METHOD OF CHAINS

G. E. Pukhov

Although circuits with variable parameters \(r(t)\), \(L(t)\), \(C(t)\) occur no less often than circuits whose parameters do not depend on time, methods for their calculation have been developed sufficiently fully only for a few special cases, which, to be sure, have very great practical significance. We have in mind circuits with rotating three-phase machines [1], microphone circuits [2—4], and a number of others.

The state of a circuit whose parameters depend only on time is described by a system of linear integro-differential equations with variable coefficients. Therefore its calculation essentially reduces to finding the solution of a system of equations with variable coefficients. In three-phase circuits, by a change of variables it is often possible to transform the original equations into equations with constant coefficients [1]. When, however, the corresponding substitution is difficult to find, one must resort to the direct solution of the original equations, which greatly complicates the problem. Even in such a comparatively simple circuit as a transformer with variable mutual inductance, the calculation turns out to be too cumbersome [5] and, moreover, not transparent, since the desired currents and voltages are expressed in the form of infinite Fourier series.

The approximate calculation method considered below is based on the possibility, noted earlier in [9], of applying the well-developed theory of chain electrical circuits [6—9] to the solution of equations with variable coefficients. The established mathematical analogy makes it possible to reduce the calculation of a circuit with variable parameters in any regime to the calculation of a certain “time chain circuit.” This method is therefore hereafter called the “method of chains.” From the mathematical point of view it is a small generalization of the so-called continuation formula, known in the theory of differential equations, and of the method of averaging coefficients [10].

1. The coefficients of the system of equations of a circuit with variable parameters may, in a sufficiently small time interval \(\tau_s=t_s-t_{s-1}\), be regarded approximately as constant. Therefore the solution of such a system, by virtue of the assumed linearity of the circuit, can be represented in the form:

\[ \begin{aligned} \rho_1(t_s) &= \zeta_{11}(\tau_s)\rho_1(t_{s-1}) +\zeta_{12}(\tau_s)\rho_2(t_{s-1}) +\ldots+ \zeta_{1n}(\tau_s)\rho_n(t_{s-1}) +\zeta_{10}(\tau_s),\\ \rho_2(t_s) &= \zeta_{21}(\tau_s)\rho_1(t_{s-1}) +\zeta_{22}(\tau_s)\rho_2(t_{s-1}) +\ldots+ \zeta_{2n}(\tau_s)\rho_n(t_{s-1}) +\zeta_{20}(\tau_s), \tag{1.1}\\ \rho_n(t_s) &= \zeta_{n1}(\tau_s)\rho_1(t_{s-1}) +\zeta_{n2}(\tau_s)\rho_2(t_{s-1}) +\ldots+ \zeta_{nn}(\tau_s)\rho_n(t_{s-1}) +\zeta_{n0}(\tau_s), \end{aligned} \]

where \(\rho_i(t_{s-1})\) and \(\rho_i(t_s)\) are the \(i\)-th current or voltage of the circuit, respectively, at the beginning and at the end of the \(s\)-th time interval, whose duration is equal to \(\tau_s=t_s-t_{s-1}\); \(\zeta_{ik}(\tau_s)\) are certain coefficients depending on the length of the interval \(\tau_s\) and on time (more precisely, on the number of the interval), which are obtained by solving the given system of integro-differential equations under the assumption that the coefficients of the equations are constant and that, at \(t=t_{s-1}\) (the beginning of the interval), the currents and voltages of the circuit are equal to \(\rho_1(t_{s-1}),\rho_2(t_{s-1}),\ldots,\rho_n(t_{s-1})\). The number of equations (1.1) is equal to the order of the circuit \(n\) ¹). It is convenient to write equation (1.1) more briefly, using matrix notation, in the form:

\[ \rho(t_s)=\zeta(\tau_s)\rho(t_{s-1})+\zeta_0(\tau_s), \tag{1.2} \]

