Analysis of Stationary Processes in a Ferroresonant Dissipative Series Circuit

A. I. Kutyavin

(Presented by the scientific seminar of the Department of TOE)

I. When considering processes in a dissipative ferroresonant circuit, as is known, it is necessary to take account of losses in the steel. In this case, in the presence of constant bias magnetization, determining the regions of existence of relay characteristics of various types is difficult and is practically possible only after a numerical solution of the problem. Meanwhile, on the basis of the theory of branching of solutions of nonlinear equations [1], they can be found without resorting to a numerical solution of the entire problem. According to this theory, an equation may have more than one solution if, in the problem of implicit functions, a partial derivative or a Jacobian becomes zero at the corresponding point, which is true on the stability boundary [2]. On the stability boundary, the equation of the system may be replaced by branching equations, which make it possible to obtain the regions of existence of relay characteristics much more simply than the original equation.

Fig. 1. Equivalent circuit of the circuit and resonance characteristics with variation of constant bias magnetization and fixed supply voltages

The aim of the present work is to analyze stationary processes in a dissipative ferroresonant circuit by the proposed quadratic-form method, which is based on branching theory. Before proceeding to it, let us establish the relations that follow from the equations of harmonic and energy balance when the circuit operates at the points of relay effects.

II. Consider the equivalent circuit of the ferroresonant circuit (Fig. 1), assuming that the fundamental harmonic cannot flow through the constant-bias circuit. The harmonic-balance equation then takes the form

\[ (-U_L + U_C)^2 + (U'_R + U_r)^2 = U^2 . \tag{1} \]

The condition for the existence of relay effects is the equality

\[ \frac{\partial U}{\partial I}=0, \]

which follows from the Routh–Hurwitz criterion [2], that is

\[ (-x+x_c)(-x_d+x_c)+R_\Sigma R_{\Sigma d}=0, \tag{2} \]

where \(R_\Sigma=R+r\), \(R_{\Sigma d}=R_d+r\), and \(x, x_d\) are the static and differential inductive resistances of the choke;

\(R, R_d\) are the static and differential active resistances of the choke, caused by losses in the steel;

\(r\) is the resistance of the load and of the choke copper. If equality (2) is solved with respect to \(x_c\), we obtain an equation determining the locus of points at which jump-like changes in current begin [4]:

\[ x_c=0.5(x+x_d)\pm \sqrt{0.25(x-x_d)^2-R_\Sigma R_{\Sigma d}}. \tag{3} \]

The critical load resistance, a slight increase of which leads to the absence of the bistable zone (relay effects), can be found from (3) by setting the expression under the radical equal to zero,

\[ r_{\mathrm{cr}}=\sqrt{0.25(R-R_d)^2+(x-x_{\mathrm{cr}})^2}-0.5(R+R_d), \tag{4} \]

where \(x_{c.\mathrm{cr}}\) is the critical capacitive reactance,

\[ x_{c\mathrm{cr}}=0.5(x+x_d). \tag{5} \]

Consideration of the energy-balance equation gives no new, additional information

\[ (-Q_L+Q_C)^2+P^2=S^2. \tag{6} \]

Expressions (1–5), in analyzing a circuit with a symmetric nonlinear characteristic, make it possible to obtain the parameters with an error admissible for engineering practice [4], which is partially consistent with the data of [3]. In the presence of constant biasing, the use of expressions (1–5) is difficult; therefore, in order to obtain new information, we shall consider the energy-balance equation as a quadratic form.

III. According to [5], the extremal values of a quadratic form coincide with the roots of the Lagrange equation and can be obtained without calculating the coordinates of the points at which they are attained. But in the case of the canonical form of the quadratic form, the roots enter directly into (6); therefore we shall try to replace (6) by another quadratic form, more convenient for analysis. We shall proceed from the fact that if two quadratic forms have one and the same invariant (discriminant), then their roots are identical; therefore one may replace the expression

\[ Q=-Q_L+Q_C=\pm \sqrt{S^2-P^2} \]

by the formal branching equations

\[ \left. \begin{aligned} Q&=\lambda+S,\\ f(\lambda,I)&=\lambda^2+2S\lambda+P^2=0 \end{aligned} \right\}. \tag{7} \]

We shall call \(\lambda=\psi(I_0)\) the response of the oscillatory circuit to the control signal, i.e., the dc bias current. We shall be interested only in those values of the roots of the equation for which the quadratic form attains extrema. Two cases may be distinguished: 1) the output signal is maximal; 2) the conditions determining the critical parameters of the system, when operation takes place at the stability boundary.

