Influence of Time Variations in Dynamic Conductances on the Characteristics of a Video-Frequency Amplifier with a Correcting Capacitance in the Cathode Circuit and Parallel Inductive Correction in the Anode Circuit

I. A. Suslov, D. I. Sviryakin

(Presented by the scientific seminar of the Radio Engineering Faculty)

Introduction

In practice, wide use has been made of a video-amplifier circuit in which inductive correction in the anode circuit is supplemented by correction using a relatively small capacitance in the cathode circuit (Fig. 1). In such a circuit the automatic-bias circuit does not distort the peaks of pulses, and distortions due to the screen-grid circuit are substantially weakened by the action of negative feedback. This noticeably reduces the difficulties associated with correcting distortions of the peaks of long-duration pulses and, ultimately, leads to a reduction in the dimensions of the amplifier.

Fig. 1.

Questions of the theory and calculation of the circuit in Fig. 1 were considered in works [1, 2]. However, the data contained in them may have only limited application, since the cited works do not take into account the time dependence of the dynamic capacitances that are included in the anode load of the stage.

The input capacitance of the stage according to the circuit of Fig. 1 is equal to

\[ C_{\mathrm{in}}(t)=C_2[1-K_k(t)], \tag{1} \]

where \(K_k(t)\) is the transmission coefficient from the grid to the cathode, and \(C_2\) is the input capacitance of the tube. The change of \(C_{\mathrm{in}}\) with time when a unit voltage is applied to the amplifier input is shown in Fig. 2. At the initial instant, when the capacitance \(C_k\) has not yet been charged, \(K_k(0)=0\) and \(C_{\mathrm{in}}(0)=C_2\).

When the voltage at the cathode has become established, then

\[ K_k(\infty)=\frac{S R_k}{1+S R_k} \]

and

\[ C_{\mathrm{in}}(\infty)=\frac{C_2}{1+S R_k}. \]

The input capacitance of a cathode follower changes analogously.

Fig. 2.

In analyzing all these circuits, \(C_{\mathrm{in}}\) is treated as a constant quantity. However, different authors proceed in different ways. Some take as \(C_{\mathrm{in}}\) its initial value \([1]^1\), others its steady-state value \([2, 3]\), and still others an averaged value \([4, 5]\). Most often the question of the value of \(C_{\mathrm{in}}\) is not touched upon at all \([6, 7, 8,\) etc.]. Through the capacitance \(C_2\), the stage under consideration is connected with the cathode circuit of the next stage, and the latter, through the tube and the anode–cathode capacitance, with its anode circuit. This circumstance makes it difficult to divide the amplifier into separate stages, since it leads to the necessity of taking into account, in the operation of the stage under consideration, the influence of the parameters and transient processes in the next stage, which determine the magnitude \(K_k(t)\). As a result, an analytical investigation of the amplifier properties encounters great difficulties. To this it should be added that the dynamic conductance \(Y_{\mathrm{out}}\), determined by the capacitance \(C_1\) (Fig. 1), also depends on time. Therefore, the main problem in analyzing the amplifier under consideration is to obtain such an equivalent circuit as will make it possible to take into account the variation of both capacitances and to carry out a separation of the stages, i.e., to reflect in a given stage the influence of the following one.

\(^1\) This follows from the comparison, given in [1], of the circuit of Fig. 1 with the usual circuit with parallel inductive correction.

Construction of the Equivalent Circuit

To obtain the equivalent circuit of the amplifier in Fig. 1, let us first determine the nature of the change in the dynamic output capacitance \(C_{\text{out}}\) caused by \(C_1\). Let \(Z'_a\) denote the resistance of the anode load without the tube output capacitance \(C_1\). In this case the equivalent circuit of the stage may be represented in the form of Fig. 3a or 3b, depending on whether the capacitance \(C_1\) is connected between the anode and ground or between the anode and cathode.

Fig. 3.

