STABILITY OF AN ANTENNA–TRANSMITTER SYSTEM WITH REACTIVE FEEDBACK

I. A. Veselkov

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

The use of reactive feedback makes it possible to ensure high stability of the phase of the current in the antenna [1]. This article investigates the stability of an antenna–transmitter system covered by reactive feedback. It is assumed that the antenna has a large capacitive load at its end, which permits it to be regarded as an asymmetric grounded dipole [2].

The antenna and the transmitter output stage are covered by reactive feedback

The equivalent circuit of the output stage for the case of underloaded and critical operating modes, with inductive coupling of the transmitter tank circuit to the antenna, can be represented in the form of Fig. 1.

Here \(L_1, C_1, R_1\) are the parameters of the tank circuit; \(L\) is the inductance of the coupling coil; \(R_{\mathrm{in}}\) and \(C_A\) are, respectively, the active input resistance and the total capacitance of the antenna.

The voltage transfer coefficient of the feedback loop, on the basis of the equivalent circuit in Fig. 1, can be represented by the expression

Fig. 1.

\[ K_{1\beta} = \frac{ jK_{01}\beta x(1+\eta_0^2)\dfrac{R_{\mathrm{in}0}}{R_{\mathrm{in}}} }{ (1+j\alpha_1) \left(1+j\alpha_A\dfrac{R_{\mathrm{in}0}}{R_{\mathrm{in}}}\right) +\eta_0^2 x^2\dfrac{R_{\mathrm{in}0}}{R_{\mathrm{in}}} }, \tag{1} \]

where

\[ K_{01}=\frac{SM\omega_0 L}{C_1R_1R_{\mathrm{in}0}(1+\eta_0^2)}, \]

\[ x=\frac{\omega}{\omega_0},\qquad \eta_0=\frac{\omega_0M}{\sqrt{R_1R_{\mathrm{in}0}}}, \]

\(R_{\mathrm{in}0}\) is the active component of the antenna input resistance at the tuning frequency,

\(R_{\mathrm{in}x}\) is the same at frequency \(f\),

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

\(\beta\) is the feedback coefficient.

It is assumed that \(\beta\) is a real quantity. If the antenna efficiency is taken as constant for the wavelength range within which self-excitation of the system is possible, then the expression for the active component of the input resistance of an asymmetrically grounded dipole, on the basis of Rodenberg’s formula, is represented by the expression [2]

\[ R_{\mathrm{in}x}=\frac{40\omega^2h^2}{\eta_A C^2}. \]

Consequently,

\[ \frac{R_{\mathrm{in}0}}{R_{\mathrm{in}x}}=\frac{\omega_0^2}{\omega^2}. \tag{2} \]

Let us denote

\[ \alpha_A=l\alpha_1. \tag{3} \]

Then, substituting (2) and (3) into (1), after simple transformations we obtain

\[ \dot K\beta= \frac{ K_{01}\beta\,\dfrac{1}{x}(1+\eta_0^2) \left[ \alpha_A\left(\dfrac{1}{l}+\dfrac{1}{x^2}\right) +j\left(1+\eta_0^2-\dfrac{\alpha_A^2}{x^2l}\right) \right] }{ \left[ \alpha_A\left(\dfrac{1}{l}+\dfrac{1}{x^2}\right) \right]^2+ \left( 1+\eta_0^2-\dfrac{\alpha_A^2}{x^2l} \right)^2 }. \tag{4} \]

The values of \(\alpha_A\) corresponding to the points of intersection of the real axis of the complex plane \(R_e\dot K\beta,\ J_m\dot K\beta\) by the vector \(\dot K\beta\) are determined from the equation

\[ J_m\dot K\beta=0. \]

Hence

\[ \alpha_{A1,2}=\pm x\sqrt{(1+\eta_0^2)l}, \]

\[ \alpha_{A3,4}=\pm\infty. \]

\(\alpha_{A3,4}\) correspond to the point at the origin of the complex plane.

Substituting the value \(\alpha_{A1}\) into (4) and neglecting unity in comparison with \(\eta_0^2\), we have

\[ \dot K\beta=\frac{K_{01}\beta\eta_0\sqrt{l}}{x^2+l}. \tag{5} \]

The value \(K_{01}\beta\) corresponding to the self-excitation boundary is determined from the expression

\[ K_{01}\beta=\frac{x^2+l}{\eta_0\sqrt{l}}. \tag{6} \]

With a sufficiently high \(Q\) of the antenna circuit, the change in \(x\) in comparison with the change in \(a_A\) may be neglected, and in expression (6) one may set \(x = 1\).

It follows from formula (6) that, for strong coupling between the circuit and the antenna, \(\eta_0^2 \gg 1\), and for values \(1 + l < \eta_0\), the limiting permissible values of \(K_0\beta\) in the case where the antenna and one transmitter stage are enclosed by reactive feedback turn out to be small.

The antenna and two transmitter stages are enclosed by reactive feedback

In this case the voltage transfer coefficient of the feedback loop is represented by the expression

\[ K\beta = \frac{ K_0\beta \dfrac{1}{x} (1+\eta_0^2)(A+jB) }{ A^2+B^2 }, \tag{7} \]

where

\[ A = a_A\left[ \frac{m}{l}(1+\eta_0^2)+\frac{x^2+l}{x^2l} -a_A^2\frac{m}{x^2l} \right], \]

\[ B = (1+\eta_0^2)-a_A^2\left( \frac{1}{x^2l}+\frac{m}{x^2l}+\frac{m}{l^2} \right), \]

\[ m=\frac{a_2}{a_1}, \qquad K_0=K_{01}\cdot K_{02}; \]

\(a_2\) is the generalized detuning of the circuit of the driver stage.

From the equation \(B=0\) we find

\[ a_{A1,2}=\pm xl\sqrt{\frac{1+\eta_0^2}{\,l+m(x^2+l)\,}}. \]

Substitution of \(a_{A1}\) into (7), after simple transformations and with the adopted assumptions \((\eta_0^2 \gg 1\) and \(x^2=1)\), gives

\[ K\beta = \frac{ K_0\beta \sqrt{\,l(m+1)+m\,} }{ m\eta_0\left[ 1-\dfrac{1}{1+m\left(1+\dfrac{1}{l}\right)} \right] }. \tag{8} \]

Table 1 gives the limiting values of \(K_0\beta\) calculated by formula (8) (found from the condition \(K\beta=1\)) for various \(l\) and \(m\) at \(\eta_0=10\).

Table 1

From the data of Table 1 it follows that, in the case where the antenna and two transmitter stages are enclosed by reactive feedback, it is possible to po-

to obtain sufficiently large feedback depths, which in most cases do not exceed the optimal ones [1]. From the point of view of improving the stability conditions, it is necessary to increase the \(Q\) of the tank circuit of the penultimate stage.

REFERENCES

1. Veselkov I. A. Stabilization of the phase of the current in a transmitter antenna by means of reactive feedback at high frequency, Izvestiya TPI, vol. 100, 179, 1962.
2. Nadenenko S. I. Antennas, Svyazizdat, Moscow, 1950.