Proceedings
of the Tomsk Order of the Red Banner of Labor Polytechnic
Institute named after S. M. Kirov

Volume 116 1962

On the Maximum Output Voltage in a Two-Stage Broadband and Pulse Amplifier

R. A. Smirnov

(Presented by the scientific seminar of the radio-engineering faculty)

In the literature [1, 2] there is an indication of the possibility of increasing the output voltage in two-stage pulse amplifiers by increasing the anode load of the second stage and correspondingly decreasing the load of the first stage. However, there are no data on the magnitude of the maximum possible gain in output voltage when amplifying harmonic and pulse signals, nor on the optimal amplifier parameters that provide this gain. Let us consider these questions as applied to several specific circuits.

When harmonic signals are amplified, by the maximum output voltage of a multistage amplifier \(U_{\text{out max}}\) at a given frequency \(\omega\) we shall mean the voltage obtained at the amplifier output when a voltage \(U_{\text{in}}(\omega)=U_{\text{in max}}\) is applied to the input. In this case the value \(U_{\text{in max}}\) is chosen so that the voltages on the grids of the amplifier tubes do not go beyond the limits of the grid characteristic over the entire range of amplified frequencies.

Similarly, when pulses are amplified, by the maximum output voltage of a multistage amplifier \(U_{\text{out max}}\) we shall mean the voltage obtained when a rectangular voltage pulse \(U_{\text{in max}}\) is applied to the input. The value \(U_{\text{in max}}\) must be chosen so that the voltage overshoots arising on the grids of the amplifier tubes do not go beyond the limits of the grid-characteristic expansion.

Determination of the Maximum Output Voltage When Amplifying a Harmonic Signal

Let the greatest permissible change in voltage on the grid of the second stage, at which the nonlinearity of the tube may be neglected, be \(\Delta U'_{g2\text{ max}}\). Then a voltage of any frequency within the passband may be applied to the amplifier input with amplitude

\[ U_{\text{in max}}= \frac{\dfrac{1}{2}\Delta U'_{g2\text{ max}}}{K'_{1\text{ max}}(\omega)} \, , \]

without fear of overloading the second stage. Here \(K'_{1\max}(\omega)\) is the largest value of the gain of the first stage. In this case

\[ U'_{\text{out max}}(\omega) = U'_{\text{in max}}K'(\omega) = \frac{1}{2}\Delta U'_{c\max} \frac{K'(\omega)}{K'_{1\max}(\omega)} , \]

where \(K'(\omega)\) is the gain of the amplifier at the given frequency. If the anode loads in both stages are changed, then

\[ U''_{\text{out max}}(\omega) = \frac{1}{2}\Delta U''_{c\max} \frac{K''(\omega)}{K''_{1\max}(\omega)} . \]

Then the ratio of the output voltages at frequency \(\omega\) is

\[ \lambda(\omega) = \frac{U''_{\text{out max}}(\omega)}{U'_{\text{out max}}(\omega)} = \frac{\Delta U''_{c\max}K''(\omega)K'_{1\max}(\omega)} {\Delta U'_{c\max}K'(\omega)K''_{1\max}(\omega)} . \tag{1} \]

In the case of using pentodes, the grid-characteristic span may be considered independent of frequency, since the condition \(Z \ll R_i\) is satisfied. Therefore formula (1) assumes the form

\[ \lambda(\omega) = \frac{K''(\omega)K'_{1\max}(\omega)} {K'(\omega)K''_{1\max}(\omega)} . \tag{2} \]

Let us apply formula (2) to a two-stage amplifier with parallel inductive correction in both stages (Fig. 1).

Fig. 1.

