CALCULATION OF AN ELECTRICAL CIRCUIT WITH STEPWISE VARIABLE RESISTANCE

G. E. PUKHOV

Below we consider the general problem of determining currents and voltages in a linear electrical circuit with lumped parameters, one of whose resistances varies with time in the following way:

\[ r(t)= \begin{cases} r_1 & \text{from } t_0 \text{ to } t_1,\\ r_2 & \text{” } t_1 \text{ ” } t_2,\\ \ldots\ldots\ldots\ldots\ldots\\ r_k & \text{” } t_{k-1} \text{ ” } t_k,\\ & \text{etc.} \end{cases} \tag{0.1} \]

and also the particular problem corresponding to the case of cyclic variation of \(r(t)\) according to the law of a periodic step function:

\[ r(t)= \left\{ \begin{array}{llll} r_1 & \text{from } t_0 & \text{to} & t_1,\\ r_2 & \text{” } t_1 & \text{”} & t_2,\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots & & &\\ r_s & \text{” } t_{s-1} & \text{”} & t_s=T,\\[2mm] r_1 & \text{from } T & \text{to} & T+t_1,\\ r_2 & \text{” } T+t_1 & \text{”} & T+t_2,\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots & & &\\ r_s & \text{” } T+t_{s-2} & \text{”} & T+t_s=2T,\\[2mm] r_1 & \text{from } 2T & \text{to} & 2T+t_1,\\ r_2 & \text{” } 2T+t_1 & \text{”} & 2T+t_2,\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots & & & \end{array} \right. \begin{array}{l} \left.\begin{array}{l}\\ \\ \\ \end{array}\right\} \text{ First cycle}\\[9mm] \left.\begin{array}{l}\\ \\ \\ \end{array}\right\} \text{ Second cycle}\\[9mm] \left.\begin{array}{l}\\ \\ \end{array}\right\} \text{ etc.} \end{array} \tag{0.2} \]

Here \(t_0=0\) is the origin of the time \(t\), \(T\) is the period of variation of \(r(t)\), or, equivalently, the duration of a cycle, and \(s\) is the number of time steps of the cycle.

During each time step \(\tau_k=t_k-t_{k-1}\) \((k=1,2,\ldots,s)\), the resistance \(r(t)\) is constant and equal to \(r_k\). For some time steps the resistance may become infinite (circuits with interrupters).

The stepwise variation \(r(t)\) corresponding to (0.1) and (0.2) is, in practice, realized by electronic and mechanical switching devices.

1. The study of processes in a circuit with stepwise variable resistances is usually carried out by means of the matching method, proposed—

introduced as early as 1912 by Academician N. D. Papaleksi [1] for the calculation of a circuit with a rectifier, that is, a nonlinear circuit. The application of this method to the solution of the linear problem considered here, and therefore of a fundamentally simpler problem, leads to somewhat less complicated but nevertheless rather cumbersome calculations, and may be briefly reduced to the following.

During the time interval $\tau_k$ the resistance of the circuit remains constant and equal to $r_k$, and therefore its state is described by a system of integro-differential equations with constant coefficients. Having solved this system in general form for the $k$-th time step and putting $k=1,2,\ldots$, one can then, knowing the initial conditions for the first step, step by step, that is, “matching” to the first step the second, to the second the third, etc., determine the required currents and voltages at any instant of time.

Being sufficiently general, the matching method also has disadvantages, of which the following two are essential for the problem under consideration: 1) determining the state of the circuit in a given time step is connected with the need for a preliminary calculation of the currents and voltages of all preceding steps, and 2) in the case of a stepwise-periodic change of resistance according to (0.2), the matching method makes it possible to obtain only an approximate solution if the established regime is being investigated.

Apparently, it was precisely these shortcomings that led to the appearance of other methods of calculation, in particular those based on the use of finite-difference equations. In [2; 3], for example, it is indicated how, with the aid of difference equations, to calculate a sawtooth-voltage generator; in [4] methods are described for calculating certain processes of discontinuous regulation; in [5], where the difference equations are algebraized by applying the so-called discrete Laplace transform, the fundamental circuit of an inverter is calculated, etc.

