Correction of Nonuniformity in the Frequency Characteristics of a Two-Stage Resonant Amplifier with Negative Feedback
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Correction of the Nonuniformity of the Frequency Characteristics of a Two-Stage Resonant Amplifier with Negative Feedback
I. A. Veselkov
(Presented by the scientific seminar of the Radio Engineering Faculty)
Negative feedback is widely used to improve the parameters of amplifier circuits. When negative feedback is used to stabilize phase relationships, it is desirable to have as great a feedback depth as possible, since as the depth increases the phase stability increases \[3, 4, 5\]. At the same time, however, “humps” in the frequency characteristic of the amplifier grow, which makes additional correction necessary.
A Two-Stage Resonant Amplifier with Negative Feedback
The frequency characteristic of a two-stage resonant amplifier with negative feedback, for equal quality factors of the circuits, may be represented by the expression
\[ M_{oc}=\frac{A}{\sqrt{(A-\delta^{2})^{2}+4\delta^{2}}}, \tag{1} \]
where
\[ M_{oc}=\frac{K_{oc}}{K_{ooc}}; \]
\[ A=1+K_{0}\beta \]
is the feedback depth;
\[ \beta \]
is the feedback coefficient;
\[ K_{0} \]
is the gain coefficient of the two-stage amplifier without negative feedback at the tuning frequency,
\[ K_{ooc} \]
is the same in the presence of negative feedback encompassing both stages,
\[ K_{oc} \]
is the gain coefficient of the amplifier with negative feedback at frequency \(f\),
\[ \delta=Q\left(\frac{f}{f_{0}}-\frac{f_{0}}{f}\right) \]
is the generalized detuning of the circuits.
It is assumed that \(\beta\) is a real quantity.
Figure 1 shows frequency characteristics calculated from formula (1) for \(A=1\), i.e., without negative feedback, and for \(A=10, 20, 50\). It is evident from the curves in the figure that the introduction of negative feedback leads to a deterioration of the amplifier selectivity and to an increase in the nonuniformity of the frequency characteristic, the greater the deeper the negative feedback. At a depth \(A=50\), the nonuniformity reaches \(357\%\).
Fig. 1.
The large nonuniformity of the frequency characteristic is due to the fact that, for frequencies different from the resonant frequency, the circuits create additional phase shifts, making the phase relations for these frequencies close to those characteristic of positive feedback. The nonuniformity of the frequency characteristic can therefore be reduced if an element is introduced into the circuit which, under detuning, creates phase shifts opposite in sign to the shift created by the circuits [1, 2].
The role of such an element, in the case of a change in the amount of generalized detuning due to instability of the input-signal frequency, may be played by a series oscillatory circuit included in the amplifier path.
One or several correcting circuits may be used.
Case of One Correcting Circuit
The frequency characteristic of a two-stage resonant amplifier with negative feedback and one correcting circuit is represented by the expression
\[ M_{oc}= \frac{A\sqrt{1+a^{2}\delta^{2}}} {\sqrt{(A-\delta^{2})^{2}[2+a(A-1)]^{2}}}, \tag{2} \]
where \(\alpha=\dfrac{Q_k}{Q}\); \(Q_k\) is the quality factor of the correcting circuit; \(Q\) is the quality factor of the amplifier circuits. The remaining notation is the same as in formula (1).
Let us determine the value of \(\alpha\) for which the resonance curve will be single-humped (there is one maximum of the resonance curve).
Taking the first derivative of expression (2) with respect to \(\delta\) and equating it to zero, we obtain the equation
\[ \delta\{\alpha^2\delta^4+2\delta^2+[\alpha^2(1-2A)+4\alpha(A-1)-2(A-2)]\}=0. \tag{3} \]
The roots of this equation give the values \(\delta=\delta_k\) at which function (2) has extrema.
One root of the equation will be equal to zero; the others are determined from the expression
\[ \delta_k^2= \frac{-1\pm\sqrt{1-\alpha^2[\alpha^2(1-2A)+4\alpha(A-1)-2(A-2)]}}{\alpha^2}. \tag{4} \]
The characteristic will be single-humped (\(\delta_k\) will be an imaginary quantity) in the case when the second term under the radical in expression (4) is greater than or equal to zero.
For the limiting case (the indicated second term is equal to zero) we obtain
\[ \alpha^2[\alpha^2(1-2A)+4\alpha(A-1)-2(A-2)]=0. \tag{5} \]
Equation (5) has the roots
\[ \alpha_1=\alpha_2=0;\quad \alpha_3=0.812;\quad \alpha_4=1.136. \]
The first two roots correspond to the absence of correction and, in the case under consideration, are of no interest. Of the other two, it is necessary to choose the smaller one, since with a lower quality factor of the correcting circuit its frequency characteristic will be more uniform and will degrade the overall frequency characteristic of the feedback loop to a lesser extent.
