Correction of Pulse-Top Distortion in Amplifier Stages with an Unbypassed Cathode Resistor When the Screen Grid Is Supplied from the Anode Filter

I. A. Suslov

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

Supplying the screen grid from the anode filter makes it possible to simplify the circuit of the amplifier stage, since it eliminates the need to place a damping resistor and a bypass capacitor in the screen-grid circuit. The calculation of low-frequency correction for such a circuit has been described in the literature [1]. It was noted that a disadvantage of the circuit is the dependence of the correction conditions on the parameters of the tube. Further simplification is obtained if the capacitor in the cathode circuit is omitted or replaced by a small capacitor for correction in the high-frequency region. This yields, in each stage, a saving of two capacitors, the dimensions of which—especially the cathode capacitor—may be substantial when amplifying pulses of sufficiently long duration.

Fig. 1

Let us derive formulas for calculating low-frequency correction in such a circuit (Fig. 1). To do this, we write the expressions for the anode and

of the screen-grid currents, assuming that \(R_a \ll R_g\) and the tube permeability \(D \ll 1\).

\[ \dot i_a = S\left[\dot U_1-(\dot i_a+\dot i_e)R_k - D_e(\dot i_a+\dot i_e)(Z_\phi+R_k)\right], \tag{1} \]

\[ \dot i_e = S_e\left[\dot U_1-(\dot i_a+\dot i_e)R_k - D_e(\dot i_a+\dot i_e)(Z_\phi+R_k)\right]. \]

Here

\[ Z_\phi=-\frac{R_\phi}{1+j\omega C_\phi R_\phi}, \]

and \(D_e\) and \(S_e\) are, respectively, the permeability of the screen grid and the transconductance of the screen current.

Solving the system of equations (1), we find

\[ \dot i_a = \frac{S\dot U_1} {1+\dfrac{S_k(1+\mu_e)}{\mu_e}R_k+\dfrac{S_k}{\mu_e}Z_\phi}, \]

\[ \dot i_e = \frac{S_e\dot U_1} {1+\dfrac{S_k(1+\mu_e)}{\mu_e}R_k+\dfrac{S_k}{\mu_e}Z_\phi}, \]

where

\[ S_k=S+S_e \qquad\text{and}\qquad \mu_e=\frac{1}{D_e}. \]

The alternating voltage at the anode with respect to the common conductor will be

\[ \dot U_2=\dot i_a R_a+(\dot i_a+\dot i_e)Z_\phi, \]

and the gain from grid to anode is

\[ \dot K = \frac{\dot U_2}{\dot U_1} = \frac{SR_a+S_k Z_\phi} {1+\dfrac{S_k(1+\mu_e)}{\mu_e}R_k+\dfrac{S_k}{\mu_e}Z_\phi}. \tag{2} \]

The gain at middle frequencies is then equal to

\[ K_0= \frac{SR_a} {1+\dfrac{S_k(1+\mu_e)}{\mu_e}R_k}, \]

and the frequency-distortion coefficient is

\[ \dot M = \frac{\dot K}{K_0} = \frac{ j\omega+ \left(\dfrac{S_k}{S}+\dfrac{R_a}{R_\phi}\right)\dfrac{1}{\tau_a} }{ j\omega+ \dfrac{1}{\tau_a} \left\{ \dfrac{R_a}{R_\phi} + \dfrac{S_kR_a} {\mu_e\left[1+\dfrac{S_k}{\mu_e}(1+\mu_e)R_k\right]} \right\} }, \tag{3} \]

where

\[ \tau_a=C_\phi R_a. \]

An analogous formula for an ordinary stage having capacitors \(C_e\) and \(C_k\) in the screen-grid and cathode circuits will be

\[ \dot M= \frac{ j\omega+\dfrac{1}{\tau_{a1}}\left(1+\dfrac{R_a}{R_{\phi 1}}\right) }{ j\omega+\dfrac{1}{\tau_{a1}}\cdot\dfrac{R_a}{R_{\phi 1}} } \tag{4} \]

Here \(R_{\phi 1}\) and \(\tau_{a1}=C_{\phi 1}R_a\) are the parameters of an equivalent (with respect to distortion) circuit with \(C_k\) and \(C_e\). This circuit will evidently be identical with the circuit of Fig. 1 if

\[ \left(1+\frac{R_a}{R_{\phi 1}}\right)\frac{1}{\tau_{a1}} = \frac{1}{\tau_a}\left(\frac{S_k}{S}+\frac{R_a}{R_\phi}\right), \tag{5} \]

and

\[ \frac{R_a}{R_{\phi 1}}\cdot\frac{1}{\tau_{a1}} = \frac{1}{\tau_a}\cdot\left(\frac{S_k}{S}\cdot\frac{K_0}{\mu}+\frac{R_a}{R_\phi}\right). \tag{6} \]