Using the nodal-voltage method, we find that the voltage transfer coefficient for the circuit of Fig. 3a is equal to

\[ \dot K = \frac{ -S\left(\dfrac{1}{R_k}+j\omega C_k\right) }{ \left(S+\dfrac{1}{R_k}+j\omega C_k\right) \left(\dfrac{1}{Z'_a}+j\omega C_1\right) }. \tag{2} \]

Similarly, for the circuit of Fig. 3b we have

\[ \dot K = \frac{ -S\left(\dfrac{1}{R_k}+j\omega C_k\right) }{ \left(\dfrac{1}{R_k}+j\omega C_k\right) \left(\dfrac{1}{Z'_a}+j\omega C_1\right) +\dfrac{1}{Z'_a}\left(S+j\omega C_1\right) }. \tag{3} \]

Expression (3) may be written in the form

\[ \dot K = \frac{ -S\left(\dfrac{1}{R_k}+j\omega C_k\right) }{ \left(S+\dfrac{1}{R_k}+j\omega C_k\right) \left[ \dfrac{1}{Z'_a} +j\omega C_1 \frac{ \dfrac{1}{R_k}+\dfrac{1}{Z'_a}+j\omega C_k }{ S+\dfrac{1}{R_k}+j\omega C_k } \right] }. \tag{3'} \]

Comparing (3′) with (2), we find that the dynamic output capacitance of the circuit of Fig. 3b is equal to

\[ C_{\text{out}} = C_1 \frac{ \dfrac{1}{R_k}+\dfrac{1}{Z'_a}+j\omega C_k }{ S+\dfrac{1}{R_k}+j\omega C_k }. \tag{4} \]

Hence, taking into account that \(\dfrac{1}{R_k} \gg \dfrac{1}{Z'_a}\), we obtain

\[ C_{\text{out}} = C_1 b \cdot \frac{1 + j q \Omega}{1 + j b q \Omega}, \]

where

\[ b = \frac{1}{1 + S R_k}, \qquad q = \frac{C_k R_k}{C_0 R_a}, \qquad \Omega = \omega C_0 R_a, \qquad C_0 = C_1 + C_2 + C'_0 . \]

For the input capacitance of the following stage, neglecting the reaction of the anode load and the coupling of the cathode circuit of this stage with its anode circuit through the capacitance \(C_1\), we obtain

\[ C_{\text{in}} = C_2 \left( 1 - \frac{S R_k}{1 + S R_k + j q \Omega} \right) = C_2 b \cdot \frac{1 + j q \Omega}{1 + j b q \Omega}. \tag{5} \]

The similar structure of the expressions for the input and output dynamic capacitances makes it possible, in the case of identical stages, to replace the influence of one of them by the action of the other, correspondingly increased. In the equivalent circuit of Fig. 4, simplified in this way, the action of the following stage is replaced by equal changes in the stage under consideration; the stages are separated.\(^1\)

Fig. 4

Formulas (4) and (5) were obtained with certain neglects. Exact expressions for the input and output conductances are given in the Appendix. With their aid, the frequency characteristics of the capacitive and negative active components of the indicated conductances were calculated for the particular case when

\[ x = \frac{C_1}{C_0} = 0.18, \qquad b = 0.4, \qquad q = 0.392, \]

\[ m = \frac{R_k}{R_a} = 0.08, \qquad k = \frac{L_a}{C_0 R_a^2} = 0.22, \qquad y = \frac{C_2}{C_0} = 0.393. \]

The results of the calculations are presented in Figs. 5 and 6. Here curves 1 denote, respectively, \(\dfrac{C_{\text{out}}}{C_1}\) and \(\dfrac{Y_{\text{out}r}}{Y_a}\) as functions of frequency; curves 2 denote analogous dependences for \(\dfrac{C_{\text{in}}}{C_2}\) and \(\dfrac{Y_{\text{in}r}}{Y_a}\). \(Y_{\text{out}r}\) and \(Y_{\text{in}r}\) are the active components of the output and input conductances, and \(Y_a = \dfrac{1}{R_a}\) is the conductance of the anode load.

Comparison of curves 1 and 2 shows that replacing the capacitance \(C_2\) by a correspondingly increased \(C_1\) is a quite satisfactory approximation.

\(^1\) The quantity \(C_2\) is included here in \(C_1\).

The node-voltage equations for the circuit of Fig. 4 are written in the form

\[ \dot U_k\left[S+\frac{1}{R_k}+j\omega(C_1+C_k)\right]-\dot U_2 j\omega C_1=S\dot U_1, \]

Fig. 5.

