附录 I：谐振包（对）
换能器可以被视为一个电输入、机械输出的黑匣子。就主谐振及其相关响应而言，该黑匣子可以近似表示为一个集总等效电路。人们已提出各种等效电路，其中最简单的如下所示。若各参数为常数且与频率无关，则该电路有效。在谐振频率附近的狭窄频率范围内，且当所观察的振动模态与其他模态充分隔离时，这一条件成立。（稍后将扩展该电路以克服后一限制。）
该电路有两条支路 —
- 机械支路 — \( L_{M} \, C_{M} \, R_{M} \)。如果陶瓷不具有压电特性，这条支路即可表征换能器。
- 电气支路 — \( C_{0} \, R_{0} \)。这条支路用于表征压电特性。
先看左侧支路，通过该支路的电流为 —
\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}
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 —
- The mechanical branch — \( L_{M} \, C_{M} \, R_{M} \). This branch would characterize the transducer if the ceramics had no piezoelectric properties.
- 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}