UNIQUENESS OF THE REGION OF OPTIMAL PARAMETERS OF FERRORESONANT VOLTAGE STABILIZERS

E. S. GITMAN, S. A. GORBATKOV, A. A. LEDIN, S. P. RYABOV

(Presented by the scientific-technical seminar of the Department of Electrical Equipment)

For the calculation of ferroresonant voltage stabilizers, it is important to establish the uniqueness of the region of optimal parameters under various requirements for the stabilization level and for deviations of the input voltage from the nominal value, with various kinds of loads.

In the present work, methods of the mathematical theory of experiment [1] are used, as set forth in application to the choice of optimal parameters of a ferroresonant stabilizer in [2], where the circuit of Fig. 1 was considered. The maximum output power was adopted as the optimality criterion. Under the requirements

Fig. 1

\[ \left. \begin{aligned} u_1 &= 220^{+20}_{-80}\ \text{V};\\ u_2 &= 220^{+12}_{-12}\ \text{V};\\ f &= 50\ \text{Hz};\\ \text{saturated core} &\text{— OL-128/80. 40, steel E 340;}\\ \text{load} &\text{ active} \end{aligned} \right\} \tag{1} \]

the following values of the optimal parameters were obtained:

\[ c = 70\ \mu\text{F}; \qquad x_s = 52.6\ \Omega . \]

Let us consider the operating modes of ferroresonant voltage stabilizers most frequently encountered in practice.

Active load

We solve the stated problem of determining the uniqueness of the region of optimal parameters as follows: under conditions (1), we impose new requirements

\[ \begin{aligned} u_1 &= 220 \pm 20\% \ \mathrm{V};\\ u_2 &= 220 \pm 2\% \ \mathrm{V} \end{aligned} \tag{2} \]

and compare the optimal parameters for both cases.

The planning matrix and the results of steepest ascent are presented in Tables 1 and 2, respectively.

Table 1

Table 2

The optimal parameters are: \(c = 73\) µF; \(x_s = 48{,}1\) ohm—and are close to the optimal parameters for conditions (1). The results obtained make it possible to conclude that the optimal parameters of ferroresonant voltage stabilizers for an active load are unique. In this case

\[ x_s/\omega c \simeq 1, \tag{3} \]

where \(\omega\) is the angular frequency.

Let us note that the power delivered by the stabilizer to the load depends mainly on the specified limits of variation of the input voltage and, within known limits, depends little on the level of stabilization of the output voltage. We shall show this by means of a two-factor experiment, in which the independent variables are taken to be: \(\pm \Delta u_1\)—the range of variation of the input voltage; \(\pm \Delta u_2\)—the range of variation of the output voltage. The independent variables are varied over wide limits (Tables 3 and 4).

Table 3

Table 4

The regression coefficients are: $b_0=713.75$; $b_1=-164$; $b_2=8.75$; $b_{12}=3.75$.

Since $b_2$ and $b_{12}$ are statistically insignificant in comparison with $b_0$ and $b_1$, one may write:

\[ P=P_0+\alpha\left(|\Delta u_{10}|-|\Delta u_1|\right), \tag{4} \]

where $P$ is the delivered power for the specified limits of variation of the input voltage $\pm \Delta u_1$; $P_0$ and $\Delta u_{10}$ are the delivered power and the limits of variation of the input voltage at the center of the experiment; $\alpha$ is a constant coefficient.

Active-inductive load

Let us consider the case of an active-inductive load with $\cos\varphi=0.8$. The results obtained below are also valid for other load values; in particular, a load with $\cos\varphi=0.6$ was checked experimentally.

The results of setting up the factorial experiment and the steepest ascent for conditions (2) are given in Tables 5 and 6.

The optimal parameters of the stabilizer are given in the next-to-last row of Table 6.

Table 5

From a comparison of the obtained results with the optimal parameters from Table 2, the following conclusions follow:

1. The optimal value of the active component of the load does not depend on $\cos\varphi$.

2. The optimal values of $x_s$ and $c$ for an inductive load can be obtained from the relations for an active load

\[ x_{su}=x_{sa}\cos\varphi;\qquad c_u=c_a/\cos\varphi, \tag{5}, (6) \]

where the subscript “и” refers to the case of an inductive load, and the subscript “а” to the case of an active load.

Table 6

REFERENCES

1. V. V. Nalimov, N. A. Chernova. Statistical Methods of Experiment Planning. Moscow, “Nauka,” 1965.

2. A. S. Gitman, E. S. Gitman. Selection of the optimal parameters of a ferroresonant stabilizer by the method of steepest ascent. Proceedings of the Tomsk Polytechnic Institute, vol. 172, 1967.