DISCRETE CONTROL OF FERROMAGNETIC DEVICES WITHOUT DC BIASING

A. V. KOZEV, V. P. OBRUSNIK, M. G. SAVKOV

(Presented by the scientific and technical seminar of the Department of Static Converters of the Research Institute of AEM at TPI)

Often the properties of semiconductor and ferromagnetic devices complement each other very successfully. In most cases, the use of semiconductor-magnetic devices in regulation systems—voltage stabilization—makes it possible to solve the technical problem most simply, reduce weight and dimensions, and increase reliability [2, 3]. An example of the simplest of them is shown in Fig. 1, a. In [2, 3] these devices are called thyristor-magnetic amplifiers, TMU. A characteristic feature of such apparatus is the absence of DC biasing and discrete control by periodic closing—opening, by means of a fast-acting switch \(K\), of an additional winding or part of the working winding.

The basic properties and characteristics of thyristor-magnetic devices depend to a considerable extent on the law of control of the switch \(K\). The most widespread method is one in which the switch (thyristor or transistor) is turned on with an adjustable delay in angle relative to the half-wave of the supply voltage (see \(\alpha_k\) in Fig. 2) and “naturally extinguishes” when the voltage passes through zero. In the other half-period the choke is naturally magnetized and saturates at the angle \(\beta_k = \alpha_k + \pi\). In the load there is then formed a voltage whose diagram for an arbitrary angle \(\alpha_k\) is shown in Fig. 2.

The amplitude \(A_{1m}\) and phase \(\varphi_1\) of the first harmonic of this voltage for different switching angles \(\alpha_k\) are determined by the expressions:

\[ A_{1m} = \frac{U_{2m}}{\pi} \sqrt{(\pi-\alpha_k)^2 + (\pi-\alpha_k)\sin 2\alpha_k - \sin^2 \alpha_k}, \tag{1} \]

\[ \varphi_1 = \operatorname{arc\,tg} \frac{-\sin^2 \alpha_k} {\pi-\alpha_k+\frac{1}{2}\sin 2\alpha_k}. \tag{2} \]

Here

\(U_{2m}\) is the amplitude of the sinusoidal output voltage in the mode of maximum output \((\alpha_k = 0)\).

Analysis of expression (2) shows that when \(\alpha_k\) changes, the angle \(\varphi_1\) necessarily changes. For many installations, however, it is necessary that the output voltage of the regulator not change in phase. Such a need arises, for example, in three-phase static voltage converters operating on an unbalanced lo—

Fig. 1

Fig. 2

loads, with their phase-by-phase stabilization [4], in the construction of telemechanical voltage regulators-stabilizers, in amplitude regulation of the speed of asynchronous two-phase motors, etc.

The authors have established that the desired result can be achieved practically without any special complication of the device if the ferromagnetic regulator is controlled in accordance with the method described in detail in [1]. In brief, it consists in the fact that the control winding of the ferromagnetic device is short-circuited by a semiconductor switch at the beginning of each period for a time not exceeding a half-period.

Below an analysis is given of the physical processes and of the principal quantities for ferromagnetic devices with discrete control, the circuit variants of which are shown in Fig. 1, while control is carried out by the method of [1].

In Fig. 1, a an electrical circuit of a known thyristor-magnetic amplifier is presented. Here a nonlinear choke, consisting of a single core with a winding, is connected in series with the load, and the semiconductor switch \(K\) closes and opens a portion of the choke turns (a tapping of the winding turns is needed only in the case when the permissible voltage of the switching element is less than the line voltage). If it is necessary to match the load voltage to the supply without galvanic isolation, the circuit in Fig. 1, b may be used. Here there are already two magnetic cores, the main core \(I\) and the shunt \(III\) (they may be structurally combined), and there are transformer-coupled windings \(W_2\) and \(W_1\). An important advantage of this circuit is the possibility of regulating, within certain limits, the output voltage at idle—

at no load. The circuit in Fig. 1, c has the same capabilities, but here galvanic isolation of all three circuits is additionally provided: the load, the switching element, and the network. Windings $W_2$ and $W_y$ are best placed in the developed window, enclosing the primary winding $W_1$, but when the secondary winding is the high-current one, then, in order to improve its cooling conditions, it may be made enclosing, as is shown in Fig. 1, d. However, if cores $\Gamma$ and $\mathrm{Ш}$ are not joined, then with such a circuit it is no longer possible to regulate the output voltage at no load.

