MATHEMATICAL MODEL OF PULSE MODULATION FOR AUTOGENERATORS WITH NEGATIVE DIFFERENTIAL RESISTANCE

Abstract

A consistent mathematical model of an autogenerator with pulse modulation and negative differential conductivity of the active element has been developed. Based on the RLC electrical circuit with an equivalent “negative resistor” (cubic/PWL approximation of the characteristic), the model is reduced to the class of Lienar equations (canonical case of van der Pol) and supplemented with two physically different modulation channels: additive pulses and parametric “windows” of change in the equivalent parameters of the circuit. For periodic modes, a formal approach to stability verification using a strobe map has been formulated; in this way, conditions have been obtained that allow distinguishing between the preservation and destruction of the limit cycle during pulse pumping. The numerical part is implemented as a fourth-order Runge-Kutta event scheme with accurate alignment of pulse moments on the integration grid and reproducible metrics: the amplitude is determined by the peaks after “warm-up,” frequency is determined by interpeak intervals, and the stability indicator is determined by the relative dispersion of peaks. Typical time realizations, phase portraits, and generalized dependencies of amplitude and frequency on the excitation parameter for modes with and without additive pumping are presented. it is shown that parametric “windows” change the effective stiffness and are capable of both amplifying and destabilizing oscillations depending on their depth and duration. In the sensory context, a frequency model with dependence on effective capacitance is presented, and sensitivity derivatives are derived, which are directly used for calibrating temperature and thickness transducers. The proposed macromodel combines negative conductivity physics, pulse modulation, and strict stability criteria in a single device, while the numerical method provides reliable reproduction and engineering interpretation of characteristics relevant to integrated LC solutions and practical stabilization circuits.