On the Critical Mode in a Two-Stage Pulse Amplifier with a Parallel Correction Circuit for Nonidentical Stages

L. I. Sharygina

(Presented at the scientific-technical conference of the radio-engineering faculty, dedicated to the 60th anniversary of the institute)

In designing multistage pulse amplifiers, the question arises of how to distribute the distortions allowed for the entire amplifier among its individual stages. In the case of nonmonotonicity of the transient response of at least one of the stages, this problem has not found a definitive solution in the theory of pulse amplifiers.

In the literature [1, 2] there are indications that, in order to obtain optimum modes in a two-stage system, the stages must be identical. However, this proposition has not been substantiated with sufficient rigor.

In order to establish certain regularities and, in particular, in order to approach the solution of the indicated problem, let us consider the critical mode of a two-stage amplifier with a parallel correction circuit in the case of nonidentical stages.

In operator form, the expression for the normalized transfer coefficient of one stage with a parallel correction circuit has the form

\[ K_{1\,\text{par.}}=\frac{p+k}{1+p+p^2k} =\frac{p-n_1}{(p-p_1)(p-p_2)}, \tag{1} \]

where \(k\) is the correction coefficient, and \(p\) is the normalized Laplace operator. The zeros (roots of the numerator) and poles (roots of the denominator) of this expression are, respectively,

\[ n_1=z_1=-\frac{1}{k}, \qquad p_{1,2}=\frac{-1\pm\sqrt{1-4k}}{2k}. \]

It is evident that the transient response corresponding to expression (1) can have three variants of solutions: a) the poles are real and unequal—the monotonic mode; b) the poles are real and equal—the critical mode; c) the poles are complex conjugate—the nonmonotonic mode.

The total transient response of a two-stage amplifier has twice the number of zeros and poles. If the total mode is cri-

…tic, then one of the stages included in the system must have a correction coefficient greater than the critical value (the first and second poles are complex conjugates)

\[ n_1=\lambda_1=-\frac{1}{\kappa_1}, \qquad p_{1,2}=-\alpha_1 \pm j\omega_1 = -\frac{1}{2\kappa_1}\pm j\frac{\sqrt{4\kappa_1-1}}{2\kappa_1}; \]

the second must be less than the critical value (the third and fourth poles are real)

\[ n_2=\lambda_2=-\frac{1}{\kappa_2}, \qquad p_3=\gamma_3=\frac{-1+\sqrt{1-4\kappa_2}}{2\kappa_2}; \]

\[ p_4=\gamma_4=\frac{-1-\sqrt{1-4\kappa_2}}{2\kappa_2}. \]

In the particular case, the second stage may be uncorrected:

\[ n_2=0, \qquad p_3=\gamma_3=-1, \qquad p_4=\gamma_4=0. \]

The location of the poles and zeros of a two-stage amplifier with correction coefficients \(\kappa_1>0.25,\ \kappa_2<0.25\) is shown in Fig. 1. The corresponding normalized transient response has the form:

Fig. 1.

\[ U_{\mathrm{en}}(t)= 1-Ae^{-\alpha_1 t} \left[ \frac{c}{L_{01}}\sin(\omega_1 t+\xi_1) -\frac{a}{\gamma_3}e^{-(\gamma_3-\alpha_1)t} -\frac{b}{\gamma_4}e^{-(\gamma_4-\alpha_1)t} \right]. \]

In this expression

\[ \xi_1=\mu_1+\eta_{11}+\eta_{12}-\delta_{13}-\delta_{14}, \]

\[ A=\frac{L_{01}^{2}\gamma_3\gamma_4}{\lambda_1\lambda_2|\gamma_3-\gamma_4|}, \qquad b=\frac{|\lambda_1-\gamma_4|\,|\lambda_2-\gamma_4|}{\gamma_{14}^{2}}, \]

\[ a=\frac{|\lambda_1-\gamma_3|\,|\lambda_2-\gamma_3|}{\gamma_{13}^{2}}, \qquad c=\frac{|\gamma_3-\gamma_4|\csc\mu_1\,\lambda_1\lambda_2}{L_{01}\gamma_{13}\gamma_{14}}, \]

the remaining notation is clear from the figure.

If the correction coefficient of the second stage is equal to zero, the normalized transient response takes the form

\[ U'_{\Sigma H}(t)=1-De^{-\alpha_1 t} \left[ \sin(\omega_1 t+\xi_1)-L_{01}\frac{d}{\gamma_3}e^{-(\gamma_3-\alpha_1)t} \right]. \]

In this expression

\[ D=\frac{\lambda_{11}\gamma_3}{\lambda_1\gamma_{13}}, \qquad d=\frac{|\lambda_1-\gamma_3|L_{01}^{2}}{\lambda_1\gamma_{13}^{2}}, \qquad \xi_1=\mu_1-q_{11}-\varepsilon_1 . \]

The arrangement of the poles and zeros for this case is shown in Fig. 2. In expressions (2) and (3), \(t\) denotes the dimensionless time

\[ t=\frac{t'}{RC}, \]

where \(t'\) is the time in seconds, \(R\) is the load resistance of the stage, and \(C\) is the load capacitance of the stage.

Mullidzhon and M. M. Aizinov [3, 4] showed that the transient response can be monotonic if at least one real pole is located between the first pair of complex-conjugate poles. Coincidence of the abscissas of the first pair of complex-conjugate poles and the first real pole \((\gamma_3=\alpha_1)\) corresponds to the critical mode, provided that there are no other poles or that their influence can be neglected. Otherwise the indicated condition is necessary but not sufficient.

