Optimization of the Normalized Transient Function of a Linear Circuit Described by a Fourth-Order Characteristic Equation

D. I. Sviryakin

(Presented by the scientific seminar of the Research Institute of Electronic Introscopy)

The study of the time characteristics of amplifiers has always been of great importance, both theoretically and practically, since without it the full use of amplifier properties is unthinkable. One means of improving the pulse characteristics of amplifiers is correction in the region of small times, which leads to an increase in the degree of the characteristic equation describing the circuit. A complete analysis is given in the literature [1, 2, 3] for amplifier circuits only with quadratic equations, or with the reduction of higher-degree equations to the second degree. Such a calculation is permissible in those cases when, during adjustment of the amplifier, tuning and selection of its elements are possible. Modern methods of producing integrated circuits, however, require a more accurate calculation of amplifiers with degrees of characteristic equations higher than the second, since adjustment of circuits after manufacture is practically excluded. The extremely limited number of works devoted to the development of such calculation methods indicates that the problem of determining parameters that ensure the minimum settling time of the transient characteristic, for specified or minimum overshoots, is one of the most difficult extremum problems in amplifier technology. In the presence of several variable parameters (correction coefficients), i.e., for characteristic equations higher than the second degree, the problem becomes considerably more complicated, and it is extremely difficult to solve it by random variation or by machine enumeration of combinations of parameters [4]. At the same time, extremum problems are not new to mathematics; they have been encountered and solved throughout the entire history of mathematics. But their intensive and systematic study began relatively recently, when, on the one hand, the demands of economics and automatic control made the solution of these problems urgent, and, on the other hand, the appearance of electronic computers (ECMs) gave researchers a powerful tool with which the solution could be carried through to the final result—a numerical result.

The well-known method of equal moduli [1] does not ensure the best of all possible results; approximate methods of analysis require preliminary determination of the monotonicity condition for the course of the transient process [2], which in the general case is a very difficult problem, and they do not guarantee sufficiently high accuracy [5].

Most often, in calculating an amplifier it is necessary to find not only the pa

parameters of a nonmonotonic transient characteristic, but also to determine the values of the correction coefficients that minimize the settling time for given overshoot values and a maximum gain coefficient. There is no analytical solution to such a problem, and without the use of computer methods and facilities it is practically impossible to solve it by the methods of numerical analysis either.

Methods for analyzing amplifiers with a third-degree characteristic equation, described in the literature [3], make it possible to estimate the nature of the transient process and the region of stable operation; however, these methods are unsuitable for determining the optimal parameters of circuits. In the present work, a procedure and an algorithm are presented for the numerical solution, on a digital electronic computer (ECM), of the problem of determining the optimal values of the correction coefficients for known initial parameters of an amplifier having a normalized transfer coefficient of the form

\[ M(p)= \frac{1+a_1p+a_2p^2+a_3p^3} {1+b_1p+b_2p^2+b_3p^3+b_4p^4} \]

and the transient characteristic

\[ h(\tau)=1+A_1e^{-\alpha_1\tau}\sin(\omega_1\tau+\varphi) +A_2e^{-\alpha_2\tau}\sin(\omega_2\tau+\psi), \]

where \(p=j\omega C_0R_a,\ j=\sqrt{-1}\), \(\omega\) is the angular frequency;

\[ \alpha_1,\ \alpha_2,\ \omega_1,\ \omega_2 \]

are the real and imaginary parts of the roots

\[ p_{1,2}=-\alpha_1\pm j\omega_1,\qquad p_{3,4}=-\alpha_2\pm j\omega_2 \]

of the characteristic equation

\[ 1+b_1p+b_2p^2+b_3p^3+b_4p^4=0. \]

The product \(C_0R_a\) is the time constant of the anode (collector) circuit of the active element of the amplifier stage being calculated.

