Stabilization of the Current Phase in a Transmitter Antenna by Means of Reactive Feedback at High Frequency

I. A. Veselkov

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

Introduction

Certain types of phase radionavigation systems, as well as systems used to investigate the change in phase of long waves during propagation, impose stringent requirements on the stability of the current phase in the transmitter antenna \[1, 9\].

The main cause of phase instability of the current in the antenna of a long-wave transmitter is the change in the antenna parameters caused by the swaying of the antenna in the wind and by changes in the condition of the soil.

To increase phase stability, the method of strong coupling between the intermediate and antenna circuits is applied \[5\]; the antenna is fed through a feeder tuned in the appropriate manner \[7\]; and negative feedback is also used \[6\].

The limiting stability of the current phase when the first method is used is restricted by the choice of the output-stage operating mode that is optimal with respect to power.

Large feedback depths, which provide high phase stability in a system with negative feedback, create difficulties in ensuring transmitter stability and also lead to a considerable reduction in the gain of the path encompassed by the feedback.

The use of reactive feedback makes it possible to obtain high phase stability at small feedback depths.

Effect of Reactive Feedback on the Stability of the Current Phase in the Antenna

We shall consider the case in which the output stage of the transmitter operates in an overvoltage mode.

The equivalent circuit of the output stage with inductive coupling between the anode and antenna circuits may be represented in the form of Fig. 1.

Here \(R_1, L_1, C_1\) are the parameters of the anode circuit, \(R_2, L_2\) of the antenna circuit; \(R_{\mathrm{in}}, X_{\mathrm{in}}\) are the input active and reactive resistances of the antenna.

Usually a long-wave antenna has a large capacitive load at the end. The reactive input resistance of such an antenna is capacitive \([3, 4]\).

Fig. 1.

Then the equivalent circuit assumes the form of Fig. 2. Here \(R = R_2 + R_{\mathrm{in}}\). In what follows, we shall neglect the change in \(R\) under detunings of the antenna circuit (valid for small detunings).

Fig. 2.

The expression for the current in the antenna circuit, on the basis of the equivalent circuit of Fig. 2, may be represented in the form

\[ I_2 = I_{20}\, \frac{1+\eta_0^2}{(1+j a_1)(1+j a_A)+\eta^2}. \tag{1} \]

where

\[ I_{20}=\frac{I_{a1}M}{C_1R_1R(1+\eta_0^2)}, \]

\[ a_1=Q_1\left(\frac{\omega}{\omega_{01}}-\frac{\omega_{01}}{\omega}\right); \]

\[ a_A=\frac{\omega L}{R}\left(1-\frac{\omega_2^2}{\omega^2}\right), \tag{1′} \]

\[ \eta=\frac{\omega M}{\sqrt{R_1R}}, \]

\(\omega_2\) is the angular frequency corresponding to the detuning of the antenna circuit,

\(I_{a1}\) is the first harmonic of the anode current.

The voltage transfer coefficient of the final stage is represented by the expression

\[ \dot K_1=-\frac{jK_{01}(1+\eta_0^2)x^2}{(1+j\alpha_1)(1+j\alpha_A)+\eta^2}. \]

Here

\[ K_{01}=\frac{I_{20}}{U_g\omega_0 C_{A0}}, \]

\[ x^2=\frac{\omega_2^2}{\omega_{02}^2}, \]

\(C_{A0}\) is the capacitance of the antenna when tuned to resonance;
\(U_g\) is the excitation voltage corresponding to the overvoltage mode.

It is advisable to apply the feedback voltage to the stages preceding the final stage [2].

Let us consider the case in which two stages are covered by reactive feedback.

The voltage transfer coefficient in this case may be written in the form

\[ \dot K_p=-\frac{jK_0(1+\eta_0^2)x^2}{(1+j\alpha_2)\left[(1+j\alpha_1)(1+j\alpha_A)+\eta^2 \pm jK_0\beta(1+\eta_0^2)x^2\right]}, \]

where

\[ K_0=K_{01}\cdot K_{02}, \]

\(K_{02}\) is the voltage transfer coefficient of the driver stage at the tuning frequency,

\(\beta\) is the feedback coefficient.

If the driver-stage circuit and the anode circuit of the final stages retain their tuning, and only the antenna circuit is detuned, then

\[ \dot K_p=-\frac{jK_0x^2}{1+j\frac{\alpha_A}{1+\eta_0^2}\pm jK_0\beta x^2}. \]

The phase instability caused by detuning of the antenna circuit is determined from the expression

\[ \operatorname{tg}\varphi_p=-\frac{1+\eta_0^2}{\alpha_A\pm K_0\beta(1+\eta_0^2)x^2}. \tag{2} \]

Analysis of expression (2) shows that, from the standpoint of phase-stabilizing properties, the optimum case is the one corresponding to the positive sign in the denominator of (2). Indeed, when the antenna circuit is detuned, an increase in the value of \(\alpha_A\) corresponds, as follows from \((1')\), to a decrease in the resonant frequency of the circuit \(f_2\), but at the same time the modulus of the second term in the denominator of (2) decreases.

Consequently, the change in the detuning of the circuit is compensated by a change in the feedback factor.

It follows from expression (2) that in this case, for \(\alpha_A=0\),

\[ \operatorname{tg}\varphi_{p0}=-\frac{1}{K_0\beta}. \tag{3} \]

i.e., the magnitude of the initial phase-shift angle \(\varphi_{ro}\) is determined by the value \(K_0\beta\).