Fig. 6.

\[ -\dot U_k(S+j\omega C_1)+\dot U_2 \left[ \frac{1}{R_a+j\omega L_a}+j\omega(C_1+C_0') \right] =-S\dot U_1. \]

Hence the voltage transfer coefficient \(\dot K=\dfrac{\dot U_2}{\dot U_1}\) is obtained as equal to

\[ \dot K = \frac{-S R_a(1+j\omega C_k R_k)\left(1+j\omega \dfrac{L_a}{R_a}\right)} {(1+S_k)\left[1+j\omega \dfrac{(C_1+C_k)R_k}{1+S R_k}\right](1+j\omega C'_0 R_a-\omega^2 L_a C'_0)+} \ldots \]

\[ \ldots\frac{} {+j\omega C_1(1+j\omega C_k R_k)\dfrac{R_a+j\omega L_a}{1+S R_k}} \tag{6} \]

Using the notation introduced above, we obtain the normalized transfer coefficient

\[ \dot M=\frac{\dot K}{K_0}= \frac{1+j(\kappa+q)\Omega+j^2\kappa q\Omega^2} {1+j[b(q+xm)+1-x+xb]\Omega+[\,\kappa(1-x)+} \ldots \]

\[ \ldots +(1-x)(q+xm)b+xb(q+\kappa)]j^2\Omega^2 +b\kappa[q+x(1-x)m]j^3\Omega^3}. \tag{7} \]

To investigate the capabilities of the circuit of Fig. 1, we shall first use the frequency method.

Determination of the optimal parameters by the frequency method

Denote in (7)

\[ \kappa+q=a_1,\qquad \kappa q=a_2,\qquad b(q+xm)+1-x+xb=b_1, \]

\[ \kappa(1-x)+(1-x)b(q+xm)+xb(q+\kappa)= \]

\[ =b_2,\qquad b\kappa[q+x(1-x)m]=b_3. \]

Then equation (7) is rewritten in the form

\[ \dot M= \frac{1+ja_1\Omega+j^2a_2\Omega^2} {1+jb_1\Omega+j^2b_2\Omega^2+j^3b_3\Omega^3}, \]

and the square of the modulus \(\dot M\) is expressed by the formula

\[ M^2= \frac{1+(a_1^2-2a_2)\Omega^2+a_2^2\Omega^4} {1+(b_1^2-2b_2)\Omega^2+(b_2^2-2b_1b_3)\Omega^4+b_3^2\Omega^6}. \]

To determine the parameters optimal from the standpoint of the shape of the frequency characteristic, we obtain from (7), in accordance with the method of G. V. Braude, two equations

\[ \kappa^2+q^2=[b(q+xm)+1-x(1-b)]^2 -2[\kappa+b(q+xm)- \]

\[ -\kappa x(1-b)-x^2mb], \tag{9} \]

\[ \kappa^2q^2=[\kappa+b(q+xm)-\kappa x(1-b)-x^2mb]^2 -2\kappa b[q+(1-x)xm]\cdot \]

\[ \cdot [b(q+xm)+1-x(1-b)]. \tag{10} \]

Here the unknowns are the parameters \(\kappa\) and \(q\).

Finding a solution of the system of equations (9) and (10) in general form is associated with great difficulties. Therefore, for selected values of the parameters \(b\), \(x\), and \(m\), we shall seek the solution graphically, assigning values to \(\kappa\) and computing \(q\), first from equation (9), then from equation (10). The results of such calculations are presented in Fig. 7. Here curves 1 were obtained for \(b=0.4,\ x=0.4,\ m=0.08\), and curves 2 for \(b=0.4,\ x=0.615,\ m=0.08\).

Fig. 7.

The curves in Fig. 7 show that equations (9) and (10) are incompatible, as a result of which it is not possible fully to realize the capabilities of G. V. Braude’s method. We shall therefore regard as optimal those values of the parameters \(\kappa\) and \(q\) that satisfy equation (9) and for which the difference between the right- and left-hand sides of equation (10), i.e.,

\[ P_2(k,q)-P_1(k,q)=\left[k^2+b(q+xm)-kx(1-b)-x^2mb\right]^2- \]

\[ {}-2kb\left[q+(1-x)xm\right]\left[b(q+xm)+1-x(1-b)\right]-k^2q^2, \]

will be minimal\(^1\).

