CORRECTION OF A VIDEO AMPLIFIER STAGE BY A COMPLEX FEEDBACK CIRCUIT IN THE CATHODE

D. I. Sviryakin

(Presented by Professor-Doctor V. I. Gorbunov)

The analysis of high-frequency cathode correction in video-amplifier stages has been carried out by many authors. As correcting elements, most of them used \(R_k C_k\) circuits, and in some works [1,2] \(L_k C_k R_k\) circuits were used (Fig. 1). In the present work, the circuit of cathode correction by an \(L_k C_k R_k\) circuit is analyzed once again; the basic functional dependences between the parameters of the stage and the parameters of the transient response are established, and the circuit under analysis is compared with the correction circuit using an \(R_k C_k\) network. The results obtained in this work differ in many respects from the results of works [1,2], since in the present case the analysis was carried out with allowance for the influence of the time variation of the input and output dynamic capacitances on the transient response. This latter circumstance somewhat complicated the analysis of the circuit, but it made it possible to obtain results that agree better with the experimental results.

The basic and equivalent circuits of the stage are shown, respectively, in Figs. 1 and 2. The analysis is carried out under the assumption that the output signal from the anode of the first tube is applied to the circuit of the control grid of the second tube (of the same type as the first), whose cathode circuit contains the same elements as the first. This accounts for the identical nature of the time dependences of the capacitances \(C_{sk}\) and \(C_{ak}\), and makes it possible to combine the input dynamic capacitance \(C_{in}\) of the second tube with the output dynamic capacitance \(C_{out}\) of the first [3], and to denote them in the equivalent circuit of Fig. 2 as a single capacitance \(C_1\). The total stray capacitance in the circuits is denoted by \(C_0\). The correcting elements in the first stage are the capacitor \(C_k\), the inductance \(L_k\), and the resistor \(R_k\) (in the basic circuit, respectively, \(C_{k1}\), \(L_{k1}\), and \(R_{k1}\)). The value of the resistor \(R_k\) is determined by the required bias voltage; therefore, in the further discussion the value \(R_k\) will be assumed constant, while the correction parameters will be varied by changing \(L_k\) and \(C_k\).

For the equivalent circuit of Fig. 2, by the same methods as in [3], the operational image of the transient response was obtained:

\[ M(p)=\frac{A(p)}{B(p)}=\frac{1+a_1p+a_2p^2}{1+b_1p+b_2p^2+b_3p^3}, \tag{1} \]

where \(p=j\omega C_0 R_a\), \(j=\sqrt{-1}\), \(\omega\) is the angular frequency,

\[ a_1=q,\qquad a_2=q\cdot k_k, \]

\[ \begin{gathered} b_1 = 1 - x + \kappa_k(1-b) + b(q+xm) + xb,\\ b_2 = \kappa_k[1-x(1-b)] + b\kappa_k(q+xm-1) + b[q+xm(1-x)],\\ b_3 = b\kappa_k[q+xm(1-x)]. \end{gathered} \]

In turn,

\[ b=\frac{1}{1+SR_k}, \qquad m=\frac{R_k}{R_a}, \qquad x=\frac{C'_1}{C_0}, \]

\[ \kappa_k=\frac{L_k}{C_0R_aR_k}, \qquad q=\frac{C_kR_k}{C_0R_a}, \]

\[ C_0=C'_0+C'_1, \qquad C'_1=C_{\text{out}}+C_{\text{in}}, \qquad C'_0=C_{\text{out}}+C_{\text{in}}. \]

The transient response of the stage, according to the expansion theorem, is determined by the expression

\[ h(\tau)=\frac{A(0)}{B(0)}+\sum_1^{\kappa} e^{p_k\tau}\cdot \frac{A(p_k)}{p_k B'(p_k)}, \]

where

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

is the relative time (\(t\) is time in seconds),

\[ p_k \]

is the \(k\)-th root of the characteristic equation

\[ 1+b_1p+b_2p^2+b_3p^3=0, \]

\(\kappa\) is a positive integer, taking in the present case the values \(1, 2, 3\).

\(A(0)\) and \(B(0)\) are the polynomials of the numerator and denominator, respectively, of expression (1) at \(\omega=0\),

\[ B'(p_k) \]

is the derivative of the polynomial \(B(p_k)\).