1. The output signal in a dissipative system is customarily considered maximal at \(U=R_\Sigma I\), that is, when the derivative with respect to the control signal vanishes (Fig. 1, curve II). Since the roots of the quadratic equation in (7) are a function of the control signal, the behavior of the system will obviously be characterized by the equalities

\[ \left. \begin{aligned} \frac{\partial f}{\partial \lambda} &= \lambda+S=0,\\[4pt] \frac{\partial f}{\partial I_0} &= (\lambda+S)\lambda+P\frac{\partial U_R'}{\partial I_0}=0. \end{aligned} \right\} \tag{8} \]

They indicate that, when the resonance curve reaches a maximum, one more relation is valid,

\[ \frac{\partial U_R'}{\partial I_0}=0, \]

which holds when the active component of the voltage across the choke, due to losses in the steel, is also extremal (Fig. 2).

Fig. 2. Dependence of the active component of the voltage across the choke under simultaneous magnetization by constant and alternating fields

1. The critical parameters of the system in the general case will depend on the nature of the variation of the active resistance of the circuit. Of greatest interest are the cases when the resistance is constant, and also when the total differential active resistance may be positive or negative. At the stability boundary the critical parameters will obviously be determined from the expression

\[ \frac{\partial f}{\partial I}=0, \]

i.e., when

\[ \lambda_1=-\frac{R_\Sigma^{1/3}(R_{\Sigma d}+R_\Sigma)}{U}. \tag{9} \]

In the case \(R_\Sigma=\mathrm{const}\)

\[ \lambda_1=-\frac{2R_\Sigma^{1/3}}{U}. \tag{10} \]

Substitution of (10) into the quadratic equation gives \(P=\sqrt{0.75}\,S\). Then \(\lambda_1=-1.5S,\ Q=-0.5S\). By Vieta’s theorem, the roots of the reduced quadratic equation are related to its coefficients by the following relations:

\[ \lambda_1+\lambda_2=-2S,\qquad \lambda_1\lambda_2=P^2. \]

As a result of the solution for \(R_\Sigma=\mathrm{const}\), we have:

\[ \left. \begin{aligned} Q&=-0.5S=\lambda_2,\qquad P=\sqrt{0.75}\,S=\sqrt{\lambda_1\lambda_2},\\[4pt] \text{and}\quad \varphi&=\operatorname{arctg}\frac{-x+x_c}{R_\Sigma}=-\frac{\pi}{6}. \end{aligned} \right\} \tag{11} \]

For \(R_{\Sigma d}>0\), from (2) and (5) we obtain that

\[ R_\Sigma R_{\Sigma d}=(-x+x_{c\,\mathrm{cr}})^2. \]

Then

\[ \lambda_1=-S,\qquad Q=0,\qquad \varphi=0. \]

In the case when \(R_{\Sigma d}<0\), \((-x+x_c)(-x_d+x_c)\) must be greater than zero, which leads to the expression

\[ R_{\Sigma}R_{\Sigma d}=-(-x+x_{c\mathrm{cr}})^2 . \tag{12} \]

Substituting into the quadratic equation (7) and (9), as well as (12), we obtain

\[ S=\pm 2Q,\quad \text{then } P=\sqrt{0.75}\,S,\quad \lambda_1=-0.5S . \tag{13} \]

The first equation of system (7), taking (13) into account, makes it possible to determine the character of the circuit reactance: \(Q=0.5S\). Finally, for \(R_{\Sigma d}<0\) we have

\[ \left. \begin{aligned} Q&=0.5S=|\lambda_1|,\quad \lambda_2=-1.5S,\\ P&=\sqrt{0.75}\,S=\sqrt{\lambda_1\lambda_2}\quad \text{and}\quad \varphi=\frac{\pi}{6}. \end{aligned} \right\} \tag{14} \]

The components of the apparent power can, obviously, be expressed through the roots of the quadratic equation in another way than in (14),

\[ Q=\sqrt{0.75}\,S,\quad P=0.5S\quad \text{and}\quad \varphi=\frac{\pi}{3}. \tag{15} \]