Despite the unquestionable successes achieved in the use of the apparatus of difference equations, it is necessary to bear in mind that this apparatus is completely unsuitable for solving the aforementioned general problem, which, according to (0.1), corresponds to a nonperiodic process, and is also unsuitable for determining currents and voltages within the cycle of a periodic process. As is clear from [2], the process within the cycle, even when finite-difference equations are used, is investigated by the matching method.

The proposed method of calculation is based on the observed analogy between the equations of so-called transmission network circuits, that is, circuits whose constituent links (subcircuits) can be described only by input and output quantities [6], and the equations of the process in the circuit under consideration with stepwise-variable resistance. Such a process corresponds to a “time” network circuit, the subcircuits of which are the time steps connected in series (the “output” of a given step is directly connected with the “input” of the next step).

The analogy between an ordinary network circuit and a “time” network circuit becomes most complete if it is agreed to call the end of any time step its “input,” and the beginning its “output.” Then the “input” of the “time” network circuit will correspond to the time instant under consideration, and the “output” to the beginning of the process. The time network circuit is shown in Fig. 1.

Let us note that the calculation expressions (2.13) and (2.14) obtained below are, in essence, a certain generalization of the formula of continuation, known in the matrix theory of differential equations [8].

1. Let us consider a method for calculating the circuit for the general case, when the resistance \(r(t)\) varies according to (0.1). To shorten the notation for the law of variation of \(r(t)\), we shall use the concept of the unit function \(\sigma_0(t)\) and introduce the notation:

\[ \gamma_k=\sigma_0(t-t_{k-1})-\sigma_0(t-t_k)= \begin{cases} 0 & \text{for } t<t_{k-1},\\ 1 & \text{for } t_{k-1}\le t\le t_k,\\ 0 & \text{for } t>t_k. \end{cases} \tag{2.1} \]

Obviously, \(\gamma_k\) represents a rectangular unit pulse of duration \(\tau_k=t_k-t_{k-1}\), equal to unity in the time interval from \(t_{k-1}\) to \(t_k\), and equal to zero in the remaining intervals. By means of \(\gamma_k\), dependence (0.1) is written more briefly:

\[ r(t)=r_k\gamma_k\quad (k=1,2,\ldots). \tag{2.2} \]

As the variables of the circuit state equations we shall choose variables that cannot change by a jump. Then, as in an ordinary chain circuit, the output quantities of the given link (time step) will simultaneously be the input quantities for the next link. If the inductances of the coils and the capacitances of the capacitors of the circuit are constant, then the currents in the coils and the voltages on the capacitors will be continuous. We shall call these currents and voltages continuous variables and denote them by \(\rho_1,\rho_2,\ldots,\rho_m\).

Fig. 1

It is convenient to regard the variables \((k=1,2,\ldots,m)\) as the components of an \(m\)-dimensional vector of the form

\[ \rho=\rho(t)= \begin{bmatrix} \rho_1=\rho_1(t)\\ \rho_2=\rho_2(t)\\ \cdot\\ \rho_m=\rho_m(t) \end{bmatrix}. \tag{2.3} \]

The order of the vector \(\rho\) is equal to the order of the differential equation describing the state of the circuit or, which leads to the same result, to the number of independent stores of electric and magnetic energy in the circuit. In the general case, when each of the \(B\) branches of the circuit contains both induct-

...inductances and capacitances, the order of the vector \(\rho\) will be equal to \(2B\). For example, for the circuit of Fig. 2 the vector \(\rho\) will be two-dimensional:

\[ \rho(t)= \begin{vmatrix} \rho_1=u(t)\\ \rho_2=i(t) \end{vmatrix}, \tag{2.4} \]

where \(u(t)\) is the voltage across the capacitance, and \(i(t)\) is the current in the inductance.

By virtue of the assumed linearity of the circuit, the law of variation of the continuous variables in the time interval from \(t_{k-1}\) to \(t_k\) must be expressed by the relation

\[ \rho(t)=\gamma_k \zeta(t-t_{k-1})\,\rho(t_{k-1})+\gamma_k \zeta_0(t-t_{k-1}), \tag{2.5} \]

where \(\zeta(t-t_{k-1})\) and \(\zeta_0(t-t_{k-1})\) are respectively a matrix and a vector whose components depend on time, and \(\rho(t_{k-1})\) is the value of the vector \(\rho(t)\) at the beginning of the \(k\)-th step, that is, at \(t=t_{k-1}\) \((k=1,2,\ldots)\).