where

\[ \rho(t_s)= \begin{vmatrix} \rho_1(t_s)\\ \rho_2(t_s)\\ \cdot\\ \rho_n(t_s) \end{vmatrix}, \qquad \rho(t_{s-1})= \begin{vmatrix} \rho_1(t_{s-1})\\ \rho_2(t_{s-1})\\ \cdot\\ \rho_n(t_{s-1}) \end{vmatrix}, \tag{1.3} \]

\[ \zeta(\tau_s)= \begin{vmatrix} \zeta_{11}(\tau_s) & \zeta_{12}(\tau_s) & \cdot & \zeta_{1n}(\tau_s)\\ \zeta_{21}(\tau_s) & \zeta_{22}(\tau_s) & \cdot & \zeta_{2n}(\tau_s)\\ \cdot & \cdot & \cdot & \cdot\\ \zeta_{n1}(\tau_s) & \zeta_{n2}(\tau_s) & \cdot & \zeta_{nn}(\tau_s) \end{vmatrix}, \qquad \zeta_0(\tau_s)= \begin{vmatrix} \zeta_{10}(\tau_s)\\ \zeta_{20}(\tau_s)\\ \cdot\\ \zeta_{n0}(\tau_s) \end{vmatrix}. \]

Let us dwell on certain properties of the matrix \(\zeta(\tau_s)\) and the vector (column matrix) \(\zeta_0(\tau_s)\). We shall agree to call \(\zeta(\tau_s)\) the matrix parameter, and \(\zeta_0(\tau_s)\) the vector parameter of the process in the circuit, relating to the interval \(\tau_s\). It is not difficult to observe that the components \(\zeta_{i0}(\tau_s)\) of the vector parameter represent the currents and voltages of the circuit at the end of the small interval \(\tau_s\), determined under the assumption of zero initial conditions for this interval, when \(\rho_i(t_{s-1})=0\) \((i=1,2,\ldots,n)\). Therefore \(\zeta_{i0}(\tau_s)\) will depend both on the parameters of the circuit in the interval \(\tau_s\) and on the electromotive forces of the sources. As for the components \(\zeta_{ik}\tau_s\) of the matrix parameter \(\zeta(\tau_s)\), their values will not depend on the form of the electromotive forces acting in the circuit, since the term \(\zeta(\tau_s)\rho(t_{s-1})\) in equation (1.2) determines those components of the currents and voltages which are due only to the stock of energy in the circuit at the beginning of the interval.

Equation (1.2), and the system of equations (1.1) equivalent to it, under certain conditions may be interpreted as the equation of a link of a certain chain network. Indeed, if as the variables \(\rho\) one chooses continuous currents and voltages of the circuit, namely the currents in inductances and the voltages across capacitors, then \(\rho(t_{s-0})=\rho(t_{s+0})\), that is, the currents and voltages at the end of a given time interval will be equal to the currents and voltages at the beginning of the next interval. In an ordinary chain network an analogous condition is satisfied with respect to spatial coordinates: the currents and voltages at the end of a link are equal to the currents and voltages at the beginning of the next link.

Thus, if as \(\rho\) one chooses continuous currents and voltages, then equations (1.1) and (1.2) may be interpreted as the equation of a link of a “time chain network,” the input quantities of which are the in—

¹) By the order of a circuit \(n\) we mean the order of the differential equation used to describe the state of the circuit. If in each of the \(b\) branches there are both inductances and capacitances, the order of the circuit is \(n=2b\). For example, a contour consisting of \(r\), \(L\), and \(C\) represents a circuit of the second order.

discontinuous currents and voltages \(\rho(t_s)\) at the end of the time interval \(\tau_s\), and as the input quantities—the currents and voltages \(\rho(t_{s-1})\) at the beginning of \(\tau_s\), when \(t=t_{s-1}\). The “time chain” is shown schematically in Fig. 1.

The noted analogy makes it possible, almost without change, to apply the known methods for calculating ordinary chain circuits to determine currents and voltages in circuits with variable parameters. Let us obtain the corresponding calculation equations.

