Some Problems in the Theory of Phase Stabilization in Amplifier Stages by Means of Feedback

I. A. Veselkov

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

For purposes of phase stabilization, negative feedback is widely used \[2, 3, 4\]. The phase instability of an amplifier with negative feedback, as is known \[5\], can be estimated on the basis of the expression

\[ \tan \varphi_n = \frac{\tan \varphi}{1 + \dfrac{K\beta}{\cos \varphi}} . \tag{1} \]

For small angles \(\varphi\) (\(\cos \varphi \approx 1\)), the phase instability is reduced by a factor of \(1 + K\beta\).

The phase-stabilizing properties of complex negative feedback are determined both by the value of the loop gain \(K\beta\) and by the magnitude of the phase angle of the feedback circuit \(\varphi_\beta\).

It is of interest to estimate the phase-stabilizing properties of negative feedback for various values of \(\varphi_\beta\) and \(K\beta\).

In this article such an estimate is made for the following two cases:

1) the phase instability is the result of the instability of several parameters of the amplifier;

2) the phase instability is caused only by a change with time in one of the reactive elements of the load.

I. The Case of Instability of Several Parameters

The modulus of the gain coefficient of an amplifier with frequency-dependent complex feedback and its phase characteristic are represented by the expression \[1, 6\]

\[ K_{\mathrm{fb}} = \frac{K}{\sqrt{1 - 2K\beta \cos(\varphi + \varphi_\beta) + (K\beta)^2}}, \tag{2} \]

\[ \varphi_{\mathrm{fb}} = \operatorname{arctg} \frac{\sin \varphi + K\beta \sin \varphi_\beta} {\cos \varphi - K\beta \cos \varphi_\beta}. \tag{3} \]

Here \(\varphi\) is the phase angle between the amplifier output voltage \(U_2\) and the voltage at its input \(U\).

Analysis of expression (3) shows that there is, in principle, a possibility of phase stabilization for various values of \(\varphi_\beta\) and \(K\beta\). In each case there will be its own “zero” phase level, relative to which stabilization is carried out.

Indeed, from (3), for \(\varphi_\beta = 180^\circ\) we obtain expression (1). If \(\varphi_\beta = 90^\circ\) or \(270^\circ\), then

\[ \varphi_p = \operatorname{arc\,tg}\frac{\sin\varphi \pm K\beta}{\cos\varphi}. \]

In this case the “zero” phase level is determined by the value \(K\beta\) (for \(\varphi = 0\), \(\varphi_{p0} = \operatorname{arc\,tg}\pm K\beta\)).

With complex feedback, \(\varphi_{k0}\) will be determined by the values \(K\beta\) and \(\varphi_\beta\).

In what follows, we shall retain the indices “\(n\),” “\(p\),” and “\(k\)” to denote, respectively, negative, reactive, and complex feedbacks.

It is obvious that the choice of one or another type of feedback for the purpose of phase stabilization is determined by three basic requirements:

1) the degree of phase stabilization,
2) the amplifier gain,
3) the stability of the amplifier with feedback.

Let us consider these three types of feedback from the standpoint of the first two requirements. An analysis of stability when using one or another type of feedback should appropriately be carried out for a specific circuit.

It follows from (3) that, for \(\varphi = 0\),

\[ \varphi_{k0} = \operatorname{arc\,tg}\frac{K\beta\cdot \sin\varphi_\beta}{1-K\beta\cdot \cos\varphi_\beta}. \tag{4} \]

Investigation of the expression \(\varphi_k-\varphi_{k0}\) for extrema with respect to the variable \(K\beta\) shows that this difference tends to zero as \(K\beta \to \infty\).

Thus, for any of the three types of feedback, high phase stability can be ensured only at large feedback depths. However, for certain depths the degree of phase stabilization when using one or another type of feedback does not remain the same.

Let us introduce the coefficients of relative phase stabilization

\[ a_1=\frac{|\varphi_n|-|\varphi_{n0}|}{|\varphi_p|-|\varphi_{p0}|} \]

and

\[ a_2=\frac{|\varphi_n|-|\varphi_{n0}|}{|\varphi_k|-|\varphi_{k0}|}, \]

which characterize the effectiveness, respectively, of reactive and complex feedbacks in comparison with negative feedback.

Figure 1 gives calculated plots \(a_2=f_1(\varphi)\) (for \(\varphi_\beta=80^\circ\) and \(\varphi_\beta=100^\circ\)) and \(a_1=f_2(\varphi)\) for values \(K\beta=5\) and 10.

