Transient Response of a Two-Stage Amplifier with Parallel Feedback and Simple Inductive Correction

R. A. Smirnov

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

Introduction

To characterize the quality of a pulse amplifier, the pulse figure of merit is used, defined as the ratio of the gain coefficient \(K_0\) at mid-range frequencies to the rise time of the pulse front \(t_y\). Such an index fully characterizes a single-stage amplifier, since both \(K_0\) and \(t_y\) are directly proportional to the value of the load resistance of the stage \(R\); as a result of a change in the load, the figure of merit \(d = \dfrac{K_0}{t_y}\) remains constant. In a multistage amplifier, as the load resistances increase, the gain coefficient increases faster than the settling time, and \(d\) depends on the load. Therefore, the pulse figure of merit of a multistage amplifier is not an unambiguous characteristic of its pulse properties, and still less of the quality of the circuit, since this parameter reflects the data of the tubes used and changes depending on the specified \(t_y\).

It is expedient to introduce a new parameter—the relative pulse figure of merit \(D_0\), for a specified overshoot \(\delta\), as the ratio of the pulse figure of merit of the given \(n\)-stage amplifier, having overshoot \(\delta\), to the pulse figure of merit of an \(n\)-stage rheostatic amplifier with identical load resistances in all stages, having the same rise time, parasitic capacitances, and tube transconductance as the amplifier under consideration.

\[ D_0=\frac{d}{d_p}. \]

Let us apply this parameter to two-stage circuits. A rheostatic amplifier with identical parasitic capacitances in both stages has a rise time \(t_y = 3.35\,R_1 C_1\), whence

\[ K_0 = K_1 \cdot K_2 = S_1 S_2 R_1^2 = \frac{S_1 S_2 t_y^2}{3.35^2 C_1^2}. \tag{1} \]

If the capacitances are different and \(\dfrac{C_2}{C_1}=m\), then, finding the rise time from the transient characteristics, one can establish that

\[ K_0 \simeq \frac{S_1 S_2 t_y^2}{3.35^2 m C_1^2}. \tag{2} \]

For \(1<m<1.5\), equality (2) is valid to an accuracy of 5%, and for \(m=1\) it becomes (1).

The pulse merit factor of the rheostatic amplifier is

\[ d_p=\frac{K_0}{t_y}=\frac{S_1 S_2 t_y}{11.2\,m C_1^2}. \]

A two-stage amplifier with inductive correction and different time constants in the two stages has a rise time \(t_y=yR_1C_1\), where \(y\) is the generalized rise time.

Let the ratio of the time constants of the stages be:

\[ \frac{\tau_2}{\tau_1}=\frac{C_2R_2}{C_1R_1}=x;\qquad \text{then } R_2=\frac{x}{m}R_1;\qquad R_1=\frac{t_y}{yC_1};\qquad R_2=\frac{xt_y}{ymC_1}. \]

Let us determine the amplifier gain at medium frequencies,

\[ K_0=K_1\cdot K_2=S_1S_2R_1R_2= \frac{S_1S_2xt_y^2}{y^2mC_1^2}, \]

its pulse merit factor,

\[ d=\frac{K_0}{t_y}=\frac{S_1S_2xt_y}{y^2mC_1^2}, \]

and the relative pulse merit factor

\[ D_0=\frac{d}{d_p}=x\left(\frac{3.35}{y}\right)^2 . \tag{3} \]

For example, for a rheostatic amplifier with identical stages \(D_0=1\); with unequal time constants \(D_0<1\). For two-stage amplifiers with identical stages, in the case of simple inductive correction \(D_0=2.14\) at \(\delta=0\); in the case of series inductive correction at the critical overshoot \(\delta_{kp}=4.4\%\), \(D_0=4.84\), etc.

A sufficiently complete picture of the pulse properties of a circuit can be obtained by plotting the curves \(D_0=f(\delta)\), which for some circuits are given in Fig. 1.

Fig. 1. Relative pulse merit factor of two-stage amplifiers:
1—with parallel inductive correction in both identical stages; 2—with parallel voltage feedback in the second stage; 3—with parallel voltage feedback in the second stage and simple inductive correction in the first stage.