Further,

\[ b_1=1-x+(1-b)k_2+b(q+xm)+xb, \]

\[ b_2=(k_1+k_2)[1-x(1-b)]+bk_2(q+xm-1)+b[q+xm(1-x)], \]

\[ b_3=(1-x)(1-b)k_1k_2+b(k_1+k_2)[q+xm(1-x)], \]

\[ b_4=bk_1k_2[q+xm(1-x)], \]

\[ A_1=\frac{1}{\omega_1b_4} \sqrt{\frac{T_1^2+\omega_1^2S_1^2}{R_1PQ}}, \]

\[ A_2=\frac{1}{\omega_2b_4} \sqrt{\frac{T_2^2+\omega_2^2S_2^2}{R_2PQ}}, \]

\[ T_1=1-\alpha_1a_1+(\alpha_1^2-\omega_1^2)a_2+(-\alpha_1^3+3\alpha_1\omega_1^2)a_3, \]

\[ T_2=1-\alpha_2a_1+(\omega_2^2-\omega_2^2)a_2+(-\alpha_2^3+3\alpha_2\omega_2^2)a_3, \]

\[ S_1=a_1-2\alpha_1a_2+(3\alpha_1^2-\omega_1^2)a_3, \]

\[ S_2=a_1-2\alpha_2a_2+(3\alpha_2^2-\omega_2^2)a_3, \]

\[ R_1=\alpha_1^2+\omega_1^2, \]

\[ R_2=\alpha_2^2+\omega_2^2, \]

\[ P=(\alpha_2-\alpha_1)^2+(\omega_1-\omega_2)^2, \]

\[ Q=(\alpha_2-\alpha_1)^2+(\omega_1+\omega_2)^2, \]

\[ \varphi=\operatorname{arctg}\frac{\omega_1S_1}{T_1} -\operatorname{arctg}\frac{\omega_1}{\alpha_1} -\operatorname{arctg}\frac{\omega_1-\omega_2}{\alpha_2-\alpha_1} -\operatorname{arctg}\frac{\omega_1+\omega_2}{\alpha_2-\alpha_1}, \]

\[ \psi=\operatorname{arctg}\frac{\omega_2S_2}{T_2} -\operatorname{arctg}\frac{\omega_2}{-\alpha_2} -\operatorname{arctg}\frac{\omega_2-\omega_1}{\alpha_1-\alpha_2} -\operatorname{arctg}\frac{\omega_1-\omega_2}{\alpha_1-\alpha_2}, \]

\[ a_1=q+k_1, \]

\[ a_2=q(k_1+k_2), \]
\[ a_3=qk_1k_2. \]

The initial parameters of the stage are \(b, m, x\); the sought correction coefficients are \(k_1, k_2\), and \(q\). For example, for a pentode stage [6] these parameters and coefficients are written in terms of the expressions

\[ b=\frac{1}{1+SR_k},\quad m=\frac{R_k}{R_a},\quad x=\frac{C_{\mathrm{out}}+C_{\mathrm{in}}}{C_0}, \]

\[ k_1=\frac{L_a}{C_0R_a^2},\quad k_2=\frac{L_k}{C_0R_aR_k},\quad g=\frac{C_kR_k}{C_0R_a}, \]

where \(S\) is the transconductance of the pentode anode current;
\(C_0=C_{\mathrm{out}}+C_{\mathrm{in}}+C_m\); \(C_{\mathrm{out}}, C_{\mathrm{in}}, C_m\) are, respectively, the capacitances of the output, the input of the pentode, and the wiring;
\(R_a\) and \(R_k\) are resistors in the anode and cathode circuits;
\(C_k\) is the correcting capacitor in the cathode;
\(L_a\) and \(L_k\) are the correcting inductances in the anode and cathode circuits.

To determine the normalized settling time \(y\), the expression \(y=\tau_{0.9}-\tau_{0.1}\) is used. Here \(\tau_{0.1}\) and \(\tau_{0.9}\) are the instants of time at which the function takes values respectively equal to 0.1 and 0.9. The first and second overshoots are respectively equal to \(\delta_1=h(\tau_{m1})-1\) and \(\delta_2=h(\tau_{m2})-2\), where \(\tau_{m1}\) and \(\tau_{m2}\) are the instants of time corresponding to the extrema of the function \(h(\tau)\) for \(\tau>0\). Here: \(\tau=\dfrac{t}{C_0R_a}\), \(t\) is time in seconds.

