FERRORESONANT CIRCUITS AS CIRCUITS WITH NEGATIVE RESISTANCE

A. I. Kutyavin

(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering)

1. The problems of nonlinear systems are incomparably more complex than those of linear systems; a solution of one problem or another in general form, even approximately, is possible only very rarely. Among such rare cases one may include obtaining, in general form, the operating conditions of ferroresonant circuits under a jump-like change of electrical parameters (with relay or trigger effects) at the frequency of the power source, if one proceeds from the distinctive features of circuits with negative differential resistance [1].

Let us clarify certain regularities of ferroresonant circuits under resonance at the fundamental harmonic and under variation of the supply voltage. We shall find the conditions for multistable operation and the boundary of transition from bistable operation to monostable operation, when a jump-like change of parameters is impossible. Such limiting parameters of the system will be called, as in [2], critical.

II. In a ferroresonant parallel circuit, when operating from a current source, relay effects occur. When operating from a voltage source they are absent; the system has, on the integral volt-ampere characteristic, a section with negative differential conductance, and therefore its analysis should preferably be carried out in terms of conductance [1]. Let us find the critical parameters of the circuit when an insignificant change in the source resistance \(r\) leads to the absence of relay effects, i.e., to the conversion of the circuit supply from a nonideal current source to a nonideal voltage source.

For the equivalent circuit of the circuit (Fig. 1, a), where \(r\) is the resistance of the source; \(b(u)\), \(g(u)\), and \(b_c\) are the static or integral conductances of the choke (reactive and active) and of the capacitance, respectively, one may write the equation

\[ ri+u=u_k . \tag{1} \]

We shall take the solution of equation (1) in the form \(u=U_{1m}\sin \omega t\) and, in what follows, for simplicity of notation we shall omit the subscript 1. Then

\[ \left. \begin{aligned} u_k&=U_{km}\sin(\omega t+\varphi);\quad i=i_L+i_R+i_c;\\ i_c&=b_c U_m\cos\omega t;\quad i_R\simeq g(U_m)U_m\sin\omega t;\\ i_L&=f(\psi)=-b(U_m)U_m\cos\omega t,\quad \text{since}\\ \frac{d\psi}{dt}&=U_m\sin\omega t,\qquad \text{and }\psi=-\frac{U_m}{\omega}\cos\omega t. \end{aligned} \right\} \tag{2} \]

Using the harmonic-balance method, equation (1), taking (2) into account, can be represented as follows:

\[ r^2(-I_{Lm}+I_{cm})^2+(rI_{Rm}+U_m)^2=U_{km}^2, \tag{3} \]

where

\[ I_{Lm}=b(U_m)U_m;\quad I_{cm}=b_cU_m;\quad I_{Rm}=g(U_m)U_m \]

are the fundamental-harmonic currents flowing through the parallel branches.

In the case of relay effects, the dependence between the supply voltage and the voltage on the parallel branches of the circuit may be represented by the curve in Fig. 1,b. At the points where \(\dfrac{dU_{km}}{dU_m}=0\), a transition of the system from one stable state to another is possible, as follows from the Routh–Hurwitz criterion [3]. Differentiating (3), we obtain

\[ r^2(-b+b_c)(-b_d+b_c)+(rg+1)(rg_d+1)=0 \tag{4} \]

where \(b, g\) are the integral conductances (reactive and active) of the choke, determined from the fundamental harmonic;

Fig. 1. Equivalent circuit of the parallel circuit a) and its characteristic b).

\(b_d, g_d\) are the reactive and active differential conductances of the choke.

Solving (4) with respect to the capacitive conductance,

\[ b_c=\frac{r(b+b_d)\pm\sqrt{r^2(b-b_d)^2-4\left[r^2gg_d+r(g+g_d)+1\right]}}{2r}, \tag{5} \]

we see that bistable regions can occur only when the expression under the radical is greater than zero, i.e., when

\[ r>r_{\mathrm{cr}}= \frac{(g+g_d)+\sqrt{(g-g_d)^2+4(b-b_{c\mathrm{cr}})^2}} {2\left[(b-b_{c\mathrm{cr}})^2-gg_d\right]}, \]

where \(b_{c\mathrm{cr}}=\dfrac{b+b_d}{2}\) is the critical capacitive conductance;

\(r_{\mathrm{cr}}\) is the critical source resistance.

