A generalized feedback oscillator consisting of an ideal nonlinear amplifier and a linear, passive or active LMCR network is considered. The network is of second order and has a ‘biquadratic’ voltage transfer function.



Though such a system represents a large family of important oscillators, it appears that an explicit nonlinear differential equation for it is not known. Consequently, a general nonlinear differential equation, valid for all possible second-order oscillators is derived. This contains explicitly all the network and amplifier parameters. The van der Pol equation is shown to be a special case of the general equation and to represent the simplest second-order oscillator. Transient and steady-state solutions of the general equation are obtained for the case of nearly sinusoidal oscillations. The solutions provide explicit expressions for the transient rise time, steady-state amplitude, harmonic content and the frequency instability of the oscillator output waveform. The expressions can be used to analyse and evaluate all possible types of second-order oscillators. A figure of merit for the oscillator network is defined, and the conditions to be met by the amplifier and the network in order to produce oscillations are determined. Validity of the formulas is verified experimentally by means of two representative oscillators.

General nonlinear equation for LMCR oscillators and its nearly sinusoidal solutions

The method of harmonic linearization is employed to analyse a class of oscillators containing a non-linear resistive two-port network. The first approximation, where non-linear two-port input and output voltages are assumed sinusoidal, and the improved first approximation, which takes into account the intermodulation effects, are obtained. The expressions for frequency and amplitude sensitivity are derived from both, the first and the improved first approximation. The method is applied to a class of oscillators whose non-linear quadrupole consists of a resistive element with voltage current characteristic, i=f(v), and of a controlled source of the typo, i c = Kci, where Kc is a constant. Such a non-linear qundrupole is a good model for a vacuum tube or transistor. Measurements carried out on practical LC oscillator circuits are in reasonable agreement with the theoretical predictions.

M. Vehovec

Papers

On oscillators containing non-linear resistive quadrupoIes†