A complex correction circuit described by a fourth-degree equation can be represented as a combination of two simpler circuits, each of which is obtained by disconnecting the correcting elements and setting respectively the coefficients \(k_1\) or \(k_2\) equal to zero. (As will be shown below, for convenience of computation it is better to take not \(k_1=0\) or \(k_2=0\), but to set them equal to a very small quantity, for example, \(k_1=1\cdot10^{-5}\) or \(k_2=1\cdot10^{-5}\).) Therefore a program compiled for calculating the basic parameters of the transient response \(y, \delta_1\), and \(\delta_2\) is suitable both for the complex circuit and for simpler circuits. Preliminary investigations carried out with the aid of a computer showed that in a circuit with two correction coefficients \(q\) and \(k_1\) there arises the problem of determining the optimal values of these coefficients in order to obtain the minimum settling time for a given permissible value of \(\delta_1\), since for any combination of the values \(b, m, x\) one can obtain an overshoot equal to the specified value for a large number of combinations of the coefficients \(q\) and \(k_1\), but at a far from minimal settling time (Fig. 1). A circuit with coefficients \(q\) and \(k_2\) can yield a transient response in which the second overshoot is comparable with, or even greater than, the first. But in this circuit each combination of the values \(q\) and \(k_2\) provides only one combination of the overshoot magnitudes \(\delta_1\) and \(\delta_2\); therefore, knowing the limiting permissible values of \(\delta_1\) and \(\delta_2\), one can determine the values of \(q\) and \(k_2\).

With three correction coefficients \(q, k_1\), and \(k_2\), the dependencies between them and the parameters of the transient response become more complicated, and here, in searching for the optimal values of \(q, k_1\), and \(k_2\), it is necessary to ensure not only the minimum settling time, but also overshoots whose magnitudes do not exceed the prescribed values \(\delta_{13}\) and \(\delta_{23}\). For known values of the initial parameters \(b, x, m\), the search for the optimal correction coefficients is carried out according to the difference between the values obtained for randomly taken combinations of \(q, k_1, k_2\) and the prescribed permissible values of the overshoots \(\delta_{13}, \delta_{23}\), and according to the “descent” of the increment of the settling time-

Fig. 1. Curves of the dependence \(Y=f(q)\) for different values of \(k_1\), with curves of constant overshoots \(\delta_1\) plotted on them (dashed), illustrating the existence of \(Y_{\min}\) for optimal combinations of \(q\) and \(k_1\).

…of regulation \((\Delta Y)\) in the successive transition from one variant to another, where overshoots equal to the specified ones were obtained. Calling the main program for determining the parameters \(Y\), \(\delta_1\), and \(\delta_2\) by means of a computer [6], the search for the optimal values of the coefficients \(q\), \(k_1\), \(k_2\) is carried out as follows.

I. Determine the initial parameters of the cascade \(b\), \(x\), \(m\), using the value of the gain coefficient at medium frequencies and the nameplate parameters of the selected active element.

II. a) For the initial values \(b\), \(x\), \(m\) and the specified permissible value of the overshoot \(\delta_{1z}\), using the main program and taking \(k_2=1\cdot 10^{-5}\), take three values of \(q\) (in the range \(q=0.3\div 0.8\)), differing from one another by \(\Delta q\) (for example, \(\Delta q=0.05\)), and, according to the results of the calculations of the main program, check whether the inequality \(Y<2b\) is satisfied for each value of \(q\), taking \(k_1=0.1\div 0.3\).

b) If the inequality is not satisfied, increase \(q\) by 0.1 and repeat everything from the beginning.