Fig. 2

The relation (2.5) for each particular circuit is found by setting up and solving its integro-differential equations for the \(k\)-th time step. Thus, for the circuit of Fig. 2 with \(\mathcal{E}=\mathrm{const.}\), after solving the obvious equations

\[ u+L\frac{di}{dt}=\mathcal{E} \quad\text{and}\quad C\frac{du}{dt}+\frac{u}{r}=i, \tag{2.6} \]

we obtain

\[ \begin{vmatrix} u(t)\\ i(t) \end{vmatrix} = \gamma_k \zeta(t-t_{k-1}) \begin{vmatrix} u(t_{k-1})\\ i(t_{k-1}) \end{vmatrix} + \gamma_k \zeta_0(t-t_{k-1}), \tag{2.7} \]

where

\[ \zeta(t-t_{k-1})= \begin{vmatrix} \sin\!\left[\omega_k(t-t_{k-1})-\Theta_k\right] & \dfrac{\sin \omega_k(t-t_{k-1})}{\omega_0 C} \\[1.2em] -\dfrac{\sin \omega_k(t-t_{k-1})}{\omega_0 L} & \sin\!\left[\omega_k(t-t_{k-1})-\Theta_k\right] \end{vmatrix} \frac{e^{-\delta_k(t-t_{k-1})}}{\sin \Theta_k}, \tag{2.8a} \]

\[ \zeta_0(t-t_{k-1})= \begin{vmatrix} \displaystyle \mathcal{E}-\mathcal{E}e^{-\delta_k(t-t_{k-1})} \frac{\sin \omega_k(t-t_{k-1})+\Theta_k}{\sin \Theta_k} \\[1.2em] \displaystyle \frac{\mathcal{E}}{r}-\frac{\mathcal{E}}{r}e^{-\delta_k(t-t_{k-1})} \frac{\sin\!\left[\omega_k(t-t_{k-1})+2\Theta_k\right]}{\sin 2\Theta_k} \end{vmatrix}, \tag{2.8b} \]

\[ \delta_k=\frac{1}{2r_k C},\quad \omega_k=\sqrt{\omega_0^2-\delta_k^2},\quad \omega_0=\frac{1}{LC}, \]

\[ \Theta_k=\operatorname{arctg}\frac{\omega_k}{\delta_k} \tag{2.8b} \]

(when carrying out the calculation it was assumed that \(\omega_0>\delta_k\)).

The general equation (2.5) makes it possible to express the continuous variables at the end of a time step in terms of their values at the beginning of the step. Substituting in (2.5), instead of \(t\), the time \(t_k\) corresponding to the end of the \(k\)-th step, and noting that \((\gamma_k)_{t=t_k}=1\), we obtain:

\[ \rho(t_k)=\zeta(\tau_k)\rho(t_{k-1})+\zeta_0(\tau_k). \tag{2.9} \]

Let us note that (2.9) corresponds to a system of ordinary algebraic equations with constant coefficients. For example, for the circuit of Fig. 2, according to (2.7) we may write:

\[ u(t_k)=\zeta_{11}(\tau_k)u(t_{k-1})+\zeta_{12}(\tau_k)i(t_{k-1})+\zeta_{10}(\tau_k), \]

\[ i(t_k)=\zeta_{21}(\tau_k)u(t_{k-1})+\zeta_{22}(\tau_k)i(t_{k-1})+\zeta_{20}(\tau_k). \tag{2.10} \]

Equation (2.9) is constructed in such a way that the input variables of a step are expressed in terms of the output variables. Since the equations of ordinary chain circuits are drawn up according to the same principle, the theory of the latter can, to a considerable extent, be used in the study of processes in a circuit with stepwise-variable resistance. In particular, the process in the circuit of Fig. 2 may be studied on the basis of the theory of a chain circuit consisting of four-terminal networks, since equations (2.10) coincide in form with the equations of an active four-terminal network [6;7].

It is natural to call relation (2.9) the equations of the \(k\)-th link of a “time” chain circuit, and \(\zeta(\tau_k)\) and \(\zeta_0(\tau_k)\), respectively, the matrix and vector parameters of such a link.

