OBTAINING DEEP NEGATIVE FEEDBACK IN THE RESONANT AMPLIFIER OF A LOW-POWER TRANSMITTER

P. A. SUSLOV

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

Obtaining deep negative feedback at high frequency in the resonant stages of a low-power transmitter is associated with the need to apply special measures to ensure stability with sufficient margins. A known difficulty is also posed by introducing the feedback voltage. If this voltage is applied to the input by means of simple circuits similar to those used in aperiodic amplifiers, then adjustment of the feedback depth leads to a change in the phase relationships in the amplifier, which complicates its tuning.

The indicated difficulty can be eliminated by using a subtracting amplifier to introduce the feedback voltage [1]. The voltage at the output of such an amplifier is equal to

\[ K_1(u_1-u_3), \tag{1} \]

where \(u_1\) is the voltage being amplified, \(u_3\) is the feedback voltage, and \(K_1\) is the gain of the subtracting stage. Let \(u_2\), \(\beta u_2\), and \(K_2\) be the gain of the following stages; then

\[ u_2=(u_1-u_3)K_1\cdot K_2=\frac{u_3}{\beta}. \tag{2} \]

From (1) and (2) we obtain

\[ K_c=\frac{u_2}{u_1}=\frac{K_1K_2}{1+\beta K_1K_2}=\frac{K}{1+\beta K}, \]

i.e., the usual expression for the gain with feedback.

As the subtracting amplifier it is more convenient to use a circuit with cathode coupling (Fig. 1). The voltages \(u_1\) and \(u_3\), in the case of negative feedback, have the same phase at the transmitter tuning frequency. With deep feedback they are comparatively large and differ little from one another. The circuit is therefore similar to a cathode follower with respect to input resistance and the slope of the dynamic characteristic.

To correct the phase characteristics of the amplifier, one may use frequency-dependent current feedback (low-\(Q\) circuits in the cathode). Such correction is convenient in that it does not require additional correcting stages.

Fig. 1.

To reduce the number of additional correcting stages, it is also desirable to apply correction in the subtracting amplifier. The correcting circuit should be connected between the cathode of the tube to which the feedback voltage is applied and the common cathode coupling resistance. The equivalent circuit of the subtracting amplifier will then have the form shown in Fig. 2.

Fig. 2.

The node-potential equations for this circuit are written in the form:

\[ \dot u_{k1}\left(S+\frac{1}{R_k}+\frac{1}{Z_k}\right) -\dot u_{k2}\cdot \frac{1}{Z_k} = S\dot u_1, \]

\[ -\dot u_{k1}\cdot \frac{1}{Z_k} +\dot u_{k2}\left(S+\frac{1}{Z_k}\right) = S\dot u_2, \]

\[ -S\dot u_{k2}+\dot u_2\cdot \frac{1}{Z_a}=-S\dot u_3. \]

Hence

\[ \dot u_2=\frac{SZ_a}{Z_k}\cdot \frac{ \begin{vmatrix} S+\dfrac{1}{R_k} & \dot u_1\\ S & \dot u_3 \end{vmatrix} }{ S^2+\dfrac{S}{R_k}+\dfrac{2S}{Z_k}+\dfrac{1}{Z_kR_k} }. \]

where \(Z_a\) and \(Z_k\) are the impedances of the anode and cathode circuits.

The expression obtained shows that the inclusion of the correcting circuit has no effect on the operation of the amplifier with cathode coupling as a subtracting device. It further follows from it that the feedback coefficient is increased by the subtracting amplifier

\[ \frac{S+\dfrac{1}{R_k}}{S} \]

(fig. 1) times.

The gain of the cascade with cathode correction is equal to

\[ \dot K=\frac{SZ_a}{1-SZ_k} \simeq \frac{Z_a}{Z_k} = \frac{Z_{a0}}{Z_{k0}}\cdot \frac{1+j\delta_k}{1+j\delta_a}. \]

Here \(Z_{a0}\) and \(Z_{k0}\) are the resonant impedances, \(\delta_a\) and \(\delta_k\) are the generalized detunings,

\[ \delta_a=Q_a\left(x-\frac{1}{x}\right), \qquad \delta_k=Q_k\left(x-\frac{1}{x}\right), \]

\(Q_a\) and \(Q_k\) are the \(Q\)-factors of the anode and cathode circuits, \(x=f/f_0\) is the relative frequency, and \(f_0\) is the tuning frequency of the circuits.

Let the characteristic impedances of the circuits be identical and let \(Q_k=\alpha Q_a=\alpha Q\). Then

\[ \dot K=\frac{1}{\alpha}\cdot \frac{1+j\alpha\delta}{1+j\delta} \tag{3} \]

and the phase shift is \(\varphi=\operatorname{arctg}\alpha\delta-\operatorname{arctg}\delta\).