The graphs of \(P_2(k,q)-P_1(k,q)\) under condition (9) are presented in Fig. 8. Curve 1 was obtained for \(b=0.4,\ x=0.4,\ m=0.08\), and curve 2 for \(b=0.4,\ x=0.615^{2}),\ m=0.08\). The graphs in Fig. 8 indicate that the conditional minimum of the function \(P_2(k,q)-P_1(k,q)\) is sufficiently flat and depends little on the value of \(x\). Thus, for \(x=0.4\) we obtain a minimum at \(k=0.22\), and in the case \(x=0.615\), at \(k=0.187\). The corresponding values of \(q\) are 0.307 and 0.28.

\(^1\) Since \(P_2>P_1\), this will approximately also correspond to the minimum of \(P_2\).

\(^2\) The experimental prototype had this value of \(x\).

Fig. 8.

Fig. 9.

In Fig. 9 the normalized frequency characteristics are given for the parameters used above. Curve 1 was obtained for \(b = 0.4,\ x = 0.4,\ m = 0.08,\ k = 0.22,\ q = 0.392\); curve 2—for \(x = 0.615,\ b = 0.4,\ m = 0.08,\ k = 0.187,\ q = 0.28\). For comparison, the frequency characteristic of an amplifier taken from [1] and obtained for \(k = 0.2;\ q = 0.807;\ b = 0.373\) is also shown here (curve 3). Comparison of curves 1 and 2 with 3 shows that, when variations of the dynamic capacitances are taken into account, the frequency characteristic for small \(\Omega\) (of order 1 and less) proves to be more uniform and is distinguished by a more rapid fall-off at high frequencies \((\Omega \gg 1)\). The leveling of the initial portion of the frequency characteristic is associated with a decrease of the dynamic capacitances by a factor of \(1 + S R_k\) at low frequencies \((\Omega < 1)\). The fall-off at high frequencies is explained by the fact that the dynamic capacitances approach the static ones and, at the same time, the time constant of the anode circuit increases owing to the shunting of the load by the negative active components of the dynamic conductances (Fig. 6).

If only the dynamic capacitances are taken into account, the frequency characteristic should lie above that calculated for the static capacitances, merging with it as \(\Omega \to \infty\), and the pass band is obtained as noticeably larger. This means that the widespread neglect of the active components of the dynamic conductances leads to substantial errors.

The equivalent circuit of Fig. 4 makes it possible to take into account both the action of the capacitive components and that of the negative active components.

Transient characteristics of the circuit of Fig. 1

Replacing \(j\Omega\) in (7) by the operator \(p\), we obtain the operational image of the transient characteristic in the form

\[ M(p)= \frac{1+(k+q)p+kq p^2} {1+[b(q+xm)+1-x(1-b)]p+[k(1-x)+(1-x)b(q+xm)+xb(q+k)]p^2+bk[q+x(1-x)m]p^3} = \frac{1+a_1p+a_2p^2} {1+b_1p+b_2p^2+b_3p^3}. \tag{12} \]

In this case the transient characteristic itself will be a function of the relative time

\[ \tau=\frac{t}{C_0R_a}. \]

Using the expansion theorem, we find for the transient characteristic the expression

\[ h(\tau)=1-A_1e^{-\beta\tau}+Ae^{-\alpha\tau}\sin(\omega\tau+\varphi). \tag{13} \]

Here \(\beta\) is the real root of the characteristic equation, \(-\alpha \pm j\omega\) are its conjugate complex roots,

\[ \varphi = \operatorname{arctg} \frac{\omega(a_1-2\alpha a_2)} {1-\alpha a_1+(\alpha^2-\omega^2)a_2} - \operatorname{arctg}\frac{\omega}{-\alpha} - \operatorname{arctg}\frac{\omega}{\beta-\alpha} = \varphi_1-\varphi_2-\varphi_3, \]

\[ A=\frac{1}{\omega b_3}\cdot \sqrt{\frac{\left[1-\alpha a_1+(\alpha^2-\omega^2)a_2\right]^2+\omega^2(a_1-2\alpha a_2)^2} {(\alpha^2+\omega^2)\left[(\beta-\alpha)^2+\omega^2\right]}}, \]

\[ A_1=\frac{1-\beta a_1+a_2\beta^2} {\beta b_3\left[(\beta-\alpha)^2+\omega^2\right]}. \]

Unlike the case in which the parasitic capacitances of the circuit are considered as static [1], when the dynamic conductances are taken into account an oscillatory regime will occur.

