SHOCK EXCITATION OF A VACUUM-TUBE SELF-OSCILLATOR

N. A. LASHUK, B. A. SOLNTSEV

In amplitude modulation of a vacuum-tube self-oscillator by a sequence of rectangular pulses, the shape of the envelope of the oscillations at its output, as is known, differs substantially from a rectangular one. This circumstance leads to a number of undesirable consequences. Thus, for example, the pulse duration is shortened, its leading edge is degraded, and the starting of the generator becomes unstable. In accordance with the existing theory, the self-oscillator is described by a nonlinear second-order differential equation

\[ \frac{d^{2}U}{dt^{2}}+\frac{dU}{dt}\,2a_{\mathrm{e}}+\omega_{0}^{2}U=0, \]

where \(U\) is the voltage across the oscillator tank circuit,

\[ \omega_{0} \]

is the natural frequency of the oscillator tank circuit.

The equivalent damping of the system \(a_{\mathrm{e}}\) consists of two parts—the intrinsic damping of the tank circuit \(a_{\mathrm{k}}\), taking into account the shunting effect of the tube, and the damping introduced by the tube through the feedback circuit. The latter is proportional to the tube transconductance and consists of linear and nonlinear parts \(\alpha_{1}\) and \(\alpha_{2}\), respectively. Thus,

\[ \alpha_{\mathrm{e}}=\alpha_{\mathrm{k}}-(\alpha_{1}+\alpha_{2}). \]

For small oscillations the nonlinear part of the introduced damping may be neglected, \(\alpha_{\mathrm{e}}=\alpha_{\mathrm{k}}-\alpha_{1}<0\), and the oscillations are a sinusoid with an exponentially increasing amplitude:

\[ U(t)=\hat{U}(0)e^{(\alpha_{\mathrm{k}}-\alpha_{1})|t|}\sin(\omega_{0}t-\varphi_{0}). \]

The amplitude \(U_{0}\) and the phase \(\varphi_{0}\) are determined by the initial conditions. As the amplitude grows, the nonlinear term \(\alpha_{2}\) begins to have an ever greater effect, and upon reaching the stationary amplitude of oscillations \(U_{\mathrm{st}}\), \(\alpha_{\mathrm{e}}=0\). The amplitude of the oscillations in the steady state is determined by the well-known amplitude-balance condition

\[ SKR_{0e}=1. \]

In the process of establishing the oscillations, the amplitude varies according to the law \([1, 2, 3]\)

\[ U(t)= \frac{U_{\mathrm{st}}} {\sqrt{\,1-\left(1-\dfrac{U_{\mathrm{st}}^{2}}{U(0)^{2}}\right)e^{-2|\alpha_{\mathrm{e}}|t}\,}}. \]

Fig. 1

Fig. 2

The change in the phase of the oscillations is accompanied by a change in frequency from \(\omega_{\mathrm{n}}=\sqrt{\omega_0^2+\alpha_3^2}\) to \(\omega_{\mathrm{k}}=\omega_0\). Thus, the physical cause of the change in both amplitude and phase in the process of establishment of the oscillations is the change in the equivalent damping \(\alpha_3\). If, when switching on the self-oscillator, it is placed under such conditions that \(\alpha_3=0\), then stationary oscillations with constant amplitude and phase are established in the self-oscillator almost instantaneously. Such conditions can be realized, for example, by creating, at the moment the generator is switched on, oscillations in the circuit with an amplitude equal to the amplitude in the established regime. For this purpose it is sufficient to charge or discharge the capacitance through an external circuit, or to obtain an induced voltage on the coupling element of the circuit with the external circuit. The circuits for these two cases are shown in Figs. 1,a and 1,b. Figure 2 gives oscillograms of the oscillations in a generator assembled according to the circuit of Fig. 1,a.

The oscillation frequency is \(420\ \text{kHz}\), and the duration of the modulating pulse is \(62\ \mu\text{s}\). A thyratron TG-1 \(0.1/1.3\) was used as the switch. The oscillograms were taken for the cases \(E=0\), \(E<U_{\mathrm{st}}\), \(E=U_{\mathrm{st}}\), and \(E>U_{\mathrm{st}}\). As expected, at \(E=U_{\mathrm{st}}\) the transient process is absent.

REFERENCES

1. I. S. Gonorovsky. Fundamentals of Radio Engineering. Svyazizdat, 1958.
2. I. M. Kapchinsky. Method of the Theory of Oscillations in Radio Engineering. Gosenergoizdat, 1954.
3. L. A. Bessonov. Nonlinear Electrical Circuits. Vysshaya shkola, 1964.