IV. For the most complete consideration of the expressions obtained above, it is useful to turn to experimental data. Figure 2 shows the variation of the active component of the voltage across the choke for different dc, \(h_0\), and ac, \(h_m\), magnetization in relative units. As follows from the figure, at \(h_m\simeq h_0\) the differential resistance becomes negative. Depending on the supply voltage, when the dc magnetization is varied there may be several “input--output” resonance characteristics (Fig. 1), with

\[ U_1<U_{\mathrm{II}}<U_{\mathrm{III}}<U_{\mathrm{IV}}. \]

In the figure: vertical axis \(\varphi\) (degrees); horizontal axes \(U\) (V) and \(r\) (cm); curves marked \(\varphi_1\), \(\varphi_2\), \(\varphi_3\); groups \(\varphi=f(u)\) and \(\varphi=f(r)\).

Fig. 3. Change of phase relationships as a function of the supply voltage \(\varphi=f(U)\) and resistance \(\varphi=f(r)\).

At points 1 and 2 of the resonance curves the reactance of the circuit has a capacitive character, and at point 3 an inductive character. Data on determining the phase relationships at these points as functions of the supply voltage and load resistance are given in Fig. 3. The curves \(\varphi=f(U)\) were obtained for different fixed values of the voltage and changes in the dc magnetization; that is, from them one can observe how the angle between the current and the supply voltage changes as the latter increases. At \(U\leq 7\) V the resonance curve 1 (Fig. 1) is single-valued, and the phase relationships are critical, since a slight increase in the load resistance leads to the absence of relay effects at this voltage. \(\varphi_3\) at \(U\) equal to 16 and 35 V is also critical. The curves \(\varphi=f(r)\) were obtained at \(U=\mathrm{const}\) and with changes in the dc magnetization. At \(r=122\ \Omega\)

relay effects in the system are impossible at the given supply voltage.

It follows from the figures presented, and also from the quadratic-form method, that the critical phase relations may be regarded as constant and equal to \(\simeq \dfrac{\pi}{6}\) and \(\simeq \dfrac{\pi}{3}\). The minimum change in phase at the critical load is about \(30^\circ\), with the character of the circuit reactance unchanged. At point 3 the resonance curves can be calculated under the condition that \(R_\Sigma=\mathrm{const}\). At points 1 and 2 expressions (14) and (15) are valid; for this it is necessary to have \(x_d<0\), \(R'_d<0\), and \(x<x_c\). The presence of negative differential resistances is possible only in circuits with dc biasing.

Fig. 4. “Input–output” characteristic of the amplifier–inverter type

One of the principal difficulties in calculating resonance curves of type II (amplifier—inverter) is the “matching” of the characteristics of simultaneous magnetization \(U_L=f(I,I_0)\) and \(U_R=f(I,I_0)\). But according to the quadratic-form method, calculations in the region of possible relay effects can be carried out at constant phase relations; therefore there is no need always to take \(U_R\) into account. This question will be considered below.

V. Suppose it is required to find the region of existence of the characteristic of a circuit of the “amplifier—inverter” type (Fig. 4). When the inductance varies according to the law \(B_{1,2}=\varepsilon H_i+a\arctg \beta H_i\), where \(H_i=H_m\sin\omega t\pm H_0\) is the field strength in each of the cores of the magnetic amplifier, the harmonic-balance equation after normalization takes the form:

\[ (-A+ah_m)^2+(\delta h_m)^2=E^2 . \tag{16} \]

Here \(A\) is the normalized first harmonic of the reactive component of the voltage across the choke, obtained in [6]:

\[ A=\gamma h_m-\frac{2-2\sqrt{1+0.5\left(\eta+\sqrt{\eta^2+4h_m^2}\right)}}{h_m}; \]

\[ \gamma=\frac{\varepsilon}{a\beta};\quad \eta=h_m^2-h_0^2-1,\quad h_m=\beta H_m,\quad h_0=\beta H_0; \]

\[ \alpha=\frac{\omega_0^2}{\omega^2};\quad \omega_0^2=\frac{1}{L_0C};\quad L_0=\frac{2W^2S}{l}\cdot 10^{-8}\alpha^2; \]

\[ \delta=\frac{R_\Sigma}{\omega L_0};\quad E=\frac{\beta U_m}{\omega L_0}\cdot\frac{W}{l};\quad R_\Sigma=R+r. \]

The active component of the voltage across the choke can be determined only experimentally and is represented empirically as

\[ U_R=\frac{h_m}{\kappa_1h_m^2+\kappa_2h_m+\kappa_3}, \]

where \(K_1\ldots K_3\) are third-degree polynomials in the dc field strength.