If the anode load of the first stage is decreased by \(n\) times

\[ \left(R_1=\frac{R}{n}\right), \]

and that of the second is increased by the same factor \((R_2=nR)\), without changing the inductances, then we shall have:

\[ K'(\omega)=S_1S_2R^2M'(\omega), \]

\[ K''(\omega)=S_1S_2R^2M''(\omega), \]

\[ K'_{1\max}(\omega)=S_1RM'_{1\max}(\omega), \]

\[ K''_{1\max}(\omega)=S_1\frac{R}{n}M''_{1\max}(\omega). \tag{3} \]

Here \(M(\omega)\) is the overall frequency response of the amplifier, and \(M_{1\max}\) is the maximum of the frequency response of the first stage. Substituting (3) into (2), we obtain

\[ \lambda(\omega)= \frac{n M'_{1\max}(\omega) M''(\omega)} {M''_{1\max}(\omega) M'(\omega)} . \tag{4} \]

The frequency characteristic of this amplifier has the form

\[ M(\Omega)= \sqrt{ \frac{ 1+k^2\Omega^2\left(n^2+\frac{1}{n^2}\right)+k^4\Omega^4 }{ (1-k^2\Omega^2)^4+\Omega^2(1-k^2\Omega^2)^2 \left(n^2+\frac{1}{n^2}\right)+\Omega^4 } }, \tag{5} \]

where \(\Omega=\omega RC\) and \(k=\dfrac{L}{R^2C}\). The frequency characteristic of the first stage has a maximum whose magnitude is

\[ M_{1\max}= \frac{kn^2} {\sqrt{2n\sqrt{2k+(kn^2)-2kn^2-1}}}. \tag{6} \]

Let us determine the optimum values of \(n\) at which the greatest gain in output voltage at mid-frequencies is obtained. Substituting (5) and (6) into (4), we obtain at mid-frequencies

\[ \lambda(O)= \frac{ \sqrt{\sqrt{8kn^2+4(kn^2)^2}-2kn^2-1} }{kn}. \tag{7} \]

Expression (7) has a maximum \(\lambda_{\max}(0)=\dfrac{1}{\sqrt{2k}}\) (8) when \(kn^2=\dfrac{2}{3}\), whence

\[ n_{\mathrm{opt}}=\sqrt{\frac{2}{3k}}. \tag{9} \]

It follows from (8) and (9) that, in order to increase \(U_{\text{out max}}\), it is advantageous to use small values of \(k\) and, correspondingly, large values of \(n\). However, all frequency characteristics determined by formula (5) intersect at the point with coordinates \(M=2k,\ \Omega=\dfrac{1}{\sqrt{2k}}\), independently of \(n\).

When \(k\) is decreased, the point of intersection shifts toward larger \(\Omega\) and smaller \(M\), and the shape of the frequency characteristic deteriorates sharply. Therefore, to increase \(U_{\text{out}}\), one can use only values of \(k\) in the range \(0.3\text{–}0.414\). In this case the gain in output voltage is \(10\text{–}29\%\).

It is known [3,4] that the optimum frequency characteristic of a two-stage amplifier is provided when the parameters of the two stages are unequal, namely: \(n=1.5913,\ R_1=0.6284R,\ R_2=1.5913R,\ k_1=\dfrac{L_1}{R_1^2C}=1.1188,\ k_2=\dfrac{L_2}{R_2^2C}=0.2696\). In this case the passband at the 0.7 level increases by 17%. In this case \(\lambda(0)=1\), i.e., at mid-frequencies such an amplifier gives no gain in output voltage in comparison with two identical stages with \(k=0.414\). At high frequencies, a certain gain is obtained owing to the greater voltage applied to the grid of the second tube. Instead of widening the passband by 17%, it is sometimes better to increase the load resistances of both amplifier stages by the same amount. Then the output voltage at low frequencies will increase by 17%, and the gain coefficient of the pair of stages by 37%.