The transient responses of the stage under consideration may be with overshoots or without overshoots. Only responses having overshoots are of interest, since in these cases the settling time is less than for responses without overshoots. In the overwhelming majority of cases, overshoots in transient responses appear when the roots of the characteristic equation are complex. Real negative roots indicate mainly the monotonic character of the transient process, i.e., the oscillatory character of the response in this case is minimal. The expressions for transient responses say the same thing. Thus, with two conjugate complex roots and one real root \(p_{1,2}=-\alpha\pm j\omega\), \(p_3=-\beta\), the transient response has the form

\[ h(\tau)=1-A_0e^{-\beta\tau}+Ae^{-\alpha\tau}\sin(\omega\tau+\varphi), \tag{2} \]

where

\[ A_0=\frac{1-a_1\beta+a_2\beta^2}{b_3\beta[(\beta-\alpha)^2+\omega^2]}, \]

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

\[ \varphi=\operatorname{arc\,tg} \frac{\omega(a_1-2\alpha a_2)} {1-\alpha a_1+(\alpha^2-\omega^2)a_2} -\operatorname{arc\,tg}\frac{\omega}{\alpha} -2\operatorname{arc\,tg}\frac{\omega}{-\alpha}. \]

In the case of real (negative) roots, the transient response is determined by the expression:

\[ h(\tau)=1+A_1e^{p_1\tau}+A_2e^{p_2\tau}+A_3e^{p_3\tau}, \tag{3} \]

where

\[ A_1=\frac{1+a_1p_1+a_2p_1^2}{b_3p_1(p_1-p_2)(p_1-p_3)}, \qquad A_2=\frac{1+a_1p_2+a_2p_2^2}{b_3p_2(p_2-p_1)(p_2-p_3)}, \]

\[ A_3=\frac{1+a_1p_3+a_2p_3^2}{b_3p_3(p_3-p_1)(p_3-p_2)} . \]

In calculating the parameters of the transient responses (2) and (3), the initial parameters of the stage were \(b\), \(m\), and \(x\), and the correction parameters were \(\kappa_k\) and \(q\). The only sought quantities adopted were the parameters of the transient responses: the normalized settling time \(Y=\dfrac{t_y}{C_0R_a}\), where \(t_y\) is the settling time in seconds, the first \(\delta_1\), second \(\delta_2\), and third \(\delta_3\) overshoots, and the normalized times of occurrence of these overshoots, respectively, \(\tau_{m1}\), \(\tau_{m2}\), \(\tau_{m3}\).

In addition, for each variant the gain was calculated from the formula

\[ B=\left(\frac{2.26}{Y}-1\right)\cdot 100\%. \]

The values of the initial parameters of the stage were as follows: \(b=0.4;\ 0.5;\ m=0.2;\ x=0.4\). The correction parameters were varied within the following discrete values: \(\kappa_k=0.2;\ 0.3;\ 0.4;\ 0.6;\ 0.8;\ 1.0;\ q=0.5;\)

Fig. 1. Schematic circuit of a stage with a cathode-correction network

\(0.6;\ 0.7;\ 0.8;\ 0.9;\ 0.95;\ 1.0;\ 1.05;\ 1.15;\ 1.2;\ 1.25;\ 1.4;\ 1.6;\ 2.0;\ 2.5;\ 3.0.\) Thus, the number of variants calculated was equal to the number of combinations of \(\kappa_k\) and \(q\). Analysis of the data obtained made it possible to determine both the optimal combinations of correction parameters and those close to them.

In addition, by calculation the dependences of the above-listed parameters of the transient responses on the initial parameters \(b\), \(m\), \(x\) were identified, with the latter varied over fairly wide ranges. The discrete values of these parameters for which the calcula-

... were as follows: \(b=0.2;\ 0.3;\ 0.4;\ 0.5;\ 0.6;\ 0.7;\ m=0.02;\ 0.04;\ 0.08;\ 0.2;\ 0.4;\ 0.6;\ 1.0;\ x=0.2;\ 0.3;\ 0.4;\ 0.5;\ 0.6;\ 0.7;\ 0.8\). All calculations were carried out on the M-20 electronic computer.