Despite the variety of circuit and structural variants of semiconductor-magnetic regulators (SMR), the processes in them proceed practically identically.

Below, primary attention should be paid to the specifics of control by the method of [1]. It is assumed that the magnetization curve of the cores is ideal, the active and inductive resistances due to leakage fluxes are zero, the switch is ideal, and the load is active. All physical processes in an SMR are explained by the circuit in Fig. 1, c.

It is convenient to begin consideration of the essence of discrete control of an SMR according to [1] from the instant $\alpha = 0$, when switch $K$ is closed, the induction in the shunt does not change and is equal to $+\mathrm{B_s}$ (Fig. 3, b). The output voltage in this case is determined by the transformation ratio:

$$ U_{\mathrm{н}} = U_1 \frac{W_2}{W_1}. \tag{3} $$

Since the control circuit is transformer-coupled with the working windings, a current flows in it

$$ I_y = I_{\mathrm{н}} \frac{W_2}{W_y}. \tag{4} $$

At the time $\alpha = \alpha_k + 2\pi \cdot n$ the switch $K$ opens. The load current and the current in the control winding become equal to zero, and the shunt core begins to be remagnetized by the primary voltage:

$$ U_1 = W_1 S_{\mathrm{ш}} \frac{dB_{\mathrm{ш}}}{dt}, \tag{5} $$

$$ \Delta B_{\mathrm{ш}} = \frac{1}{W_1 S_{\mathrm{ш}}} \int_{\alpha_k}^{\pi} U_1 dt. \tag{6} $$

At $\alpha = \beta_k$ it becomes saturated (Fig. 3, b), and a voltage appears at the output

$$ U_{\mathrm{н}} = U_1 \frac{W_2}{W_1}, $$

whose instantaneous value is again determined by the transformation ratio. In the saturated state the core will remain until the end of this half-period. During magnetization

$$ \Delta B_{\mathrm{ш}} = \frac{1}{W_1 S_{\mathrm{ш}}} \int_{\pi}^{\beta_k} U_1 dt. \tag{7} $$

Since the total change of induction over a period is zero, one may obtain

Fig. 3.

\[ \cos \alpha_{\mathrm{k}}=\cos \beta_{\mathrm{k}} \quad \text{and} \quad \beta_{\mathrm{k}}=2\pi\cdot n-\alpha_{\mathrm{k}} . \tag{8} \]

As a result of such forced-natural remagnetization, a voltage is formed at the output (Fig. 3, c), whose first harmonic has no phase shift relative to the line voltage.

The coefficients and phase shifts of the first and other harmonics can be found by expanding the curve (Fig. 3, c) into a harmonic series:

\[ S(\alpha)=\frac{a_0}{2}+A_1\sin(\alpha+\psi_1)+A_2\sin(2\alpha+\psi_2)+\ldots \tag{9} \]

\[ A_n=\sqrt{a_n^2+b_n^2} \quad \text{and} \quad \tg \psi_n=\frac{a_n}{b_n}, \tag{10} \]

where \(a_n\) and \(b_n\) are the Fourier coefficients of the given function.

Since the function (Fig. 3, c) is odd, that is

\[ f(-\alpha)=-f(\alpha), \]

then \(a_n=0\)

\[ b_n=\frac{2}{\pi}\int_{0}^{\pi} f(\alpha)\sin n\alpha\,d\alpha . \tag{11} \]

It follows from this that, for all existing harmonics,

\[ b_n\ne 0;\quad a_n=0;\quad \tg\psi_n=0 \quad \text{and} \quad \psi_n=0, \]

that is, all harmonics of the output voltage have no phase shift relative to the line voltage.