For each particular circuit, the parameters \(\zeta(\tau_k)\) and \(\zeta_0(\tau_k)\) of all steps can be calculated very simply if equation (2.5) has been compiled. Thus, for example, for the circuit of Fig. 2 the components of these parameters for any time step are determined by formulas (2.8) by replacing \(t\) with \(t_k\).

It should be noted that the matrix parameters do not depend on the form of the e.m.f.’s acting in the circuit, but only on its primary parameters \((p_k, L, C)\) and on the duration of the time step \(\tau_k\).

The matrix and vector parameters of a “time” chain circuit consisting of several time steps are found in the same way as for a chain circuit whose links are characterized only by input and output quantities [6]. Thus, for a “time” circuit consisting of two links, the matrix parameter is equal to the product of the matrix parameters of the component links, while the vector parameter is the sum whose terms are the vector parameter of the second link and the product of its matrix parameter by the vector parameter of the first link, that is,

\[ \zeta(t_2)=\zeta(\tau_2)\zeta(\tau_1), \tag{2.11} \]

\[ \zeta_0(t_2)=\zeta_0(\tau_2)+\zeta(\tau_2)\zeta_0(\tau_1). \tag{2.12} \]

For a “time” circuit consisting of \(k-1\) links, the matrix and vector parameters will be equal to:

\[ \zeta(t_{k-1})=\zeta(\tau_{k-1})\zeta(\tau_{k-2})\ldots\zeta(\tau_1), \tag{2.13} \]

\[ \zeta_0(t_{k-1})=\zeta_0(\tau_{k-1})+\zeta(\tau_{k-1})\zeta_0(\tau_{k-2})+ \]

\[ +\ldots+\zeta(\tau_{k-1})\zeta(\tau_{k-2})\ldots\zeta(\tau_{k-2})\zeta_0(\tau_1). \tag{2.14} \]

The parameters \(\zeta(t_{k-1})\) and \(\zeta_0(t_{k-1})\) make it possible, in the simplest way, to obtain the value of the vector \(\rho(t_{k-1})\), characterizing the beginning of the \(k\)-th stage. Obviously, this vector is equal to

\[ \rho(t_{k-1})=\zeta(t_{k-1})\rho(t_0)+\zeta_0(t_{k-1}). \tag{2.15} \]

This equation may be called the equation of the “time” network, consisting of \(k-1\) links (time stages).

Thus, in contrast to the fitting method, the values of the continuous variables at the beginning of the \(k\)-th stage are obtained, according to (2.15), directly from the matrix and vector parameters of the time interval equal to \(t_{k-1}\), and from the known values of the variables at the beginning of the first stage.

In the particular case where the continuous variables, that is, the currents in the inductances and the voltages on the capacitors of the circuit, are equal to zero at the beginning of the first stage, the vector

\[ \rho(t_{k-1})=\zeta_0(t_{k-1}), \tag{2.16} \]

since \(\rho(t_0)=0\).

Having determined the vector \(\rho(t_{k-1})\), it can then be substituted into the general equation (2.5), and thereby all currents and voltages of the circuit in the time interval from \(t_{k-1}\) to \(t_k\) can be determined.

Let us note that finding the parameters of the network may be reduced exclusively to the operation of multiplying cell matrices of the form

\[ \xi(\tau_k)= \begin{bmatrix} \zeta(\tau_k) & \zeta_0(\tau_k)\\ 0 & 1 \end{bmatrix}, \tag{2.17} \]

since by direct calculation it is easy to prove that

\[ \xi(t_{k-1})=\prod_{\nu=k-1}^{1}\xi(\tau_\nu)= \begin{bmatrix} \zeta(t_{k-1}) & \zeta_0(t_{k-1})\\ 0 & 1 \end{bmatrix}. \tag{2.18} \]

The practical use of such “extended” parameters is quite convenient. It is seen from (2.18) that, in order to find the extended parameter of a “time” network, it is sufficient to multiply the extended parameters of its links.

