Methods for Reducing the Settling Time of Oscillations in a Volume Resonator

N. A. LASHUK, B. A. SOLNTSEV

(Presented by the scientific seminar of the Research Institute of Nuclear Physics)

For volume resonators used in physical installations to obtain strong alternating electric fields, contradictory requirements are imposed in a number of cases. On the one hand, volume resonators must have a high shunt resistance \(R_{\mathrm{э}}\), so that the power required for excitation does not exceed reasonable limits. On the other hand, the settling time of the oscillations must be less than some previously specified value, and this value can often be considerably smaller than that determined by the \(Q\) factor of the system. The usual methods recommended for reducing the settling time are associated either with shunting the resonator or with detuning it relative to the excitation frequency; in the present case they cannot be implemented, since they lead either to a decrease in the amplitude of the oscillations in the steady-state regime at a given excitation power, or to an increase in the power of the feeding generator if the amplitude of the stationary oscillations is specified.

Oscillations in a resonator, as in any oscillatory system to whose input a sinusoidal voltage is applied as a step, develop according to the law [1]:

\[ U(t)=\sqrt{2PR_{\mathrm{э}}}\left(1-e^{-\alpha t}\right)\cos \omega t, \tag{1} \]

where \(P\) is the supplied power, and \(\alpha=\dfrac{\omega}{2Q}\) is the damping of the system.

From this expression it is evident that the voltage reaches a prescribed value the faster, the greater the supplied power. If, after the prescribed amplitude has been reached, the power is reduced to the value sufficient to compensate the losses, the amplitude of the oscillations will not change. Thus, by increasing the power introduced at the initial instant, it is possible to reduce the rise time of the oscillation envelope. Varying the magnitude of the initial power pulse \(P_g\) and its duration makes it possible to regulate the magnitude of the rise time \(\tau_{\phi}\). The dependence of the additional power on the magnitude of the required rise time is

\[ \frac{P_g}{P}=\frac{0.81}{\left(1-e^{-2.3\tau_{\phi}/\tau}\right)^2}. \tag{2} \]

Here \(\tau = \dfrac{4.6Q}{\omega}\) is the “natural” time for the establishment of oscillations to the level \(0.9\,U_{\max}\). This dependence is shown in Fig. 1. If, for exciting the resonator, a generator with independent excitation is used, the required initial power \(P_g\) can be obtained by pulse-amplitude modulation, for example by introducing a powerful voltage pulse into the anode circuit of the tubes of the generator output stage. In some cases it may prove more expedient to use additional pulsed generator tubes. Such tubes can operate directly into the resonator, forming together with it a self-oscillating system. It is known that the establishment of oscillations in an oscillator is determined by the initial voltage on the circuit \(U_0\) and by the equivalent damping \(\alpha_{\mathrm{e}}\) [2].

Fig. 1.

\[ U(t)=\frac{U}{\sqrt{1-\left(1-\frac{U^{2}}{U_0^{2}}\right)e^{-2\alpha_{\mathrm{e}}t}}}. \tag{3} \]

The equivalent damping takes account of the influence of the tube and is equal to

\[ \alpha_{\mathrm{e}}=\frac{1}{2C}\left[S\left(|K_{\mathrm{oc}}|-D\right)-\frac{1}{R_{\mathrm{e}}}\right], \tag{4} \]

where \(C\) is the equivalent capacitance of the resonator, \(K_{\mathrm{oc}}\) is the feedback coefficient, and \(D\) is the tube perveance.

After the specified amplitude has been reached, the pulsed tubes are switched off and the subsequent oscillations are maintained by the main generator. In both of the cases considered, in order to obtain a substantial shortening of the leading edge, large pulse powers are required at the initial instant. In addition, here a very precise tuning of the resonator to the excitation frequency is required. Even slight detunings of the resonator lead to considerable changes of phase during the process of establishing the oscillations.

A mismatch between the phases of the main and auxiliary self-oscillators in the second case leads to distortion of the oscillation envelope, and when the phase difference is \(180^\circ\) the oscillations, after the self-oscillator is switched off, are completely disrupted and then arise again, but now under the action of the main generator [1].

In the case where a self-oscillator is used to excite the resonator, a more effective method is one based on its shock excitation. It is already clear from expression (3) that the time for the establishment of oscillations in the generator depends substantially on the amplitude of the initial oscillations \(U_0\). If the resonator is the circuit of the generator, then the initial voltage can be obtained by rapidly charging or discharging the capacitance of the resonator, or by passing a current pulse ...

through a loop inductively coupled to the resonator. These two cases are shown schematically in Fig. 2. If, for example, in the circuit of Fig. 2, a, the source \(E\) charges the resonator capacitance to a voltage equal to the voltage amplitude in the steady state, or if the current generator of Fig. 2, b, provides a current whose magnitude is equal to the current amplitude in the steady state, then, with simultaneous switching on of the anode current and ignition of the thyratron in the self-oscillator, there will be no transient process in amplitude—the stationary oscillations will be established instantaneously.

Fig. 2.

An analysis of the processes occurring in this case shows that there will likewise be no transient process in phase. This is also understandable, since the principal cause of both the change in amplitude and the change in phase during the process of establishing oscillations in a self-oscillator is the change in the equivalent damping \(\alpha_{\mathrm{e}}\). In the case under consideration, the self-oscillator is immediately placed under conditions for which \(\alpha = 0\), which precisely corresponds to the condition for a stationary regime.

REFERENCES

1. A. S. Vinitskii, P. I. Evdokimov. “Features of transient processes in a resonant circuit when manipulating the phase of oscillations and when manipulating the parameters of the circuit.” Radiotekhnika, vol. 14, No. 7, 1959.

2. I. S. Gonorovskii. Fundamentals of Radio Engineering. Svyazizdat, 1957.