# Appendix I: Resonance packet (pair)

A transducer can be considered to be a black box with electrical input and mechanical output. In-so-far as the primary resonance and its associated response are concerned, the black box can be approximately represented as a lumped equivalent electrical circuit. Various equivalent circuits have been proposed but the simplest is shown below. This circuit is valid if the parameters are constant and independent of frequency. This will be true for a narrow frequency range around the resonant frequency and if the observed vibration mode is sufficiently isolated from other modes. (The circuit will be expanded later to overcome the latter limitation.)

This circuit has two branches —

1. The mechanical branch — $$L_{M} \, C_{M} \, R_{M}$$. This branch would characterize the transducer if the ceramics had no piezoelectric properties.
2. The electrical branch — $$C_{0} \, R_{0}$$. This branch accounts for the piezoelectric properties.

Starting first with the left branch, the current through this branch is given by —

\begin{align} \label{eq:10027z} I_0 &=E \, \frac{1}{\left[R_0 + \large\frac{-j}{\omega C_0} \right] } \\[0.7em]%complex_eqn_interline_spacing &=E \, \frac{1}{\left[R_0 + \large\frac{-j}{\omega C_0} \right] } \left\{\frac{\left[R_0 - \large\frac{-j}{\omega C_0} \right]}{\left[R_0 - \large\frac{-j}{\omega C_0} \right] }\right\} \nonumber \\[0.7em]%complex_eqn_interline_spacing &=E \, \frac{\left[R_0 - \large\frac{-j}{\omega C_0} \right]}{\left[R_{0}^{\;2} + \left(\large\frac{1}{\omega C_0}\right)^2 \right] } \nonumber \\[0.7em]%complex_eqn_interline_spacing &=E \left[\frac{R_0}{Z_0^2}\right] + j E \left[\frac{1}{Z_0^2}\left(\frac{1}{\omega C_0}\right) \right] \nonumber \\[0.7em]%complex_eqn_interline_spacing &= I_{R_0} + j I_{C_0} \nonumber \end{align}