There is an increasing reliance on mathematical modelling to assist in the design of piezoelectric ultrasonic transducers since this provides a cost-effective and quick way to arrive at a first prototype. Given a desired operating envelope for the sensor the inverse problem of obtaining the associated design parameters within the model can be considered. It is therefore of practical interest to examine the well-posedness of such models. There is a need to extend the use of such sensors into high temperature environments and so this paper shows, for a broad class of models, the well-posedness of the magneto-electro-thermo-elastic problem. Due to its widespread use in the literature, we also show the well-posedness of the quasi-electrostatic case.