Let us consider the possibility of increasing the output voltage in a two-stage amplifier with parallel feedback (Fig. 2). The frequency characteristic of this amplifier has the form [5]

\[ M(\omega)=\frac{1}{\sqrt{1+b_1\omega^2+b_2\omega^4}}, \]

where

\[ b_1=\left[ \frac{(mR_1+R_2)R_1C-(R_1-R_2)mxR_1C}{R_1+R_2+mR_1+mS_2R_1^2} \right]^2 -2\frac{mxR_1^2C^2R_2}{R_1+R_2+mR_1+mS_2R_1^2}, \]

\[ b_2= \frac{m^2x^2R_1^2C^2R_2^2}{R_1+R_2+mR_1+mS_2R_1^2}, \qquad m=\frac{R_2}{R_1}. \]

Fig. 2.

For an optimal frequency characteristic we have the equalities \(b_1=0\) and \(b_2\omega_0^4=1\), which form a system of equations for determining \(R_1\), \(R_2\), and \(m\).

Since there is a system of two equations with three unknowns, one of them may be chosen arbitrarily. It is most convenient, in determining the maximum output voltage, to vary \(m\). In this case, for a given \(\omega_0\), a frequency characteristic is obtained that is the same for any \(m\). As \(m\) is increased, only the gain of the entire amplifier decreases slightly. Therefore formula (2) in the present case takes the form

\[ \gamma(\omega)= \frac{K_{02}M''(\omega)K_{01}M'_{1\max}(\omega)} {K_{02}M''(\omega)K_{01}M'_{1\max}(\omega)} = \frac{K_{02}M'_{1\max}(\omega)} {K_{02}M''_{1\max}(\omega)}, \tag{10} \]

where

\[ K_0= \frac{mS_1R_1^2(S_2R_2-1)} {R_2+R_1(1+m)+mS_2R_1^2} \]

is the gain of the entire amplifier at mid-frequencies,

\[ K_{01}= \frac{S_1R_1(mR_1+R_2)} {R_2+R_1(1+m)+mS_2R_1^2} \]

is the gain of the first stage at mid-frequencies,

\[ K_{02}= \frac{mR_1(S_2R_2-1)} {mR_1+R_2} \]

is the gain of the second stage at mid-frequencies. \(\tag{11}\)

In the case of the optimal frequency characteristic of the entire amplifier, the frequency characteristic of the first stage has the form

\[ M_1(\omega)= \frac{ 1+\left(\frac{mxR_1R_2C}{mR_1+R_2}\right)^2\omega^2 }{ 1+\left(\frac{mxR_1^2R_2C^2}{R_1+R_2+mR_1-mS_2R_1^2}\right)^2\omega^4 }. \tag{12} \]

From the first derivative of this expression we determine the frequency at which the maximum rise of the frequency characteristic of the first stage occurs, and then the magnitude of this rise:

\[ M_{1\max}= \sqrt{ \frac{1}{2} \left( 1+ \sqrt{ 1+ \left[ \frac{mxR_2\left(R_1+R_2+mR_1-mS_2R_1^2\right)} {\left(mR_1+R_2\right)^2} \right]^2 } \right) }. \]

Solving the system of equations \(b_1=0\) and \(b_2\omega_b^4=1\) with respect to \(R_1\) and \(R_2\), we obtain

\[ R_1= \frac{ (\varepsilon-\sqrt{2})(1+xm) + \sqrt{ (\varepsilon-\sqrt{2})^2(1+xm)^2 + 4x(1+xm^2)(\varepsilon\sqrt{2}-1-x) } }{ 2xmC\omega_b(\sqrt{2}\varepsilon-1-x) }; \]

\[ R_2= \frac{mR_1(1+x)} {\sqrt{2}\,mx\omega_bCR_1-1-xm}, \]

where

\[ \varepsilon=\frac{S_2}{C\omega_b}. \]

Analysis of the expressions for \(K_{02}\) and \(M_{1\max}\) shows that, when the values \(R_1\) and \(R_2\) are substituted into (11) and (12), the quantity \(C\omega_b\) cancels everywhere. Thus, \(K_{02}\) and \(M_{1\max}\) are functions only of the dimensionless parameters \(\varepsilon\), \(x\), and \(m\). The results obtained below with specific data \(S_2\), \(C\), and \(\omega_b\) remain valid for other values of these parameters corresponding to the same value of \(\varepsilon\).

