of an FBD is currently used in a number of industrial installations; it provides them with high reliability and good operating qualities.

Below, an analysis is given of the physical processes, energy relations, and other characteristics of pulse-biased FBDs, with the aim of their theoretical substantiation, explaining the essence of the phenomena occurring, identifying the possibility of optimization, and obtaining practical conclusions.

The reasoning is carried out as applied to the circuit of one of the variants of an FBD in Fig. 1, a, where quantization of the biasing voltage is performed by the fast-acting switch \(K\). For this circuit, three circuits can be distinguished, each of which is characterized by a definite law of current variation: the switch circuit, in which the current \(i_y\), consumed from the biasing source, flows; the circuit of the winding \(W_y\) with the biasing current \(i_d\); and the circuit of the diode \(D\), in which the current \(i_D\) flows.

On the active resistance of the winding \(W_y\), when the current \(i_d\) flows,

power \(P_d\) is dissipated. In relation to the load power \(P_{\mathrm{н}}\), it is characterized by the power-transfer coefficient of the FCTU itself, equal to

\[ K'_p = \frac{P_{\mathrm{н}}}{P_d}. \tag{1} \]

Fig. 1

This coefficient \(K'_p\) depends on the design parameters of the FCTU and does not depend on the method of control.

From the magnetization source, a power \(P_y\) is consumed, which characterizes the power-transfer coefficient in the magnetization circuit,

\[ K_p'''=\frac{P_d}{P_y}. \tag{2} \]

The coefficient \(K_p'''\) is characteristic only for pulsed magnetization and therefore will be analyzed in more detail below.

It is obvious that the overall power gain coefficient of an MPU controlled from an intermediate amplifier will be equal to the product of \(K_p'\) and \(K_p'''\): \(K_p=K_p'\cdot K_p'''\).

Since the inclusion of diode \(D\) provides the MPU with an operating mode with free even current harmonics, the conclusions known from the theory of MAs [6] will be used in constructing the equivalent circuit. Thus, the saturation interval of core \(A\) may be represented by the circuit in Fig. 1, \(b\)—under the assumption that the numbers of turns of the working winding and of the control winding of the MPU are the same. When core \(A\) is saturated, its working winding is, as it were, short-circuited, while the unsaturated core \(B\) is shunted by the control circuit. If the condition \(R_n \ll R_y\) is adopted, then the current in the control circuit will be completely determined by the working circuit and in this half-period is equal to the load current: \(i^*=i_n\). In the next half-period the cores exchange places, and this interval may be represented by the circuit (Fig. 1, \(v\)). Here terminal \(b\) is transferred to point \(v\), while the current \(i^*=i_n\) still flows from terminal \(a\) to terminal \(b\). Thus, a rectified load current flows through the control circuit, and the working circuit may be considered, with respect to the control circuit, as a generator of even-current harmonics. The phenomena described are well reflected by the equivalent circuit of the MPU magnetization loop in Fig. 2, \(a\), where

\(E_u\) — the emf of the magnetization source,

\(R_u\) — the internal resistance of this source,

\(R_y\) — the resistance of the control winding,

\(L_y\) — the inductance of the unsaturated core with winding \(W_y\),

\(i_{2n}\) — the currents of the even-harmonic generators for \(n=1, 2, 3\ldots\)

In what follows, for the purpose of simplification and better clarification of the physical nature of the processes, only the double-frequency current generator \(i_2 \sim i_n\) is taken into consideration, and for the circuit with pulsed control the circuit in Fig. 2, \(b\), is obtained, where elements \(K\) and \(D\) have been transferred from the circuit in Fig. 1, \(a\). Here, when switch \(K\) is closed, the current flows through the magnetization source \(E_u, R_u\), and when it is open—through the return valve \(D\). Since a diode with current may be regarded as an element conducting in both directions, the circuit of Fig. 2, \(b\) is transformed into the circuit of Fig. 2, \(v\), which is adopted as the calculation circuit.