The method of composing the extended parameters of the links is clear from consideration of (2.17). In particular, for the circuit of Fig. 2 the extended parameter of the \(k\)-th link is equal to

\[ \xi(\tau_k)= \begin{bmatrix} - e^{-\delta_k\tau_k}\dfrac{\sin(\omega_k\tau_k-\Theta_k)}{\sin\Theta_k} & e^{-\delta_k\tau_k}\dfrac{\sin\omega_k\tau_k}{\omega_0 C\sin\Theta_k} & \mathcal{E}-\mathcal{E}e^{-\delta_k\tau_k}\dfrac{\sin(\omega_k\tau_k+\Theta_k)}{\sin\Theta_k} \\[1.2em] - e^{-\delta_k\tau_k}\dfrac{\sin\omega_k\tau_k}{\omega_0 L\sin\Theta_k} & e^{-\delta_k\tau_k}\dfrac{\sin(\omega_k\tau_k+\Theta_k)}{\sin\Theta_k} & \dfrac{\mathcal{E}}{r_k}-\dfrac{\mathcal{E}}{r_k}e^{-\delta_k\tau_k}\dfrac{\sin(\omega_k\tau_k+2\Theta_k)}{\sin 2\Theta_k} \\[1.2em] 0 & 0 & 1 \end{bmatrix}. \tag{2.19} \]

In writing (2.19) we have made use of expressions (2.8) and (2.18).

The following technique may be recommended for calculating a circuit with stepwise-variable resistance by the proposed method:

a) having solved the integro-differential equations of the circuit, written with respect to the continuous variables, and having represented the solution in the form

(2.9), one can determine the matrix \(\zeta(\tau_k)\) and vector \(\zeta_0(\tau_k)\) parameters for each link of the “time” chain circuit, putting \(k=1,2,\ldots\);

b) from expressions (2.13), (2.14), or (2.18) one should find the matrix \(\zeta(t_{k-1})\) and vector \(\zeta_0(t_{k-1})\) parameters of the time interval \(t_{k-1}\) corresponding to \(k-1\) links;

c) from equations (2.5), determine the continuous variables, and then all the remaining currents and voltages of the circuit in the given time interval from \(t_{k-1}\) to \(t_k\).

The calculation method described can, without major changes, also be applied to a circuit with several stepwise variable resistances. In this case, a step of time should be understood as a time interval during which all the circuit resistances remain constant.

1. Let us now consider the case when the stepwise variable resistance \(r(t)\) changes periodically with time, that is, according to (0.2).

We shall call the period of variation of \(r(t)\) the duration of the cycle, and assume that in each cycle the laws of variation of the circuit emf are identical. Suppose further, in accordance with (0.2), that the cycle consists of \(s\) steps, in the general case of unequal duration. The parameters of the cycle can be determined by the method indicated in Sec. 2. Since the laws of variation of the resistance \(r(t)\) and of the circuit emf are the same in all cycles, the parameters of all cycles will likewise be the same.

In view of what has been said, let us find the matrix \(\zeta(nT)\) and vector \(\zeta_0(nT)\) parameters of the time interval \(nT\), equal to \(n\) cycles (\(T\) is the duration of a cycle).

On the basis of expressions (2.13) and (2.14) we find

\[ \zeta(nT)=[\zeta(T)]^n, \tag{3.1} \]

\[ \zeta_0(nT)=[1+\zeta(T)+\zeta^2(T)+\ldots+\zeta^n(T)]\zeta_0(T)= \]

\[ =[1-\zeta(T)]^{-1}[1-\zeta^n(T)]\zeta_0(T), \tag{3.2} \]

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

These formulas for the special case when the circuit contains only one capacitor are given in [3], where they were obtained by means of difference equations.

Knowing the parameters \(\zeta(nT)\) and \(\zeta_0(nT)\), it is easy to determine the continuous currents and voltages at the beginning of the \((n+1)\)-st cycle:

\[ \rho(nT)=\zeta(nT)\rho(t_0)+\zeta_0(nT)= \]

\[ =\zeta^n(T)\rho(t_0)+[1-\zeta(T)]^{-1}[1-\zeta^n(T)]\zeta_0(T), \tag{3.3} \]

and then, from the general equation (2.5), the laws of their variation during the entire cycle.

Thus, for a given number \(n\) and the parameters of only one cycle, without any intermediate calculations, the sought currents and voltages of the circuit are determined directly from equations (3.3) and (2.5).