If correction is carried out in two cascades, then \(\varphi=2\operatorname{arctg}\alpha\delta-2\operatorname{arctg}\delta\). We find the maximum phase shift by setting

\[ \frac{d\varphi}{d\delta}=0. \]

We obtain

\[ \delta=\frac{1}{\sqrt{\alpha}}. \]

Let a stability margin in phase be specified,

\[ \Delta\varphi=\frac{\pi}{n}. \]

Then

\[ \gamma_{\max}=2\arctg\sqrt{\alpha}-2\arctg\frac{1}{\sqrt{\alpha}}-\pi+\frac{\pi}{n}. \]

The angles \(\arctg\sqrt{\alpha}\) and \(\arctg \dfrac{1}{\sqrt{\alpha}}\) complement one another to \(\dfrac{\pi}{2}\). Replacing \(\arctg \dfrac{1}{\sqrt{\alpha}}\) by \(\dfrac{\pi}{2}-\arctg\sqrt{\alpha}\), we obtain

\[ \alpha \doteq \left(\tg\frac{\pi}{4n}\right)^2 . \]

Usually the correcting circuit has a considerably lower quality factor than the circuit being corrected, and \(\alpha \ll 1\). Therefore \(\alpha \simeq \dfrac{\pi^2}{16n^2}\).

Let us compare cathode correction with the results obtained by applying the method of successive switching-out of stages [2]. The block diagram of an amplifier constructed by this method is shown in Fig. 3. The sign “\(+\)” denotes summing stages. The amplification coefficient after the first summation is equal to

Fig. 3.

\[ \dot K_1+\dot K_{11} = \frac{K_{10}}{1+j\delta} + \frac{K_{110}}{1+j\varepsilon} \simeq \frac{K_0(1+j\gamma\delta)}{1+j\delta}. \tag{4} \]

Here \(\varepsilon\) is the generalized detuning of the aperiodic stage \(\dot K_{11}\),

\[ K_0=\frac{K_{10}}{K_{10}+K_{110}}, \qquad \gamma=\frac{K_{110}}{K_{10}+K_{110}} . \]

Comparing (4) and (3), we find that the method of “switching out” stages is equivalent to cathode correction, although its circuit implementation is much more complicated.

To correct the characteristic of the output stage, one can use two stages with aperiodic loads and symmetrically detuned correcting circuits. In this case it is possible to obtain a smaller attenuation at the amplification frequency and a droop of the characteristic at the frequencies of the humps of the coupled circuits of the output stage. To reduce positive feedback, incomplete connection of the cathode circuits is used with such a connection coefficient \(p\) that the depth of positive feedback at the transmitter frequency is of the order of 10. At \(p=1\) a large loss of gain is obtained; at small \(p\) the range of detunings for which correction is sufficiently effective is reduced.

The cathode circuits of the correcting stages are loaded by resistances connected in parallel with the output taps. These resistances are chosen so that the \(Q\)-factors of the correcting circuits are approximately equal to the average \(Q\)-factors of the circuits of the output stage, taking into account radiation losses and the shunting action of the tube on the intermediate circuit.

The generalized detuning for the correcting circuit with the lower natural frequency may be written in the form

\[ Q\left(\frac{\omega}{\omega_0-\Delta}-\frac{\omega_0-\Delta}{\omega}\right) = Q\left|x\left(1+\frac{\Delta}{\omega_0}\right) -\frac{1}{x}\left(1-\frac{\Delta}{\omega_0}\right)\right| =\delta-\delta_0, \]

where

\[ \delta_0=\frac{2\Delta}{\omega_0}. \]

When \(x\) varies within the limits from \(0.8\) to \(1.2\) (\(\delta\) in this case, for \(Q=100\), varies from \(-37\) to \(37\)), the quantity \(x+\dfrac{1}{x}\) changes by only \(1.5\%\). Therefore \(\delta_0\) may be regarded as constant. Similarly, for the circuit with the higher natural frequency we obtain

\[ Q\left(\frac{\omega}{\omega_0+\Delta}-\frac{\omega_0+\Delta}{\omega}\right)=\delta-\delta_0. \]

Here we neglect the changes in \(Q\) as \(x\) changes.

The equation of the frequency-phase characteristic of a two-stage amplifier with correcting symmetrically detuned circuits will be

\[ K_{\mathrm{cor}}K_{\mathrm{ob.cor}} \,[1+j(\delta-\delta_0)]\,[1+j(\delta+\delta_0)]. \]

The loop gain of an amplifier with two resonant stages corrected by loaded circuits and with two stages for correcting the output stage is equal to

\[ K\beta = \frac{K_0(1+j a\delta)^2}{(1+j\delta)^2} \cdot \frac{\left[(1+j\delta_1)^2+\delta_{10}^{\,2}\right]\beta} {\left[(1+j\delta_1)^2+\eta^2\right] \left[(1+j\delta)^2+\delta_{20}^{\,2}\right]} \]

\[ = \frac{K_0\beta(1+j a\delta)} {(1+j\delta)^2\left[(1+j\delta_2)^2+\delta_{20}^{\,2}\right]}, \]

where \(\eta\) is the coupling parameter and \(\delta_1\) is the mean generalized detuning of the circuits of the output stage,

\[ \delta_2= \frac{\delta_1}{1+p^2 Z_{\mathrm{r.cor.}}S}. \]

Thus, as a result of the correction, the characteristics of the output stage are replaced by the characteristics of circuits with \(Q\)-factors \(1+Sp^2Z_{\mathrm{r.cor.}}\) times worse, while cathode correction in the first stages creates the necessary phase-stability margin. To compensate the \(90^\circ\) phase shift produced by the output stage, it is necessary to introduce a phase inverter into the circuit.

REFERENCES

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

2. Polonnikov D. E. Broadband decision-making (operational) amplifiers, Automation and Remote Control, No. 12, 1960.