# Analytical approach to piezoelectric model synthesis with the use of Cauer’s method for system design

## TL;DR

This paper introduces a new method for modeling piezoelectric systems using Cauer’s approach, showing promising results in approximating system behavior with limited data.

## Contribution

A novel synthesis method for piezoelectric models using Cauer’s method and a non-standard analysis technique with edge graphs and structural numbers.

## Key findings

- The model achieved a 5% error in resonance frequency prediction.
- The displacement peak exceeded the physical limits of a 3.5mm piezoelectric plate.
- The new analysis method produced identical resonance frequencies but varied vibration magnitudes by 50–58%.

## Abstract

Background: Piezoceramic materials have unique property which enables direct and bilateral conversion between mechanical and electrical energy. This ability facilitates significant miniaturisation of technology and opens many opportunities in design of new actuators and energy harvesters. Mathematical modelling of piezoelectric modules is notoriously hard due to complex constitutive equations defining mechanical and electrical energy conversion.

Methods: The article presents research on a new synthesis method based on the Cauer’s method for electrical and mechanical system design. Mechanical damping is introduced with the use of Rayleigh’s approximation. A discrete electromechanical model is formed based on the Mason’s piezoelectric model. As an additional element of the research, a non-standard model analysis method using edge graphs and structural numbers is employed to investigate potential reductions in computational requirements for model analysis.

Results: The electromechanical model analysis has shown an error of 5% in relation to the resonance target. The calculated peak in magnitude of displacement at the resonant frequency was well above the capability of a 3.5mm thick piezoelectric plate. The proposed non-standard method of analysis gave identical results in terms of obtained resonance frequencies. The calculated magnitude of vibrations was varying by 50 – 58%.

Conclusions: The synthesis method presented in this paper allows an approximation of piezoelectric parameters of a real system based on the created mathematical model. Currently this method is subject to a considerable error in the determined vibration amplitudes of the system, but it allows a coarse approximation of the parameters while having a very limited number of input data. The additional method of analysis based on structural numbers offers a promising alternative to matrix calculations but requires a more thorough investigation of the computational power required to determine whether it can compete with existing algorithms.