As is seen from expressions (2.8), all components of the matrix \(\zeta(nT)\) contain exponential functions with a negative exponent, owing to which

\[ \lim_{n\to\infty}\zeta^n(T)\to 0. \tag{3.4} \]

This makes it possible to solve the important problem of the distribution of currents and voltages in the circuit in the steady-state regime. Indeed, putting \(n=\infty\) in (3.3), we find the continuous variables at the beginning of the cycle of the steady-state regime:

\[ \rho_0=[1-\zeta(T)]^{-1}\zeta_0(T), \tag{3.5} \]

—and then, substituting \(\rho_0\) into equation (2.5) instead of \(\rho(t_{k-1})\), one can determine the laws of variation of \(\rho(t)\) over all steps of the cycle (in this case it is expedient to count the time \(t\) from the beginning of the cycle).

Let us note that \(\rho_0\) does not depend on \(n\). Therefore, in the steady-state regime the continuous currents and voltages at the beginning and end of the cycle will take the same values, and, consequently, the process in the circuit will be periodic in the full sense, since \(\rho(t)=\rho(t+T)\).

1. Let us give examples of calculating circuits with stepwise variable resistances.

Example 1. Determine the current of the steady-state regime in the circuit shown in Fig. 3, for the case when the e.m.f. of the source and the resistance depend on time as follows:

\[ u(t)=14.1\sin(314\,t)\ \text{V},\qquad r(t)= \begin{cases} r_1=1\ \Omega, & \text{when } u(t)<0,\\ r_2=6\ \Omega, & \text{when } u(t)>0. \end{cases} \]

\[ \text{Inductance } L=\frac{8}{314}\ \text{H}. \]

At the \(k\)-th time step the current of the circuit is obviously equal to:

\[ i(t)=\gamma_k e^{-\delta_k(t-t_{k-1})} i(t_{k-1}) +\gamma_k I_{m(k)} \left\{ \sin\left[\omega(t-t_{k-1})-\varphi_k\right] +e^{-\delta_k(t-t_{k-1})}\sin\varphi_k \right\}, \tag{4.1} \]

where

\[ \delta_k=\frac{r_k}{L},\qquad I_{m(k)}=\frac{u_m}{\sqrt{r_k^2+(\omega L)^2}},\qquad \varphi_k=\operatorname{arctg}\frac{\omega}{r_k}L. \tag{4.2} \]

Substituting \(t=t_k\) in (4.1), we obtain the equation of the \(k\)-th link of the “time” circuit diagram:

\[ i(t_k)=e^{-\delta_k\tau_k}i(t_{k-1}) +I_{m(k)}\left[\sin(\omega\tau_k-\varphi_k)+e^{-\delta_k\tau_k}\sin\varphi_k\right]. \tag{4.3} \]

Fig. 3

Comparing this dependence with the general equation (2.9), we conclude that in the case under consideration the parameters of the \(k\)-th link are equal to

\[ \zeta(\tau_k)=e^{-\delta_k\tau_k}, \tag{4.4} \]

\[ \zeta_0(\tau_k)=I_{m(k)}\left[\sin(\omega\tau_k-\varphi_k)+e^{-\delta_k\tau_k}\sin\varphi_k\right]. \]

Putting \(k=1\) and \(2\) and noting that \(\tau_1=\tau_2=\dfrac{T}{2}=0.01\ \text{s}\) and \(\omega\tau_k=\pi\),

after substituting the numerical values in (4.4), we obtain the parameters of the links of the cycle:

\[ \zeta(\tau_1)=e^{-\delta_1\tau_1}=e^{-0.392}=0.677,\qquad \zeta(\tau_2)=e^{-\delta_2\tau_2}=0.0944, \tag{4.5} \]

\[ \zeta_0(\tau_1)=I_{m(1)}\sin\varphi_1\left(1+e^{-\delta_1\tau_1}\right)\simeq 2.91\ \text{A}, \]

\[ \zeta_0(\tau_2)=I_{m(2)}\sin\varphi_2\left(1+e^{-\delta_2\tau_2}\right)\simeq 1.24\ \text{A}. \]

The cycle parameters, according to (2.11) and (2.12), are

\[ \zeta(T)=\zeta(\tau_2)\zeta(\tau_1)=0.0639, \tag{4.6} \]

\[ \zeta_0(T)=\zeta_0(\tau_2)+\zeta(\tau_2)\zeta_0(\tau_2)=1.513\ \text{a}. \]

