VIDEO AMPLIFIER WITH CATHODE CORRECTION

I. A. SUSLOV, D. I. SVIRYAKIN

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

A number of works \([1, 2, 3, 4]\) have been devoted to the problem of correction by means of a capacitance in the cathode circuit. However, in all these works the time variations of the input and output dynamic conductances were not taken into account, which leads to appreciable errors in determining the optimum value of the correcting capacitance. In the present article a refinement is made of the theory of a stage with cathode correction (Fig. 1) by taking into account the influence of the indicated variations.

Fig. 1. Schematic diagram of the stage.

In the circuit of the stage, the output capacitance is made up of the capacitances \(C_{ak}\)—anode–cathode, \(C_{a3}\)—anode–screen grid, and the wiring capacitance of the output of the first stage \(C_{m1}\). Here the capacitance \(C_{m1}\) is connected directly between the anode and ground, while the capacitance \(C_{a3}\) is connected through the large filter capacitor of the screen grid, \(C_{3}\). The magnitudes of these capacitances do not depend on time, i.e., they may be taken as constant. The capacitance \(C_{ak}\) is connected between the anode and cathode; therefore, as the signal passes, it varies with time depending on the voltage across the capacitance \(C_{k}\). Consequently, the capacitance \(C_{ak}\) is a function of time (in frequency-method analysis, a function of frequency).

The input capacitance of the following stage is composed of the wiring capacitance of the input \(C_{m2}\) and the capacitance \(C_{gk}\) (control grid—cathode). For convenience of analysis we proceed from the assumption that the circuit of the second stage is analogous to the first. The wiring capacitance \(C_{m2}\) is connected between the control grid of the second tube and ground; therefore it is constant. The capacitance \(C_{gk}\), however, depends on the potential at the cathode of the second tube, i.e., it is a function of time. Analysis shows that the character of the time variation of the capacitance \(C_{gk}\) is close to the time dependence of the variation of the capacitance \(C_{ak}\). Therefore, with sufficient accuracy, these two capacitances may be combined into one,

\[ C'_1 = C_{ak} + C_{gk}, \]

and the latter may be considered connected between the anode and cathode of the first tube.

Fig. 2. Equivalent circuit of the stage.

Let us consider the equivalent circuit of the stage (Fig. 2). Here

\[ C'_1 = C_{ak} + C_{gk}, \]

\[ C'_o = C_{m1} + C_{m2} + C_{ag} = C_m + C_{ag}. \]

The action of the screen grid of the tube is taken into account by an equivalent current generator \(S_e \cdot U_g\), where \(S_e\) is the transconductance of the tube with respect to the screen grid. The slope of the cathode-current characteristic is expressed as the sum

\[ S_k = S + S_e. \]

For the circuit of Fig. 2 one can write a system of two nodal-potential equations. Solving the latter, we find the frequency-distortion coefficient in the form

\[ \dot M = \frac{1 + j q \Omega} {1 + j\Omega\,[\,b(q + xm) + 1 - x + bx\,] + j^2 \Omega^2 b(q + xm - x^2m)} . \tag{1} \]

where

\[ q = \frac{C_k R_k}{C_o R_a}, \qquad \Omega = \omega C_o R_a, \qquad C_o = C'_1 + C'_o, \]

\[ b = \frac{1}{1 + S_k R_k}, \qquad m = \frac{R_k}{R_a}, \qquad x = \frac{C'_1}{C'_1 + C'_o} = \frac{C'_1}{C_o}. \tag{2} \]

Denoting

\[ a = q,\qquad b_1 = b(q + xm) + 1 - x + bx,\qquad b_2 = b(q + xm - x^2m), \]

we obtain

\[ \dot M = \frac{1 + ja\Omega} {1 + jb_1\Omega + j^2 b_2\Omega^2}. \tag{3} \]

Hence the modulus of the frequency-distortion coefficient is equal to

\[ M = \sqrt{ \frac{1 + \Omega^2 a^2} {1 + \Omega^2(b_1^2 - 2b_2) + \Omega^4 b_2^2} }. \tag{4} \]

To obtain an expression for the transient response, in expression (3) we replace \(j\Omega\) by \(p\). Then we obtain

\[ M(p)=\frac{1+ap}{1+b_1p+b_2p^2} =\frac{1+ap}{b_2\left[p^2+\frac{b_1}{b_2}p+\frac{1}{b_2}\right]} . \tag{5} \]

The roots of the characteristic equation will be

\[ p_{1,2}=-\frac{b_1}{2b_2}\pm \sqrt{\left(\frac{b_1}{2b_2}\right)^2-\frac{1}{b_2}} . \]

We are interested only in complex roots, i.e., the case when the transient response is oscillatory in character. Therefore we denote

\[ \frac{b_1}{2b_2}=\alpha,\qquad \sqrt{\left(\frac{b_1}{2b_2}\right)^2-\frac{1}{b_2}}=j\omega . \]

Then

\[ p_{1,2}=-\alpha\pm j\omega . \]