Fig. 3. Dependence \(\lambda=f(m)\) and \(y=f(m)\) for a two-stage amplifier with parallel feedback under amplification of harmonic signals. Curve 1—for \(\varepsilon=2\), curve 2—for \(\varepsilon=4\), curve 3—for \(\varepsilon=40\).

A calculation was carried out for a two-stage amplifier with parallel feedback using tubes with transconductance

\[ S_1=S_2=5\cdot10^{-3}\,\frac{a}{v}, \quad C= \]

… \(2\cdot 10^{11}\) F, \(x=1\) with bandwidths of 1, 10, and 20 MHz. In this case the range of variation of the parameter \(\varepsilon\) is taken within \(2\div 40\), which is quite sufficient for obtaining the general regularities. The calculation was carried out for \(m=1, 2, 5, 10\), and \(\infty\). First \(R_1\) and \(R_2\) were determined, then \(K_0\), \(K_{02}\), and \(M_{1\max}\). In addition to the gain in output voltage, \(\gamma\) was determined—the ratio of the gain coefficient of the whole amplifier for the chosen \(m\) to the gain coefficient for \(m=1\).

From the graphs of Fig. 3 it can be seen that the maximum gain in output voltage in the case of large values of \(\varepsilon\) is attained at \(m=\infty\) and amounts to 20–30%. For small \(\varepsilon\) (a wide passband), the maximum gain is already attained at \(m=5\text{–}10\) and amounts to 10–15%. For \(m>5\), \(U_{\text{out max}}\) changes little. The drop in the overall gain coefficient at \(m=\infty\) is insignificant and amounts to 2–5%.

Therefore it may be assumed that the optimum results, from the point of view of the greatest output voltage, are obtained over a wide interval of values of \(m\), from 5 to \(\infty\).

The choice of the value of \(m\) in this interval may be made on the basis of other considerations (tube power-supply conditions, low-frequency correction, etc.).

Determination of the maximum output voltage when amplifying rectangular pulses

If the span of the grid characteristic of the tube in the second stage of the amplifier is equal to \(\Delta U'_{c2\max}\), then at the input of such an amplifier one may apply a rectangular voltage pulse

\[ \Delta U'_{\text{in}}= \frac{\Delta U'_{c2\max}}{K'_{01}(1+2\delta'_1)}, \]

where \(\delta_1\) is the overshoot in the transient response of the first stage.

In this case, at the output of the amplifier we obtain a voltage pulse

\[ \Delta U'_{\text{out}}= \frac{\Delta U'_{c2\max}K'_0}{K'_{01}(1+2\delta'_1)} = \frac{\Delta U'_{c2\max}K'_{02}}{1+2\delta'_1}. \]

For an arbitrary change of the amplifier parameters,

\[ \Delta U''_{\text{out}}= \frac{\Delta U''_{c2\max}K''_{02}}{1+2\delta''_1}. \]

Then the ratio of the output voltages, under the condition that the span of the grid characteristic remains unchanged, is

\[ \lambda= \frac{K''_{02}(1+2\delta'_1)} {K'_{02}(1+2\delta''_1)}. \tag{13} \]

In what follows we shall consider two-stage pulse amplifiers with an overshoot \(\delta=1\%\) in the output-voltage pulse. The amplifier of Fig. 1 has \(\delta=1\%\) and a normalized rise time

\[ t'_y=\frac{t_y}{RC}=1.872 \]

for identical stages and correction parameter \(k=0.342\).

However, the rise time can be improved without reducing the gain coefficient by making the stages unequal [3, 4]. With \(R_1=0.482R\), \(R_2=2.073R\), and correction coefficients \(k_1=1.4\) and \(k_2=0.325\), the amplifier has an overshoot of 1% and a normalized rise time

\[ t'_y=1.58, \]

which is 15.6% less than with identical stages.

