ON THE CALCULATION OF TRANSIENT PROCESSES IN LINEAR CIRCUITS UNDER COMPLEX EXCITATIONS

P. P. Galinskii

(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering)

Transient processes in linear circuits with sources of emf and current, described by smooth or piecewise-smooth functions, are usually investigated by means of superposition integrals (Duhamel’s, Fourier’s), or by the operational method. The methods named are universal for their areas of application. In a number of cases, these universal methods lead to cumbersome notation and calculations. This circumstance justifies the search for methods of direct integration of the differential equations of state of electrical circuits (the classical method), not only for cases of constant and harmonic sources, but also in the case of sources described by other functions.

In the classical method, as is known, the solution is found by the method of superposition in the form of the sum of a particular solution of the complete equation (or system of equations)—the forced component of the solution—and the general solution of the homogeneous equation—the free component of the solution.

For arbitrary excitation functions, the constants of the forced solution may be found separately when the functions of the forced and free responses of the circuit do not coincide. When the functions of the forced and free components of the solution coincide, the constants of integration are found jointly for both components of the solution, as is customary in the classical method under constant and harmonic excitations for the constants of the free solution.

The form of the forced-response function is determined by the form of the disturbance function (emf, current source). In most cases the form of the forced response can be determined by the sum of the disturbance function and its derivatives. This is true for disturbance functions represented by integral polynomials, exponential, hyperbolic, harmonic functions, and their pairwise products. For example, for the disturbance \(e(t)=mt\), the sum \(e(t)+e'(t)=mt+m\) is formed. The second and higher derivatives are equal to zero. From the obtained sum the forced-response function is formed: \(i_{\text{pr}}(t)=at+b\), where \(a\) and \(b\) are constants.

For \(e(t)=Et e^{-\alpha t}\), one may form the sum \(e(t)+e'(t)=Et e^{-\alpha t}-\alpha Et e^{-\alpha t}+Ee^{-\alpha t}\). Higher derivatives do not yield functions of a new form; consequently,
\(i_{\text{pr}}(t)=a t e^{-\alpha t}+b e^{-\alpha t}\), where \(a, b\) are constants.

Similarly, for \(e(t)=mt^2\), the forced-response function is determined in the form of a quadratic parabola of general form

\[ i_{\mathrm{pr}}(t)=at^2+bt+c. \]

For an \(n\)-th order circuit, after determining the form of the forced response, a system of \(n\) equations is written for the independent variables—the currents in the inductances and the voltages on the capacitances—for the time of the transient process \(t\geq +0\):

\[ \vec{\vec{a}}\,\vec{i}=\vec{e}. \tag{1} \]

The initial values of the indicated variables and of \(n-1\) of their first derivatives are determined. From the principal determinant the characteristic equation is formed and its roots are found. The subsequent procedure of calculation differs depending on whether the functions of the free and forced responses coincide or do not coincide. If these functions do not coincide, then the forced response is determined by independently substituting the forced-response function into the system of equations (1):

\[ \vec{\vec{a}}\,\vec{i}_{\mathrm{pr}}=\vec{e}. \tag{2} \]

Equating in the obtained system (2) the coefficients of identical functions on both sides of the equality (functional balance), we obtain a linear algebraic system of equations with respect to the unknown constants \(\vec{d}\) of the forced response

\[ \vec{\vec{a}}_{1}\,\vec{d}=\vec{e}_{1}, \tag{3} \]

from which these constants are determined. Finally, the constants of the free response are found from the initial values, as usual in the classical method.

If, however, some excitation function coincides with a free-response function, then this is equivalent to the appearance of an (additional) multiple root. In that case it is not possible to separate out the forced response; the constants of the forced and free responses are determined jointly, as in the case of multiple roots. But even in this case, substitution into system (1) of the sum of the terms of the forced and free solutions that are connected by the multiplicity of the root gives the missing linear algebraic equations for the joint determination of the constants of the free and forced solutions.

