Calculating multimode generation in BAW transducers and resonators

A matrix method is given for calculating the frequency responses for the excitation amplitudes of the three acoustic modes in transducer-substrate geometries and in resonators. The effectiveness of the method is illustrated for typical resonator and transducer structures. Transducers and resonators can be multilayer; viscosity are taken into account. A bulk-acoustic-wave transducer used to excite one of the three bulk modes of a substrate material to which it is coupled is equivalent to a mechanically loaded resonator. In practice all three acoustic bulk modes get generated due to orientation tolerances in transducer and substrate cuts and alignment errors with respect to the substrate. The matrix method for finding the acoustic mode amplitudes in transducers or resonator geometries allows an explicit calculation of: 1. The electrical driving point impedance, insertion loss, and scattering coefficients; 2. The electrical power supplied by the source; 3. The amplitudes of the three bulk modes excited in the substrate; 4. The total acoustic power and the power for each mode in the substrate. These quantities are calculated as a function of frequency using formulas which follow directly from the electrical and mechanical boundary condition matrix equations for the structure. The computation uses only the orientations and thicknesses of the materials, their thermodynamic constants, and the device area.