By the superposition method, the circuit in Fig. 2, \(v\) is represented by its two components: one in Fig. 2, \(g\), where only the emf source \(E_u\) acts and the terminals of the current generator are open (its internal resistance \(R_g\) is taken equal to \(\infty\)); and the other in Fig. 2, \(d\), where only the current generator acts, while the terminals of the emf source are short-circuited (its internal resistance is equal to zero).

For the circuit of Fig. 2, \(g\), the following equations may be written: for the time interval when the voltage of the control pulse is applied to winding \(W_y\), that is, switch \(K\) is in position 1, and \(0\leq \alpha \leq \alpha_n\),

\[ E_u - R_u\cdot i_y - R_y\cdot i_d - L_y\frac{di_L}{dt}=0. \tag{3} \]

Taking into account that \(\alpha=\omega t\) and in the interval under consideration \(i_y'=i_d=i_2\), one can obtain

Figure 2.

\[ \omega L_y \frac{di_L}{d\alpha}+R i_L=E_u, \tag{4} \]

where

\[ R=R_u+R_y. \]

For the pause interval, when the dc voltage source is disconnected, switch \(K\) is in position 2 and \(\alpha_u \le \alpha \le \alpha_{\mathrm{t}}\):

\[ \omega L_y \frac{di_L}{d\alpha}+R_y i_L=0. \tag{5} \]

Solving (4), (5) and matching the solutions, we obtain, for

\[ i_L=K_1 e^{-\frac{R}{\omega L_y}\alpha}+\frac{E_u}{R}, \tag{6} \]

where

\[ k_1= \frac{ \dfrac{E_u}{R} \left( e^{-\frac{R_y}{\omega L_y}\alpha_{\mathrm{t}}} \cdot e^{\frac{R_y}{\omega L_y}\alpha_u} -1 \right) }{ 1- e^{-\frac{R}{\omega L_y}\alpha_u} \cdot e^{\frac{R_y}{\omega L_y}\alpha_u} \cdot e^{-\frac{R_y}{\omega L_y}\alpha_{\mathrm{t}}} }, \tag{7} \]

For \(\alpha_u \le \alpha \le \alpha_{\mathrm{t}}\):

\[ i_L=k_2 e^{-\frac{R_y}{\omega L_y}\alpha}, \qquad i'_y=0,\quad i'_d=i_L, \tag{8} \]

where

\[ k_2= \frac{ \dfrac{E_u}{R} e^{\frac{R_y}{\omega L_y}\alpha_u} \left( 1-e^{-\frac{R}{\omega L_y}\alpha_u} \right) }{ 1- e^{-\frac{R}{\omega L_y}\alpha_u} \cdot e^{\frac{R_y}{\omega L_y}\alpha_u} e^{-\frac{R_y}{\omega L_y}\alpha_{\mathrm{t}}} }. \tag{9} \]

The power dissipated in the active resistance of the control winding will be equal to

\[ \begin{aligned} P_d &= \frac{1}{\alpha_{\mathrm{t}}} \int_0^{\alpha_{\mathrm{t}}} R_y i_d^2(\alpha)\,d\alpha = \left(\frac{E_u}{R}\right)^2 R_y \gamma + \frac{R_y}{\alpha_{\mathrm{t}}} k_1^2 \frac{\omega L_y}{2R} \left( 1-e^{-\frac{2R}{\omega L_y}\alpha_u} \right) \\ &\quad + \frac{R_y}{\alpha_{\mathrm{t}}} 2k_1\frac{E_u}{R}\cdot\frac{\omega L_y}{R} \left( 1-e^{-\frac{R}{\omega L_y}\alpha_u} \right) + \frac{R_y}{\alpha_{\mathrm{t}}} k_2^2 \frac{\omega L_y}{2R_y} \left( e^{-\frac{2R_y}{\omega L_y}\alpha_u} - e^{-\frac{2R_y}{\omega L_y}\alpha_{\mathrm{t}}} \right). \tag{10} \end{aligned} \]

Here

\[ \gamma=\frac{\alpha_u}{\alpha_{\mathrm{t}}} \]

is the relative duration of the control pulse.

