One Engineering Method for Calculating a Series Ferroresonant Circuit

V. N. Stanevko, E. I. Goldshtein

(Presented by the scientific-technical seminar of the Faculty of Automatic Systems)

At the present time a number of ways of investigating and calculating ferroresonant circuits are known, but most of them have not been developed into engineering calculation methods [1].

In Fig. 1,a a ferroresonant circuit is shown, consisting of an active resistance, an inductance, and a capacitance.

In Fig. 1,b the well-known dependence \(U=f(I)\) is shown, which may be obtained, for example, experimentally. Here also

Fig. 1. \(a\) — stable region, \(b\) — unstable region

the regions of stable and unstable equilibrium are shown. Points 1 and 2 lie on the boundary between the zones of stable and unstable equilibrium. In carrying out practical calculations of nonlinear electric circuits whose equivalent circuits reduce to a series ferroresonant circuit of the type shown in Fig. 1,a, the problem often arises of determining the relation of the circuit parameters to the values of \(U\) and \(I\) for the critical points 1 and 2.

To solve the stated problem one may use the results of [2], in which, for the equation of a ferroresonant circuit,

\[ \omega L \frac{dI}{d\tau} + rI + \frac{1}{\omega C}\int I\,d\tau = U \tag{1} \]

the determinant of the system describing the behavior of the operating point of the circuit under study is obtained.

The determinant has the form

\[ D(p)=-(r+2\omega L_{0}p)^{2}-\frac{1}{I_{0}}\,U_{x0}\,\frac{dU_{x}}{dI}. \tag{2} \]

The roots of equation (2) are determined from the expression

\[ p_{1,2}=\frac{1}{2\omega L_0}\left(-r\pm \sqrt{-\frac{1}{I_0}U_{x0}\frac{dU_x}{dI}}\right), \tag{3} \]

where \(I_0\) is the value of the current in the circuit (Fig. 1, a) for one of the stable states,

\[ U_x=I\left(\omega L-\frac{1}{\omega C}\right)=I\cdot X, \]

\[ U_{x0}=I_0\left(\omega L-\frac{1}{\omega C}\right). \]

In order that the point with coordinates \(U, I\) (Fig. 1, b) lie in the zone of a stable state, it is necessary, as was noted in [2], that the condition

\[ \frac{1}{I_0}U_{x0}\frac{dU_x}{dI}>0 \tag{4} \]

be satisfied.

It should be added that for the stable region the condition

\[ \frac{1}{I_0}U_{x0}\frac{dU_x}{dI}=0 \tag{5} \]

is also valid.

Moreover, when the condition

\[ \frac{1}{I_0}U_{x0}\frac{dU_x}{dI}<0 \tag{6} \]

is satisfied, a state of stable equilibrium is possible for

\[ \left(\sqrt{-\frac{1}{I_0}U_{x0}\frac{dU_x}{dI}}\right)<r \tag{7} \]

and a state of unstable equilibrium for

\[ \left(\sqrt{-\frac{1}{I_0}U_{x0}\frac{dU_x}{dI}}\right)>r. \tag{8} \]

From the theory of stability of motion it is known that the system will be on the stability boundary if at least one of the roots is either purely imaginary or zero. The former is possible when condition (4) is satisfied and \(r=0\). Since in real systems it is impossible to achieve the equality \(r=0\), this case is of no interest. A zero root may occur when condition (6) is satisfied and, simultaneously, condition (9), which follows from the condition that expression (3) be equal to zero, is satisfied:

\[ \frac{1}{I_0}U_{x0}\frac{dU_x}{dI}=-r^2. \tag{9} \]

If we take into account that the quantity \(r^2\) is always greater than zero, then the left-hand side of expression (9) will coincide with (6), i.e., it may be said that condition (9) is not only necessary but also sufficient for the characteristic of the stability boundary (Fig. 1, b).

