CALCULATION OF RELAY EFFECTS IN A SERIES FERRORESONANT CIRCUIT

A. I. Kutyavin

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

At the present time interest has increased in ferroresonant circuits, which have found wide application in the construction of voltage stabilizers, frequency dividers and multipliers, contactless relays, and certain other elements of automation.

Although the first works on the theory and application of the series ferroresonant circuit under variation of the supply voltage appeared about fifty years ago, there is still no simple engineering method for calculating relay effects and the critical active resistance of the circuit, above which relay effects are absent. If, in determining the current at the points of trigger effects without taking account of hysteresis losses, an admissible error is obtained [1], this cannot be said of the supply voltage during the reverse jump of the current, and still less of the critical resistance. Taking account of losses in the steel somewhat complicates the problem, but makes it possible to find all the parameters to be determined [2]. In connection with the fact that in an oscillatory circuit, in the general case, the induction curve is close to a sinusoid, whereas the current curve is not, it is necessary to specify the parameters of the choke as functions of the induction. This is precisely what was done in most works on the theory of ferroresonant devices [3, 4]. The conditions for the existence of relay effects, when losses in the steel are taken into account, are then expressed through rather cumbersome expressions. In this case the physical picture of the conditions proves less convenient for understanding than when dependences on current are introduced, allowing one to operate with the concepts of the static and differential resistance of the choke [1]. In contrast to [1], the graph-analytical method for calculating relay effects set forth below takes into account the active component of the voltage on the choke, caused by losses in the steel, and, evidently, makes it possible to find all the parameters to be determined, as well as the critical active resistance of the circuit.

Let us introduce the following assumptions, simplifying the calculation:

1) in determining the parameters of the nonlinear inductance, the rms values of the electrical quantities are measured;

2) the active and reactive components of the voltage on it are found from the phase relations determined for the fundamental harmonic;

3) the active resistance of the choke winding is included in the load resistance, \(r\).

Since at resonance the share of higher harmonics is small in comparison with the fundamental, in the investigation of the simplest phenomena in ferroreso-

ferroresonant circuits, volt-ampere characteristics for rms values [5] are widely used; therefore, the first two assumptions should not significantly affect the accuracy of the calculation.

Fig. 1. Equivalent circuit of the ferroresonant circuit

Fig. 1 shows the equivalent circuit of a series ferroresonant circuit. The dependences of the active and reactive components of the voltage across the choke on the current, as well as the total active resistance of the circuit, \(R_\Sigma = R + r\), are presented in Fig. 2, a.

Fig. 2. Characteristics of the choke (a) and of the ferroresonant circuit (b)

For the circuit under consideration, the following expression is valid:

\[ (U_L-U_C)^2+(U_R+U_l)^2=U^2. \tag{1} \]

The condition for the existence of relay effects in the circuit is the equality

\[ \frac{dU}{dI}=0, \]

which follows from the Routh–Hurwitz criterion [4].

Differentiating (1) with respect to the current, we obtain

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

where

\[ \begin{aligned} x,\ x_d &\text{ are the static and differential reactances of the reactor, respectively;}\\ R,\ R_d &\text{ are the static and differential active resistances of the reactor, respectively;}\\ x_c &\text{ is the static capacitive reactance.} \end{aligned} \]

Expression (2) can be solved with respect to

\[ x_c=\frac{x+x_d}{2}\pm \sqrt{\frac{(x-x_d)^2}{4}-R_\Sigma(R_d+r)}. \tag{3} \]

The right-hand side of (3) depends on the characteristics of the reactor and on the magnitude of the load resistance; the left-hand side does not depend on them. By plotting the left- and right-hand sides separately and finding the points of their intersection for \(x_c=\mathrm{const}\), one can obtain the circuit parameters corresponding to current jumps (Fig. 2, b). The variation of the active differential resistance of the reactor with current, \(R_d\), is also presented there. Because of the smallness of the load resistance (\(r=13\ \Omega\)) and of \(R\) in comparison with the reactive components, the system, as in [1], differs little from a conservative one, since the upper and lower branches expressing the right-hand side of equality (3) almost everywhere coincide with the values of the static or differential inductive reactance. Indeed, if \(R_\Sigma=0\), the curve \(x_c=x_d\) is the locus of points where a jump-like increase of current begins, and the curve \(x_c=x\) is the locus of points where a jump-like decrease of current begins.