We obtain the transient response with the aid of the expansion theorem

\[ h(t)=\frac{A(0)}{B(0)}+\sum_{1}^{k} e^{p_k t}\frac{A(p_k)}{p_k B'(p_k)} =1+e^{(-\alpha+j\omega)t}\cdot \frac{1-a\alpha+ja\omega}{(-\alpha+j\omega)b_2\cdot 2j\omega}+ \]

\[ +e^{(-\alpha-j\omega)t}\cdot \frac{1-a\alpha-ja\omega}{(-\alpha-j\omega)b_2(-2j\omega)} =1+A\cdot e^{-\alpha t}\cdot \sin(\omega t+\varphi), \]

where

\[ A=\frac{1}{\omega b_2} \sqrt{\frac{(1-a\alpha)^2+(a\omega)^2}{\frac{1}{b_2}}}, \qquad \varphi=\varphi_1+\varphi_2, \]

\[ \varphi_1=\operatorname{arc\,tg}\frac{a\omega}{1-a\alpha}, \qquad \varphi_2=\operatorname{arc\,tg}\frac{\omega}{-\alpha}. \]

In order to determine the magnitude of the first overshoot of the transient response, it is necessary to set equal to zero the expression for the first derivative of this response

\[ h'(t)=-A\alpha e^{-\alpha t}\sin(\omega t+\varphi) +A\omega e^{-\alpha t}\cos(\omega t+\varphi)= \]

\[ =A\sqrt{\alpha^2+\omega^2}\,e^{-\alpha t}\sin(\omega t+\varphi_1)=0. \]

The derivative will be equal to zero under the condition \(\sin(\omega t+\varphi_1)=0\). Hence \(\omega t+\varphi_1=n\cdot\pi\), where \(n\) is an integer. The first overshoot of the transient response will correspond to the time

\[ t_{\max 1}=\frac{\pi-\varphi_1}{\omega}, \]

the second (negative) overshoot occurs

Fig. 3. Dependence of the overshoot magnitude on the parameters \(q\) and \(x\) for \(b = 0.4\) and \(m = 0.08\).

Fig. 4. Dependence of the overshoot magnitude on the parameters \(q\) and \(x\) for \(b = 0.5\) and \(m = 0.08\).

Fig. 5. Dependence of the overshoot magnitude on the parameters \(q\) and \(x\) for \(b = 0.6\) and \(m = 0.08\).

for

\[ t_{\max 2}=\frac{2\pi-\varphi_1}{\omega} \]

and so on. The second overshoot is small. For the first overshoot we obtain the expression

\[ \delta = h(t_{\max 1})-1 = A e^{-a t_{\max 1}}\sin(\pi-\varphi_2). \]

The settling time of the transient characteristic is determined by the trial method in combination with interpolation [5].

In the manner indicated, the dependences of the overshoot magnitude \(\delta\) and of the generalized settling time \(y\) on the parameters \(b\), \(x\), and \(q\) were calculated for \(m=0.08\). The results obtained are given in Tables 1 and 2. From the tables the graphs in Figs. 3, 4, 5, and 6 were constructed.

Fig. 6. Dependence of the generalized settling time on the parameters \(q\) and \(x\) for \(b=0.4;\ 0.5;\ 0.6;\) and \(m=0.08\).

Using the tables or the graphs, one can, for a specified overshoot \(\delta\) and known \(b\) and \(x\), determine the parameter \(q\) and the generalized settling time. Then, from formulas (2), the magnitudes of the circuit elements can be found.

Comparison of the graphs in Figs. 3, 4, 5, and 6 with the results obtained in [3] shows that, for identical overshoots, when the time variations of the capacitances are taken into account, smaller values of the parameter \(q\) correspond than in [3]. The use of parameters found from the graphs of [3] leads to the appearance of an overshoot much larger than that adopted in the calculation. Thus, a stage calculated for an overshoot \(\delta=1\%\), when variations of the dynamic capacitances are taken into account, has an overshoot of \(12\%\).

In a two-stage experimental model, when the value of \(q\) obtained for \(\delta=1\%\) on the basis of our graphs was used in the circuit,

Overshoots of the transient characteristic

Table 1

Generalized rise times \(y = \dfrac{t_y}{\tau_a}\)

Table 2

when, the overshoot of the transient characteristic on the screen of the IPKh-57 was approximately equal to 1%. When, however, \(C_k\), found from the graphs [3], was used, the overshoot was about 12%. The experimental check thus confirms the results of the present article, as well as the conclusion regarding the presence of significant overcorrection in the amplifier calculated from the materials of [3].

REFERENCES

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

2. Tsykin G. S., Calculation of Stages of Broadband and Pulse Amplification with Simple High-Frequency Correction. Svyazizdat, 1955.

3. Lisogurskii V. I., A gain stage with cathode correction. Collection of Scientific Papers of the Leningrad Electrotechnical Institute of Communications named after Prof. M. A. Bonch-Bruevich, issue 1 (34), Leningrad, 1958.

4. Kreitser V. P., Video Amplifiers. Sovetskoe Radio Publishing House, 1952.

5. Suslov I. A., Sviryakin D. I. The influence of time variations of dynamic conductances on the characteristics of a video-frequency amplifier. Proceedings of TPI, vol. 105, Tomsk, 1960.