Using the dependence \(D_0=f(\delta)\), one can conveniently compare various circuits with one another and choose the circuit that provides the required gain with the smallest number of correcting parameters. For this, from the specified overshoot \(\delta\), using the graph \(D_0=f(\delta)\), \(D_0\) is determined and the gain coefficient is found from the formula

\[ K_0=D_0K_{0p}=\frac{D_0S_1S_2}{m}\left(\frac{t_y}{3.35\,C_1}\right)^2 . \tag{4} \]

If the gain is insufficient, it is necessary to switch to another circuit that provides a larger value of \(D_0\).

Let us consider the amplifier circuit of Fig. 2 with parallel voltage feedback, supplemented by V. V. Anikeev [1] with simple inductive correction in the first stage. To analyze the quality of the circuit and to calculate it, we shall use the parameter \(D_0\).

Fig. 2.

Investigation of the Transient Characteristics of the Amplifier

The equivalent circuits of the amplifier under consideration are shown in Fig. 3.

Solving the system of nodal-potential equations for the circuit of Fig. 3б, one can determine the gain factor of the entire amplifier:

\[ K=\frac{I_3 Z_3}{U_1} = \frac{S_1 Z_1 Z_3 (S_2 Z_2 - 1)} {Z_1 + Z_2 + Z_3 + S_2 Z_1 Z_3}. \]

Fig. 3.

The gain factor of the first stage will then be equal to

\[ K_1=\frac{I_1 Z_1}{U_1} = S_1 Z_1 \frac{Z_2 + Z_3}{Z_1 + Z_2 + Z_3 + S_2 Z_1 Z_3}, \]

and for the second stage we obtain

\[ K_2=\frac{K}{K_1} = Z_3 \frac{S_2 Z_2 - 1}{Z_2 + Z_3}. \]

At medium frequencies,

\[ K_0=\frac{S_1 R_1(S_2 R_2 - 1)}{1 + S_2 R_1}, \qquad K_{01}=\frac{S_1 R_1}{1 + S_2 R_1}. \]

\[ K_{02}=S_2R_2-1. \]

The operational images of the transient characteristics of the individual stages are obtained by replacing the resistances \(Z_n\) by their operational images

\[ Z_1(p)=\frac{R_1+pZ}{1+pR_1C_1+p^2ZC_1}; \]

\[ Z_2(p)=R_2;\qquad Z_3(p)=\frac{1}{pC_2}. \]

Then

\[ H_1(p)=\frac{K_1(p)}{K_{01}}= \frac{(1+\kappa p)(1+xp)} {1+b_1p+b_2p^2+b_3p^3}, \tag{5} \]

\[ H_2(p)=\frac{K_2(p)}{K_{02}}=\frac{1}{1+xp} \tag{6} \]

and the image of the transient characteristic of the entire amplifier is

\[ H(p)=\frac{1+\kappa p}{1+b_1p+b_2p^2+b_3p^3}, \tag{7} \]

where

\[ b_1=\gamma(m+1+x-\kappa)+\kappa;\qquad b_2=\gamma(m\kappa+\kappa+x);\qquad b_3=\kappa\gamma x, \]

\[ \kappa=\frac{Z}{R_1^2C_1},\qquad \gamma=\frac{1}{1+S_2R_1},\qquad m=\frac{C_2}{C_1},\qquad x=\frac{\tau_2}{\tau_1};\qquad p=j\omega R_1C_1. \]

The relative impulse merit of minimum-phase circuits with inductive correction is determined mainly by the degree of the characteristic equation and increases as the order of this equation is raised. The situation is somewhat different in circuits using negative feedback.

For example, a circuit with parallel voltage feedback (the circuit of Fig. 2 for \(Z=0\)) is described by a characteristic equation of the second degree, but, owing to the feedback, it makes it possible to obtain complex roots of the characteristic equation with only one reactance in each stage. This leads to a reduction in the rise time. In a noncascade treatment this circuit has a pole of one stage coincident with a zero of the other. It is completely analogous to the circuit of anti-noise correction by an inductance with a shunting resistance, if the input circuit is regarded as a preceding rheostat stage with a large time constant. Both the circuit with parallel voltage feedback and the circuit of anti-noise correction have identical relative impulse merits and the same dependence of the gain on the upper cutoff frequency,

\[ K_0=\frac{S_1S_2\sqrt{1-M_i^2}}{C_1C_2\omega_{\mathrm{в}}^2M_i}, \]

although the first circuit has two reactive elements and the second three.