The power consumed from the biasing source:

\[ P_y= \frac{1}{\alpha_{\mathrm{t}}} \int_0^{\alpha_{\mathrm{t}}} E_u \cdot i'_y(\alpha)\,d\alpha = \frac{E_u^2}{R}\gamma + \frac{E_u}{\alpha_{\mathrm{t}}}k_1 \frac{\omega L_y}{R} \left( 1-e^{-\frac{R}{\omega L_y}\alpha_u} \right). \tag{11} \]

For clarity of analysis and conclusions the assumptions adopted are: \(L_y \to \infty\); \(R_u=0\). Then

\[ i'_y=i'_d=i_L=\lim_{L_y\to\infty} k_1\cdot e^{-\frac{R_y}{\omega L_y}\alpha} +\lim_{L_y\to\infty}\frac{E_u}{R_y} =\frac{E_u}{R}\gamma;\quad 0\leq \alpha\leq \alpha_u \tag{12} \]

\[ \left. \begin{aligned} i'_y&=0,\\ i'_d&=i_L=\lim_{L_y\to\infty} k_2 e^{-\frac{R_y\alpha}{\omega L_y}} =\frac{E_u}{R}\gamma \end{aligned} \right\} \quad \alpha_u\leq \alpha\leq \alpha_\tau . \tag{12a} \]

\[ \left. \begin{aligned} P_d&=\lim_{L_y\to\infty} P_d(L_y)=\frac{E_u^2}{R}\gamma^2,\\ P_y&=\lim_{L_p\to\infty} P_y(L_y)=\frac{E_u^2}{R}\gamma^2 \end{aligned} \right\}. \tag{13} \]

As was to be expected, in the absence of the current generator and with \(R_u=0\), the power consumed from the bias source \(P_y\) is equal to the power dissipated in the active resistance of the control winding \(P_d\), that is, the power-transfer coefficient in the bias circuit is equal to unity.

In the circuit of Fig. 2, \(d\), at double frequency \(W L_y\gg R\) always, and in the interval \(0\leq \alpha\leq \alpha_u\),

\[ i''_y=i''_d=I_m\sin(\alpha+\psi),\quad i_L=0, \tag{14} \]

and in the pause interval, when \(\alpha_u\leq \alpha\leq \alpha_\tau\), one can obtain

\[ i_L=0,\quad i''_y=0,\quad i''_d=I_m\sin(\alpha+\psi). \tag{15} \]

Then, by the method of superposition: \(i_y=i'_y+i''_y;\quad i_d=i'_d+i''_d\).

\[ i_d=i_y=k_1 e^{\frac{R}{\omega L_y}\alpha}+\frac{E_u}{R}+I_m\sin(\alpha+\psi) \tag{16} \]

for \(0\leq \alpha\leq \alpha_u\),

\[ i_y=0;\quad i_d=k_2 e^{-\frac{R}{\omega L_y}\alpha_u}+I_m\sin(\alpha+\psi) \tag{17} \]

for \(\alpha_u\leq \alpha\leq \alpha_\tau\).

\[ P_y=\frac{1}{\alpha_\tau}\int_0^{\alpha_\tau} E_u\cdot i_y(\alpha)d\alpha =\frac{E_u^2}{R}\gamma+\frac{E_u}{\alpha_\tau}k_1\frac{\omega L_y}{R} \left(1-e^{-\frac{R}{\omega L_y}\alpha_u}\right) - \]

\[ -\frac{E_u}{\alpha_\tau}I_m[\cos(\alpha_u+\psi)-\cos\psi]. \tag{18} \]

\[ P_d=\frac{E_u^2}{R^2}\cdot R_y\cdot \gamma+\frac{1}{2}R_y I_m^2 +\frac{R_y}{\alpha_\tau}k_1^2\frac{\omega L_y}{2R} \left(1-e^{-\frac{2R}{\omega L_y}\alpha_u}\right)+ \]

