STABILIZATION OF THE CURRENT PHASE IN THE ANTENNA OF A LONG-WAVE TRANSMITTER BY MEANS OF COMPLEX HIGH-FREQUENCY FEEDBACK

I. A. VESELKOV

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

It is known [1] that, when negative feedback is used, the stability of the current phase in the antenna increases as the depth of feedback increases. However, at large feedback depths difficulties arise in ensuring transmitter stability, which considerably limits the possibilities of this stabilization method. The use of complex feedback makes it possible to ensure high phase stability with small feedback depths.

1. The influence of complex feedback on the stability of the current phase in the antenna

Let us consider the most common case of inductive coupling of the anode circuit with the antenna circuit. In this case the equivalent circuit of the output stage in under-voltage and critical operating modes may be represented as in Fig. 1.

Fig. 1.

Here \(R_1, L_1, C_1\) are the parameters of the anode circuit;
\(L_{cv}\) and \(L_n\) are, respectively, the inductances of the coupling coil and of the antenna tuning coil.
\(C_A\) is the total capacitance of the antenna.
\(R = R_2 + R_{\mathrm{in}A}\),

\(R_2\) — active resistance of the inductance coils,
\(R_{\mathrm{in}A}\) — active input resistance of the antenna.

If the antenna circuit and the final stage of the transmitter are enclosed in complex feedback, taking the feedback voltage, for example, from the inductive branch of the antenna circuit, then the voltage gain coefficient can be represented by the expression:

\[ K_{oc}= \frac{jK(1+\eta_0^2)x^2} {1+\eta_0^2-\alpha_1\alpha_A+K\beta(1+\eta_0^2)x^2\sin\varphi_\beta +j\left[\alpha_1+\alpha_A-K\beta(1+\eta_0^2)x^2\cos\varphi_\beta\right]}, \tag{1} \]

where \(\alpha_1\) and \(\alpha_A\) are the generalized detunings of the anode and antenna circuits,

\[ x=\frac{f_A}{f_c}, \]

\(f_A\) — resonant frequency of the antenna circuit,
\(f_c\) — signal frequency,

\[ \eta_0=\frac{\omega_c M}{\sqrt{R_1R}}, \]

\[ \beta=\frac{u_\beta}{u_L}, \]

\(\varphi_\beta\) — phase angle of the feedback circuit.

In deriving formula (1), the change in \(R\) was neglected. Such a neglect is valid because the causes responsible for the instability of the antenna parameters (swaying of the antenna by the wind, changes in the state of the soil) do not lead to rapid and significant changes in the total active resistance of the antenna circuit. Slow changes in \(R\) cause only a shift of the “zero” phase level, which is not accompanied by significant phase distortions.

The phase instability caused by detuning of the circuits is determined from the expression:

\[ \tg\varphi = -\frac{\alpha_1+\alpha_A-K\beta(1+\eta_0^2)x^2\cos\varphi_\beta} {1-\alpha_1\alpha_A+\eta_0^2+K\beta(1+\eta_0^2)x^2\sin\varphi_\beta}. \]

In the case of detuning of only the antenna circuit,

\[ \tg\varphi = -\frac{\alpha_A-K\beta(1+\eta_0^2)x^2\cos\varphi_\beta} {(1+\eta_0^2)(1+K\beta x^2\sin\varphi_\beta)}. \tag{2} \]

For \(\alpha_A=0\),

\[ \tg\varphi_0= \frac{K\beta\cos\varphi_\beta} {1+K\beta\sin\varphi_\beta}, \tag{3} \]

The value \(\varphi_0\) in this case determines the “zero” phase level relative to which stabilization is carried out.

Equating (2) to (3) and solving the equation with respect to \(K\beta\), we obtain:

\[ (K\beta)_{opt} = \frac{-Q_A} {Q_A\sin\varphi_\beta+(1+\eta_0^2)\cos\varphi_\beta}, \tag{4} \]

where

\[ Q_A=\frac{\omega_c L}{R}, \qquad L=L_{cv}+L_n . \]

For known values of \(Q_A\) and \(\eta_0\), expression (4) makes it possible to determine the value of \(K^\beta\) at which the oscillations of the current phase in the antenna, caused by detuning of the antenna circuit, will be minimal.

The minus sign in formula (4) is due to the choice of initial conditions in deriving expression (1), corresponding to positive feedback.

Analysis of (4) shows that such a \(\varphi_\beta\) can always be chosen for which, for any values of \(Q_A\), the quantity \((K^\beta)_{\text{opt}}\) is small, which is very important from the standpoint of ensuring the stability of the system.