By formula (3.5) we determine the current at the beginning of the cycle in the steady-state regime:

\[ i_0=[1-\zeta(T)]^{-1}\zeta_0(T)=1.615\ \text{a}. \tag{4.7} \]

If the origin of time is made to coincide with the beginning of the cycle of the steady-state regime, then the laws of variation of the current in the first and in the second steps are expressed by the equations:

a) in the first step \((0\leq t\leq 0.01\ \text{sec})\)

\[ \begin{aligned} i(t)&=i_0 e^{-\vartheta_1 t}+I_{m(1)}\left[\sin(\omega t-\varphi_1)+e^{-\vartheta_1 t}\sin\varphi_1\right] \\ &=1.615 e^{-39.2t}+1.75\sin(18000t-83^\circ10')\ \text{a}; \end{aligned} \tag{4.8} \]

b) in the second step \((0.01\leq t\leq 0.02\ \text{sec})\)

\[ \begin{aligned} i(t)&=i(\tau_1)e^{-\vartheta_2(t-\tau_1)}+I_{m(2)} \left\{\sin[\omega(t-\tau_1)-\varphi_2]\right.\\ &\qquad\left.+e^{-\vartheta_2(t-\tau_2)}\sin\varphi_2\right\} \\ &=54.5e^{-236t}+1.41\sin(18000t-233^\circ)\ \text{a}. \end{aligned} \tag{4.9} \]

From (4.8) and (4.9) it is seen that the stepwise-variable resistance of the circuit behaves in this case as a rectifier, since the current \(i(t)>0\) throughout the entire cycle.

Fig. 4

Example 2. Let the resistance \(r(t)\) in the circuit of Fig. 2 vary during two cycles as shown in Fig. 4, and then remain constant and equal to \(r_{13}=50\ \Omega\). Assuming zero initial conditions, let us find the currents and voltages in the thirteenth time step \((t\geq 0.012\ \text{sec})\) for the case when

\[ \mathcal{E}=100\ \text{v}, \qquad L=0.1\ \text{H}, \qquad C=25\ \mu\text{F}. \]

By formulas (2.19) and (2.8b) we determine the augmented parameters of the cycle stages¹:

\[ \xi(\tau_1)= \left| \begin{array}{ccc} 0{,}3341 & 25{,}8738 & 23{,}9210\\ -0{,}0065 & 0{,}7608 & 0{,}9463\\ 0{,}0000 & 0{,}0000 & 1{,}0000 \end{array} \right|, \quad \left| \begin{array}{ccc} 0{,}5209 & 30{,}9387 & 16{,}9020\\ -0{,}0077 & 0{,}8310 & 0{,}9414\\ 0{,}0000 & 0{,}0000 & 1{,}0000 \end{array} \right| =\xi(\tau_2), \]

\[ \xi(\tau_3)= \left| \begin{array}{ccc} 0{,}6068 & 32{,}7559 & 14{,}9310\\ -0{,}0082 & 0{,}8507 & 1{,}0040\\ 0{,}0000 & 0{,}0000 & 1{,}0000 \end{array} \right|, \quad \left| \begin{array}{ccc} 0{,}6478 & 34{,}2467 & 17{,}6632\\ -0{,}0085 & 0{,}8234 & 0{,}9316\\ 0{,}0000 & 0{,}0000 & 1{,}0000 \end{array} \right| =\xi(\tau_4), \]

\[ \xi(\tau_5)= \left| \begin{array}{ccc} 0{,}6068 & 32{,}7559 & 14{,}9310\\ -0{,}0082 & 0{,}8507 & 1{,}0040\\ 0{,}0000 & 0{,}0000 & 1{,}0000 \end{array} \right|; \quad \left| \begin{array}{ccc} 0{,}5209 & 30{,}9387 & 16{,}9020\\ -0{,}0077 & 0{,}8310 & 0{,}9414\\ 0{,}0000 & 0{,}0000 & 1{,}0000 \end{array} \right| =\xi(\tau_6). \]

Multiplying the augmented parameters of the stages, we find the augmented parameter of the cycle

\[ \xi(t_6)=\xi(\tau_6)\xi(\tau_5)\ldots\xi(\tau_1)= \left| \begin{array}{ccc} -0{,}0514 & -7{,}5791 & 133{,}5724\\ 0{,}0012 & -0{,}4156 & 0{,}7040\\ 0{,}0000 & 0{,}0000 & 1{,}0000 \end{array} \right|. \]