Subtracting (6) from (5), we obtain

\[ \tau_{a1}= \frac{\tau_a}{\dfrac{S_k}{S}\left(1-\dfrac{K_0}{\mu_e}\right)}. \tag{7} \]

Dividing (5) by (6), we find

\[ R_{\phi 1}=R_\phi \frac{ \dfrac{S_k}{S}\left(1-\dfrac{K_0}{\mu_e}\right) }{ 1+\dfrac{S_k}{S}\cdot\dfrac{K_0}{\mu_e}\cdot\dfrac{R_\phi}{R_a} }. \tag{8} \]

Formulas (7) and (8) show that, for \(K_0>\mu_e\), correction, as in the case when there is no capacitor in the cathode circuit [1], is impossible. Experimental verification shows that in this case, instead of raising the pulse top, connecting the anode filter leads to a droop, the greater the smaller \(C_\phi\) is.

If the capacitor \(C_\phi\) is connected between the screen grid and the cathode (Fig. 2), then, to determine the anode and screen currents, we have the relations

\[ \dot I_a=S\left[\dot U_1-(\dot I_a+\dot I_e)R_k-D_e(\dot I_a+\dot I_2)(R_\phi+R_k)\right], \]

\[ \dot I_e=S_e\left[\dot U_1-(\dot I_a+\dot I_e)R_k-D_e(\dot I_a+\dot I_2)(R_\phi+R_k)\right], \tag{9} \]

\[ \dot I_e=\dot I_1+\dot I_2, \]

\[ \dot I_1 \cdot \frac{1}{j\omega C_{\phi}}=(\dot I_a+\dot I_2)(R_k+R_{\phi}). \]

From equations (9) one can find \(\dot I_a\) and \(\dot I_3=\dot I_a+\dot I_2\).

Fig. 2

\[ \dot I_a= \frac{ S\dot U_1\left(R_k+R_{\phi}+\dfrac{1}{j\omega C_{\phi}}\right) }{ R_k+R_{\phi}+\dfrac{1}{j\omega C_{\phi}} \left( 1+S_k\frac{\mu_3+1}{\mu_3}R_k+\frac{S_k}{\mu_3}R_{\phi} \right) }, \]

\[ \dot I_3= \frac{ S_k\dot U_1\cdot \dfrac{1}{j\omega C_{\phi}} }{ R_k+R_{\phi}+\dfrac{1}{j\omega C_{\phi}} \left( 1+S_k\frac{\mu_3+1}{\mu_3}R_k+\frac{S_k}{\mu_3}R_{\phi} \right) }. \]

The gain from the grid to the anode will then be

\[ \dot K=\frac{\dot U_2}{\dot U_1} = \frac{\dot I_a R_a+\dot I_3 R_{\phi}}{\dot U_1} = \]

\[ = \frac{ S\left[ R_a\left(R_k+R_{\phi}+\dfrac{1}{j\omega C_{\phi}}\right) +\dfrac{S_k}{S}\cdot \dfrac{R_{\phi}}{j\omega C_{\phi}} \right] }{ R_k+R_{\phi}+\dfrac{1}{j\omega C_{\phi}} \left( 1+S_k\frac{\mu_3+1}{\mu_3}R_k+\frac{S_k}{\mu_3}R_{\phi} \right) }. \]

Hence, for \(\omega\to\infty\), we obtain that the gain at medium frequencies is equal to \(K_0=SR_a\), i.e., that the action of the negative feedback in the given circuit is completely compensated by the positive feedback (Fig. 2).