The amplitude of the harmonic components can be found from the general expression:

\[ A_n=\frac{U_{2\mathrm{m}}}{\pi} \left\{ [-\sin(1+n)\alpha_{\mathrm{k}}]\cdot\frac{1}{1+n} + [\sin(1-n)\alpha_{\mathrm{k}}]\frac{1}{1-n} \right\}, \tag{12} \]

where \(A_n\) is the amplitude of the \(n\)-th harmonic of the output voltage,
\(n\) is the harmonic number,
\(\alpha_k\) is the switching angle of the key.

The presence of the third and especially the second harmonics leads to a deterioration of the power factor \(K_{\mathrm{m}}\):

\[ K_{\mathrm{m}}=\cos \psi_1 \cdot \pi, \tag{13} \]

where \(\psi_1\) is the phase-shift angle of the first harmonic.

It should be noted that, since \(\cos \psi_1 = 1\), \(k_{\mathrm{m}}\) is determined by the distortion factor, which may be found from the known expression:

\[ \lambda = \frac{2mI}{\sqrt{U_{2\mathrm{mI}}^{2}+U_{2\mathrm{mII}}^{2}+\ldots+U_{2\mathrm{mn}}^{2}}} \cdot \frac{U_{2\mathrm{mI}}}{\phantom{\sqrt{U_{2\mathrm{mI}}^{2}+U_{2\mathrm{mII}}^{2}+\ldots+U_{2\mathrm{mn}}^{2}}}}, \tag{14} \]

Fig. 4

Fig. 5

Using expressions (12), (14), Fig. 4 plots the dependence \(\lambda=f(\alpha_k)\). As can be seen, the distortion factor \(\lambda\) varies over wide limits as the switching angle changes: from \(\lambda=1\) in the maximum-output mode \((\alpha_k=\pi)\) to \(\lambda<0.1\) at \(\alpha_k \approx 0\). Accordingly, the power factor

changes within the same limits; it becomes especially poor under deep control. However, the comparison that was carried out of the method proposed in [1] with the known methods, where the angle \(\psi\) is lagging and varies during control, shows that in the compared variants the numerical values of the power factor \(k_{\mathrm{M}}\) are practically identical over the entire control range. If, however, the additional effect of stabilizing the phase of the output voltage in the first case is taken into account, then the prospects become evident for using PMR controlled by the method of [1] in various automation devices, including high-power ones. It is often necessary to know the nonlinear-distortion factor \(k_{\mathrm{f}}\) at the output of the regulator. In this case

\[ k_{\mathrm{f}}= \frac{ \sqrt{\sum_{n=2}^{\infty} U_{2mn}^{2}} }{ U_{2m1} } \tag{15} \]

From expressions (14) and (15) one can establish the relation between \(k_{\mathrm{f}}\) and \(\lambda\):

\[ k_{\mathrm{f}}=\sqrt{\frac{1}{\lambda^{2}}-1}. \tag{16} \]

Using (16) and the dependence \(\lambda=f(\alpha_{\mathrm{k}})\) (Fig. 4), the curve \(k_{\mathrm{f}}=f(\alpha_{\mathrm{k}})\) is also constructed here. Such a large value of the nonlinear-distortion factor under deep control (\(k_{\mathrm{f}}>3\)) (Fig. 4) is not always permissible, especially in telemechanics devices. Therefore the problem often arises of reducing \(k_{\mathrm{f}}\), which in the present case reduces to the problem of filtering the output voltage. The most successful is the use of a resonant filter according to the circuit of Fig. 8 [5]. In [5] a convenient engineering procedure is given for calculating such a filter, where it is necessary to use design curves that depend on the waveform of the filter output voltage, and these curves are presented for a rectangular voltage. Below are given expressions for analogous curves, taking into account the waveform of the output voltage of the PMR controlled by the method of [1]. The nonlinear-distortion factor at the filter output can be determined as

\[ k_{\mathrm{f1}}= \frac{ \sqrt{\sum_{n=2}^{\infty}\left(k_{un}\cdot U_{2mn}\right)^{2}} }{ k_{u1}\cdot U_{2m1} }. \tag{17} \]