If, as the optimum transient response, one takes that for which the first overshoot is equal to 2%, and the second overshoot does not exceed 1%, with the smallest settling time for these overshoot values, then, for the stage parameters \(b=0.4;\ x=0.4;\ m=0.2\), the most successful combination of correction coefficients, according to the computational materials, should be regarded as \(\kappa_k=0.3\) and \(q=1.1\). In this case \(Y=0.06817,\ \delta_1=2.162\%,\ \delta_2=0.142\%,\ \tau_{m1}=1.516,\ \tau_{m2}=3.449,\ B=29.08\%\).

Fig. 2. Equivalent circuit

A comparative analysis of the parameters of the transient responses obtained as a result of the calculations showed that, for the above initial parameters, of practical interest are mainly the variants obtained with combinations of values of the correction parameters varied within the ranges \(q=0.9 \div 2.0,\ \kappa_k=0 \div 0.4\).

For \(\kappa_k=0\), the stage being analyzed is transformed into a stage with capacitive correction in the cathode, the results of whose investigation are presented in [4]. Therefore, comparison of the results of calculations performed for the variants of the present problem with \(\kappa_k\) with the results of [4] served as a check on the correctness of the formulas obtained above and of the program for calculations on the M-20 computer. In addition, this comparison showed that correction by the \(L_k C_k R_k\) circuit is more effective than correction by the \(R_k C_k\) circuit only for small overshoots. For large overshoots \((\delta_1>30\%)\), both correction circuits give almost identical results.

Thus, for example, in the variant with \(\kappa_k=0\) and \(q=0.8\), the gain is 17.8% for overshoots \(\delta_1=2.69\%\) and \(\delta_2\approx 0\). Almost the same overshoot, \(\delta_1=2.16\%\), and a barely noticeable second overshoot, \(\delta_2=-0.14\%\), in the variant \(\kappa_k=0.3\) and \(q=1.1\), make it possible to obtain a gain \(B=29.1\%\). That is, at small values of the first overshoot, the correction circuit \(L_k C_k R_k\), owing to the appearance of a small second overshoot, gives, in comparison with correction by the \(R_k C_k\) circuit, an increase in gain of more than 12%.

Let us now consider the region of large overshoots. To do this, let us compare two variants: \(\kappa_k=0,\ q=2.5\), where \(B=83.4\%,\ \delta_1=38.2\%,\ \delta_2=-0.09\%\), and \(\kappa_k=0.6,\ q=3.0\), where \(B=87.3\%,\ \delta_1=39.8\%,\ \delta_2=-0.56\%\).

We see that the substantial increase in gain in the second variant is mainly explained by the somewhat larger overshoot \(\delta_1\). The second overshoot can practically be neglected in both variants. Thus, introducing the inductance \(L_k\) into the cathode circuit in combination with the capacitor \(C_k\) is effective only for small overshoots of the stage transient response. In stages where large overshoots are permissible, one may confine oneself to introducing only the correcting circuit \(R_k C_k\).

On the basis of the calculation results, graphs were plotted. Thus, Fig. 3 shows the dependences \(B=f(\delta_1)\) and \(\delta_2=f(\delta_1)\) for values of the coefficient \(\kappa_k=0;\ 0.2;\ 0.3;\ 0.4\) and for the coefficient \(q\) varying within the range \(0.7 \div 3.0\). In order to have an idea of the magnitude of the coefficient \(q\) for a specified value of the overshoot \(\delta_1\), curves of equal values of \(q\) are plotted on the graphs \(B=f(\delta_1)\) and \(\delta_2=f(\delta_1)\). The graphs show that, with increa-

As the permissible value of the overshoot \(\delta_1\) increases, the gain provided by this correction circuit increases. The second overshoot at \(\delta_1 > 5\%\), even for \(\kappa_k = 0.4\), is comparatively small and may be neglected. Increasing the correction coefficient \(\kappa_k\) for small overshoots \(\delta_1\) leads

Fig. 3. Plots of \(B=f(\delta_1)\) and \(\delta_2=f(\delta_1)\) for \(b=0.4,\ x=0.4,\ m=0.2\).

to a noticeable increase in gain, but this occurs at the expense of increased oscillatory behavior of the transient response, or, in other words, at the expense of increasing the second (negative) \(\delta_2\), as well as the third (positive) \(\delta_3\), overshoots.

