Correction of the Phase Characteristic of a Two-Circuit Resonant System

I. A. Suslov

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

When deep feedbacks are used in transmitters at high frequency, the absence of self-excitation and the necessary stability margin can be ensured by means of phase-correcting circuits. If the output stage contains antenna and intermediate circuits, its frequency response will have two humps, and the phase characteristic (after subtracting \(90^\circ\)) has the form shown in Fig. 1. To correct the phase shifts of the output stage, it is desirable to have a circuit whose phase characteristic has the opposite form, i.e., differs by phase shifts of opposite sign and by slope. In this case, because of the great steepness of the phase characteristics of the circuits of the output stage, the correcting characteristic must also have a sufficiently large slope. An example of a circuit possessing such properties is the chain shown in Fig. 2. It is a divider in which one arm is formed by a parallel circuit and the other by a series circuit. The circuits are coupled to each other by a sufficiently strong coupling, which, by increasing the damping of the circuits, leads to an increase in the transfer coefficient of the divider and to a displacement of the portions of the phase characteristic that have maximum steepness.

Fig. 1.

The equations of the circuit currents for the circuit under consideration are written in the form:

\[ \dot{u}_1 = i_1 \left( 2j\omega L + 2R + 2j\omega M + \frac{1}{j\omega C} \right) + i_2(j\omega L + R + j\omega M). \]

\[ 0=i_1(j\omega L+R+j\omega M)+i_2\left(j\omega L+R+\frac{1}{j\omega C}\right). \tag{1} \]

Hence the currents \(i_1\) and \(i_2\) are determined by the expressions

\[ i_1=\frac{\dot u_1}{\Delta}\left(j\omega L+R+\frac{1}{j\omega C}\right), \]

\[ i_2=-\frac{\dot u_1}{\Delta}(j\omega L+R+j\omega M), \tag{2} \]

where \(\Delta\) is the determinant of system (1), equal to

\[ \Delta= \begin{vmatrix} j\omega L+R+j\omega M+\dfrac{1}{j\omega C}, & j\omega M-\dfrac{1}{j\omega C} \\[6pt] j\omega L+R+j\omega M, & j\omega L+R+\dfrac{1}{j\omega C} \end{vmatrix}. \tag{3} \]

Fig. 2.

The output voltage is equal to

\[ \dot u_2=i_1\left(j\omega L+R+\frac{1}{j\omega C}\right)+(i_1+i_2)j\omega M. \tag{4} \]

Substituting into (4) \(i_1\) and \(i_2\) from (2) and \(\Delta\) from (3), we obtain

\[ \dot u_2= \frac{ R^2(1+j\delta)^2+\dfrac{M}{C}+\omega^2M^2 }{ R^2(1+j\delta)^2+\dfrac{2M}{C}+\dfrac{L}{C}+\omega^2M^2 }\,\dot u_1, \]

where \(\delta\) is the generalized detuning.
Hence the transmission coefficient will be

\[ K= \frac{(1+j\delta)^2+\kappa(1-\kappa x^2)Q^2} {(1+j\delta)^2+(1+2\kappa+\kappa^2x^2)Q^2}. \tag{5} \]

Fig. 3.

Fig. 4.

For not very large detunings, the quantity \(x\) differs little from 1. Further, in formula (5), \(x^{2}\) is multiplied by \(\kappa\) and by \(\kappa^{2}\). In the case \(\kappa \ll 1\), the influence of changes in \(x\) is thereby weakened to an even greater degree. Setting \(x=1\), we rewrite (5) in the form

\[ \dot K = \frac{(1+j\delta)^2+\kappa(1+\kappa)Q^2} {(1+j\delta)^2+(1+\kappa)^2Q^2}. \tag{6} \]

For the equation of the frequency response of the correcting circuit of Fig. 2, we obtain the expression

\[ K = \sqrt{ \frac{\left[1-\delta^2+\kappa(1+\kappa)Q^2\right]^2+4\delta^2} {\left[1-\delta^2+(1+\kappa)^2Q^2\right]^2+4\delta^2} }. \tag{7} \]

The characteristic (7) is shown in Fig. 3 (right half). The equation of the phase characteristic of the circuit of Fig. 2 in accordance with (6) will be

\[ \varphi = \arctan \frac{2\delta}{1-\delta^2+\kappa(1+\kappa)Q^2} - \arctan \frac{2\delta}{1-\delta^2+(1+\kappa)^2Q^2}. \tag{8} \]

The graph of characteristic (8) is presented in Fig. 4. It makes it possible to compensate the phase shifts of the output stage up to values of \(\delta\) on the order of \(\delta=100\). The rise of the frequency response (7) at large generalized detunings \(\delta\) will not create serious difficulties, since it will be eliminated by the dips produced by the two circuits of the output stage and by the circuits of the preliminary amplifier.

Experimental verification of the frequency and phase characteristics showed that they correspond in the main to those calculated by formulas (7) and (8), and differ only by a certain asymmetry caused by the influence of changes in \(x\).

The proposed correction circuit is most effective with sufficiently strong coupling between the circuits (\(\kappa\) on the order of 0.1).