Similarly, the circuit of Fig. 2 may also be regarded as a “degenerate” fourth-order circuit with a coincident pole and zero (formulas 5 and 6). Therefore its impulse properties should be close to those of a two-stage amplifier with simple inductive correction, having a fourth-degree characteristic equation.

By means of the residue theorem we determine the transient response corresponding to the transform (7):

\[ h(t)=\frac{1}{2\pi j}\int\limits_{c-j\infty}^{c+j\infty} \frac{(1+kp)e^{pt}\,dp}{1-b_1p+b_2p^2+b_3p^3} = 1-Ae^{-\beta t'}-Be^{-\alpha t'}\sin(\omega t'+\varphi), \tag{8} \]

where

\[ A=\frac{1-\beta k}{b_3\left[(\beta-\alpha)^2+\omega^2\right]}, \qquad t'=\frac{t}{R_1C_1}, \]

\[ B=\frac{1}{\omega b_3} \sqrt{ \frac{(1-\alpha k)^2+(\omega k)^2} {(\alpha^2+\omega^2)\left[(\beta-\alpha)^2+\omega^2\right]} }, \]

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

To investigate the properties of the circuit for various overshoots, we use O. B. Lur’e’s method of equal moduli [2]. For this purpose we normalize the transform of the transient response (8). Setting

\[ p=\frac{p_1}{\sqrt[3]{b_3}}, \]

we obtain the expression

\[ H(p_1)=\frac{1+gp_1}{1+d_1p_1+d_2p_1^2+p_1^3}, \tag{9} \]

in which

\[ d_1=\frac{b_1}{\sqrt[3]{b_3}}, \qquad d_2=\frac{b_2}{\sqrt[3]{b_3^2}}, \qquad g=\frac{k}{\sqrt[3]{b_3}}. \]

Let us denote the poles of the normalized transform by

\[ p_1=-\beta' \quad\text{and}\quad p_{2,3}=-\alpha'\pm j\omega'. \]

Proceeding from the condition of equality of the moduli, we have

\[ d_1d_2=1+2\alpha', \tag{10} \]

\[ \alpha'^2+\omega'^2=\beta'^2=1. \]

We assign a series of values of the oscillatory parameter

\[ G=\frac{\omega'}{\alpha'} \quad (0;\;0.8;\;1;\;1.2\ \text{etc.}), \]

by means of which we determine the values of the complex roots

\[ \alpha'=\frac{1}{\sqrt{1+G^2}}, \qquad \omega'=G\alpha'. \]

Then expression (10) gives a system of two equations with four unknowns \(\kappa\), \(\gamma\), \(x\), and \(m\). The numerical value of \(m\) is determined by the amplifier design and in the subsequent calculations is taken equal to 1. By specifying the quantity \(x\), one can solve the system (10) and find the parameters \(\kappa\) and \(\gamma\) corresponding to the given oscillatory parameter \(G\). The values \(\alpha\), \(\omega\), and \(\beta\) needed for calculating the transient response (8) are obtained by multiplying \(\alpha'\), \(\omega'\), and \(\beta'\) by the normalizing factor

\[ \sqrt[3]{\frac{1}{b_3}} = \frac{1}{\sqrt[3]{\kappa_1 x}}. \]

Table 1

For \(C=0\) the image of the transient response has a multiple pole \(p_1=-1\), and formula (8) becomes unsuitable. In this case we shall use the formula

\[ h(t_1)=1-\left(1+t_1+\frac{1-g}{2}\,t_1^2\right)e^{-t_1}, \]

where

\[ t_1=\frac{t}{\sqrt[3]{krx}}, \qquad g=\frac{k}{\sqrt[3]{krx}} . \]

From the obtained transient responses, the values of the generalized rise time \(y=\dfrac{t_y}{R_1C_1}\) and overshoot \(\delta\) were calculated. The results of the calculations are summarized in Table 1.

Let us determine the relative impulse merit factor of the given circuit. Since \(S_2R_1\gg 1\), the direct passage of the signal from the first stage of the amplifier to its output may be neglected and we may write

\[ K_0 \simeq \frac{S_1R_1S_2R_2}{1+S_2R_1} =\frac{S_1S_2xr\,t_y^2}{y^2mC_1^2}. \]

Then the relative impulse merit factor of the amplifier is obtained as

\[ D_0=\frac{K_0}{K_{op}}=xr\left(\frac{3.35}{y}\right)^2 . \tag{11} \]

In the absence of feedback, \(\gamma=1\), and formula (11) reduces to (3). From the data of Table 1 the graphs in Fig. 4 and Fig. 5 were plotted. It may be

Fig. 4.

noted that when \(x\) is varied within the limits \(x=5\div 15\), \(D_0\) changes only by 3–4%. Therefore, to a first approximation, the relative impulse merit factor of the circuit at a given overshoot may be regarded as independent of \(x\). From the standpoint of frequency-phase characteristics this is analogous to the assumption that, in the formula expressing the dependence of the gain on the passband

Fig. 5.