Judging from the plots in Fig. 3, the best, in comparison with the above, value of the correction-factor gain \(K_k\) is \(0.35\) at \(q=1.12\). The gain at \(\delta_1=2\%\) for \(K_k=0.35\) is approximately \(31\%\), and the second overshoot is \(\delta_2\approx -0.9\%\).

Similar plots are also given in Fig. 4, not only for \(b=0.5\). With these plots, as in Fig. 3, the dependences are presented

Fig. 4. Plots of \(B=f(\delta_1)\) and \(\delta_2=f(\delta_1)\) for \(b=0.5,\ x=0.4,\ m=0.2\)

\(B=f(\delta_1)\) and \(\delta_2=f(\delta_1)\), on which curves of equal values of the coefficient \(q\) are plotted with dashed lines. The plots show that, even for \(b=0.5\), the first extremum of the transient characteristics for \(K_k=0.5\) may fail to reach the unit level. In the plot \(B=f(\delta_1)\), this is represented by negative values of \(\delta_1\). Thus, for example, at \(q=1.2\) and \(K_k=0.5\), the first overshoot is \(\delta_1=-1.56\%\), the gain is \(B=21.1\%\), and the second overshoot is \(\delta_2=-2.37\%\) (in the plot it has gone beyond the limits of the figure). It should be noted that, both for \(b=0.5\) (Fig. 4) and for \(b=0.4\) (Fig. 3), transient characteristics with negative values of \(\delta_1\) are practically never used,

since the gain in these cases, despite the significant magnitude of \(\delta_2\), is always less than for positive values of the first overshoot.

Guided by the previously adopted optimality criterion for the transient response and taking into account the plots in Fig. 4, one may regard, for \(b=0.5\), \(x=0.4\), \(m=0.2\), the correction-coefficient values \(q=1.33\) and \(\kappa_k=0.45\) as close to optimal, since in this case \(\delta_1=-2\%\), \(B\approx31\%\), and \(\delta_2=-0.85\%\).

The gain obtained by increasing the permissible overshoot \(\delta_1\) may also be much larger than \(30\div32\%\). As is seen from the plots \(B=f(\delta_1)\) in Figs. 3 and 4, this can be achieved by increasing the coefficient \(q\), and, if an increase in the overshoot \(\delta_2\) is also permissible, by increasing the coefficient \(\kappa_k\).

Let us note that increasing the gain by increasing \(\kappa_k\) with \(q\) unchanged is impossible, since in this case, as the figures show, an increase in \(\kappa_k\) leads to an increase in \(\delta_2\), a decrease in \(\delta_1\), and a reduction in the gain. That is, if we wish to increase the gain, then, simultaneously with increasing \(\kappa_k\), we must increase \(q\), or else increase the coefficient \(q\) while leaving \(\kappa_k\) unchanged. In the latter case, along with an increase in \(B\), the overshoot \(\delta_1\) increases, while the overshoot \(\delta_2\) decreases or remains approximately constant.

How the magnitudes of the overshoots \(\delta_1\) and \(\delta_2\) vary with the coefficient \(q\) for certain constant values of \(\kappa_k\) is shown in Fig. 5. Here the plot \(\delta_2=f(q)\) for \(\kappa_k=0\) is absent altogether, since correction by the circuit \(R_k C_k\) gives a very small second overshoot. The plot \(\delta_1=f(q)\) for \(\kappa_k=0\), on the other hand, lies above all analogous plots for \(\kappa_k>0\). This indicates that introducing the inductance \(L_k\) into the cathode, with \(q\) unchanged, reduces \(\delta_1\) at the expense of the appearance of \(\delta_2\). Moreover, the second overshoots have their maximum magnitude at the minimum values of the first overshoots (\(\delta_1=1\div5\%\)), except in cases where \(\kappa_k\leq0.2\).