c) If the inequality is satisfied for each of the adopted values of \(q\), determine the value of \(k_1\) at which the overshoot \(\delta_1\) is obtained equal to the specified one (i.e., \(\delta_1=\delta_{1z}\)). For this purpose, specify the value \(\Delta k_1=0.001\div 0.005\) and, using the main program, compute at unchanged \(q\) two variants: 1) at \(k_1\), where the overshoot \(\delta_1'\) is obtained; 2) at \(k_1'=k_1+\Delta k_1\), where \(\delta_1''\) is obtained. Having determined the algebraic difference \(\Delta\delta_1=\delta_1''-\delta_1'\), find the required value \(k_1\) by the formula

\[ k_1'' = k_1 + \Delta k_1 \frac{\delta_{1z}-\delta_1'}{\Delta\delta_1}, \]

d) Check the found value \(k_1''\) by the difference \(\delta_{1z}-\delta_{1p}\), where \(\delta_{1p}\) is the overshoot obtained by the main program for \(k_1''\).

If \(\delta_{1z}-\delta_{1p}>\pm\varepsilon\), then, taking as initial the found value \(k_1''\) and the previous value of \(q\), and decreasing the increment \(\Delta k_1\) (for example, by a factor of 5), repeat the calculations until the condition \(\delta_{1z}-\delta_{1p}\leq \pm\varepsilon\) is satisfied, where \(\varepsilon\) is the specified error.

III a) Having three variants with identical overshoot values, but with different settling times and values of \(q\) and \(k_1\), determine

algebraic values of the increments of the settling time and of the coefficient.

\[ \begin{array}{ll} 1) & q_{\mathrm{I}},\ k''_{1,\mathrm{I}},\ \delta_{1\mathrm{I}}=\delta_{13},\ Y_{\mathrm{I}}\\ 2) & q_{\mathrm{II}},\ k''_{1,\mathrm{II}},\ \delta_{1\mathrm{II}}=\delta_{13},\ Y_{\mathrm{II}}\\ 3) & q_{\mathrm{III}},\ k''_{1,\mathrm{III}},\ \delta_{1\mathrm{III}}=\delta_{13},\ Y_{\mathrm{III}} \end{array} \left\{ \begin{array}{l} \Delta k_1' = k''_{1,\mathrm{II}}-k''_{1,\mathrm{I}};\quad \Delta Y'=Y_{\mathrm{II}}-Y_{\mathrm{I}};\\[2mm] \Delta k_1'' = k''_{1,\mathrm{III}}-k''_{1,\mathrm{II}};\quad \Delta Y''=Y_{\mathrm{III}}-Y_{\mathrm{II}}. \end{array} \right. \]

b) Using the curve of equal overshoots corresponding to the value \(\delta_{13}\), determine, from the change in the value \(\Delta Y\) in passing from variant to variant, the desired values of \(q\) and \(k_1\), at which a minimum of the settling time will be observed, by the formulas

\[ q_{\mathrm{sought}}=q_{\mathrm{III}}+\Delta q\,\frac{\Delta Y''}{\Delta Y'-\Delta Y''}, \]

\[ k_{1\,\mathrm{sought}}=k''_{1,\mathrm{III}}+ \frac{\Delta Y''}{\dfrac{\Delta Y'}{\Delta k_1'}-\dfrac{\Delta Y''}{\Delta k_1''}}. \]

c) For the values \(q_{\mathrm{sought}}\) and \(k_{1\,\mathrm{sought}}\), compute the parameters of the transient characteristic according to the main program and send them to the storage device.

IV a) Taking the found values \(q_{\mathrm{sought}}\) and \(k_{1\,\mathrm{sought}}\) as initial ones and decreasing the increment \(\Delta q\) (for example, by a factor of 5), repeat the search, beginning with item Ia.

b) Compare the results of the computations of stages III and IVa with respect to the values \(\delta_1\) and \(Y\).

Repeat the computations until, with the prescribed accuracy, the values \(\delta_1\) and \(Y\) obtained at the last stage coincide with the values of these same parameters from the preceding stage.