If expression (16) is written in the form

\[ -A+a h_m=\pm \sqrt{E^2-(\delta h_m)^2}, \]

then in the left-hand side of the equality the variables can be separated and, by the method of iteration, the dependence \(h_0=f(h_m,F,h_0)\) can be found:

\[ h_0=\frac{F}{2}\sqrt{\frac{4h_m}{F^2-4}}-1, \tag{17} \]

where \(F=\gamma h_m^2+2\pm h_m\sqrt{E^2-(\delta h_m)^2}\) is an auxiliary function,

\[ \nu=a-\gamma . \]

Expression (17) is the basic equation of the “input—output” characteristic of the ferroresonant circuit in the presence of constant biasing. Figure 4 presents an experimentally obtained dependence—curve 1—and the dependence obtained by numerical solution of the problem—curve 2, when, from the standpoint of choosing the magnitude of the supply voltage, the characteristic is optimal. For larger \(U\) a transition to a dependence of type III is possible (Fig. 1); for smaller values the realizable output signal decreases. As the experimental data show, the resonance curves attain their maximum value at somewhat larger \(h_m\) than magnetic amplifiers have at the same \(U\) and \(h_0\). Since, in the regime with suppressed even harmonics, the equality of the magnetomotive forces in average values is valid in the control circuit for a magnetic amplifier, the expression

\[ h_m^2-h_0^2-1=\gamma_1=0, \tag{18} \]

obviously, may be regarded as the optimality condition. It must also follow from the expression

\[ \frac{\partial U_R}{\partial I_0}=0. \]

Equality (18), for

\[ E=\delta h_m, \]

makes it possible to find the maximum output signal if the capacitance is known,

\[ \gamma^2 h_m^3+4\gamma h_m-4=0 \]

or the capacitance for a given alternating-field intensity,

\[ C=\frac{h_m^2}{\omega^2 L_0\left[\gamma h_m^2+2\left(-1\pm\sqrt{1+h_m}\right)\right]}. \]

To determine the relay characteristic, we shall use the method of the quadratic form. Let curve 3 (Fig. 4) be constructed from equation (17), when the auxiliary function

\[ F=\gamma h_m^2+2-0.5Eh_m, \]

which is equivalent to satisfying identity (14), or, in other words,

\[ (-U_L+U_C)=0.5U. \tag{19} \]

The curve is the locus of points of jump-like change of alternating current at the critical load. Then \(R_0\), when the current in the working circuit decreases, can be determined by taking \(h_m\) equal to the maximum output signal. In a jump-like increase of \(h_m\), in addition to (14), the following is also satisfied:

\[ \frac{\partial h_0}{\partial h_m}=0. \]

Figure 4 also presents experimental points of breakdown and excitation of oscillations at various load resistances. They indicate that the curve constructed from equation (19),

is the “skeletal” curve. It makes it possible to determine all possible resonance phenomena for a given voltage and a change in the load.

In conclusion, we note that the materials of this article were reported in 1971 at the Fourth All-Union Conference on the Theory and Methods of Calculation of Nonlinear Electrical Circuits (Tashkent).

Conclusions

1. The quadratic-form method, being one variety of bifurcation theory, makes it possible to determine the critical phase relations in a series ferroresonant circuit.

2. The phase relations obtained are constants, which substantially simplifies finding the region of existence of relay effects.

3. The possibility of obtaining satisfactory data in determining resonance curves by the fundamental harmonic is confirmed.

References

1. M. M. Vainberg, V. A. Trenogin. Theory of Branching of Solutions of Nonlinear Equations. “Nauka,” 1969.

2. T. Hayashi. Nonlinear Oscillations in Physical Systems. “Mir,” 1968.

3. S. N. Ginzburg. Nonlinear Circuits and Their Functional Characteristics. GEI, 1958.

4. A. I. Kutyavin. Calculation of relay effects in a series ferroresonant circuit. Izvestiya TPI, vol. 244.

5. R. Bellman. Introduction to Matrix Theory. “Nauka,” 1969.

6. L. S. Goldfarb, G. R. Herzenberg. Determination of harmonics of current and voltage in electrical circuits containing iron. Elektrichestvo, 1938, No. 1.