The overshoot of the transient response (13) can be determined analytically. To do this, we set the derivative equal to zero:

\[ h'(\tau)=A_1\beta e^{-\beta\tau} +A\sqrt{\alpha^2+\omega^2}\cdot e^{-\alpha\tau}\cdot \sin(\omega\tau+\varphi_1-\varphi_3)=0. \tag{14} \]

One of the columns of Table 2 below contains the values of the quantity \(100A_1 e^{-\beta\tau_m}\), where \(\tau_m\) is the relative time corresponding to the overshoot. The figures in this column show that the term \(A_1 e^{-\beta\tau_m}\) in expression (13) is very small and can change the magnitude of the overshoot only by thousandths of a percent or less. Therefore, in equation (14) the first term on the left-hand side may be omitted. Then the condition for overshoots will be \(\omega\tau+\varphi_1-\varphi_3=n\pi\), where \(n\) is an integer. In particular, the first overshoot will occur at

\[ \tau_m=\frac{\pi+\varphi_3-\varphi_1}{\omega}. \]

The magnitude of this overshoot will be equal to

\[ \delta=Ae^{-\alpha\tau_m}\cdot \sin\varphi_2. \tag{15} \]

Using the formulas given above, the overshoots and rise times were calculated for transient responses corresponding to various \(x\) and \(q\) at the correction parameter \(\kappa=0.2\). This \(\kappa\) was chosen on the basis of the results obtained in analyzing the operation of the circuit by the frequency-response method. The value of \(q\) was varied within certain limits around the values satisfying equation (9) at \(\kappa=0.2\) and the selected \(x\) ¹. The values of \(x\) used and the corresponding solutions of equation (9) are given in Table 1.

Table 1

Table 2 gives the values of the circuit parameters used in the calculations, as well as the values calculated for them of \(\beta\), \(\alpha\), \(\omega\), the overshoots \(\delta\), the normalized rise times

\[ y=\frac{t_y}{C_0R_a} \]

and of individual quantities that determine the shape of the transient response.

From the data of this table, the graphs in Figs. 10 and 11 were plotted; they express the dependence of the overshoot and of the normalized rise time on the circuit parameters \(x\) and \(q\) at \(\kappa=0.2\).

Table 2

The curves of Fig. 10 show that the optimal value of \(q\) depends on \(x\). The larger \(x\) is, the smaller \(q\) is. For example, for \(x = 0.5\) and an overshoot \(\delta = 1.2\%\), we obtain \(q = 0.276\), instead of \(q = 0.807\) for the same overshoot in the case where the variations of the dynamic conductances are not taken into account [1]. In determining the rise time, such \(\tau_1\) and \(\tau_2\) were first selected by trial so that \(h(\tau_1)\) was close to 0.1 and \(h(\tau_2)\) close to 0.9.

Fig. 10.

Fig. 11.

Then, by interpolation, the values of \(\tau_1\) corresponding to \(h(\tau)=0.1\) and \(h(\tau)=0.9\) were determined, and the generalized rise time \(y\) was calculated. The curves of Figs. 10 and 11 make it possible, for a given overshoot and known \(x\), to find the parameter \(q\) and the generalized rise time \(y\), and then to determine the main parameters of the circuit \(R_a, L_a, C_k\). In addition, they make it possible to compare the effectiveness of the circuit under consideration

with other amplifier circuits. For this, by specifying a definite value of the overshoot \(\delta\), we find the pulse figure of merit of the circuit of Fig. 1

\[ D=\frac{bK_0}{t_y}=\frac{bK_0}{y\tau_a}, \qquad \text{where } \tau_a=C_0R_a \text{ and } K_0=SR_a, \]

and the pulse figure of merit of the ordinary circuit with parallel correction

\[ D_{p.k.}=\frac{K_0}{y_{p.k.}\tau_a}. \]

The ratios \(\dfrac{D}{D_{p.k.}}\) for overshoots \(\delta=1\%\) and \(\delta=3\%\) are given in Table 3.

Table 3

The figures in Table 3 show that the effectiveness of the circuit under consideration increases somewhat as the permissible overshoot is increased.