\[ +\frac{R_y}{\alpha_\tau}\cdot\frac{E_u}{R}\,2k_1\frac{\omega L_y}{R} \left(1-e^{-\frac{R}{\omega L_y}\alpha_u}\right) -\frac{R_y}{\alpha_\tau}2k_1 I_m\cdot \frac{e^{-\frac{R}{\omega L_y}\alpha_u}} {\left(\frac{R}{\omega L_y}\right)^2+1}. \]

\[ \begin{aligned} &\left[\frac{R}{\omega L_y}\sin(a_{\mathrm{u}}+\psi)+\cos(a_{\mathrm{u}}+\psi)\right] +\frac{R_y}{\alpha_{\mathrm{T}}}\,2k_1 I_{\mathrm{m}}\, \frac{1}{\left(\dfrac{R}{\omega L_y}\right)^2+1}\times \\[4pt] &\quad{}\times \left[\frac{R}{\omega L_y}\sin\psi+\cos\psi\right] -\frac{R_y}{\alpha_{\mathrm{T}}}\cdot\frac{E_{\mathrm{u}}}{R}\cdot 2I_{\mathrm{m}}\cos(a_{\mathrm{u}}+\psi) +\frac{R_y}{\alpha_{\mathrm{T}}}\cdot\frac{E_{\mathrm{u}}}{R}\,2I_{\mathrm{m}}\cos\psi + \\[4pt] &\quad{}+ \frac{R_y}{\alpha_{\mathrm{T}}}k_2^{\prime}\frac{\omega L_y}{2R_y} \left(e^{-\frac{2R_y}{\omega L_y}a_{\mathrm{u}}} -e^{-\frac{2R_y}{\omega L_y}\alpha_{\mathrm{T}}}\right) \frac{R_y}{\alpha_{\mathrm{T}}}\,2k_2 I_{\mathrm{m}}\, \frac{e^{\frac{R_y}{\omega L_y}\alpha_{\mathrm{T}}}} {\left(\dfrac{R_y}{\omega L_y}\right)^2+1}\times \\[4pt] &\quad{}\times \left[\frac{R_y}{\omega L_y}\sin(a_{\mathrm{T}}+\psi)+\cos(a_{\mathrm{T}}+\psi)\right] +\frac{R_y}{\alpha_{\mathrm{T}}}\,2k_2 I_{\mathrm{m}}\, \frac{e^{-\frac{R_y}{\omega L_y}a_{\mathrm{u}}}} {\left(\dfrac{R_y}{\omega L_y}\right)^2+1}\times \\[4pt] &\quad{}\times \left[\frac{R_y}{\omega L_y}\sin a_{\mathrm{u}}+\psi)+\cos(a_{\mathrm{u}}+\psi)\right]. \end{aligned} \tag{19} \]

For further analysis of the processes, the relation is established between the maximum value of the current of the second harmonic \(I_{\mathrm{m}}\) and the average value of the bias-winding current \(I_{d\mathrm{cp}}\):

\[ I_{d\mathrm{cp}}=\frac{1}{\alpha_{\mathrm{T}}}\int_{0}^{\alpha_{\mathrm{T}}} i_d(\alpha)\,d\alpha, \tag{20} \]

\[ I_{\mathrm{m}}=K_{\mathrm{M}2}\cdot I_{d\mathrm{cp}}, \tag{21} \]

where

\[ K_{\mathrm{M}2} \]

is the coefficient of modulation of the bias current by the second harmonic; in the general case it depends nonlinearly on \(I_{d\mathrm{cp}}\).

From expressions (16) and (17), taking into account the assumptions made above, one can find:

for \(0\le \alpha\le a_{\mathrm{u}}\)

\[ i_d=\lim_{L_y\to\infty} i_d(L_y)=\frac{E_{\mathrm{u}}}{R}\gamma+I_{\mathrm{m}}\sin(\alpha+\psi); \tag{22a} \]

for \(a_{\mathrm{u}}\le \alpha\le \alpha_{\mathrm{T}}\)

\[ i_d=\lim i_d(L_y)=\frac{E_{\mathrm{u}}}{R}\gamma+I_{\mathrm{m}}\sin(\alpha+\psi). \tag{22b} \]

\[ \alpha_y\to\infty. \]