Using the notation employed in writing (3), expression (9) can be represented in the form

\[ \frac{1}{I_0}\left(U_{L0}-\frac{I_0}{\omega C}\right)\frac{d}{dI}\left(U_L-\frac{I}{\omega C}\right)=-r^2. \tag{10} \]

where

\[ U_{L0}=I_0\omega L,\qquad U_L=I\omega L. \]

To approximate the dependence \(U_L(I)\), we propose the expression

\[ U_L(I)=\frac{U_{L\mathrm{n}}SI}{IS-1}, \tag{11} \]

where \(U_{L\mathrm{n}}\) is the saturation voltage,
\(S\) is the approximation parameter.

This expression makes it possible to carry out the calculation using rms values of current and voltage for the electric circuit (Fig. 1, \(a\)).

Using expression (11) and noting that \(\dfrac{dU_L}{dI}\) is the derivative at \(I=I_0\), expression (10), after simple transformations, takes the form

\[ \left[\frac{U_{L\mathrm{n}}S}{SI_0-1}-\frac{1}{\omega C}\right] \left[-\frac{U_{L\mathrm{n}}S}{(SI_0-1)^2}+\frac{1}{\omega C}\right] =-r^2. \tag{12} \]

Expression (12) determines the conditions for finding the system at the stability boundary. The validity of this expression was checked by us and confirmed experimentally.

Let us reduce expression (12) to the form

\[ [U_{L\mathrm{n}}S\omega C-(SI-1)] [U_{L\mathrm{n}}S\omega C+(SI-1)^2] =(r\omega C)^2(SI-1)^3. \tag{13} \]

To solve the problem posed above, it is necessary to solve equation (13), which is rather difficult; moreover, equation (13) does not give a clear relationship between the parameters and the electrical quantities of the circuit (Fig. 1, \(a\)).

Introduce the notation:

\[ \begin{aligned} U_{L\mathrm{n}}S\omega C&=A,\\ (SI-1)&=B,\\ (r\omega C)^2&=D. \end{aligned} \tag{14} \]

Then equation (13) takes the form

\[ (A-B)(A+B^2)=-DB^3. \tag{15} \]

Let us assume that the parameters \(A\), \(B\), \(D\) are coordinates along the axes of a three-dimensional space (Fig. 2). Equation (15) will correspond to a certain surface. If, for a fixed value \(A=A_i\), the space is sectioned by a plane parallel to the plane \(BOD\), then in the section of this plane we obtain a curve whose equation has the form

Fig. 2

\[ D=1-\left(\frac{A_i}{B}-\frac{A_i}{B^2}+\frac{A_i^2}{B^3}\right). \tag{16} \]

By making a set of sections for different values of \(A_i\), one can obtain a family of curves described by equation (16). For

in the presence of such a family, calculation of the ferroresonant circuit is possible without cumbersome computations. Thus, if the circuit parameters are specified, the value \(A_i\) is determined from (14); then, for the obtained \(A_i\), the corresponding curve is selected in the \(BOD\) plane. From the found \(D\), \(B_i\) is determined, and then the value \(I_i\) is found from the formula

\[ I_i=\frac{B_i+1}{S}. \tag{17} \]

The proposed calculation procedure was tested in the analysis of a successive ferroresonant circuit. The results of calculation and experiment are summarized in Table 1.

Table 1

Conclusions

1. The proposed method provides sufficient accuracy for engineering calculations.

2. Using an electronic digital computer, one can construct a nomogram of a family of curves \(D=f(B)\) for many values of \(A\). Such a nomogram will make the proposed method convenient for the analysis and synthesis of electrical circuits (Fig. 1, a).

3. To obtain good results by the proposed method, it is necessary to take into account losses in the steel.

Literature

1. L. A. Bessonov. Nonlinear Electric Circuits. “Vysshaya shkola,” Moscow, 1964.

2. K. M. Polivanov. Determination of stable equilibrium points in ferroresonance. Izv. vuzov, “Elektromekhanika,” Moscow, No. 11, 1965.