Let us determine the gain in output voltage in the case of unequal stages when rectangular pulses are applied to the input. In the case of identical stages, the first stage gives an overshoot \(\delta'_1 = 0.81\%\), whereas in the above-mentioned amplifier with different stages \(\delta'_1 = 46.4\%\). Therefore the use of formula (13) gives \(\lambda = 1.093\). If the load resistances are increased by \(15.6\%\), then the output voltage will increase by a factor of \(1.26\), and the gain by a factor of \(1.33\), in comparison with identical stages at the same rise time. The gain in output voltage in pulse amplification is obtained as somewhat larger than in amplification of a harmonic signal, since the spectrum of a rectangular pulse decreases at high frequencies, which permits a certain increase in \(U_{\mathrm{in}}\).

In practice, one can obtain a gain in output voltage considerably greater than \(26\%\), since the pulses applied to the grid of the first stage cannot have a front duration equal to zero, and the overshoot produced by the first stage will be less than \(46.4\%\). To estimate the gain in output voltage for a finite duration of the pulse front, it is necessary to choose a method of approximating the pulse front. The often-used trapezoidal approximation gives too crude an approximation to the actual character of the pulse rise. Better results are obtained with an exponential approximation and with approximation by a pulse with \(1\%\) overshoot [6]. The latter type of approximation gives the best approximation to reality, but considerably complicates the calculations. The exponential approximation conveys well the main features of the input pulse: the finite duration of the front and the smooth transition to the flat part of the pulse, without substantially reducing the accuracy of the result obtained. Therefore, in all subsequent calculations the exponential approximation is used. Suppose that a unit exponential pulse

\[ f(t) = 1 - e^{-t/T}. \]

is applied to the input of the amplifier under consideration. Then the operational image of the voltage on the grid of the second stage will be

\[ U_{c_2}(p) = \frac{1+k_1 p_1}{(pn+1)(1+p_1+k_1p_1^2)} , \tag{14} \]

where

\[ n=\frac{T}{R_1C}, \qquad p_1=pR_1C. \]

Using the residue theorem, we determine the original corresponding to image (14). It has the form

\[ U_{c_2}(t') = 1 - A e^{-\beta t'} + B e^{-\alpha t'} \sin(\omega t' + \varphi), \tag{15} \]

where \(-\beta\) is the real pole and \(-\alpha \pm j\omega\) are the conjugate complex poles of the image,

\[ t'=\frac{t}{R_1C} \]

is the relative time.

\[ A = \frac{1-\beta k_1}{k_1\left[(\beta-\alpha)^2+\omega^2\right]}, \qquad B = \frac{1}{\omega k_1 n} \sqrt{ \frac{(1-\alpha k_1)^2+(\omega k_1)^2} {(\alpha^2+\omega^2)\left[(\beta-\alpha)^2+\omega^2\right]} }, \]

\[ \varphi = \arctg \frac{\omega k_1}{1-\alpha k_1} - \arctg \frac{\omega}{-\alpha} - \arctg \frac{\omega}{\beta-\alpha}. \]

To determine the instant at which overshoot occurs, \(t'_{\mathrm{в}}\), we take the derivative of (15) and set it equal to zero. Then \(t'_{\mathrm{в}}\) is determined by the iteration method from the equation

\[ t'_{\mathrm{в}} = \frac{\vartheta-\varepsilon}{\omega} - \frac{ \arcsin \left|\frac{[[unclear: coefficient]]}{B} e^{(\alpha-\beta)t'_{\mathrm{в}}}\right| }{\omega}, \]

where

\[ \vartheta=\operatorname{arctg}\frac{\omega}{\alpha}. \]

Next we determine the overshoot

\[ \delta_1 = B e^{-\alpha t'_{\mathrm{в}}}\sin(\omega t'_{\mathrm{в}}-\varepsilon)-A e^{-\beta t'_{\mathrm{в}}}, \]

and the gain in output voltage by formula (13), supplemented by the factor \(1.156\). The results are given in Table 1, and the graph in Fig. 4 has been plotted.