Fig. 1

In order to obtain more convenient calculation formulas, equations (1.1) should be replaced by the equations:

\[ \begin{aligned} \rho_1(t_s) &= \zeta_{11}(\tau_s)\rho_2(t_{s-1})+\zeta_{12}(\tau_s)\rho_2(t_{s-1})+\ldots+\zeta_{1n}(\tau_s)\rho_n(t_{s-1})+\zeta_{10}(\tau_s)1,\\ \rho_2(t_s) &= \zeta_{21}(\tau_s)\rho_1(t_{s-1})+\zeta_{22}(\tau_s)\rho_2(t_{s-1})+\ldots+\zeta_{2n}(\tau_s)\rho_n(t_{s-1})+\zeta_{20}(\tau_s)1,\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ \rho_n(t_s) &= \zeta_{n1}(\tau_s)\rho_1(t_{s-1})+\zeta_{n2}(\tau_s)\rho_2(t_{s-1})+\ldots+\zeta_{nn}(\tau_s)\rho_n(t_{s-1})+\zeta_{n0}(\tau_s)1,\\ 1 &= 0\,\rho_1(t_{s-1})+0\,\rho_2(t_{s-1})+\ldots+0\,\rho_n(t_{s-1})+1\cdot 1, \end{aligned} \tag{1.4} \]

which, using the notation (1.3), can be written more briefly:

\[ \begin{bmatrix} \rho(t_s)\\ 1 \end{bmatrix} = \zeta(\tau_s) \begin{bmatrix} \rho(t_{s-1})\\ 1 \end{bmatrix} = \begin{bmatrix} \zeta(\tau_s) & \zeta_0(\tau)\\ 0 & 1 \end{bmatrix} \times \begin{bmatrix} \rho(t_{s-1})\\ 1 \end{bmatrix} \tag{1.5} \]

or, finally, in an even more compact form,

\[ R(t_s)=\xi(\tau_s)R(t_{s-1}), \tag{1.6} \]

where

\[ R(t_s)= \begin{bmatrix} \rho_1(t_s)\\ \rho_2(t_s)\\ \cdot\\ \rho_n(t_s)\\ 1 \end{bmatrix}, \qquad R(t_{s-1})= \begin{bmatrix} \rho_1(t_{s-1})\\ \rho_2(t_{s-1})\\ \cdot\\ \rho_n(t_{s-1})\\ 1 \end{bmatrix}, \tag{1.7} \]

\[ \xi(\tau_s)= \begin{bmatrix} \zeta_{11}(\tau_s) & \zeta_{12}(\tau_s) & \cdot & \zeta_{1n}(\tau_s) & \zeta_{10}(\tau_s)\\ \zeta_{21}(\tau_s) & \zeta_{22}(\tau_s) & \cdot & \zeta_{2n}(\tau_s) & \zeta_{20}(\tau_s)\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \zeta_{n1}(\tau_s) & \zeta_{n2}(\tau_s) & \cdot & \zeta_{nn}(\tau_s) & \zeta_{n0}(\tau_s)\\ 0 & 0 & \cdot & 0 & 1 \end{bmatrix}. \tag{1.8} \]

The vector \(R(t)\) is naturally called the extended vector of continuous variables, and the matrix \(\xi(\tau_s)\) the extended parameter of a link of the time chain. For \(t=t_s\) the vector \(R(t)=R(t_s)\), for \(t=t_{s-1}\), \(R(t)=R(t_{s-1})\), and so on.

Equation (1.6) represents the extended equation of a link of the time chain. Replacing in this equation \(s\) by \(s-1\), \(s-2\), etc., it is easy to obtain the dependences

\[ \begin{aligned} R(t_s) &= \xi(\tau_s)R(t_{s-1}),\\ R(t_{s-1}) &= \xi(\tau_{s-1})R(t_{s-2}),\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ R(t_1) &= \xi(\tau_1)R(t_0), \end{aligned} \tag{1.9} \]

from which it follows that

\[ R(t_s)=\xi(\tau_s)\xi(\tau_{s-1})\ldots \xi(\tau_1)R(t_0)=\xi(t_s)R(t_0). \tag{1.10} \]