Substitution of the value \((K^\beta)_{\text{opt}}\), corresponding to \(\varphi_\beta=90^\circ\), into expression (1) shows that in the case of negative feedback there are no optimal conditions corresponding to this stabilization principle. In the case of reactive feedback \((\varphi_\beta=180^\circ)\),

\[ (K^\beta)_{\text{opt}}=\frac{Q_A}{1+\eta_0^2}. \]

It follows from this that if the damping of the antenna is sufficiently small, it is expedient to use complex feedback.

As in the case of reactive feedback, so also with complex feedback, there arises the need to stabilize the “zero” level. From expression (4) it follows that the stability of \(\varphi_0\) is determined by the stability of \(K^\beta\) and \(\varphi_\beta\). Ensuring constancy of \(\varphi_\beta\) is usually not difficult.

There are also effective methods for stabilizing the voltage gain coefficients of both high-power [5] and low-power stages [6].

Thus, stabilization of the “zero” phase level can be accomplished without significantly complicating the transmitter circuit.

Formula (4) was obtained for the case of operation of the final stage in underloaded or critical modes.

However, it will also be valid for the overvoltage mode, since the frequency structure of the voltage transfer coefficients is the same in both cases.

2. Stability of the antenna–transmitter system with complex feedback.

We shall investigate stability using the Nyquist criterion. It is known [2,4] that electronic amplifiers with feedback are in most cases designed to be absolutely stable. Therefore, for the system under consideration we shall impose the requirement of absolute stability.

The principle of phase stabilization by means of complex feedback implies the presence, in the feedback circuit, of a phase shifter by means of which the required phase shift can be obtained between the feedback voltage and the input voltage.

In what follows we shall assume that the magnitude and phase of the voltage transfer coefficient of the phase shifter remain constant up to the frequencies at which self-excitation of the system occurs.

In analyzing stability we shall also assume that, when the signal frequency changes, the efficiency of the antenna remains constant. This

This assumption is valid because self-excitation of the system, even when feedback embraces only the final stage, occurs at frequencies sufficiently close to resonance. In such a small frequency range the efficiency of the antenna remains practically constant.

a) The antenna circuit and the final stage of the transmitter are included in the feedback loop

In this case the loop gain, on the basis of the equivalent circuit in Fig. 1, is represented by the expression:

\[ \dot K_\beta = \frac{ K_\beta (1+\eta_0^2)\dfrac{1}{x} \left[ A\cos\varphi_\beta - B\sin\varphi_\beta + j\left(A\sin\varphi_\beta + B\cos\varphi_\beta\right) \right] }{ A^2+B^2 }. \]

Here

\[ A=\alpha_A\left(\frac{l+1}{l}\right), \]

\[ B=1+\eta_0^2-\frac{\alpha_A^2}{l}, \]

\[ l=\frac{a_A}{a_1}. \]

In what follows we shall assume that \(x=1\), since the change in \(x\) at large \(Q_A\) is negligibly small in comparison with the change in \(\alpha_A\).

From the expression \(\operatorname{Im}\dot K_\beta=0\) we find:

\[ \alpha_{A1,2}= \frac{l+1}{2}\operatorname{tg}\varphi_\beta \pm \sqrt{ \left(\frac{l+1}{2}\right)^2 \operatorname{tg}^2\varphi_\beta + (1+\eta_0^2)\,l }, \]

\[ \alpha_{A\,3,4}=\pm\infty. \]

Table 1 gives the calculated values of \(\alpha_{A1}\), as well as the values of \(K_\beta\) found from the condition \(\dot K_\beta=1\), for various \(l\) and \(\eta_0\).

Table 1

The angle \(\varphi_\beta\) was chosen within the range \(0 \div 90^\circ\). In this case the phase angle of the loop gain varies from \(90^\circ\) to \(180^\circ\).

Calculations for values \(\varphi_\beta = 90^\circ \div 180^\circ\) are not given, since the diagram is symmetric.

From the data in the table it is clear that, when feedback encompasses one stage, fairly large depths can be realized for values \(\varphi_\beta > 20^\circ\) and larger \(l\). As \(\eta_0\) increases, the permissible feedback depth decreases.