Attention should be paid to the fact that when the coefficient \(\kappa_k\) is increased above values of 0.2, the transient response of the corrected stage may, for some values of \(q\), have a negative first overshoot. That is, the numerical value of the normalized transient response at the point of the first inflection does not reach the unit level, and the magnitude of the first overshoot indicates, in percent, how much this value is less than unity. Combinations of correction coefficients that produce negative first overshoots are almost never used in practice; therefore the plots in Fig. 5 for \(\kappa_k>0.2\) are bounded by the \(q\)-axis.

What values of \(q\) and \(\kappa_k\), then, ensure a transient response with optimal parameters? Let us turn to Fig. 6, which gives the plots \(Y=f(q)\) for several values of \(\kappa_k\). On the plot \(Y=f(q)\) the curves of equal overshoots are plotted for several values of \(\delta_1\) (1, 5, 10, 15, 20%). An increase in \(\kappa_k\), as the figure shows, causes the curves \(Y=f(q)\) to shift upward and to the right, i.e., at constant \(q\) it causes an increase in \(Y\). But this does not mean that introducing the inductance \(L_k\) into the cathode has led to a deterioration in the parameters of the stage transient response. In percentage terms, the settling time has not increased as strongly as the magnitude of the first overshoot has decreased. This, for example, can be traced from Figs. 5 and 6 for the case when \(q=1.2\), and \(\kappa_k\) changes abruptly from 0 to 0.2.

If one considers the relative position of the curves of equal overshoots (Fig. 6), it may be noted that the greatest correction effect is obtained when \(\kappa_k\) and \(q\) are increased simultaneously. This is especially noticeable for small \(\delta_1\). Thus, for example, for \(\kappa_k=0\) and \(q=0.9\), the settling time is 0.693 at \(\delta_1=5\%\), while for \(\kappa_k=0.8\) and \(q=1.27\) the settling time ...

Visible graph labels:

• Left vertical axis: $\delta_1,\%$
• Right vertical axis: $\delta_2,\%$
• Horizontal axis: $q$
• Parameters inside the graph: $\beta = 0,4,\quad x = 0,4,\quad m = 0,2$

Curve labels:

• $\delta_1$ at $k_i = 0,0$
• $\delta_1$ at $k_i = 0,2$
• $\delta_1$ at $k_i = 0,4$
• $\delta_1$ at $k_i = 0,6$

• $\delta_1$ at $k_i = 0,8$

• $\delta_2$ at $k_i = 0,8$

• $\delta_2$ at $k_i = 0,6$
• $\delta_2$ at $k_i = 0,4$
• $\delta_2$ at $k_i = 0,2$

Fig. 5. Graphs of the dependence of the magnitudes of the first and second overshoots on the coefficient $q$ for constant values of the coefficient $k$.

Fig. 6. Plots of \(Y = f(q)\) with curves of equal overshoots.

Fig. 7. Plots of \(B=f(q)\), \(\delta_1=f(q)\), and \(\delta_2=f(q)\) for
\(\theta=0.4\) and \(\chi=0.4\)

is equal to 0.64 when $\delta_1=1\%$. However, in the first case $\delta_2=0$, and in the second $\delta_2=-9.5\%$. That is, the factor limiting the increase in the correction coefficients, and consequently the gain, is the second overshoot. If the magnitudes of the first and second forward overshoots are specified

Fig. 8. Influence of the choice of the coefficient $\kappa_k$ on the shape of the curves for
$b=0.5$, $x=0.5$, and $m=0.2$, with $B=f(q)$, $\delta_1=f(q)$, and $\delta_2=f(q)$

(as are the initial parameters $b$, $x$, and $m$), then for the correction circuit being analyzed there exists only one combination of values of the correction coefficients that provides the minimum rise time of the transient response.