Comparison with the results of work [1] and experimental verification

The optimum value of the correction parameter \(k\) obtained by us was the same as in [1]. The relative value of the pulse figure of merit \(\dfrac{D}{D_{p.k.}}\) (with respect to the circuit with parallel correction) proved to be somewhat smaller (in [1] the ratio \(\dfrac{D}{D_{p.k.}}\) for \(\delta=1\%\) is equal to 0.73). The greatest discrepancy, as already noted, occurs in the value of the parameter \(q\). The optimum values of the parameter \(q\) may be several times smaller than those recommended in [1]. Use of the latter leads to overcorrection of the amplifier. In Fig. 12 are shown the transient characteristics in whose calculation the variations of the dynamic conductances were taken into account, but the value of \(q\) was taken equal to \(q=0.807\), as in [1]. Curve 1 corresponds to \(x=0.4\), curve 2 to \(x=0.615\). In the first case the overshoot is equal to 16.7%, in the second to 24.1%. The corresponding normalized rise times are \(y=0.53\) and \(y=0.45\). The frequency characteristics for the same parameter values are shown in Fig. 9 (curves 4 and 5).

In the experimental verification of the results obtained, a two-stage amplifier was assembled according to the circuit of Fig. 1. The parasitic capacitances of the circuit with “hot” tubes were measured from the frequency characteristics. With \(R_k\) shunted by capacitors \(C_k=10000\) pF and with \(L_a\) short-circuited, the total capacitance \(C_0\) was measured; then the input of the next stage was disconnected and \(C_1+C_0\) was measured. The capacitance \(C_1\) was determined from the frequency characteristic with the blocking capacitor disconnected and with such an arrangement of the anode-load resistors that its capacitance to ground was minimal. To determine the parasitic capacitance in the cathode \(C_k\), a compensated divider was assembled, one arm of which was formed by the output co-

resistance of the stage (including \(R_k\) and the capacitance being measured), and the other—with a small capacitance (previously measured with the aid of a PIMEL) and a resistance connected in parallel. By selecting the resistance, a uniform frequency characteristic was obtained.

Fig. 12.

From the condition for obtaining a uniform frequency characteristic, the value of the parasitic capacitance was determined. The input capacitance of the voltmeter was measured separately. The measurement results were as follows: \(C_1 = 3.43\ \text{pF}\), \(C_2 = 13.3\ \text{pF}\), \(C_0 = 27.18\ \text{pF}\), \(C_k = 50\ \text{pF}\), \(x = 0.615\). The frequency and transient characteristics of the circuit were recorded.

Fig. 13.

Figure 13 shows a photograph of the transient characteristic obtained on the screen of the IPKh-57 with the parameters found taking into account

of the time variations of the dynamic capacitances, and in Fig. 14 with the use of the parameters recommended in [1]. The frequency response in the first case was monotonic; in the second it had a rise of about 30%.

Fig. 14.

The results presented confirm the correctness of the conclusions obtained above concerning the influence of dynamic conductances on the characteristics of the amplifier.

Conclusion

Neglecting the time variations of the dynamic conductances in the circuit of Fig. 1 leads to a considerable overestimation of the value of the correcting capacitance in the cathode circuit. The consequence of this is overcorrection of the amplifier, as a result of which the overshoot in a single stage may reach 20% or more.

The proposed method makes it possible to take into account the influence of time variations of the active and capacitive components of the input and output dynamic conductances without any significant increase in the difficulty of the analytical determination of the elements of transient responses, and leads to results that agree well with experimental verification data.

Appendix

Let us investigate the dependence of the input conductance of the amplifier under consideration on frequency. For this purpose we shall use the equivalent circuit of the stage, shown in Fig. 15.

Fig. 15.

Here the capacitance \(C_2'\) represents the averaged (for example, by the method of E. N. Mokhov [4]) input capacitance of the stage following the one under consideration.

The equations of the node voltages for this circuit are written in the form

\[ \dot U_1 \cdot j\omega C_2-\dot U_k \cdot j\omega C_2=\dot I_1, \]

\[ -\dot U_1 \cdot j\omega C_2+\dot U_k\left[S+\frac{1}{R_k}+j\omega(C_2+C_k+C_1)\right] -\dot U_2 j\omega C_1=S\dot U_1, \]

\[ -\dot U_k(S+j\omega C_1)+\dot U_2\left[\frac{1}{R_a+j\omega L_a} +j\omega(C_1+C_2')\right]=-S\dot U_1. \]