This basic computational equation relates the values of the continuous currents and voltages at the beginning of the process \((t=t_0)\) to the values that they assume at the \(s\)-th instant of time \((t=t_s)\). Taking in (1.10) \(s=1,2,3,\ldots\), etc., one can determine the currents and voltages in the circuit for any instant of time. The extended parameters \(\xi(\tau_s)\) should be multiplied in the following order:

\[ \xi(\tau_2)\xi(t_1)=\xi(t_2),\quad \xi(\tau_3)\xi(t_2)=\xi(t_3), \]

\[ \ldots,\xi(\tau_s)\xi(t_{s-1})=\xi(t_s), \tag{1.11} \]

since in this case, from the equations

\[ \begin{aligned} R(t_1)&=\xi(\tau_1)R(t_0),\\ R(t_2)&=\xi(t_2)R(t_0),\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ R(t_{s-1})&=\xi(t_{s-1})R(t_0) \end{aligned} \tag{1.12} \]

the currents and voltages are determined not only at the instant \(t=t_s\), but also at the intermediate instants of time \(t_1,t_2,\ldots,t_{s-1}\).

The quantity

\[ \xi(t_s)=\xi(\tau_s)\xi(\tau_{s-1})\ldots \xi(\tau_1)=\prod_{\nu=s}^{1}\xi(\tau_\nu) \tag{1.13} \]

represents the extended parameter of the process in the interval from \(t_0\) to \(t_s\). It is obvious that the duration of such an interval is equal to the sum of the time intervals \(\tau_\nu\), i.e.

\[ \sum_{\nu=1}^{s}\tau_\nu=\tau_1+\tau_2+\ldots+\tau_s=t_s-t_0. \tag{1.14} \]

From (1.13) it is clear that, as for ordinary chain circuits [8], the extended parameter of a time chain consisting of \(s\) links is equal to the product of the extended parameters of the component links.

Using (1.13), it is not difficult to verify that the matrix \(\zeta(t_s)\) and vector \(\zeta_0(t_s)\) parameters for the time interval \(t_s-t_0\) will be expressed in the following way through the parameters of the component links:

\[ \zeta(t_s)=\zeta(\tau_s)\zeta(\tau_{s-1})\ldots \zeta(\tau_1)=\prod_{\nu=s}^{1}\zeta(t_\nu), \tag{1.15} \]

\[ \zeta_0(t_s)=\zeta(\tau_s)+\zeta(\tau_s)\zeta_0(\tau_{s-1})+\ldots \tag{1.16} \]

\[ \ldots+\zeta(\tau_s)\zeta(\tau_{s-1})\ldots \zeta(\tau_2)\zeta_0(\tau_1). \]

We shall use expressions (1.15) and (1.16) below in calculating circuits with periodically varying parameters.

On the basis of what has been set forth, it appears possible to formulate the following procedure for calculating circuits with variable parameters by the chain method under consideration:

a) the integro-differential equations of the circuit are written with respect to continuous variables (currents in coils, voltages across capacitors, charges, flux linkages). The time \(t\) in the coefficients of the equations is replaced by \(t_s\);

b) the resulting system of equations is solved under the assumption that the coefficients do not depend on time and that the initial conditions at \(t=t_{s-1}\) are not zero. Replacing \(t\) by \(t_s\) in such a solution and putting \(t_s-t_{s-1}=\tau_s\), we obtain equations (1.1) for the \(s\)-th link of the time chain, which it is then expedient to replace by equations of the form (1.4);

c) the extended parameters of all links of the time chain are determined; for this purpose, in the extended parameter \(\xi(\tau_s)\), formed on the basis of equations (1.4), one should put \(s=1,2\), etc.;

d) by means of the basic calculation equation (1.10), the desired currents and voltages of the circuit are determined at the prescribed instants of time.

