ELECTRICAL ENGINEERING

Corresponding Member of the Academy of Sciences of the USSR K. B. KARANDЕEV and K. M. SOBOLEVSKII

ON HOMOGENEOUS COMPENSATOR CIRCUITS

The desire to eliminate an inherent fundamental shortcoming of multistage compensator circuits with ordinary shunting and double decades \((^{1-3})\)—the wide range and large number of nominal values of the sectional resistances of the decades—led the authors of the present communication to the creation of homogeneous circuits, in which sectional resistances of a single nominal value are used. The problem of constructing homogeneous multistage compensator circuits has been solved for the general case of a numeral system with any integer base \(K\).

Let us first formulate the general principle of constructing homogeneous multistage circuits. Each digit group (in particular, for the decimal system of numeration—each decade) of such a homogeneous circuit will consist of sectional resistances \(Z_0\) equal in nominal value. To carry out readout (or regulation) in a system of numeration with base \(K\), it is necessary that the voltage drops across the sections of the digit groups of any two adjacent stages be in the ratio \(K : 1\). Given equality of the sectional resistances of both digit groups, this ratio can occur only on condition that the currents flowing through these resistances be in the same ratio. Hence two fundamental conditions follow: 1) the electrical circuit of a homogeneous multistage compensator is inevitably a branched circuit, in which each subsequent stage shunts some portion of the preceding stage; 2) the total shunting resistance of the circuit of any stage of the compensator must be \(K - 1\) times greater than the resistance of the shunted portion of the preceding stage, where \(K\) is the base of the numeral system used as the basis for constructing the compensator circuit.

As applied to cascade compensator circuits, i.e., circuits with shunting digit groups, where a branched electrical circuit is present, analysis of methods for implementing the general principle set forth shows that such circuits can most simply be constructed either by independently using two-link cascades (Fig. 1) or by combined use of intermediate two-link and final one-link cascades (Fig. 2). As is evident from the figures, in these multistage circuits, along with the number of homogeneous sectional resistances required for each stage, there are included certain additional resistances \(Z_{\mathrm{d}}^{1}, \ldots, Z_{\mathrm{d}}^{n}\) (Fig. 1) or \(Z_{\mathrm{d}}^{1}, \ldots, Z_{\mathrm{d}}^{(n-1)}\) (Fig. 2), located outside the circuit from which the required voltage drop is taken. The role of the additional resistances is to bring the total shunting resistance \(Z_{\mathrm{sh}}^{(i)}\) of the circuit of any digit group \(i\) to a value \(K - 1\) times greater than the resistance of the shunted portion of the preceding digit group.

Having drawn up a series of equations relating the corresponding total shunting resistances \(Z_{\mathrm{sh}}^{(i)}\) to the resistances of the shunted portions, it is easy to find the values of the additional resistances as functions of the value of the sectional resistance \(Z_0\) and of the base of the numeral system \(K\). Thus, for

Fig. 1. Homogeneous two-section multistage cascade compensator circuit

Fig. 2. Homogeneous single-section combined multistage cascade compensator circuit

Fig. 3. Compensator circuit with double homogeneous decade groups

of the circuit in Fig. 1 (it may be called a homogeneous two-branch multistage cascade circuit), taking into account that \(Z_{\mathrm{sh}}^{(n)}=Z_{\mathrm{sh}}^{(n-1)}=Z_{\mathrm{sh}}^{(n-2)}=\ldots=Z_{\mathrm{sh}}^{(2)}=Z_{\mathrm{sh}}^{(1)}=2(K-1)Z_0\), we obtain \(Z_{\mathrm{d}}^{(n)}=(K-2)Z_0\); \(Z_{\mathrm{d}}^{(n-1)}=Z_{\mathrm{d}}^{(n-2)}=\ldots=Z_{\mathrm{d}}^{(1)}=\dfrac{(K-2)(K-1)}{K}Z_0\). For the circuit in Fig. 2 (it may be called a homogeneous combined multistage cascade circuit), taking into account that \(Z_{\mathrm{sh}}^{(n)}=(K-1)Z_0\) and \(Z_{\mathrm{sh}}^{(n-1)}=Z_{\mathrm{sh}}^{(n-2)}=\ldots=Z_{\mathrm{sh}}^{(2)}=Z_{\mathrm{sh}}^{(1)}=2(K-1)Z_0\), we obtain \(Z_{\mathrm{d}}^{(n-1)}=\dfrac{(K-1)^2}{K}Z_0\); \(Z_{\mathrm{d}}^{(n-2)}=\ldots=Z_{\mathrm{d}}^{(1)}=\dfrac{(K-2)(K-1)}{K}Z_0\).

The application of the general principle for constructing homogeneous compensator circuits to circuits with double discharge groups is somewhat different. In this case the original circuit is a series electrical chain without any branches. Therefore, in order to obtain the required ratio of currents in the sectional resistances of the discharge groups of the corresponding stages, the use of auxiliary shunt resistances is necessary.

The schematic diagram of the optimal version of a compensator with double homogeneous discharge groups is shown in Fig. 3. In this circuit the discharge groups of the last stage consist of \(K\) sections, while the discharge groups of all the remaining stages, except the first, consist of \(K-1\) sections (the first stage may, obviously, contain—as in cascade circuits—any number of sections in the discharge groups). Such numbers of sections in the discharge groups of the individual stages follow from the condition of obtaining auxiliary shunt resistances equal to one another in nominal value, with the minimum number of sections in the discharge groups. The values of the auxiliary shunt resistances, found by solving equations that relate the corresponding total shunt resistances \(Z_{\mathrm{sh}}^{(i)}\) to the resistances of the shunted sections (i.e., in the present case, to these same sought auxiliary shunts), are equal to:

\[ Z^{(n-1)}=Z^{(n-2)}=\ldots=Z^{(1)}=\frac{K}{K-1}Z_0. \]

Institute of Automation and Electrometry
Siberian Branch of the Academy of Sciences of the USSR

Received
27 IX 1961

References

1. K. B. Karandeev, Methods of Electrical Measurements, 1952.
2. V. O. Arutyunov, Electrical Measuring Instruments and Measurements, 1958.
3. M. L. Morgan, J. C. Riley, IRE Trans. on Instrumentation, 1–9, Sept., No. 2 (1960).