The condition for the absence of phase distortions of the output signal when the antenna circuit is detuned is the constancy of the argument of expression (2) as \(\omega_2\) changes.

On the basis of (2) and (3), this condition is written in the form

\[ \frac{1+\eta_0^2}{\alpha_A+K_0\beta(1+\eta_0^2)x^2} = \frac{1}{K_0\beta}. \tag{4} \]

Solving (4) with respect to \(K_0\beta\), we obtain

\[ (K_0\beta)_{opt} = \frac{\alpha_A}{(1+\eta_0^2)(1-x^2)}. \tag{5} \]

For given values of \(Q_2\) and \(\eta_0\), expression (5) makes it possible to calculate the value of \(K_0\beta\) for which the phase oscillations of the output signal, as the natural frequency of the antenna circuit changes, will be minimal. After substituting the value of \(\alpha_A\) into (5), we obtain

\[ (K_0\beta)_{opt} = \frac{Q_2}{1+\eta_0^2}. \tag{6} \]

Formula (6) is the design formula. In high-power transmitters the values of \(Q_2\) are usually of the same order as \(\eta_0^2\). Therefore the values of \((K_0\beta)_{opt}\) are small, which is very important from the standpoint of ensuring stability.

Fig. 3.

Expression (2), after substituting (6) into it, can be represented in the form

\[ \operatorname{tg}\varphi_{ro} = - \frac{1+\eta_0^2}{\alpha_A+(K_0\beta)_{opt}(1+\eta_0^2)x^2}. \tag{7} \]

Let us estimate the effectiveness of reactive feedback with respect to increasing phase stability in comparison with degenerative feedback.

The phase instability in the case of using degenerative feedback for stabilization purposes is determined from expression [6]

\[ \operatorname{tg}\varphi_{oc} = \frac{\alpha_A}{(1+\eta_0^2)(1+K_0\beta x^2)}. \tag{8} \]

In this case the stability increases with increasing feedback depth, but the phase angles for symmetrical detunings of the antenna circuit’s natural frequency relative to the resonant frequency become unequal.

In Fig. 3 a plot of \(\varphi_{oc}=f(\alpha_A)\) is given in the detuning range \(\alpha_A=-10 \div +10\), calculated from formula (8) for the case \(Q_2=50\) and \(1+\eta_0^2=25\).

The selected parameters, on the basis of (6), correspond to \((K_0\beta)_{\mathrm{opt}}=2\). It is evident from the plot that, in the case of feedback, the phase instability reaches \(\Delta\varphi > 15^\circ\). It is easy to verify, using expression (7), that the signal phase when reactive feedback is used remains constant within the selected detunings.

The phase-stability analysis has been carried out only for the case in which two stages are enclosed by reactive feedback. However, it remains valid for any number of stages, provided that all the preliminary amplification stages are tuned to resonance.

Experimental Results

The block diagram of the experimental setup is shown in Fig. 4.

Here: \(K\) is a two-stage amplifier, the output stage of which operates in an overvoltage mode and has as its load inductively coupled circuits;

Diagram labels: \(U_1\), \(U_\beta\), \(K\), \(U_2\), DA, Ph, CF.

Fig. 4.

CF is a cathode follower, intended to avoid shunting the second circuit of the final stage by the feedback circuit;

Ph is a bridge phase shifter.

To obtain a grounded output, the bridge phase shifter is connected to the grids of the differential amplifier DA. The feedback voltage was taken from the inductance coil of the second circuit and fed to the cathode of the first amplifier stage.

The phase shifts under detuning of the second circuit were measured by the method described in [8] at a frequency of \(62.5\ \mathrm{kHz}\) (division factor of the initial frequency \(n=20\)). The parameters of the second circuit were: \(C_2=3000\ \mathrm{pF}\), \(L_2=2.15\ \mathrm{mH}\), \(1+\eta_0^2=30\), \(Q_2=30\).

The phase drift when the circuit capacitance \(C_2\) was increased or decreased by \(300\ \mathrm{pF}\) in the circuit without feedback was \(\Delta\varphi \simeq 6^\circ 20'\). In the case of reactive feedback, no phase drift was observed.

Conclusions

1. Reactive feedback makes it possible to obtain high stability of the current phase in the transmitter antenna for small feedback depths equal to the optimum values.

1. The use of small depths facilitates stable operation of the transmitter and also does not lead to a significant reduction in the gain.

References

1. Astaf’ev G. P., Shebashevich V. S., Yurkov Yu. A. Radio-Navigation Devices and Systems, Sov. Radio, Moscow, 1958.

2. Model’ Z. P., Person Z. V. Compensation of distortions in transmitting devices by the use of negative feedback, IEST, No. 7, 1938.

3. Nadenenko S. I. Antennas, Svyazizdat, Moscow, 1959.

4. Neiman M. S. Transmitting Antennas, Energoizdat, 1934.

5. Pestryakov V. B. Radio-Navigation Goniometric Systems, Gosenergoizdat, Moscow–Leningrad, 1955.

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

7. Shtridman L., Soskin G. A. Stabilization of the phases and amplitudes of currents in antennas, IEST, No. 11, 1940.

8. Shul’zhenko K. M., Silov E. N., Sun’yan V. Ya. On the measurement of phase shifts and phase modulation by means of interference patterns. Reports of the Tomsk City Scientific and Technical Conference on the topic “Automation of Production Processes,” 1959.

9. Propagation of Long and Superlong Waves. Collection of articles edited by V. B. Pestryakov, IL, Moscow, 1960.