The overlap method can further be viewed as a field-based generalization of Berlincourt’s quasi-static formulation for material coupling, which itself is defined in terms of this underlying energy-ratio. A commonly used approximation to the IEEE definition expresses coupling as a function of the resonant (fs) and anti-resonant (fp) frequencies, allowing the electromechanical coupling to be calculated from laboratory admittance measurements of a given resonator. This approximation assumes an idealized, single-mode-dominated resonator. In contrast, the energy-normalized overlap integral (overlap method) of a resonator’s electromechanical fields is also used as a measure of coupling. This method allows researchers to understand how electromechanical eigenmode shape impacts coupling, typically through simulation.
This work clarifies the relationship between these formulations across regimes of material coupling (k2t) and acoustic mismatch. For small k2t and matched acoustic impedance and velocity between layers, the resonator forced response is single-mode dominated, and both formulations are equivalent and consistent with the IEEE energy-ratio definition. However, in the presence of acoustic mismatch, the resonator’s forced response involves multimodal participation, causing the (single mode) frequency-based IEEE approximation to deviate from the energy-ratio definition, while the mode-specific overlap formulation agrees with that definition.
At large k2t, both the IEEE frequency approximation and the overlap formulation diverge from their underlying energy-ratio definition due to differences in the modal profiles at fs and fp. In this case, accurate computation of the forced response requires a multimode electromechanical expansion.
An efficient and accurate modal expansion of the resonator’s admittance and displacement responses can be obtained by introducing a semi-analytic dual-mode basis constructed from the fs and fp eigenmodes. This framework unifies the interpretation of coupling definitions and provides a practical approach to BAW resonator analysis and design.