The values of \(\alpha_{A1}\) given in the table confirm the validity of the assumptions made earlier \((\eta_A=\mathrm{const},\ x=1)\).

b) The feedback encompasses the antenna circuit and two transmitter stages

The expression for the loop gain in this case is written in the form:

\[ K_\beta = \frac{ K^3(1+\eta_0^2)\left[(D\cos\varphi_\beta-C\sin\varphi_\beta) +j(D\sin\varphi_\beta+C\cos\varphi_\beta)\right] }{ C^2+D^2 }, \]

where

\[ C=1+\eta_0^2-\alpha_A^2\frac{1+l+m}{l\cdot m}, \]

\[ D=\alpha_A\left[ \frac{m(1+l)+l(1+\eta_0^2)-\alpha_A^2}{l\cdot m} \right], \]

\[ m=\frac{\alpha_A}{\alpha_2}, \]

\(\alpha_2\) is the generalized detuning of the circuit of the penultimate stage,

\[ K=K_1K_2. \]

Table 2 gives the values of \(K_\beta\), determined by a method analogous to that for the single-stage amplifier, as well as the values of \(\alpha_A\) corresponding to the self-excitation limits, calculated for various \(l\), \(m\), and \(\eta_0\).

Table 2

From the data in the table it is clear that, as in the case of a single-stage amplifier, the permissible feedback depths here increase with increasing \(l\), and the increase becomes especially sharp as \(\varphi_\beta\) tends to \(90^\circ\).

In addition, to ensure large feedback depths, it is necessary that the quality factor of the circuit of the penultimate stage be high (large value of \(m\)).

Unlike the single-stage amplifier, here an increase in \(\eta_0\) has no significant effect on the permissible feedback depth.

Experiment

The block diagram of the experimental setup is shown in Fig. 2.

Fig. 2.

Here 1 and 2 are, respectively, the final and pre-final stages of the transmitter,

3 and 7—cathode followers,
4, 11, 12, 13, and 14—phase shifters,
5—a differential amplifier,
6—a voltage-summing stage,
8—a selective circuit,
9—a frequency divider,
10—the GSS-6 generator,
\(O_1\) and \(O_2\)—SI-1 oscillographs.

By means of \(O_2\), the phase angle of the feedback loop was monitored.

Two stages were enclosed by feedback. The circuit of the feedback chain is analogous to that described in [1].

The final stage operated with a G-807 tube. As the load, a T-shaped antenna was used; its horizontal wire and downlead were made of three iron wires of diameter \(d = 1.5\) mm, with spacing between them \(D = 1\) m.

The suspension height of the horizontal part was \(h = 25\) m, and the length of the horizontal wire was \(L = 130\) m. The antenna had a device permitting the suspension height of the horizontal wire to be varied in its middle part while the height at the end points remained unchanged.

The phase instability was measured by the method described in [3]. The frequency-division coefficient was \(n = 12\).

Figure 3 shows an oscillogram for the circuit without feedback when the antenna wire was rocked with amplitude \(\Delta h = 0.8\) m. In this case the phase fluctuation, determined from the formula,

\[ \varphi = \arcsin\left(\frac{\Delta x}{x}\cdot \sin \frac{\pi}{n}\right) \simeq 4^\circ 55' . \]

For the circuit with feedback, the oscillogram is shown in Fig. 4 \((K^{\circ} = 1.8)\). In this case there are no noticeable phase distortions. The fuzziness of the end portions of the trace in Fig. 4 characterizes the change in the amplitude of the output voltage.

Fig. 3.

Fig. 4.

Conclusions

1. The use of complex feedback makes it possible to ensure high stability of the current phase in the transmitter antenna at small optimum feedback depths.

2. The use of small feedback depths facilitates stable operation of the transmitter and also does not lead to a significant reduction in the gain of the feedback-covered path.

References

1. Suslov I. A. Stabilization of the current phase in an antenna by means of inverse coupling, Proceedings of SFTI, issue 37, 291, 1959.

2. Tsykin G. S. Electronic Amplifiers, Svyazizdat, 1960.

3. Shulzhenko K. M., Silov E. N., Supyan V. Ya. On the measurement of phase shifts and phase modulation by means of interference patterns. Reports of the Tomsk City Scientific-Technical Conference on the topic “Automation of Production Processes,” Tomsk University Publishing House, 50, 1959.

4. Erglis K. E., Stepanenko I. P. Electronic Amplifiers, Fizmatgiz, 1961.

5. Charles A. Wilkins. Voltage Regulator System, U.S. Patent, cl. 179—171 No. 2904644, 15.09.59.

6. Lorenz H. Eine Methode zur Stabilisierung der Verstärkung von Trioden und Pentoden, Nachrichtentechnik, No. 12, 534, 1960.