It follows from the plots that, for the selected \(\varphi\) and \(K\beta\), the greatest phase stability is ensured at \(\varphi_\beta=80^\circ\).

For all three values of \(\varphi_\beta\) (\(90^\circ\), \(80^\circ\), \(100^\circ\)), as \(K\beta\) increases, the effectiveness of complex and reactive feedbacks in comparison with negative feedback increases.

When complex and reactive feedbacks are used for the purpose of phase stabilization, an interesting phenomenon may be observed: for certain values of \(K\beta\) and \(\varphi_\beta\) there exists an angle \(\varphi\) for which \(\varphi_{\mathrm{fb}}=\varphi_{\mathrm{fb}0}\). Indeed, on the basis of (3) and (4), the condition for the absence of phase distortions when \(\varphi\) varies is the equality

\[ \frac{\sin \varphi + K\beta \cdot \sin \varphi_\beta} {\cos \varphi - K\beta \cdot \cos \varphi_\beta} = \frac{K\beta \cdot \sin \varphi_\beta} {1 - K\beta \cdot \cos \varphi_\beta}. \tag{5} \]

Solving (5) with respect to \(\varphi\), we obtain

\[ \varphi = - \arcsin \frac{2K\beta \cdot \sin \varphi_\beta \left(1 - K\beta \cdot \cos \varphi_\beta\right)} {1 - 2K\beta \cdot \cos \varphi_\beta + (K\beta)^2}. \tag{6} \]

Expression (6) makes it possible to determine the angle \(\varphi\) at which, for the selected values of \(K\beta\) and \(\varphi_\beta\), the best stabilization is achieved. In Fig. 1 (curves 5, 6), the cases \(\varphi_\beta=80^\circ\) and \(K\beta=5,10\), respectively, correspond to \(\varphi_1=3^\circ 4'\) and \(\varphi_2=8^\circ 33'\).

Fig. 1.
\(1\) — \(\varphi_\beta=100^\circ,\ K\beta=5;\)

\(2\) — \(\varphi_\beta=100^\circ,\ K\beta=10;\quad 3\) — \(\varphi_\beta=80^\circ,\ K\beta=10;\)

\(4\) — \(\varphi_\beta=90^\circ,\ K\beta=5;\quad 5\) — \(\varphi_\beta=80^\circ,\ K\beta=5;\)

\(6\) — \(\varphi_\beta=90^\circ,\ K\beta=10.\)

Analysis of expression (2) shows that a larger gain coefficient at the tuning frequency corresponds to complex feedback when \(\varphi_\beta<90^\circ\) and \(\varphi_\beta>270^\circ\).

In comparison with the other two types of feedback, a circuit with degenerative feedback has a smaller gain coefficient.

The principal disadvantage of the method of phase stabilization by means of complex feedback is the dependence of the stability of the “zero” phase level \((\varphi_{\mathrm{fb}0})\) on the stability of the loop gain.

However, such a dependence is characteristic only of rapid changes in \(K\beta\), which are mainly the result of instability of the supply voltages and can be eliminated.

Slow changes in \(K_\beta\), whose cause may be the “aging” of amplifier elements, do not lead to a significant instability of the zero level.

Indeed, as follows from the graphs in Fig. 1, for the case \(\varphi_\beta = 80^\circ\), a slow change in the magnitude \(K_\beta\) from 10 to 5 causes a displacement of curve 3 relative to the abscissa axis toward the position of curve 5. In this case the values of \(a_2\) remain sufficiently high.

II. Case of a Load Unstable in Time

We shall carry out the analysis using the equivalent circuit of Fig. 2. We assume that the element unstable in time is the capacitance \(C\). Such a case occurs when the amplifier has an unstable load of capacitive character.

If the voltage is taken from the capacitance, then the voltage transfer coefficient may be represented by the expression

\[ K = \frac{K x^2}{1 + j\alpha}, \tag{7} \]

where

\[ K = \frac{1}{j\omega C_0 R}, \qquad x^2 = \frac{\omega_k^2}{\omega_c^2}, \]

\[ \alpha = \frac{\omega_c a}{R}(1 - x^2), \tag{8} \]

Fig. 2.

\(C_0\) is the capacitance corresponding to tuning the circuit to the signal frequency;
\(\omega_c\) is the angular frequency of the signal;
\(\omega_k\) is the natural angular frequency of the circuit.