Example. In the circuit (Fig. 1), at the instant \(t=0\), a source of linear emf \(e(t)=Et\) and a source of exponential current \(I(t)=Ie^{-\alpha t}\) are connected. Determine the currents and voltages of the transient process.

We shall first seek the independent quantities \(i_L(t)=i\) and \(u_c(t)=u\). The forced components of the solution due to the action of the emf and of the current source will be found by the superposition method.

Those forced by the action of the emf have the form

\[ i_{\mathrm{pr}}^{e}=At+B;\qquad u_{\mathrm{pr}}^{e}=Dt+F; \]

those forced by the current source,

\[ i_{\mathrm{pr}}^{I}=Me^{-\alpha t},\qquad u_{\mathrm{pr}}^{I}=Ne^{-\alpha t}. \]

We compose a system of equations for the post-switching circuit and eliminate from it the dependent quantity \(i_R\) by the node equation.

\[ i_R=i+C\frac{du}{dt}-I(t). \]

As a result we obtain a system of equations with respect to the independent quantities \((i=i_L,\ u=u_c)\):

\[ \left. \begin{aligned} (R+r)i+L\frac{di}{dt}+RC\frac{du}{dt}&=e(t)+RI(t),\\ -ri-L\frac{di}{dt}+u&=0. \end{aligned} \right\} \tag{*} \]

The initial independent quantities are found from the precommutation circuit: \(i(0)=0,\ u(0)=0\). Substituting these values and \(t=0\) into the system \((*)\), we find the initial values of the derivatives: \(i'(0)=0,\ u'(0)=-I/C\).

The characteristic equation is obtained by setting the principal determinant of the system \((*)\) equal to zero:

\[ \left| \begin{array}{cc} R+r+Lp, & RCp\\ -r-Lp, & 1 \end{array} \right| =0,\quad (R+r+Lp)\cdot 1+(r+Lp)RCp=0. \]

From the latter we determine the roots \(P_1\) and \(P_2\).
For different roots, the free components have the form

\[ i_{\mathrm{sv}}(t)=A_1e^{P_1t}+A_2e^{P_2t}, \]

\[ u_{\mathrm{sv}}(t)=B_1e^{P_1t}+B_2e^{P_2t}; \]

for multiple roots \(P_1=P_2=-\alpha\):

\[ i_{\mathrm{sv}}(t)=(A_3+A_4t)e^{-\alpha t}; \]

\[ u_{\mathrm{sv}}(t)=(B_3+B_4t)e^{-at}. \]

The forced components due to the emf (linear functions) do not coincide with the free functions (exponentials), and therefore they can be found independently by substituting these forced components into the system \((*)\):

\[ (R+r)(At+B)+L\frac{d}{dt}(At+B)+RC\frac{d}{dt}(Dt+F)=Et; \]

\[ -r(At+B)-L\frac{d}{dt}(At+B)+Dt+F=0. \]

By the method of functional balance (equating the coefficients of identical functions), from the last system we obtain an algebraic linear system \((**)\) for the constants of the forced solution \(A,\ B,\ D,\ F\), and determine these constants.

\[ \left. \begin{array}{rl} t\ \left\{ \begin{array}{l} (R+r)A=E,\\ -rA+D=0, \end{array} \right.\\[1.2em] t^\circ\ \left\{ \begin{array}{l} (R+r)B+LA+RCD=0,\\ -rB-LA+F=0. \end{array} \right. \end{array} \right\} \tag{**} \]

The forced components due to an exponential current source, in the case where the roots \(p_1\) and \(p_2\) of the free mode do not coincide with the exponent \(-\alpha\), are...

of the desired current-source function are found independently of the constants of the free solution. In doing so one may use the system (*) and the aforementioned universal method of functional balance:

\[ (R+r)Me^{-\alpha t}+L\frac{d}{dt}(Me^{-\alpha t})+RC\frac{d}{dt}(Ne^{-\alpha t}) =R\cdot Ie^{-\alpha t}; \]

\[ -rMe^{-\alpha t}-L\frac{d}{dt}(Me^{-\alpha t})+Ne^{-\alpha t}=0. \]

Whence we obtain an algebraic system with respect to \(M\) and \(N\):

\[ (R+r-\alpha L)M-\alpha RCN=RI; \]

\[ (-r+\alpha L)M+N=0, \]

and determine these constants.

