Water Loaded Thin-Disk Piezoelectric Transducer - 2D Axi-symmetric Model
This next example looks at the piezoelectric behavior of FEWaves and compares the impedance resonance of a thin-disk transducer with that of theory. The reference used for this discussion is Principles of Acoustic Devices by V.M. Ristic, John Wiley and Sons, 1983.
A large thin piezoelectric (PZT-5H) plate is excited with a voltage Vo which results in longitudinal vibrations to create what is commonly referred to as a thickness-mode transducer. See the drawing below for the details of this model. This example is designed to resonate near 3 Mhz and impedance curves generated from FEWaves will be compared with theory.
The thickness t is much smaller than the Area of the plate, which allows for a simple finite element model.
A small disk section of the overall plate size can be used to model the thickness mode behavior of the transducer.
To calculate the current, the solution from FEWaves must be properly scaled as given by the ratio of the actual
plate area to the finite element disk area. The real plate has dimensions of 50.8mm by 16.5mm, which is an area
of 838.2 square mm. The area of our finite element model is 
= .7854 square mm. Hence, the scale factor becomes 1067.22. The resultant axi-symmetric finite element
model (2dplate.geo) becomes:
A voltage of 1.0 volts is applied across the ceramic of thickness t. Additional boundary conditions that are required are the absorbing boundary conditions on the top and bottom edges to represent infinite water loading and a the radial displacement component is constrained to zero at the edge of the ceramic which represents and infinitely large plate.
For resonance near 3 Mhz, the thickness t is set to be 0.6528 mm. The finite element model will be run from 2 to 4 Mhz. Using the standard rule of thumb of at least 16 finite elements per wavelength at the highest wavelength and at least one wavelength distance from the ceramic over the entire frequency range, we will set the thickness of the water to be 1 mm thick and have at least 30 first order elements. The finite element mesh is shown below:
The current prediction for a water loaded thin-disk resonator is given by the following matrix equation:
where
The above matrix equation can be solved for I over the frequency range of 2 - 4 Mhz. Comparison with the output from FEWaves generates the following plot:
These results are first order accurate: an improvement in accuracy would be achieved using 2nd order finite elements.