With negative differential active resistance, when \(|R_d|>|r|\), the right-hand side of equality (3) becomes greater than the static inductive reactance of the reactor and less than the differential one. As the dissipativity of the system (\(r\)) increases, the region of space bounded by the right-hand side of equality (3) decreases and degenerates into a curve that characterizes the critical value of the capacitive reactance. Relay effects can no longer exist in the circuit:

\[ x_{\mathrm{cr}}=\frac{x+x_d}{2}. \tag{4} \]

Substituting (4) into the radicand of identity (3), we obtain the critical active resistance. Taking into account that it can only be positive, we have

\[ r_{\mathrm{cr}}=-\frac{R+R_d}{2}+ \sqrt{\frac{(R-R_d)^2}{4}+(x-x_{\mathrm{cr}})^2}. \tag{5} \]

In the saturation region, the active resistance of the reactor caused by losses in the steel is small and, evidently, may be neglected. Then expression (5) is simplified to

\[ r_{\mathrm{cr}}=x-x_{\mathrm{cr}}. \]

This means that in a circuit with a linear active resistance, relay effects cannot exist if the angle between the current and the supply voltage is equal to \(\frac{\pi}{4}\). In the general case, when both linear and nonlinear active resistances are present, the critical phase relation is not constant.

Let us consider the results of calculation and experimental verification for a series ferroresonant circuit. The test specimen had a Ш-shaped core made of transformer steel and operated at a frequency \(f = 50\) Hz with a sinusoidal supply voltage. The intersections of the curve corresponding to the dependence of the right-hand side of equality (3), for \(r = 13\ \Omega\), on the current, with the straight line \(x_c = \mathrm{const}\), determine the calculated current values for direct and reverse jumps (Fig. 2, b). The experimental current values are marked in the figure by crosses.

Table 1

| No. | \(C\) (µF) | \multicolumn{2}{c}{Data for a stepwise increase of current} | \multicolumn{2}{c}{Data for a stepwise decrease of current} |
|---:|---:|---:|---:|---:|---:|
| | | \(I_{\mathrm{e}}/I_{\mathrm{p}}\) (A) | \(U_{\mathrm{e}}/U_{\mathrm{p}}\) (V) | \(I_{\mathrm{e}}/I_{\mathrm{p}}\) (A) | \(U_{\mathrm{e}}/U_{\mathrm{p}}\) (V) |
| 1 | 4 | \(\frac{0.04}{0.04}\) | \(\frac{20.5}{20.4}\) | \(\frac{0.075}{0.072}\) | \(\frac{18.4}{19.6}\) |
| 2 | 8.5 | \(\frac{0.06}{0.065}\) | \(\frac{34.5}{33.6}\) | \(\frac{0.25}{0.23}\) | \(\frac{17.1}{18.4}\) |
| 3 | 14.00 | \(\frac{1.102}{0.097}\) | \(\frac{47.7}{44.6}\) | \(\frac{0.52}{0.43}\) | \(\frac{19.7}{18.5}\) |
| 4 | 18.2 | \(\frac{0.125}{0.132}\) | \(\frac{53.7}{50.6}\) | \(\frac{0.71}{0.63}\) | \(\frac{22.5}{20.5}\) |
| 5 | 22.3 | \(\frac{0.16}{0.175}\) | \(\frac{58}{55.1}\) | \(\frac{0.87}{0.78}\) | \(\frac{25.8}{22.1}\) |

Table 2

Table 1 gives the data comparing calculation and experiment at the points of trigger effects. In the critical operating mode, owing to the difficulty of the experiment, we assumed that it occurs when the current jump ratio in the circuit does not exceed 1.4. Therefore, Table 2 gives two values of the experimentally determined current, corresponding to the current before and after the jump. As the comparison of experimental and calculated data shows, the error in determining the parameters does not exceed \(10 \div 20\%\).

Conclusions

1. At small load resistances in a series oscillatory circuit, the currents at the points of relay effects can be determined as for a conservative system. To find the other parameters, it is necessary to take into account the active component of the voltage across the choke, caused by losses in the steel, which makes it possible to carry out the calculation with sufficient accuracy.

2. In the absence of nonlinear active resistance in the circuit, the critical phase relation between the current and the supply voltage is constant and equal to $\frac{\pi}{4}$, which cannot be said when it is present.

3. The graph-analytic method considered makes it possible to obtain the conditions for relay effects as functions of the static and differential resistances, which contributes to a better understanding of the physical picture of the processes.

Literature

1. S. A. Ginzburg. Nonlinear circuits and their functional characteristics, GEI, 1958.
2. M. A. Rozenblat. Magnetic amplifiers. Vol. 1, “Sovetskoe radio,” 1960.
3. L. A. Bessonov. Electrical circuits with steel, GEI, 1948.
4. G. H. Hayt. Nonlinear oscillations in physical systems, “Mir,” 1968.
5. L. A. Bessonov. Nonlinear electrical circuits, “Vysshaya shkola,” 1964.