The voltage transfer coefficient of the system with feedback may be written in the form

\[ \dot K_{cb} = \frac{Kx^2} {(1 - K\beta x^2 \cdot \cos \varphi_\beta) + j(\alpha - K\beta x^2 \cdot \sin \varphi_\beta)} . \tag{9} \]

Hence

\[ \tg \varphi_{cb} = - \frac{\alpha - K\beta x^2 \cdot \sin \varphi_\beta} {1 - K\beta x^2 \cdot \cos \varphi_\beta}. \tag{10} \]

For \(\varphi_\beta = 90^\circ\) or \(270^\circ\),

\[ \tg \varphi_\beta = -(\alpha \pm K\beta x^2). \tag{11} \]

It follows from expression (11) that, in the case of reactive feedback, for purposes of phase stabilization one may use the effect of a change in the amplitude \(U_\beta\), which is the result of detuning of the circuit. In this case, from the point of view of phase-stabilizing properties, feedback at \(\varphi_\beta = 270^\circ\) will be optimal (the plus sign before the second term in (11)).

Indeed, when the circuit is detuned, an increase in the value of \(\alpha\) corresponds to a decrease in the natural frequency of the circuit \(f_k\), but at the same time the modulus of the second term of expression (11) decreases. Thus, the change in detuning of the circuit may be compensated by a change in the feedback factor. The optimal condition for phase stabilization is also satisfied in the case of complex feedback for certain values of \(K^\beta\) and \(\varphi_\beta\).

For \(z=0\), expression (10) is transformed into (4). As before, the condition for the absence of phase distortions under detuning of the circuit can be written in the form of the equality

\[ \frac{-z+K\beta x^{2}\sin\varphi_{\beta}} {1-K\beta x^{2}\cos\varphi_{\beta}} = \frac{K\beta\sin\varphi_{\beta}} {1-K\beta\cos\varphi_{\beta}} . \tag{12} \]

Solving (12) with respect to \(K\beta\), we obtain

\[ (K\beta)_{opt}= \frac{Q}{Q\cos\varphi_{\beta}-\sin\varphi_{\beta}} . \tag{13} \]

Here

\[ Q=\frac{\omega_c L}{R}. \]

Formula (13) makes it possible to determine the value of the loop gain at which the phase instability under detuning of the circuit will be minimal.

It follows from (9) and (13) that \(\varphi_{\beta}=0\) corresponds to positive feedback. For \(\varphi_{\beta}=180^\circ\), \((K\beta)_{opt}=-1\).

Fig. 3.

Thus, for negative feedback the condition of optimal stabilization, determined by formula (13), is absent, since the value \((K\beta)_{opt}'=-1\) means a transition to positive feedback.

Indeed, for \(a=0\) and \(\varphi_{\beta}=180^\circ\), expression (9) takes the form

\[ K_{\mathrm{svo}}=\frac{K}{1+K\beta}. \tag{9'} \]

Substitution of \(K\beta=-1\) into (9′) leads to \(K_{\mathrm{svo}}\to\infty\).

The dependence \((K\beta)_{opt}=f(\varphi_{\beta})\) for \(Q=10\) is illustrated by the graph in Fig. 3. The values of \((K\beta)_{opt}\) for larger values of \(Q\) differ only slightly from those shown in the graph.

On the basis of the graph it follows that, for values of $\varphi_\beta$ determined by the formula $\tan \varphi_\beta = Q$, $(K\beta)_{opt} \to \infty$. Consequently, these values of $\varphi_\beta$ cannot be used. For certain values of $\varphi_\beta$, as is evident from the graph, the optimum stabilization conditions can be ensured at small values of $(K\beta)_{opt}$. The choice of $\varphi_\beta$ will obviously also be determined by the stability condition of the system.

Thus, in the case of a time variation of one of the reactive elements of the load, high phase stability in a circuit with reactive feedback (if the quality factor of the circuit is not large), as well as in a circuit with complex feedback, can be ensured at small optimum feedback depths.

References

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

2. Molel Z. I. and Person S. V. Compensation of distortions in radio-transmitting devices by means of negative feedback. IEST, No. 7, 1938.

3. Rizkin A. A. Amplifiers with negative feedback. Elektrosvyaz’, No. 2, 1938.

4. Suslov I. A. Stabilization of the phase of the current in an antenna by means of negative feedback. Proceedings of SFTI, issue 37, 1959.

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

6. Tsykin G. S. Electronic Amplifiers, Svyazizdat, Moscow, 1960.