V a) Taking the values of the coefficients \(q\) and \(k_1\) found at stage IV as initial ones (while increasing, in order to shorten the search time, the value of \(q\) also found at stage IV by approximately \(0.2\div 0.4\)), assign some value of the coefficient \(k_2\) (for example, \(k_2=0.2\div 0.5\)) and the magnitudes of the increments \(\Delta q_1=0.05\div 0.1\) and \(\Delta k_2=0.01\div 0.05\), and compute, according to the main program, the parameters \(Y,\ \delta_1,\ \delta_2\) for three variants: 1) \(q,\ k_1,\ k_2\), where \(Y,\ \delta_1,\ \delta_2\) are obtained; 2) \(q'=q+\Delta q,\ k_1,\ k_2\), where \(Y',\ \delta_1',\ \delta_2'\) are obtained; 3) \(q,\ k_1,\ k_2'=k_2+\Delta k_2\), where \(Y'',\ \delta_1''\) and \(\delta_2''\) are obtained.

If in variant 1) the inequality \(Y>2b\) or \(\delta_1<0\) holds, assign a new value of \(q\) (for example, \(q+0.1\div 0.2\)) and repeat the computations from the beginning of stage Va.

b) It may turn out that the condition \(Y<2b\) is not satisfied for any of the adopted values of \(q\). In this case take the average of the tried values of \(q\) and vary the value \(k_2\).

If the condition \(Y<2b\) is again not satisfied in any of the variants, this means that, in comparison with the uncorrected cascade, the correction circuit being calculated, with the obtained combination of initial parameters \(b,\ x,\ m\), cannot provide a gain in settling time of more than 10%. Therefore, the active element should be changed and the computations begun with new values of \(b,\ x,\ m\).

c) If \(Y<2b\), continue the computations.

VI. a) Determine the increments of the overshoots when \(q\) and \(k_2\) are varied by \(\Delta q\) and \(\Delta k_2\), and also the increment \(\Delta Y_{\Delta q}\) when \(q\) is varied,

\[ \Delta\delta_{1\Delta q}=\delta_1'-\delta_1,\qquad \Delta\delta_{2\Delta q}=\delta_2'-\delta_2,\qquad \Delta Y_{\Delta q}=Y'-Y, \]

\[ \Delta\delta_{1\Delta k_2}=\delta_1''-\delta_1,\qquad \Delta\delta_{2\Delta k_2}=\delta_2''-\delta_2. \]

b) Check the ratio \(\dfrac{\Delta Y_{\Delta q}}{\Delta\delta_{1\Delta q}}\). If

\[ \frac{\Delta Y_{\Delta q}}{\Delta\delta_{1\Delta q}}>0, \]

give an increment in \(q\) and begin the computations again from item Va.

c) If

$$ \frac{\Delta Y_{\Delta q}}{\Delta \delta_{1\Delta q}}<0 $$

— continue the calculations.

VII. a) Determine the numbers of steps in \(q\) and \(k_2\) needed to obtain the values of \(q\) and \(k_2\) for which fulfillment of the equalities \(\delta_1=\delta_{13}\) and \(\delta_2=\delta_{23}\) is assumed, according to the formulas

$$ n= \frac{ \delta_{13}-\delta_1-\dfrac{\Delta\delta_{1\Delta k_2}}{\Delta\delta_{2\Delta k_2}}(\delta_{23}-\delta_2) }{ \Delta\delta_{1\Delta q}-\Delta\delta_{2\Delta q}\dfrac{\Delta\delta_{1\Delta k_2}}{\Delta\delta_{2\Delta k_2}} } \qquad \text{(number of steps in } q\text{),} $$

$$ m= \frac{\delta_{23}-\delta_2-\Delta\delta_{2\Delta q}\cdot n}{\Delta\delta_{2\Delta k_2}} \qquad \text{(number of steps in } k_2\text{).} $$