Here: \(k_{un}\) is the transmission factor of the filter at the \(n\)-th harmonic. In [5] the relation is established between the filter transmission factor \(k_u\), the characteristic impedances of the series and parallel circuits, and the harmonic number:

\[ k_u=\frac{1}{1-\dfrac{\rho_1}{\rho_2}\Delta^2}, \tag{18} \]

where

\[ \rho_1=\sqrt{\frac{L_1}{c_1}} \quad\text{— characteristic impedance of the series resonant} \]

\[ \text{circuit.} \]

\[ \rho_2=\sqrt{\frac{L_2}{c_2}} \quad\text{— characteristic impedance of the parallel resonant} \]

\[ \text{circuit.} \]

\[ \Delta=n-\frac{1}{n}, \quad \text{where } n \text{ is the harmonic number.} \]

Using (12), (17), (18), one can obtain

\[ k_f = \frac{ \displaystyle \sum_{n=2}^{\infty} \left[ \frac{1}{1+n}\,\sin(1+n)\alpha_k + \frac{1}{1-n}\,\sin(1-n)\alpha_k \right] \left( \frac{1}{1-\dfrac{\rho_1}{\rho_2}\Delta^2} \right)^2 }{ \displaystyle \alpha_k - \frac{1}{2}\sin 2\alpha_k }. \tag{19} \]

From expression (19), in Fig. 5 a family of curves has been constructed:

\[ k_f = f\!\left(\frac{\rho_1}{\rho_2},\,\alpha_k\right). \]

Here \(\alpha_k\) is the parameter; curve 1 corresponds to

\[ \alpha_k=\frac{\pi}{12},\quad 2\ —\ \alpha_k=\frac{\pi}{6},\quad 3\ —\ \alpha_k=\frac{\pi}{4},\quad 4\ —\ \alpha_k=\frac{\pi}{3}, \]

\[ 5\ —\ \alpha_k=\frac{\pi}{2},\quad 6\ —\ \alpha_k=\frac{2\pi}{3},\quad 7\ —\ \alpha_k=\frac{5\pi}{6}. \]

Fig. 6.

Fig. 7.

Using the calculated curves of Fig. 5, by the method of [5], one can calculate the filter shown in Fig. 8. Since, in a correctly designed filter, at the fundamental harmonic it provides an active load reaction for the regulator, the filter introduces no additional phase shifts.

Fig. 8

The experimental data well confirm the theoretical assumptions made in the paper. Thus, Fig. 6 gives the dependence \(\cos\psi = f(\alpha_k)\) at a fixed load resistance. Variation of \(R_{\mathrm{n}}\) does not have a substantial effect on the characteristic. A certain deviation from the theoretical dependence is observed at \(\alpha_k = 0 \div \frac{\pi}{12}\), which is explained by the additional influence of the leakage inductive resistances of the operating windings and by the residual inductance of the core.

Figure 7 gives the dependence \(\cos\psi = f(I_{\mathrm{n}})\), taken at a fixed switching angle \(\alpha_k\). Practically over the entire range of variation of the load current, \(\cos\psi\) remains constant and has a certain deviation from the theoretical dependence only at a load close to no-load operation, when the magnetizing current of the SMR has a significant influence.

Semiconductor-magnetic regulators under discrete control have high speed of response. They are practically inertia-free and are characterized, like all systems with discrete control, by a mean statistical delay, which in this case is equal to a half-period.

The analysis carried out shows that control of an SMR by the method of [1] provides them with a new quality: natural stabilization of the phase of the output voltage over the entire control range while preserving the known positive properties of such devices, which expands the possibilities for using semiconductor-magnetic regulators in automatic installations for various purposes.

REFERENCES

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

2. M. A. Rosenblat. Magnetic Elements of Automation and Computer Engineering. “Nauka,” 1966.

3. V. N. Ivanchuk, R. A. Lipman, B. Ya. Ruvinov. Thyristor Amplifiers in Electric-Drive Circuits. “Energiya,” 1966.

4. G. F. Andreev, A. I. Chernyshev. Phase-by-phase stabilization of the voltage of three-phase static converters. In: Electronic and Electromechanical Devices, “Energiya,” 1969.

5. A. I. Chernyshev. Some problems in calculating filters of the output voltage of static converters. In: Electronic and Electromechanical Devices, “Energiya,” 1969.