Such a combination, as indicated above (see Fig. 3), for $b=0.4$, $x=0.4$, and $m=0.2$ is $q=1.12$ and $\kappa_k=0.35$. Additional calculations carried out for the same values of $b$, $x$, $m$, and $\kappa_k$, but with $q=1.15$, gave the following results: $\delta_1=2.52\%$, $\delta_2=-0.55\%$, $\delta_3$ is practically equal to zero, $B=32.5\%$, and, correspondingly, $Y=0.6639$. Somewhat better

Fig. 9. Effect of varying the coefficients \(k\) and \(q\) on the values of the overshoots and the gain provided by the circuit.

the results were obtained for \(m=0.04\) with the same \(b=0.4\), \(x=0.4\), \(\kappa_k=0.35\), but for \(q=1.1\). For this variant \(Y=0.6438\), \(\delta_1=1.88\%\), \(\delta_2=-0.93\%\), \(\delta_3=0\), and \(B=36.7\%\). Fig. 7, with its plots \(\delta_1=f(q)\), \(\delta_2=f(q)\), and \(B=f(q)\), confirms this proposition. It shows curves plotted for \(m=0.04\) and \(m=0.2\). This makes it possible to determine how a change in the parameter \(m\) affects the transient response of the amplifier. We see that the transition from \(m=0.04\) to \(m=0.2\) is reflected most noticeably in the value of \(B\), especially for small permissible overshoots, i.e., in the range of \(q\) values from 1.0 to 1.2. The magnitudes of the overshoots \(\delta_1\) and \(\delta_2\) change not so strongly in this case. In addition, Fig. 7 shows that the plots of all the above dependences have maximum steepness in the interval \(q=1.0 \div 1.2\).

How the choice of the coefficient \(\kappa_k\) affects the form of the plots \(B=f(q)\), \(\delta_1=f(q)\), and \(\delta_2=f(q)\) for \(b=0.5\), \(x=0.5\), and \(m=0.2\) is shown in Fig. 8. The magnitude of the gain depends most strongly on the coefficient \(\kappa_k\) only in the region of its negative values. A negative gain indicates that, for \(q<0.9\), the stage becomes worse than the uncorrected one. In the region of \(B\) values of practical interest, the second overshoot is subject to the greatest change when passing from \(\kappa_k=0.3\) to \(\kappa_k=0.35\). The first overshoot and the gain change hardly at all in this case. But this does not mean that the introduction of the correction coefficient \(\kappa_k\) led only to the appearance of this overshoot. When this coefficient is varied over wider limits than indicated above (Fig. 9), the gain also changes appreciably. However, the overshoots \(\delta_1\) and \(\delta_2\) undergo the most substantial changes. Moreover, \(\delta_1\), like the gain, decreases as \(\kappa_k\) increases, while \(\delta_2\) increases. As can be seen from Figs. 7, 8, and 9, the appearance of the second overshoot is due not only to the magnitude of the coefficient \(\kappa_k\), but also to the values of the parameter \(m\) and the correction coefficient \(q\).

In the course of investigating the stage with the correction circuit under analysis, it was found that the appearance and magnitude of the second overshoot, as well as the magnitudes of the gain and the first overshoot, are influenced just as noticeably by the parameters \(b\) and \(x\). Therefore, in order to carry out optimal correction, for each particular combination of the parameters \(b\), \(m\), and \(x\) it is necessary to select a quite definite combination of correction coefficients. And, owing to the fact that, with the introduction of the correction coefficient \(\kappa_k\), the first overshoot in percentage terms decreases faster than the gain, while the second overshoot appears only at a certain value of \(\kappa_k\) (Fig. 9) and has a clearly expressed minimum in the plots \(\delta_2=f(q)\) (Figs. 7, 8), the gain provided by a stage with a high-frequency correction circuit \(L_k C_k R_k\) is, in some cases, twice as high as with correction by the circuit \(R_k C_k\). This difference is especially noticeable for values \(\delta_1<10\%\). The introduction of the inductance \(L_k\) into the stage naturally somewhat complicated the circuit and its adjustment.

References

1. V. E. Katyukhin. Correction in a video amplifier. Tekhnika kino i televideniya, No. 10, 1965.

2. V. E. Katyukhin, V. M. Kirpichnikov. An amplifier with complex correction in the cathode. Radiotekhnika (Proceedings of Higher Educational Institutions, Ministry of Higher and Secondary Specialized Education of the USSR), Vol. IX, No. 3, 1966.

3. I. A. Suslov, D. I. Sviryakin. The influence of changes 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. Izv. TPI, Vol. 105, 1960.

4. D. I. Sviryakin. On the calculation of a video-amplifier stage with cathode correction. Tr. TIRiETA, Vol. No. 7, 1972.