b) Change \(q\) and \(k_2\) to the values \(q''=q+n\cdot\Delta q\) and \(k_2''=k_2+m\cdot\Delta k_2\), compute again the values of \(\delta_1\) and \(\delta_2\), and, if they differ from \(\delta_{13}\) and \(\delta_{23}\) by more than the prescribed amount, then reduce the step in \(q\) and \(k_2\) to the values \(\Delta q'(0.2\div0.5)\cdot\Delta q\), \(\Delta k_2'=(0.2\div0.5)\Delta k_2\) and repeat the cycle of calculations, beginning with stage VI, until the values of \(\delta_1\) and \(\delta_2\) are obtained with the prescribed accuracy. Send the values of \(Y\), \(\delta_1\), \(\delta_2\) obtained in the last cycle to the memory device.

VIII. a) Taking the values of \(q\) and \(k_2\) obtained in the last cycle of stage VIIb, and the value \(k_1\) used in the same cycle, as the initial values, and having specified an increment \(\Delta k_1=(0.005\div0.01)\), repeat the calculations according to stages VI–VIII for the values \(k_1'=k_1-\Delta k_1\) and \(k_1''=k_1+\Delta k_1\).

b) Having three variants in which \(\delta_1=\delta_{13}\) and \(\delta_2=\delta_{23}\),

$$ \begin{aligned} 1)&\quad q^{\mathrm I},\ k_1^{\mathrm I}=k_1-\Delta k_1,\ k_2^{\mathrm I},\ Y^{\mathrm I} \\ 2)&\quad q^{\mathrm{II}},\ k_1^{\mathrm{II}}=k_1,\ k_2^{\mathrm{II}},\ Y^{\mathrm{II}} \\ 3)&\quad q^{\mathrm{III}},\ k_1^{\mathrm{III}}=k_1+\Delta k_1,\ k_2^{\mathrm{III}},\ Y^{\mathrm{III}} \end{aligned} \qquad \left\{ \begin{aligned} \Delta Y^{\mathrm I}&=Y^{\mathrm{II}}-Y^{\mathrm I},\quad \Delta k_2&=k_2^{\mathrm{II}}-k_2^{\mathrm I},\quad \Delta q^{\mathrm I}=q^{\mathrm{II}}-q^{\mathrm I},\\ \Delta Y^{\mathrm{II}}&=Y^{\mathrm{III}}-Y^{\mathrm{II}},\quad \Delta k_2^{\mathrm{II}}=k_2^{\mathrm{III}}-k_2^{\mathrm{II}},\\ \Delta q^{\mathrm{II}}&=q^{\mathrm{III}}-q^{\mathrm{II}}. \end{aligned} \right. $$

Determine the number of steps in \(k_1\) that must be made in order to reach the assumed optimal value or a value close to the optimal one.

$$ n_{k_1}=\frac{\Delta Y''}{\Delta Y'-\Delta Y''}. $$

The assumed optimal value is to be determined as

$$ k_{1\,\mathrm{opt}}=k_1+n_{k_1}\cdot\Delta k_1. $$

c) Determine the assumed optimal values of \(q\) and \(k_2\) from the formulas

$$ q_{\mathrm{opt}}=q'''+ \frac{\Delta Y''}{ \dfrac{\Delta Y'}{\Delta q'}-\dfrac{\Delta Y''}{\Delta q''} }, $$

$$ k_{2\,\mathrm{opt}}=k_2''+ \frac{\Delta Y''}{ \dfrac{\Delta Y'}{\Delta k_2'}-\dfrac{\Delta Y''}{\Delta k_2''} }. $$

d) For the found values \(q_{\mathrm{opt}}\), \(k_{1\,\mathrm{opt}}\), \(k_{2\,\mathrm{opt}}\), compute, according to the main program, the values \(Y\), \(\delta_1\), \(\delta_2\) and send them to the memory device.

IX a) Taking the values \(q_{\mathrm{opt}}\), \(k_{1\,\mathrm{opt}}\), \(k_{2\,\mathrm{opt}}\), found at stage VIII, as the initial values, repeat the cycle of calculations beginning with stage V (without increasing \(q\) by \(0.2\div0.4\)), and obtain values of the coefficients \(q'_{\mathrm{opt}}\), \(k'_{1\,\mathrm{opt}}\), \(k'_{2\,\mathrm{opt}}\) closer, in comparison with those obtained in the preceding cycle, to the optimal ones.