Graph labels: ordinate \(K\); abscissa [[unclear: axis label]], with tick marks \(0, 2, 4, 6, 8, 10\). Curves are labeled \(T=15\), \(T=10\), and \(T=5\).

\[ K_0=\frac{xS_1S_2\sqrt{1-M_b^2}}{C_1C_2\omega_b^2M_b} \]

where \(x\) is a constant. For the amplifier under consideration \(x\simeq 2\) [1].

The calculation shows that the rise time

\[ t_y=yR_1C_1=\frac{(1-\gamma)y}{\gamma}\frac{C_1}{S_2} \]

is the same (to an accuracy of \(5\%\)) for all values of oscillation for the chosen value of \(x\), and is directly proportional to the quantity \(x\).

In calculating the amplifier of Fig. 2, we first assign the value of \(x\) within the limits

\[ x=(0.65\div 0.70)\, t_y\,\frac{S_2}{C_1}, \]

and, from this choice, determine \(D_0\) and \(\kappa\) from the graphs of Figs. 4 and 5. Then the gain of the circuit is found from formula (4). From the expression

\[ K_0=\frac{S_1S_2\dfrac{x}{m}R_1^2}{1-S_2R_1} \]

we determine

\[ R_1=\frac{K_0+\sqrt{K_0^2+4K_0\dfrac{S_1}{S_2}\dfrac{x}{m}}}{2S_1\dfrac{x}{m}}. \]

Then we find \(R_2=\dfrac{x}{m}R_1\) and \(L=\kappa R_1^2C_2\).

Example of calculation. Given \(t_y=6\cdot 10^{-8}\) sec, \(\delta=5\%\), \(C_1=C_2=20\cdot 10^{-12}\,\mathrm{F}\); \(S_1=S_2=5\cdot 10^{-3}\,\dfrac{a}{v}\). We choose \(x=10\). From the graph of Fig. 4 we determine \(D_0=4\), and from the graph of Fig. 5, \(\kappa=0.347\).

The gain of the circuit is

\[ K_0=\frac{4\cdot 5\cdot 10^{-3}\cdot 5\cdot 10^{-3}}{1} \left(\frac{6\cdot 10^{-8}}{3.35\cdot 20\cdot 10^{-12}}\right)^2=80. \]

\[ R_1=\frac{80+\sqrt{80^2+4\cdot 80\cdot 10}}{2\cdot 5\cdot 10^{-3}\cdot 10} =1.78\cdot 10^3\ \Omega,\qquad R_2=17.8\cdot 10^3\ \Omega. \]

\[ L=0.347\cdot (1.78\cdot 10^3)^2\cdot 20\cdot 10^{-12} =22\cdot 10^{-6}\ \mathrm{H}. \]

A large load resistance in the second stage makes it possible to increase the output voltage considerably if pulses with a long front duration are amplified.

The minimum front duration of a pulse amplified without narrowing of the dynamic range is determined specifically for each circuit, which is a subject for separate consideration. In addition, the presence of deep negative feedback makes it possible to reduce the time constants of the transient and decoupling circuits, and even to combine the cathodes and screen grids of both tubes, bypassing them to ground with capacitors of small capacitance, as was noted in [1].

Conclusions

1. The circuit considered in Fig. 2 makes it possible to obtain a relative pulse quality factor practically the same as that of a two-stage circuit with parallel inductive correction in both stages. With

In this case, the amplifier has only one inductance and smaller capacitor dimensions.

1. The proposed parameter—the relative pulse quality factor—unambiguously characterizes the quality of a multistage pulse amplifier and can be used for engineering circuit calculations.

REFERENCES

1. Anikeev V. V. High-frequency correction in a two-stage amplifier with feedback, Proceedings of the Research Institute of the State Committee for Aviation Technology, issue XII, 1958.

2. Durye O. B. Video-frequency Amplifiers, Sovetskoe Radio Publishing House, 1955.