The choice of the time intervals \(\tau_s\) is to some extent arbitrary. As \(\tau_s\) is decreased, the accuracy of the calculation will increase. The accuracy of the calculation can also be improved somewhat if the parameters \(\xi(\tau_s)\) are taken equal to their mean values in the interval from \(t_{s-1}\) to \(t_s\), as, for example, is recommended in the method of averaging coefficients [9].

2. The variable parameters of a circuit are very often periodic functions of time. Let us consider this case, important for practice, assuming that the period of variation of the emf sources is equal to, or is a multiple of, the period of variation of the circuit parameters.

If a transient process is being studied, then the calculation method set forth above in Sec. 1 is obviously applicable to this case without any changes. Therefore we shall consider here only the calculation of the steady-state regime.

Under the assumptions made, the process in the circuit may be represented as a series of cycles of identical duration \(T\). Dividing the time \(T\) into several, say \(l\), subintervals, we can determine the extended parameter of a cycle

\[ \xi(T)=\prod_{\nu=1}^{l}\xi(\tau_\nu)= \left| \begin{array}{cc} \zeta(T) & \zeta_0(T)\\ 0 & 1 \end{array} \right|, \tag{2.1} \]

where \(\zeta(T)\) and \(\zeta_0(T)\) are the matrix and vector parameters of the cycle.

The parameters of the process in all cycles will obviously be identical. Therefore, for \(m\) cycles,

\[ \xi(mT)=[\xi(T)]^m . \tag{2.2} \]

Thus, the values of the currents and voltages at the beginning of the \((m+1)\)-st cycle will be determined by the equation

\[ R(mT)=[\xi(T)]^m R(t_0). \tag{2.3} \]

In order to pass to the steady-state regime, it is necessary in (2.3) to put \(m=\infty\). The passage to the limit, however, can be avoided if one notes that in a periodic regime the continuous variables must assume the same values \(R_0\) at the beginning of any cycle, as a result of which the matrix equality

\[ R_0=\xi(T)R_0. \tag{2.4} \]

From (2.1) and (2.4) it follows that

\[ R_0= \left| \frac{[I-\xi(T)]^{-1}\xi_0(T)}{1} \right|, \tag{2.5} \]

where \(I\) is the identity matrix.

The following technique may be recommended for calculating the steady-state regime in a circuit with periodically varying parameters:

a) the cycle is divided into \(l\) subintervals, and then the augmented parameters of the subintervals are determined by the method considered in Sec. 1;

b) the augmented parameter of the cycle is calculated as the product of the parameters of the subintervals;

c) by formula (2.5) the values of the continuous variables at the beginning of the cycle of the steady-state regime are found;

d) the currents and voltages of the circuit within the cycle are determined, for which one should use the expression

\[ R(t_s)=\xi(t_s)R_0;\qquad (s=1,2,\ldots,l). \tag{2.6} \]

1. The method considered above for calculating circuits with variable parameters leads to simple computations in the case of circuits of the first and second order, but proves practically complicated already for a third-order circuit, since in this case it becomes necessary to find the roots of a cubic characteristic equation. Such a difficulty, incidentally, also occurs in the investigation of transient regimes in circuits with constant parameters.

In those cases when the order of the circuit is greater than two, it seems expedient to clarify the possibility of finding the augmented parameters of the links of the time chain without resorting to the solution of differential equations with constant coefficients, which in any case describe the process in the given time interval only approximately. Such a possibility exists and can be realized by applying to the study of the process the well-known method of successive intervals. Let us show this.

The integro-differential equations of the circuit, as is known, can always be reduced to a system of first-order equations:

\[ \frac{d\rho_1}{dt} = a_{11}\rho_1+a_{12}\rho_2+\ldots+a_{1n}\rho_n+a_{10}, \]

\[ \frac{d\rho_2}{dt} = a_{21}\rho_1+a_{22}\rho_2+\ldots+a_{2n}\rho_n+a_{20}, \tag{3.1} \]

\[ \frac{d\rho_n}{dt} = a_{n1}\rho_1+a_{n2}\rho_2+\ldots+a_{nn}\rho_n+a_{n0}, \]

where \(\rho_i=\rho_i(t)\) are continuous variables, and \(a_{ik}=a_{ik}(t)\) are prescribed functions of time.