Fig. 2. Curves of the dependence \(B_{\max}=f(x)\) for \(q_{\mathrm{opt}}=f_{1}(x)\), \(k_{1\mathrm{opt}}=f_{2}(x)\), and overshoots \(\delta_{1}=2\%\) and \(\delta_{2}\le -0.15\%\).

Fig. 3. Curves of the dependence \(B_{\max}=f_{1}(x)\) for \(q_{\mathrm{opt}}=f(x)\), \(k_{2\mathrm{opt}}=f_{2}(x)\), and overshoots \(\delta_{1}=2\%\) and \(\delta_{2}\le -0.01\%\).

Fig. 4. Curves of the dependence \(B_{\max}=f(x)\) for \(q_{\mathrm{opt}}=f_{1}(x)\), \(k_{1\mathrm{opt}}=f_{2}(x)\), \(k_{2\mathrm{opt}}=f_{3}(x)\), and overshoots \(\delta_{1}=2\%\) and \(\delta_{2}=-0.01\%\).

b) Compare the settling times obtained in the first \((Y_1)\) and second \((Y_2)\) cycles. If \(Y_1 - Y_2 \le \pm \varepsilon\) (where \(\varepsilon\) is the specified error), \(q'_{\mathrm{opt}}, k'_{1\mathrm{opt}}, k'_{2\mathrm{opt}}\) may be regarded as the optimal values of the correction coefficients. If \(Y_1 - Y_2 > \pm \varepsilon\), or \(\delta_{13} - \delta_1 > \pm \varepsilon_1\) and \(\delta_{23} - \delta_2 > \pm \varepsilon_2\), then, taking \(q'_{\mathrm{opt}}, k'_{1\mathrm{opt}}\), and \(k'_{2\mathrm{opt}}\) as the initial values, repeat the calculations again, starting from stage V, without increasing \(q\) by \(0.2 \div 0.4\).

Automation of the search for the optimal values of the coefficients \(q, k_1, k_2\) of a complex correction circuit has made it possible to use this circuit for any values of the initial cascade parameters \(b, x, m\), and for any values (but not more than \(15 \div 20\%\)) of the first and second overshoots, without resorting to cumbersome engineering calculations or large expenditures of time.

Figures 2, 3, and 4 present graphs of the optimal values of the coefficients \(q, k_1, k_2\) (or two of them) and the resulting values of the gain \(B\) as functions of \(x\), for constant permissible values of \(\delta_1\) and \(\delta_2\).

Determination of each optimal combination of correction coefficients when working on the M-20 computer requires, depending on the complexity of the circuit, from three to thirty minutes of machine time.

REFERENCES

1. O. B. Lourie. Video-Frequency Amplifiers. Moscow, “Sovetskoe Radio,” 1961.

2. G. K. Gavrilov. Approximate Methods for Analyzing Transient Processes. Moscow, “Sovetskoe Radio,” 1966.

3. I. G. Mamonkin. Pulse Amplifiers. Moscow, Gosenergoizdat, 1953.

4. I. V. Dubrovskaya. A method for investigating the transfer function of a linear circuit in the region of small times with a characteristic equation higher than third degree. Izvestiya Vuzov SSSR, “Radioelectronics,” vol. 13, no. 6, 1970, 716–720.

5. T. M. Agakhanian. Approximate calculation of the transient response in the region of small times. Theory and Design of Pulse and Amplifier Circuits Using Semiconductor Devices. Collection of articles, issue 1. Moscow, Atomizdat, 1969.

6. D. I. Sviryakin, I. A. Suslov. Video amplifier with parallel inductive correction in the cathode and anode circuits. Proceedings of TIRiET, vol. 7. Tomsk, TSU Publishing House, 1972.