Fig. 2.
In Fig. 2 the frequency characteristic of an amplifier with one correcting circuit is shown for \(\alpha=0.812\) and feedback depth \(A=20\).
The characteristic has become single-humped, but very blunt; the selectivity of the amplifier has sharply deteriorated.
The selectivity can be increased by returning to a double-humped resonance curve with a specified nonuniformity of the frequency characteristic. Suppose that a nonuniformity of 30% is permissible. Then, substituting the value of \(\delta\) from expression (4) into expression (2), in which we set \(M_{\text{oc}}=1.3\), one can obtain an equation with respect to \(\alpha\).
In front of the root in expression (4) it is necessary to choose the plus sign. The solution of this equation gives, in principle, the possibility of determining the value of $\alpha$ corresponding to the specified depth $A$ and nonuniformity $M_{oc}$ of the amplifier characteristic. However, the unwieldiness of the resulting expression makes it difficult to solve with respect to $\alpha$.
We used the method of selecting $\alpha$, seeking successive approximation of the right-hand side of equality (2) to the specified value of nonuniformity ($M_{oc}=1.3$). For the case $A=20$ we obtained $\alpha_{opt}=0.118$.
Figure 3 shows the frequency characteristic of the amplifier corresponding to the indicated data ($A=20$, $\alpha=0.118$).
Fig. 3.
Comparison of the frequency characteristics in Fig. 3 and Fig. 1 (for $A=20$) shows that the selectivity of the corrected amplifier increased somewhat and the nonuniformity of the characteristic decreased from 230% to 30%. A further increase in selectivity can be achieved by introducing two correcting circuits.
Case of two correcting circuits
The frequency characteristic of an amplifier with two correcting circuits is determined by the expression
$$ M_{oc}= \frac{A(1+\alpha^{2}\delta^{2})} {\sqrt{\{A-\delta^{2}[1+\alpha^{2}(A-1)]\}^{2} +4\delta^{2}[1+\alpha(A-1)]^{2}}}. \tag{6} $$
The notation is analogous to that used in formula (2).
Equating the first derivative with respect to $\delta$ of expression (6) to zero, we obtain the equation
$$ \delta^{3}\{2\alpha^{2}[1+\alpha(A-1)]^{2} -A\alpha^{2}[1+\alpha^{2}(A-1)] -[1+\alpha^{2}(A-1)]^{2}\} $$
$$ =\delta\{2[1+\alpha(A-1)]^{2} -A[1+\alpha^{2}(A-1)] -\alpha^{2}A^{2}\}. \tag{7} $$
The roots of this equation will be equal to
$$ \delta_{1}=0;\quad \delta_{2,3}=\pm \sqrt{ \frac{ 2[1+\alpha(A-1)]^{2} -A[1+\alpha^{2}(A-1)] -\alpha^{2}A^{2} }{ 2\alpha^{2}[1+\alpha(A-1)]^{2} -A\alpha^{2}[1+\alpha^{2}(A-1)] -[1+\alpha^{2}(A-1)]^{2} } }. \tag{8} $$
The maxima of the curve are symmetrically located with respect to the ordinate $\delta=0$.
Analogously to the case considered of one correcting circuit, with the aid of expressions (6) and (8) one can select values of $\alpha$ that provide a specified nonuniformity of the frequency characteristics for various feedback depths.
Figure 4 gives the calculated graph of the dependence $\alpha_{opt}=f(A)$ for values of the depth $A=5\div50$ with a nonuniformity equal to 30%.
The graph makes it possible, for a selected depth of feedback, to determine \(\alpha\), which ensures a nonuniformity \(M_{oc}=1.3\), and can be used in calculations.
Fig. 5 shows the characteristics of an amplifier with two correcting circuits for depths \(A=10,\ 20,\ 50\).
Fig. 4.
Comparison of the frequency characteristic in Fig. 5 for \(A=20\) with the frequency characteristic in Fig. 3 shows that the introduction of the second correcting circuit leads to a further increase in the selectivity of the amplifier.
Fig. 5.
The results presented show that the use of correcting series circuits makes it possible to significantly reduce the deterioration of the shape of the frequency characteristic caused by the introduction of feedback.
The task of further work consists in creating more effective correcting circuits.
LITERATURE
- Kuzovkov N. T., Theory of Automatic Control, GIOP, 1957.
- Model Z. P., Person S. V., Lebedev-Karmanov A. N., and Pisarevskii A. M., Questions of the theory of deep feedback, IZEST, No. 11, 1939.
- Rizkin A. A., Investigation of selective systems with feedback, Elektrosvyaz’, No. 4, 1941.
- Suslov I. A., Stabilization of the phase of the current in an antenna by means of feedback, Proceedings of SFTI, issue 37, 1959.
- Tsykin G. S., Negative Feedback and Its Application, Svyazizdat, 1940.