The frequency-distortion coefficient for the incomplete stage circuit (without the transitional network) can now be represented in the form

\[ \dot M = \frac{ j\omega+ \dfrac{R_a+R_\phi \cdot \dfrac{S_k}{S}} {(R_k+R_\phi)\tau_a} }{ j\omega+ \dfrac{ \left( 1+S_k\dfrac{\mu_э+1}{\mu_э}R_k+ \dfrac{S_k}{\mu_э}R_\phi \right)R_a }{ (R_k+R_\phi)\tau_a } }. \tag{10} \]

Formula (10) will be identical with (4) when

\[ \tau_{a1}=\tau_a\left(1+\frac{R_k}{R_\phi}\right)\cdot \frac{1}{\dfrac{S_k}{S}(1-a)}. \tag{11} \]

and

\[ R_{\phi1}=R_\phi\cdot\frac{S_k}{S}\cdot \frac{1-a}{1+\dfrac{R_\phi}{R_a}\cdot\dfrac{S_k}{S}\cdot a}, \tag{12} \]

where

\[ a=\left(\frac{\mu_э+1}{\mu_э}\cdot\frac{R_k}{R_\phi} +\frac{1}{\mu_э}\right)K_0 . \]

From (11) and (12) it is seen that correction is feasible if

\[ K_0< \frac{1}{ \dfrac{\mu_э+1}{\mu_э}\cdot\dfrac{R_k}{R_\phi} +\dfrac{1}{\mu_э} }. \]

The dependence of the correction conditions on tube parameters that follows from formulas (7), (8), (11), and (12) apparently cannot be an obstacle to the use of the circuits under consideration. The point is that the influence of these parameters is weakened by the very structure of the indicated formulas, which contain \(S_k/S=(1+S_э/S)\) (where \(S_э\) is approximately proportional to \(S\)) and also include terms of the form \(K_0/\mu_э\) and \(a\), usually several times smaller in comparison with the nearby units. In addition, this influence is weakened by the presence of dc negative feedbacks that stabilize the tube operating point. The depth of feedback can be determined in the following way. From formula (3), the dc gain coefficient (for \(\omega\to0\)) will be

\[ K_{\,}=K_{\omega\to0}= \frac{SR_a+S_kR_\phi} {1+\dfrac{S_k(\mu_э+1)}{\mu_э}R_k+\dfrac{S_k}{\mu_э}R_\phi}. \tag{13} \]

In the absence of feedbacks this coefficient would be equal to

\[ K = SR_a + S_k R_f . \tag{14} \]

Comparing (13) and (14), we find that the depth of the dc feedback in the circuits of Figs. 1 and 2 will be

\[ 1+\dot K \xi = 1 + \frac{S_k(\mu_e+1)}{\mu_e} R_k + \frac{S_k}{\mu_e} R_f . \]

For example, for a stage using a 6Zh4 tube, with \(R_k = 0.16\ \text{k}\Omega\) and \(R_f = 12\ \text{k}\Omega\), \(1+\dot K \xi = 7\), i.e., variations of the dc components of the currents \(i_a\) and \(i_e\) are attenuated by a factor of 7.

In calculating low-frequency correction by the formulas given here, first \(R_f\) is found from the known (or chosen) voltages of the anode-supply source \((E_a)\) and the screen grid \((E_e)\). Then \(R_{f1}\), \(d_1 = R_a/R_{f1}\), are found, and the time constant is determined,

\[ \tau_{a1}=\frac{T}{8}\sqrt{\frac{d_1}{\Delta}}, \]

where \(T\) is the pulse repetition period, and \(\Delta\) is the distortion of the corrected pulse (Fig. 3). From formulas (7) or (11) we then find \(\tau_a\), and then determine

\[ C_f=\frac{\tau_a}{R_a}. \]

The obtained result is rounded to the nearest larger value of the standard capacitance, and the new values of the constants \(\tau_a\) and \(\tau_{a1}\) are calculated. Finally, from the correction condition \(\tau_{a1}=\tau_g\) we find the time constant \(\tau_g=C_gR_g\), and, having chosen \(C_g\), determine

\[ R_g=\frac{\tau_g}{C_g}. \]

Fig. 3

The use of simpler formulas [3], valid for the case

\[ \frac{t_u}{\tau_g} \ll 1, \]

leads to a less economical solution (larger values of the time constant \(\tau_g\)).

To check the formulas obtained, a three-stage amplifier was calculated and assembled according to the circuit of Fig. 1. This amplifier reproduced, without distortion, symmetrical rectangular pulses with a repetition frequency \(1/T = 25\) cycles, for whose transmission it had been calculated.

No preliminary adjustment was required.

REFERENCES

1. Lur’e O. B. Video-frequency amplifiers. Izd. “Sov. radio,” 1955.
2. Kreitser V. L. Video amplifiers. Izd. “Sov. radio,” 1952.
3. Stepanov L. V. Pulse amplifiers. Gosenergoizdat, 1954.