Using matrix notation, we write equations (3.1) in the more compact form:

\[ \frac{d\rho(t)}{dt}=a(t)\rho(t)+a_0(t). \tag{3.2} \]

According to the method of successive intervals, this equation may be replaced by the approximate one

\[ \rho(t_s)=\rho(t_{s-1})+[a(t_{s-1})\rho(t_{s-1})+a_0(t_{s-1})]\tau_s. \tag{3.3} \]

Here \(\rho(t_{s-1})\) and \(\rho(t_s)\) are used in the same sense as in all the preceding relations, and \(a'(t_{s-1})=a(t)_{t=t_{s-1}}\) and \(a(t_s)=a(t)_{t=t_s}\).

Expression (3.3) corresponds to a system of algebraic equations and can be rewritten as

\[ \rho(t_s)=\left[I+\tau_s a(t_{s-1})\right]\rho(t_{s-1})+\tau_s a_0(t_{s-1}). \tag{3.4} \]

This equation, in its form, does not differ from equation (1.2). The coefficients of (1.2) and (3.4), for a sufficiently small interval \(\tau_s\), will be approximately the same, i.e.

\[ \zeta(\tau_s)\simeq I+\tau_s a(t_{s-1}) \quad \text{and} \quad \zeta_0(\tau_s)\simeq \tau_s a_0(t_{s-1}). \tag{3.5} \]

From what has been set forth it is clear that the augmented parameters of the links of the time chain can be found by directly using the coefficients of the differential equations (3.1) or (3.2), by substituting the relations (3.5) into the expression obtained earlier

\[ \xi(\tau_s)= \begin{vmatrix} \zeta(\tau_s) & \zeta_0(\tau_s)\\ 0 & 1 \end{vmatrix}. \tag{3.6} \]

See also (1.5).

After the augmented parameters of the links of the time chain have been found, the calculation of the process is carried out as indicated in item 1, or, in the case of a periodic variation of the parameters, as in item 2.

Let us note that if it is desirable to increase somewhat the accuracy of the calculations for a given \(\tau_s\), then instead of (3.3) one should use the more exact relation

\[ \rho(t_s)=\rho(t_{s-1})+\frac{\tau_s}{2} \left[ a(t_{s-1})\rho(t_{s-1})+a_0(t_{s-1})+a(t_s)\rho(t_s)+a_0(t_s) \right]. \tag{3.7} \]

In this case the parameters \(\zeta(\tau_s)\) and \(\zeta_0(\tau_s)\) are determined by the formulas

\[ \zeta(\tau_s)= \left[ I-\frac{\tau_s a(t_s)}{2} \right]^{-1} \left[ I+\frac{\tau_s a(t_{s-1})}{2} \right], \tag{3.8} \]

\[ \zeta_0(\tau_s)= \left[ I-\frac{\tau_s a(t_s)}{2} \right]^{-1} \frac{\tau_s}{2} \left[ a_0(t_{s-1})+a_0(t_s) \right], \]

which are somewhat more complicated than (3.5).

1. In conclusion, let us note that the chain method considered differs from other approximate methods of mathematical investigation of circuits with variable parameters in that it makes it possible not only to perform numerical calculations, but also to obtain certain general formulas for the sought quantities. Illustrations of this are, in particular, formulas (2.5) and (2.6), which pertain to the established periodic regime.

If the inductances and capacitances of the circuit are constant, while the resistances are step functions of time, then the relations given in items 1 and 2 provide the exact solution of the problem. As applied to such a case they were considered earlier in [9]. The laws of variation in time of the continuous variables in the \((s+1)\)-st interval are then determined by the equation

\[ R(t)=\xi(t-t_s)\xi(t_s)R(t_0), \tag{4.1} \]

since at \(t=t_{s+1}\) the vector \(R(t)\), on the basis of (1.10), must take the value \(R(t_{s+1})=\xi(t_{s+1})R(t_0)\).

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