Fig. 4.

For convenience in the practical use of the calculation results, the graph gives the dependence of \(\lambda\) on the quantity

\[ n_1=\frac{t_{\mathrm{u\,imp}}}{t_{\mathrm{u\,amp}}}, \]

which is related to \(n\) by the simple dependence:

\[ n_1= \frac{\dfrac{2.2nR_1C}{1.58}}{0.482R_1C} = 0.671\,n. \]

As can be seen from the results obtained, the maximum output voltage can be increased by a factor of 2.43, compared with an amplifier consisting of identical stages, when the pulse-front duration is greater than the rise time of the amplifier by a factor of 1.88.

Table 1

Let us determine the optimal ratio of the resistances \(m=\dfrac{R_2}{R_1}\) for the amplifier of Fig. 2, at which the greatest undistorted input pulse is obtained. This amplifier has an overshoot \(\delta=1\%\) and a generalized rise time \(t_y=at_y=2.135\) for the value of the parameter \(k=0.305\) [7]. Here

\[ k_1=\frac{R_1+R_2+mR_1+mS_2R_1^2}{4a^2mxC^2R_1^2R_2}, \tag{16} \]

\[ a=\frac{mR_1(1-x)+R_2(1+mx)}{2mxR_1R_2C}. \tag{17} \]

Expressions (16) and (17) form a system of equations from which \(R_1\) and \(R_2\) can be found for given \(k\) and \(a\), at any chosen value of \(m\). We then find the image of the transient response of the first stage, which has the form

\[ H_1(p)=\frac{1-gp}{1+b_1p+b_2p^2}, \tag{18} \]

where

\[ g=\frac{mxR_1R_2}{mR_1+R_2}, \qquad b_1=\frac{R_1C(mxR_1)+mR_1R_2+mxR_2}{R_1(1+m)+R_2+S_2mR_1^2}, \]

\[ b_2=\frac{R_1^2R_2Cxm}{R_1(1+m)+R_2+S_2mR_1^2}. \]

The image (18) corresponds to the original function

\[ H_1(t)=1-Ae^{-\alpha t}\sin(\omega t+\psi), \]

where

\[ A=\frac{\sqrt{(g-\alpha b_2)^2+(\omega b_2)^2}}{\omega b_2}, \qquad \tan\psi=-\frac{\omega b_2}{g-\alpha b_2}, \]

and

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

are the poles of the image (18). Next we find the first overshoot of the transient response

\[ \delta_1= \sqrt{\frac{(g-\alpha b_2)^2+(\omega b_2)^2}{b_2}}\, e^{-\frac{\arctg\frac{\omega}{\alpha}+\arctg\frac{\omega b_2}{g-\alpha b_2}}{\frac{\omega}{\alpha}}} \]

and the gain in the output voltage \(\lambda\) from formula (13). The calculation was carried out for an amplifier with \(t_y=2\cdot10^{-7}\) sec, \(6\cdot10^{-8}\) sec, and \(2\cdot10^{-8}\) sec,

\[ S_1=S_2=5\cdot10^{-3}\,\frac{a}{b}, \qquad C=2\cdot10^{-11}\,\text{F}, \qquad x=1. \]

Analysis of the expressions for \(\delta_1\) and \(K_{02}\) shows that the parameters \(t_y\), \(S_2\), and \(C\), which enter them implicitly, can always be combined into the expression

\[ \xi=\frac{S_2t_y}{C}. \]

Therefore, the results obtained for an amplifier with selected parameters \(S_2\), \(C\), and \(t_y\) remain valid for different values of these parameters corresponding to the same value of \(\xi\). The calculation results are shown in Fig. 5.

The greatest gain in output voltage is obtained for \(m=5\text{--}7\) and reaches \(23\%\) for \(\xi=50\). With a further increase in \(m\), the gain decreases slightly. The output voltage can

they are significantly increased when amplifying pulses with a sufficiently long rise time. Here the indisputable advantages of the amplifier with \(m=\infty\) appear.

