X-parameters-based closed-form expressions for evaluating power-dependent fundamental negative and positive real impedance boundaries in oscillator design

In this study, an analytical design tool, applicable under large-signal operation, based on X-parameters closed-form expressions, to determine the large-signal boundary between the negative (energy delivered) and positive (energy absorbing) real input impedance regions of the non-linear oscillator block, is presented to be used prior to and/or during non-linear oscillator circuit design. The consistency of the approach is demonstrated since in small-signal operation; the proposed expressions for the gamma equal to unity loci converges to the classic expression, generally referred to as the input or output stability circles. At large-signal levels, while we are not claiming that this formulation provides by itself similar general stability guidance, it can provide relevant information for oscillator design. The validity and usefulness of the proposed expressions as a real-time design aid, thus either minimising or avoiding the necessity for complex and time-consuming harmonic balance simulations, is demonstrated in the study.