Then

\[ I_{d\mathrm{cp}}=\frac{1}{\alpha_{\mathrm{T}}}\int_{0}^{\alpha_{\mathrm{T}}} i_d(\alpha)d\alpha =\frac{E_{\mathrm{u}}}{R}\gamma. \tag{23} \]

As is seen from (23), the average value of the bias current \(I_{d\mathrm{cp}}\) does not depend on the degree of its modulation, but depends only on the circuit parameters and the control level, and expression (21) takes the form

\[ I_{\mathrm{m}}=K_{\mathrm{M}2}\cdot\frac{E_{\mathrm{u}}}{R}\gamma. \tag{24} \]

Taking into account (24), as well as the assumptions made that \(R_{\mathrm{u}}=0\), \(L_y\to\infty\), from expressions (18, 19) one can obtain convenient calculation dependences:

\[ P_y=\lim_{L_y\to\infty}P_y(L_y)=\frac{E_u^2}{R}\gamma^2-K_{M2}\frac{E_u^2}{R\alpha_T}\gamma\,[\cos(\alpha_u+\psi)-\cos\psi], \tag{25} \]

\[ P_d=\lim_{L_y\to\infty}P_d(L_y)=\frac{E_u^2}{R}\gamma^2\left(1+\frac{1}{2}K_{M2}^2\right). \tag{26} \]

Figure 3 presents time diagrams of currents in a circuit with pulsed magnetization, constructed from expressions (16, 17). The calculated currents \(i_d\), \(i_y\), and \(i_l=i_d\cdot i_y\) in this figure are very close to the time dia-

Fig. 3

oscillograms obtained experimentally in actual FBCDs, which confirms the correctness of the calculation scheme of Fig. 2, c.

Analysis of expressions (25), (26) makes it possible to draw a number of interesting conclusions.

Thus, if \(\gamma = 1\), \(a_u = a_\tau\), i.e., discrete control becomes continuous, then

\[ P_y = \frac{E_u^2}{R}, \]

\[ P_d = \frac{E_u^2}{R}\left(1+\frac{1}{2}K_{M2}^2\right) \quad \text{and} \quad k_p''=\frac{P_d}{P_y}=1+\frac{1}{2}K_{M2}^2. \]

It follows from this that if the modulation coefficient of the bias current \(K_{M2}\) is equal to 1, then the power-transfer coefficient in the bias circuit \(K_p''\) will be equal to 1.5.

If, however, \(K_M = 0\), which occurs under very strong saturation of the cores by a constant field, then \(K_p'' = 1\). Obviously, under continuous biasing the energy relations in the control circuit do not depend on the phase of the even-harmonic currents.

When the modulation coefficient is varied, the greatest possible range of variation of \(K_p''\) is from 1 to 1.5. Below, in order to assess the advantages of pulsed biasing, it is necessary to compare the values of \(K_p''\) characterizing this method with \(K_p'' = 1 \div 1.5\).

In Fig. 4 are shown the dependences \(k_p'' = f(\gamma; \Psi)\), constructed from expressions (25), (26) for the modulation coefficient \(k_{M2}=1\), and in Fig. 5—the dependence \(k_{p\max}'' = f(k_{M2})\), for \(\gamma = 1/12\) \(\left(a_u = \frac{\pi}{6}\right)\). Analysis of the curves in Figs. 4, 5 makes it possible to draw several more important conclusions.

In the figure: vertical axis \(k_p''\); horizontal axis \(\Psi\). Curve labels: \(\gamma = \frac{1}{24}\), \(\gamma = \frac{1}{12}\), \(\gamma = \frac{1}{6}\); annotation \(k_{p\max}'' = 338\).

Fig. 4

1. The power-transfer coefficient in the bias circuit \(k_p''\) depends strongly on the phase of the second-harmonic current relative to the pul-

In particular, if $\Psi = 0 \div 150^\circ$ (Fig. 4), then $K''_p$ under pulsed biasing turns out to be even smaller than under continuous biasing ($K''_p < 1.5$). At the same time, for $\Psi = 240 \div 265^\circ$, $k''_p$ increases by several tens of times.