In the present paper, the condition for the critical mode in the form \(\gamma_3=\alpha_1\) was used as sufficient. The errors arising in this case are investigated below.

From this condition the correction coefficients of one of the stages were determined for various values of the correction coefficient of the second stage.

For each combination of stages providing the critical mode, the total transient responses were calculated and the settling time was determined.

Fig. 2.

The results of the calculations are summarized in the table.

Figure 3 shows the dependence of the settling time of the total transient response on the correction coefficient of one of the stages, when the correction coefficient of the second stage provides the total critical mode.

As can be seen from the graph, the case of two identical stages, each of which has a correction coefficient equal to 0.25, is not ...

best from the point of view of the settling time of the resultant transient response. The minimum settling time corresponds to the combination of stages \(\kappa_{1\mathrm{opt.}} = 0.3869\), \(\kappa_{2\mathrm{opt.}} = 0.1750\). The gain in settling time, in comparison with the case of identical stages, is then about \(8\%\).

Table

To clarify the reasons for this phenomenon, let us turn to expression (2) for the resultant transient response of a two-stage amplifier and simplify it somewhat, taking into account that the resultant mode must be critical:

Fig. 3.

\[ U_{2H}(t)=1-Ae^{-\alpha_1 t} \left[ \frac{c}{L_{01}}\sin(\omega_1 t+\xi_1) -\frac{a}{\gamma_3} -\frac{b}{\gamma_4}e^{-(\gamma_4-\alpha_1)t} \right]. \]

Calculations show that, as \(\kappa_2\) is increased from zero to \(\kappa_{2\mathrm{opt.}} = 0.1750\), the influence of the term \(\dfrac{b}{\gamma_4}e^{-(\gamma_4-\alpha_1)t}\) may be neglected.

The transient response has the same form as in the presence of only one real pole. The condition for the critical mode in the form \(\gamma_3=\alpha_1\) is not only necessary but also sufficient. In this case \(\alpha_1\) increases from 1 to 1.2922, the exponential \(e^{-\alpha_1 t}\) decays ever more rapidly, the steepness of the transient response increases, and the settling time decreases. Beginning with \(\kappa_{2\mathrm{opt.}}=0.1750\), the transient response is also affected by the second real pole \(\gamma_4\); moreover, the closer \(\kappa_2\) is to 0.2500, the more significant this influence. When \(\kappa_2\) changes from 0.1750 to 0.2500, \(\gamma_4\) changes from 4.4221 to 2.0000; the exponential \(e^{-(\gamma_4-\alpha_1)t}\) decays ever more slowly, the steepness of the transient response decreases, and the rise time increases.

Since for \(\kappa_2>\kappa_{2\mathrm{opt.}}\) the influence of the pole \(\gamma_4\) cannot be neglected, doubt arises as to the possibility of using the condition \(\gamma_3=\alpha_1\) as the criterion for the critical mode. Taking into account the influence of the pole \(\gamma_4\) will make it necessary to choose the correction coefficient of the first stage somewhat larger in order to ensure the resultant critical mode; in this case the settling time of the resultant transient response will be somewhat less than that indicated in the table for all points except \(\kappa_1=\kappa_2=0.25\). However, this reduction is so insigni-

so slight that it cannot affect the character of the regularities that have been found.

As is seen from the graph in Fig. 3, when the correction coefficient of the second stage is varied from 0.1750 to 0.0500, the settling time of the total transient response gradually increases; for \(k_2 < 0.0500\) this increase slows, and then for \(k_2 < 0.0300\) the settling time begins to decrease. However, this decrease in the settling time is only apparent. It is explained by the fact that, for \(k_2 < 0.0500\), \(k_1 > 0.4736\), the total transient response is not strictly monotonic (Fig. 4), but has a small negative undershoot, reaching \(0.18\%\) at \(k_2 = 0.0000\), \(k_1 = 0.5000\).

Fig. 4.

The appearance of nonmonotonicity is explained by the large discrepancy between the settling times of the stages included in the system. To eliminate the nonmonotonicity of the total transient response at small \(k_2\), one should abandon the critical mode and somewhat reduce \(k_1\); in this case the settling time of the transient response will increase (the dashed curve in Fig. 3). However, the distortion of the total transient response at \(k_1 = 0.5000\), \(k_2 = 0.0000\) is very small, and in this case one may rely on the criterion of the critical mode in the form \(a_1 = \gamma_3\).

On the basis of the above, the following conclusions may be drawn.

1. The minimum rise time of the total transient response of a two-stage amplifier corresponding to the critical mode is obtained not with identical stages, but with different stages:
   \(k_1 = 0.3869,\ k_2 = 0.1750\).

2. In the case when a gain or loss in rise time of the order of \(8\%\) is not of substantial importance in a two-stage circuit, it is not necessary to achieve the critical mode in each stage; one of the stages may be made rheostatic, and the required total response may be obtained by adjusting only one stage. If both stages are corrected, then the required resultant response can be obtained by adjusting only one of the stages.

REFERENCES

1. I. G. Mamonkin. Pulse Amplifiers, Gosenergoizdat, 1958.

2. O. B. Lur’e. Video-Frequency Amplifiers, Sovetskoe Radio, 1955.

3. J. H. Mulligan, PIRE, No. 5, 1949, “The effect of pole and zero locations on the transient response of linear dynamic systems.”

4. M. M. Aizinov. On the analysis and synthesis of linear radio-engineering circuits by transient-mode parameters. Scientific Notes, Issue 9, Leningrad Higher Naval Engineering School named after Admiral Makarov, 1953.