ON PHASE STABILIZATION IN A SINGLE-STAGE RESONANT AMPLIFIER BY MEANS OF FEEDBACK

I. A. VESELIKOV

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

The use of negative feedback (degenerative feedback) makes it possible to reduce the phase shifts introduced by an amplifier stage in accordance with the formula [4]

\[ \operatorname{tg}\varphi_{oc} = \frac{\operatorname{tg}\varphi}{1+\dfrac{K_0\beta}{\cos\varphi}}, \tag{1} \]

where \(\varphi_{oc}\) is the phase-shift angle between the input and output voltages (not counting the angle \(\pi\) introduced by the tube), \(\varphi\) is the same without degenerative feedback, and \(K_0\beta\) is the feedback factor.

For small angles \(\varphi\) (\(\cos\varphi \simeq 1\)), the phase shifts introduced by the amplifier are reduced by a factor of \(1+K_0\beta\).

Thus, in order to obtain a significant reduction of phase shifts, large depths of degenerative feedback are required. However, in this case the voltage transfer coefficient decreases and the stability of the amplifier deteriorates [1, 3].

The present article considers the possibility of reducing phase shifts when wattless feedback is used, that is, feedback in which the feedback voltage is shifted relative to the input voltage by an angle

\[ \frac{\pi}{2} \]

[2].

We shall carry out a comparative evaluation of output-voltage phase stabilization by analyzing the expressions for the phase characteristics. The voltage transfer coefficients of single-stage amplifiers with degenerative feedback and with wattless feedback can be written in the form

\[ K_{oc}=\frac{K_0}{1+j\delta+K_0\beta}, \tag{2} \]

\[ K_{Б}=\frac{K_0}{1+j\delta+jK_0\beta}. \tag{3} \]

Whence

\[ \varphi_{\mathrm{oc}}=\operatorname{arc\,tg}\left(-\frac{\delta}{1+K_0\beta}\right), \tag{4} \]

\[ \varphi_{B}=\operatorname{arc\,tg}-(\delta+K_0\xi). \tag{5} \]

Here \(K_0\) is the gain at the tuning frequency,

\[ \xi=Q\left(\frac{f}{f_0}-\frac{f_0}{f}\right) \]

is the generalized detuning of the circuit,

\[ \varphi_{B} \]

is the phase-shift angle between the input and output voltages of an amplifier with wattless feedback.

From expression (5) it follows that, for \(\delta=0\), the magnitude of the initial phase-shift angle \(\varphi_{B0}\) is determined by the value of the feedback factor.

Comparison of expressions (4) and (5) shows that, for small \(\delta\), the phase stability in the case of wattless feedback will be greater than with negative feedback.

Fig. 1.

Fig. 2.

Figures 1 and 2 give, respectively, the calculated phase characteristics of amplifiers with negative feedback and with wattless feedback for \(K_0\beta=10\) and for \(\delta\) varying from 0 to 1. A comparison of the graphs in Figs. 1 and 2 shows that the phase shift caused by detuning of the oscillatory system, in the case of wattless feedback with \(K_0\beta=10\) and \(\delta=1\), is 8 times smaller than in the case of negative feedback.

Let us introduce the coefficient of relative phase stabilization

\[ a=\frac{\varphi_{\mathrm{oc}}-\varphi_0}{\varphi_{B0}-\varphi_{B}}, \]

which characterizes the effectiveness of wattless feedback in comparison with negative feedback. We shall denote the coefficient of relative stabilization for \(\delta=1\) by \(a_1\). Analyzing expressions (4) and (5), it is not difficult to note that, for a certain value of \(K_0\beta\), \(a>a_1\) in the interval \(0<\delta<1\) and \(a<a_1\) for \(\delta>1\). With an increase in the depth of feedback, the relative coefficient of phase stabilization increases.

The moduli of expressions (2) and (3) show that the voltage transmission coefficients of amplifiers with wattless feedback and with negative feedback at the tuning frequency are practically equal for equal values of \(K_0\beta\).

The stability of amplifiers with wattless feedback is a subject for special consideration.

Conclusions

1. The use of positive feedback makes it possible to obtain greater phase stability in a single-stage resonant amplifier than when negative feedback is used.

2. For the same degree of phase stability, an amplifier with positive feedback has a higher gain than an amplifier with negative feedback.

References

1. Bode, H., Network Analysis and Feedback Amplifier Design, GIIL, 1948.

2. Braude, G. V., “On Oscillatory Systems with Positive Feedback,” ZhTF, vol. 1, no. 1, 1931.

3. Kolosov, A. A., Resonant Systems and Resonant Amplifiers, Svyazizdat, Moscow, 1949.

4. Tsykin, G. S., Negative Feedback and Its Application, Svyazizdat, Moscow, 1940.