If \(r<r_{\mathrm{cr}}\), relay effects cannot exist in the system, and on the integral current-voltage characteristic of the circuit there is a section with negative differential conductance, which is equivalent to operation of the circuit from a nonideal voltage source. If \(r>r_{\mathrm{cr}}\), such effects are possible, and the source must be regarded as a nonideal current source.

Figure 2 presents the dependences \(r_{\mathrm{cr}}=f(b_{c\mathrm{cr}})\), theoretical and experimental, for the circuit operating at industrial frequency. The experimental curve lies above the theoretical one because it was recorded with an insignificant stepwise change in the parameters (the multiplicity factor, i.e., the stepwise change of the voltage across the choke, did not exceed 1.5).

In the simplest case, when the circuit is conservative (losses are absent, \(g=0\)),

Fig. 2. Dependence of the critical resistance of the source on the critical capacitive conductance.

\[ b_{c.\mathrm{cr}}=\frac{b+b_d}{2}; \qquad \frac{1}{r_{\mathrm{cr}}}=b-b_{c.\mathrm{cr}}. \tag{6} \]

III. A ferroresonant series circuit, when operating from a current source, may have a section with negative differential resistance on the integral (static) volt-ampere characteristic (Fig. 3, b); therefore, its analysis should preferably be carried out in terms of resistance [1]. Assuming that the fundamental harmonic is dominant in the circuit, in analyzing the equivalent circuit of the loop (Fig. 3, a) we determine the locus of points at which stepwise changes of current begin [4].

Fig. 3. Equivalent circuit of the series circuit (a) and its volt-ampere characteristic (b).

\[ x_c=\frac{x+x_d}{2}\pm \sqrt{\frac{(x-x_d)^2}{4}-R_\Sigma R_{\Sigma d}}, \tag{7} \]

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

\(R, R_d\) are the integral and differential active resistances of the reactor, due to losses in the steel.

According to (7), the bistable region may also be absent if the radical expression becomes zero. For multistable operation the resistance of the source (load) must be less than the critical value

\[ r<r_{\mathrm{cr}}=-\frac{R+R_d}{2}+\sqrt{\frac{(R-R_d)^2}{4}-(x-x_{c,\mathrm{cr}})^2}, \]

where \(x_{c,\mathrm{cr}}=\dfrac{x+x_d}{2}\) is the critical capacitive reactance.

For \(r<r_{\mathrm{cr}}\), relay effects are possible in the system, and it must be assumed that operation is from a nonideal voltage source. In operation from a nonideal current source \((r>r_{\mathrm{cr}})\), a section with negative differential resistance is observed on the integral volt-ampere characteristic of the circuit.

Assuming the circuit to be conservative, while \(r\) remains the source resistance, one can obtain for the critical regime:

\[ x_{c,\mathrm{cr}}=\frac{x+x_d}{2};\qquad r_{\mathrm{cr}}=x-x_{c,\mathrm{cr}}. \tag{8} \]

Comparing expressions (8) and (6), we see that for conservative circuits the critical parameters are analogous.

IV. Considering ferroresonant circuits as circuits with negative resistance makes it possible to obtain rather exhaustive information about their operating regimes, which is confirmed experimentally both for the parallel and for the series circuit [4]. The critical parameters, expressed in one case through conductances and in the other through resistances, testify to the generality of the method for calculating various ferroresonant circuits. The concepts of current source and voltage source, when considering circuits with negative resistance, become more flexible and universal.

LITERATURE

1. S. A. Goryainov, I. D. Abezgauz. Semiconductor Devices with Negative Resistance. “Energiya,” 1966.

2. L. A. Bessonov. Electric Circuits with Steel. GEI, 1948.

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

4. A. I. Kutyavin. Calculation of relay effects in a series ferroresonant circuit. Izvestiya TPI, vol. 244, Perm, TSU Publishing House, 1972.