Mathematical Model of Multi-Frequency Piezoresonance Oscillation System

The paper represents the basic model of multi-frequency piezoresonance oscillation system (MPOS) – the piezoresonance devices (PRD) core, which enables to study the processes of establishing multi-frequency oscillation mode and its stability. The basic structure of multi-channel multi-frequency PRD core, which is based on principles of filter schemes, is proposed, and the main designations are entered. The peculiarities of truncated differential equations for amplitude, phase and auto-bias voltage of MPOS for the quantity of simultaneously generated frequencies are examined. On the example of three-frequency mode of oscillation under polynomial approximation of transferable characteristics of active elements the characteristic cases of establishing oscillations in MPOS are represented. The area of a steady three-frequency oscillating behavior is defined and the assessment of time of establishment of oscillations and value of group runout of frequencies is made. Received results enable to form a new approach to construction of piezoresonance devices with controlled dynamics, which are represented in the form of adaptive controlled systems with predictive standard model and develop on its basis the new class of invariant to destabilizing disturbing PRD factors. On the basis of such approach there is the principle of using natural redundancy in multi-frequency basis of PRD core – multi-frequency oscillation system, which enables not only to synthesize the system with current identification of disturbing factors on basis of instruments of invariance theory, but also do the adaptation of PRD in accordance with their influences.