Hence the input conductance of the stage is obtained as equal to

\[ Y_{\mathrm{in}}= \frac{ \left| \begin{array}{cc} \dfrac{1}{R_a}+j\omega(C_1+C_2), & -j\omega C_1\\[6pt] -j\omega C_1, & \dfrac{1}{R_a+j\omega L_a}+j\omega(C_1+C_2') \end{array} \right| }{ \left| \begin{array}{cc} S+\dfrac{1}{R_k}+j\omega(C_k+C_1+C_2), & -j\omega C_1\\[6pt] -(S+j\omega C_1), & \dfrac{1}{R_a+j\omega L_a}+j\omega(C_1+C_2') \end{array} \right| } = \]

\[ =-j\Omega yb\, \frac{ 1+j\Omega\{q+xm+(1+j\kappa\Omega)[1+jq\Omega+jx(1-x)m\Omega]\}\ldots }{ 1+j\Omega\{(q+xm+ym)b+b(1+j\kappa\Omega)[1+ } \]

\[ \ldots \frac{ }{+j\Omega(q+ym)+(1-x)\dfrac{1-b}{b}+j\Omega mx(1-x)]\} }. \tag{16} \]

Here

\[ b=\frac{1}{1+SR_k},\qquad x=\frac{C_1}{C_0},\qquad C_0=C_1+C_0',\qquad y=\frac{C_2}{C_0},\qquad m=\frac{R_k}{R_a}, \]

\[ \Omega=\omega C_0R_a,\qquad \kappa=\frac{L_a}{C_0R_a^2}. \]

Separating in (16) the real and imaginary parts, one can determine the dynamic input capacitance and the active input conductance. We do not give the corresponding formulas because of the cumbersome form of the expressions obtained. In calculating \(\dfrac{C_{\mathrm{in}}}{C_1}(\Omega)\) and \(\dfrac{Y_{\mathrm{in}r}}{Y_a}(\Omega)\) for specific values of the parameters, it is more convenient to substitute these values into (16) and to carry out the separation of the real and imaginary parts in the resulting numerical expression.

From formula (16) it is easy to see that at low frequencies \((\Omega\to0)\) the input conductance has a purely capacitive character and will be approximately equal to \(j\omega C_2b\). At high frequencies \((\Omega\to\infty)\), \(C_{\mathrm{in}}\approx C_2\), and, in addition, a negative active conductance appears.

The output capacitance, in accordance with expression (4), is equal to

\[ C_{\mathrm{out}}= C_1 \frac{ \dfrac{1}{R_k}+\dfrac{1}{R_a+j\omega L_a} +j\omega(C_k+C_0'+C_{\mathrm{in}}) }{ S+\dfrac{1}{R_k}+j\omega C_k } = \]

\[ = C_1 b \cdot \frac{ 1 + m + j \Omega [q + (1 - x)m] + j^2 \Omega^2 \kappa [q + (1 - x)m] }{ 1 + j \Omega (\kappa + bq) + j^2 \kappa b q \Omega^2 }. \]

The separation of the real and imaginary parts here is also conveniently carried out after substituting numerical values.

REFERENCES

1. Lur’e O. B. and Sirgo A. D., A video-frequency amplifier with an anode inductance and with a correcting capacitance in the cathode circuit. Tekhnika kino i televideniya, No. 1, 1958.

2. Lisogurskii V. I., An amplifier stage with cathode correction. Collection of Scientific Works of the Leningrad Electrotechnical Institute of Communications named after M. A. Bonch-Bruevich, issue 1 (34), 1958.

3. Varshaver B. A., Design of Pulse Amplifiers. MEI, 1953.

4. Mokhov E. N., Certain questions of the theory, calculation, and high-frequency correction of a stage with a cathode load in the pulse-amplification regime (Candidate dissertation). MEIS, 1957.

5. Faizullaev B. N., On the calculation of a cathode follower in the pulse regime. Radiotekhnika, vol. 10, No. 5, 1955.

6. Stepanov D. V., Pulse Amplifiers. Gosenergoizdat, 1954.

7. Rizkin A. A., Fundamentals of the Theory of Amplifier Circuits. Sovetskoe Radio Publishing House, 1954.

8. Mamonkin I. G., Pulse Amplifiers. Gosenergoizdat, 1958.

1. In accordance with the same results, it would also have been possible to take \(q=0.33\) and determine \(\kappa\) satisfying (9) for the selected \(x\). ↩