Fig. 5

This circumstance must be taken into account when designing intermediate amplifiers for pulsed control of an FPU. In particular, taking into account that the phase of the second harmonic of the current depends on a number of factors (the degree of biasing, the nature and magnitude of the load, etc.), the amplifier must be designed so that the phase of the pulse also changes in accordance with the change in the phase of the second harmonic of the current.

1. The maximum value of the coefficient $k''_p$, and consequently also of the overall power gain coefficient of the FPU, $k''_p$, depends very strongly on the pulse width. Thus, for

\[ \alpha_u = \frac{\pi}{12} \qquad \left(\gamma = \frac{1}{24}\right) \]

\[ K''_{p\max} = 338, \]

(Fig. 4), while for

\[ \alpha_u = \frac{\pi}{3} \qquad \left(\gamma = \frac{1}{6}\right) \qquad k''_{p\max} = 34.7. \]

It follows from this that, in order to maintain the power gain coefficient at a sufficiently high level, it is necessary to limit the maximum pulse width.

Since a certain average voltage value $U_y = E_u \cdot \gamma$ is required for controlling an FPU, this important conclusion may be formulated somewhat differently: under pulsed control of an FPU it is very advantageous to sharply increase the voltage of the power supply source $E_u$. And since in this case the maximum $\gamma$ required by the regulation conditions decreases in the same proportion, $k''_{p\max}$ increases sharply. Along with the reduction in the power of the control unit, the currents in the key circuit are also reduced, which is an additional positive factor and makes it possible to considerably reduce the dimensions and weight of the intermediate amplifier.

1. The maximum value of the coefficient $k''_p$ changes sharply when the modulation coefficient of the biasing current $K_{m2}$ is changed (Fig. 5).

For example, for $\gamma = \frac{1}{12}$ and $K_{m2} = 1$, $k''_p$ will be equal to 115.5, but already for

\[ K_{m2} = 0.9,\quad k''_p = 12.7, \]

and for $K_{m2} = 0.5$, $k''_p = 2.22$.

The question of finding the optimal current modulation coefficient has not yet been sufficiently studied, but the available experimental data show that, in the nominal mode, for most designed FPUs \(K_{\mathrm{m}2}\) lies within the range \(0.85 \div 0.9\), which, with an appropriate limitation of the pulse duty factor and selection of the pulse phase within the range \((240 \div 260^\circ)\), gives a gain in the power amplification factor under pulsed control, compared with continuous control, by a factor of \(5 \div 10\) or more.

The analysis carried out of the physical processes under pulsed biasing made it possible to give a number of important practical recommendations, the use of which in the design of intermediate amplifiers will improve the characteristics of FPUs and expand the possibilities for applying the pulsed method of biasing in automatic installations for various practical purposes.

References

1. V. P. Obrusnik, A. V. Kobzev. Method of pulsed control. Inventor’s certificate No. 245842, class 21a\(^2\)18/08. Bulletin of Inventions, No. 20, 1969.

2. V. P. Obrusnik, A. V. Kobzev. Operating modes and characteristics of single-phase TRPSh under pulsed biasing. Reports of the Sixth Scientific and Technical Conference on Problems of Production Automation. Vol. II, TTU Publishing House, Tomsk, 1969.

3. V. P. Obrusnik, A. V. Kobzev. Intermediate amplifiers for biased regulators. Reports of the Sixth Scientific and Technical Conference on Problems of Production Automation. Vol. II, TTU Publishing House, Tomsk, 1969.

4. A. M. Baimas, S. V. Shapiro. Voltage and current stabilizers with biased transformers. Energiya, 1965.

5. E. S. Achildieva-Frolova. Processes in control circuits of transformers regulated by changes in biasing. Author’s abstract of candidate dissertation. Gorky, 1968.

6. M. A. Rozenblat. Magnetic elements of automation and computer engineering. Nauka, 1966.