However, to determine the forced exponential response (when the exponent \(s=-\alpha\) is not equal to the roots \(p_1\) and \(p_2\)) one can make use of the well-developed methods for calculating direct-current circuits. Since for direct current, complex harmonic current, and exponential current the basic function is the exponential, the general methods of calculation, the theory of four-terminal networks, the concept of a transfer (system) function, etc. [1, 2], are equally applicable to these currents.

Fig. 1.

Assuming that in the circuit of Fig. 1 only an exponential current source \(I(t)=Ie^{-\alpha}\) acts, we calculate the forced regime by the node-voltage method.

The voltage between the nodes, equal to the voltage on the capacitor, is

\[ N=U_c(-\alpha)= \frac{I}{\dfrac{1}{R}+\dfrac{1}{r-\alpha L}-\alpha C}. \]

The inductor current is

\[ M=I_L(-\alpha)=\frac{N}{r-\alpha L}. \]

Here the concepts of the resistances of the resistor, inductance, and capacitance for an exponential excitation are used \((R,\, sL,\, 1/sC,\ s=-\alpha)\).

It is not difficult to verify that the result obtained by the node-voltage method coincides with the result of solving the preceding system by the method of functional balance.

Having thus determined both forced components—the linear and the exponential—we then determine the constants of the free regime in the usual way for the classical method, through the initial values of the quantities and their derivatives.

Let us now consider the case in which the forced and free exponential functions coincide.

If \(p_2 \ne p_1=-\alpha\), then the free component from the root \(p_1\) and the forced component from the exponential current are associated. Their constants should be found jointly:

\[ i(p_1,\alpha)=(A_1+A_3t)e^{-\alpha t}; \]

\[ u(p_1,\alpha)=(B_1+B_3t)e^{-\alpha t}. \]

Substituting these values into system \((*)\), retaining on the right-hand side of the latter only \(RI(t)\), and equating coefficients for the function \(e^{-\alpha t}\), we obtain two additional equations for the constants \(A_1, A_3, B_1, B_3\):

\[ (R+r-\alpha L)A_1-\alpha RCB_1+LA_3+RCB_3=RI, \]

\[ -(r-\alpha L)A_1+B_1-LA_3=0. \]

The remaining four equations for the constants \(A_1, A_2, A_3, B_1, B_2, B_3\) are obtained from the initial conditions for the independent quantities

\[ i(t)=At+B+(A_1+A_3t)e^{-\alpha t}+A_2e^{p_2t}; \]

\[ u(t)=Dt+F+(B_1+B_3t)e^{-\alpha t}+B_2e^{p_2t} \]

and their derivatives.

If, however, both roots coincide with the exponent of the exponential current source, \(p_1=p_2=-\alpha\), then all three exponential terms turn out to be associated:

\[ i(p_1,p_2,\alpha)=(A_1+A_2t+A_3t^2)e^{-\alpha t}; \]

\[ u(p_1,p_2\alpha)=(B_1+B_2t+B_3t^2)e^{-\alpha t}. \]

Substitution of these values into system \((*)\) and the subsequent balance for \(e^{-\alpha t}\) gives two equations for the integration constants:

\[ (R+r-\alpha L)A_1-\alpha RCB_1+LA_2+RCB_2=RI; \]

\[ -(r-\alpha L)A_1+B_1-LA_2=0. \]

The remaining four equations, as in the preceding case, are obtained from the initial conditions for the current \(i=i_{\mathrm{pr}}+i_{\mathrm{sv}}\) and voltage \(u=u_{\mathrm{pr}}+u_{\mathrm{sv}}\).

LITERATURE

1. K. M. Polivanov. Theoretical Foundations of Electrical Engineering, Vol. 1, “Energiya,” 1972.

2. F. Reza and S. Seely. Modern Analysis of Electrical Circuits. “Energiya,” 1964.