The character of the dependence of the output voltage on the rise-time duration of the pulses at \(m=\infty\) remains the same as for the amplifier of Fig. 1 with the optimal transient characteristic of the pair of stages. But in the present case a considerably larger gain in output voltage can be obtained, which depends on the value of the resistance \(R_2\). Calculation of the value of \(\lambda\) in this case is cumbersome and is omitted as having no fundamental significance. The limiting capabilities of the circuit can readily be determined with sufficient accuracy in the following way. At \(m=\infty\) (i.e., \(R_3=\infty\)) the first stage produces a large overshoot, expressed in hundreds of percent. The second stage has the properties of a rheostat with a large time constant \(R_2C\) and smooths the overshoot down to \(1\%\). Thus, an exponential pulse with \(T=R_2C\), after being amplified by the first stage, gives at the grid of the second tube an overshoot of only \(1\%\), which can already be disregarded, and the plateau of its grid characteristic can be used completely. Therefore, the maximum output voltage that can be obtained from the given amplifier at \(m=\infty\) is equal to the product \(\Delta I_{2\max} R_2\), provided that the rise time of the pulse is not less than \(2.2 R_2C\).

Fig. 5. Dependence \(\lambda=f(n)\) and \(y=f(n)\) for a two-stage amplifier with parallel feedback when amplifying rectangular pulses. Curve 1—for \(\xi=5\), curve 2—for \(\xi=15\), curve 3—for \(\xi=50\).

Conclusions

1. The greatest output voltage of a two-stage amplifier with simple inductive correction (when amplifying a frequency band) can be obtained with unequal stages. If the frequency characteristic of the pair of stages is optimal, then the gain in output voltage is \(17\%\). A somewhat higher output voltage can be obtained, but at the cost of a deterioration in the shape of the frequency characteristic.

2. The gain in output voltage when amplifying rectangular pulses by a pair of stages with mutual pulse compensation and with the parameters given in [3] is \(26\%\). The maximum gain can be obtained when amplifying an exponential pulse with \(t_{\text{rise}} \ge 1.88\, t_{\text{settle}}\), and is equal to \(243\%\).

3. The greatest output voltage in a two-stage amplifier with parallel feedback can be obtained at \(m=\infty\) in the case of amplification of a harmonic signal, and at \(m=5\text{–}7\) when amplifying rectangular pulses. The magnitude of the gain depends on the bandwidth (rise time) of the amplifier and amounts to \(15\text{–}30\%\).

4. The value of the parameter \(m\) at which the greatest output voltage is obtained is not critical, and in practice the value of \(m\) can be used in the interval from 5 to \(\infty\).

1. For \(m=\infty\), the dynamic range of the characteristics of the second tube is used completely in the amplification of exponential pulses with \(t_{\text{imp}} \ge R_2 C\).

REFERENCES

1. G. V. Voishvillo, Low-Frequency Amplifiers Using Electron Tubes, Svyazizdat, 1959.

2. G. I. Byalik, Tube Broadband Amplifiers, Gosenergoizdat, 1960.

3. F. A. Muller, “High frequency compensation of RC amplifiers,” PIRE, No. 8, 1954.

4. V. P. Shasherin, “On the principles of designing multistage broadband and pulse amplifiers with correction,” in Proceedings of the Jubilee Session of VNITORIE Dedicated to the 100th Anniversary of the Birth of A. S. Popov, 1960.

5. V. L. Kreitser, Video Amplifiers, Sov. Radio Publishing House, 1952.

6. E. N. Mokhov, Certain Problems in the Theory, Calculation, and High-Frequency Correction of a Cathode-Loaded Stage in the Pulse-Amplification Regime (Candidate’s dissertation), MEIS, 1957.

7. O. B. Lur’e, “Nonstationary processes in broadband amplifiers,” ZhTF, Vol. XIX, No. 8, 1948.