The parameters of the cycles are identical. Therefore the augmented parameter of two cycles will be equal to the square of the parameter of one cycle, i.e.

\[ \xi(t_{12})=[\xi(t_6)]^2= \left| \begin{array}{ccc} -0{,}0065 & 3{,}5395 & 121{,}3711\\ -0{,}0006 & 0{,}1636 & 0{,}5717\\ 0{,}0000 & 0{,}0000 & 1{,}0000 \end{array} \right|. \]

Since, by assumption, there is no initial reserve of energy in the circuit, according to expression (2.16) the voltage on the capacitor and the current in the inductance will simply be equal to the vector parameter \(\zeta_0(t_{12})\), i.e. \(u(t_{12})=121{,}3711\ \mathrm{V}\) and \(i(t_{12})=0{,}5717\ \mathrm{A}\). Substituting \(u(t_{12})\) and \(i(t_{12})\) into equation (2.7) and using (2.19), we determine the voltage and current in the specified thirteenth time stage

\[ \left| \begin{array}{c} u(t)\\ i(t) \end{array} \right| = \frac{r_{13}e^{-\delta_{13}(t-t_{12})}}{\sin\Theta_{13}} \left| \begin{array}{cc} \sin\left[\omega_{13}(t-t_{12})-\Theta_{13}\right] & \dfrac{\sin\omega_{13}(t-t_{12})}{\omega_0 C}\\[1.2em] \dfrac{\sin\omega_{13}(t-t_{12})}{\omega_0 L} & \sin\left[\omega_{13}(t-t_{12})+\Theta_{13}\right] \end{array} \right| \times \]

\[ \times \left| \begin{array}{c} u(t_{12})\\ i(t_{12}) \end{array} \right| + r_{13} \left| \begin{array}{c} \displaystyle \mathcal{E}-\mathcal{E}e^{-\delta_{13}(t-t_{12})} \frac{\sin\left[\omega_{13}(t-t_{12})+\Theta_{13}\right]}{\sin\Theta_{13}} \\[1.4em] \displaystyle \frac{\mathcal{E}}{r_{13}}-\frac{\mathcal{E}}{r_{13}}e^{-\delta_{13}(t-t_{12})} \frac{\sin\left[\omega_{13}(t-t_{12})+2\Theta_{13}\right]}{\sin 2\Theta_{13}} \end{array} \right|. \]

or, after calculation,

\[ u(t)=\gamma_{13}100+\gamma_{13}2630 e^{-400t}\sin(28074t+108^\circ11')\ \text{V}, \]

\[ (t\geqslant t_{12}=0.012\ \text{s}), \]

\[ i(t)=\gamma_{13}2+\gamma_{13}341.5 e^{-400t}\sin(28074t+233^\circ46')\ \text{A}. \]

The method of “time-varying circuit diagrams” described above can also be applied to an approximate analysis of circuits whose parameters vary continuously with time.

The calculation of circuits with continuously varying parameters is set forth in the paper “The Method of Chains,” which is published in the present volume of Izvestiya TPI.

References

1. N. D. Papeleksi. On processes in an alternating-current circuit containing an electric valve. Collected Works. Publishing House of the Academy of Sciences of the USSR, 1948.

2. M. F. Gardner, J. L. Barnes. Transient Processes in Linear Systems. Gostekhizdat, 1949.

3. A. O. Gromshtein. On the approximation of the process in a linear electric circuit with variable parameters, containing one reactive element, to the steady-state regime. 11, 1950.

4. R. Oldenbourg, H. Sartorius. Dynamics of Automatic Control. Gosenergoizdat, 1949.

5. Ya. Z. Tsypkin. Transient and Steady-State Processes in Pulse Circuits. Gosenergoizdat, 1951.

6. G. E. Pukhov. Theory of the Subcircuit Method. Elektrichestvo, No. 8, 1952.

7. E. V. Zelyakh. General Theory of Linear Electrical Circuits. Publishing House of the Academy of Sciences of the USSR, 1951.

8. R. Frazer, W. Duncan, A. Collar. Matrix Theory. Foreign Literature Publishing House, 1950.

1. The computations were carried out by assistant N. P. Dovganovskii. ↩