Multi-physics power-drive-level dependant equivalent circuit models extraction for electro-mechanical resonators

This paper discusses multi-physics modeling methodology for the design and analysis of integrated crystal oscillators based on power-waves formulation. Conventional Barkhausen oscillation conditions criterion is derived in terms of reflection coefficients. Necessity of power-drive-level dependant equivalent circuit models derivation for electromechanical resonators is investigated in order to build bridging connections between more than one energy domain (mechanical, electrical, acoustics, etc...) for multi-physics co-simulation and co-analysis. Electro-mechanical coupling modules are introduced at the interfaces of different energy-domains to account for energy-conversion aspects. The proposed methodology is successfully applied to the design of low power and low noise integrated crystal oscillator.