Appendix H: Effect of stack bolt on electromechanical coupling coefficient κ
When the ceramics are prestressed by a stack bolt, the electromechanical coupling coefficient \( \kappa \) is reduced because of the stiffness of the stack bolt. This appendix derives the relationship. (Note: in this discussion, "stack bolt" refers to any means of applying a prestress to the ceramic stack. This could include a center stack bolt, peripheral stack bolts, or a peripheral shell.)
See Transducer design — stack bolt for a general discussion.
Consider a ceramic that is contained between two rigid massless platens. When a D-C voltage \( V \) is applied, energy is transferred to the ceramic. Part of this energy charges the ceramic's capacitance. This "capacitive energy" is present due to the ceramic's dielectric properties; it would be present even if the ceramics did not have piezoelectric properties. This energy is —
\begin{align} \label{eq:10028a} W_0=\small\frac{1}{2} \, C_0 \, V^2 \end{align}
where —
| \( W_0 \) | = Capacitive energy |
| \( C_0 \) | = Ceramic capacitance |
| \( V \) | = Voltage |
The rest of this energy causes the ceramic to expand; this is the desired effect by which the applied electrical energy is transformed into mechanical (strain) energy. This strain energy is —
\begin{align} \label{eq:10027a} W_k=\small\frac{1}{2} \, k \, U^2 \end{align}
where —
| \( W_k \) | = strain energy |
| \( k \) | = stiffness of ceramic stack (including the bolt) |
| \( U \) | = displacement of ceramic stack due to voltage \( V \) |
The electromechanical coupling coefficient \( \kappa \) (kappa) is given by Waanders (1), p. 12 as —
\begin{align} \label{eq:10026a} \kappa^2 &= {\left[\frac{\textsf{Energy converted}}{\textsf{Energy input}} \right]}_{\textsf {Low frequency}} \end{align}
Note the requirement of "low frequency" so that the inertial energy due to acceleration is negligible compared to the strain energy. Since \( \kappa \) is the ratio of two energies, \( \kappa \) must be dimensionless.
In the case of a driven transducer, the \( Energy \, converted \) is the energy that is converted from electrical form to mechanical form. Hence, the numerator of equation \eqref{eq:10026a} is just the strain energy \( W_k \) of equation \eqref{eq:10027a}. The denomintor is the total input (system) energy (i.e., the combined energy of \( W_k \) and \( W_0 \)). Thus, equation \eqref{eq:10026a} can be written as —
\begin{align} \kappa^2 &=\cfrac{W_k}{W_k + W_0} \nonumber \\[0.7em]%complex_eqn_interline_spacing &= \cfrac{1}{\left( \cfrac{W_0}{W_k} \right) +1} \nonumber \\[0.7em]%complex_eqn_interline_spacing \label{eq:10029a} &= \cfrac{1}{\left( \cfrac{C_0}{k} \right) {\left( \cfrac{V}{U} \right)}^2 +1} \end{align}
Equation \eqref{eq:10029a} is perfectly general. Then it can be applied when there is no stack bolt (case 1) and when a stack bolt is present (case 2).
Case 1 — no stack bolt
Assigning subscript 1 in equation \eqref{eq:10029a} to indicate that no stack bolt is present —
\begin{align} \label{eq:10030a} {\kappa_1}^2 &= \frac{1}{{\left(\Large\frac{C_0}{k_1}\right) } \left(\Large\frac{V}{U_1} \right)^2 +1} \end{align}
Here \( k_1 \) is the stiffness of ceramic alone since there is no stack bolt. For convenience we designate this as \( k_c \).
Solving for \( \left(\frac{V}{U_1}\right)^2 \) (for later use) —
\begin{align} \label{eq:10031a} \left(\frac{V}{U_1} \right)^2= \frac{\Large\frac{1}{{\kappa_1}^2} -1}{\left(\Large\frac{C_0}{k_c}\right) } \end{align}
Side note
It might appear that \( \kappa \) given in equation \eqref{eq:10030a} depends on the particular parameters of the test (e.g., \( C_0 \) and \( k_c \) which both depend on the ceramic dimensions, and also \( V \), and \( U \)). However, this is not true.
First consider how \( C_0 \) and \( k_c \) are related to the ceramic dimensions.
\begin{align} \label{eq:10031b} C_0 &= {\frac{\varepsilon \, A_c}{L_c}} \end{align}
where —
| \( \varepsilon \) | = permittivity |
| \( A_c \) | = cross-sectional area of ceramic (i.e., the area of the individual ceramic's flat face) |
| \( L_c \) | = length (height) of ceramic stack |
Case 2 — with stack bolt
Assigning subscript 2 in equation \eqref{eq:10029a} to indicate that a stack bolt is present —
\begin{align} {\kappa_2}^2 &= \frac{1}{{\left(\Large\frac{C_0}{k_2}\right) } \left(\Large\frac{V}{U_2} \right)^2 +1} \nonumber \\[0.7em]%complex_eqn_interline_spacing \label{eq:10032a} &= \frac{1}{{\left(\Large\frac{C_0}{k_2}\right) } \left(\Large\frac{V}{U_1} \right)^2 \left(\Large\frac{U_1}{U_2} \right)^2 +1} \end{align}
Since the ceramics and bolt are mechanically in parallel, the total stack stiffness \( k_2 \) just equals the ceramic stiffness \( k_c \) plus the bolt stiffness \( k_b \) —
\begin{align} \label{eq:10032b} k_2 = k_c + k_b \end{align}
Thus equation \eqref{eq:10032a} becomes —
\begin{align} \label{eq:10032c} {\kappa_2}^2 &= \frac{1}{{\left(\Large\frac{C_0}{k_c \, + \, k_b}\right) } \left(\Large\frac{V}{U_1} \right)^2 \left(\Large\frac{U_1}{U_2} \right)^2 +1} \end{align}
For a given ceramic configuration, the stack displacement \( U \) varies inversely with the stack stiffness. Thus, generally —
\begin{align} \label{eq:10033a} U \,\propto \, \frac{1}{k} \end{align}
Then the relative displacements \( U_1 \) without a stack bolt and \( U_2 \) with a stack bolt can be expressed in terms of the stiffnesses as —
\begin{align} \frac{U_1}{U_2} &= \frac{k_2}{k_1} \nonumber \\[0.7em]%complex_eqn_interline_spacing &= \frac{k_c \, + \, k_b}{k_c} \nonumber \\[0.7em]%complex_eqn_interline_spacing \label{eq:10034a} &= \frac{k_b}{k_c} + 1 \end{align}
Substituting equations \eqref{eq:10031a} and \eqref{eq:10034a} into equation \eqref{eq:10032c} and simplifying gives —
\begin{align} \label{eq:10037a} {\kappa_2}^2 &= \frac{1}{ \left({\Large\frac{1}{{\kappa_1}^2}} -1 \right) \left({\Large\frac{k_b}{k_c}} +1\right) +1} \end{align}
where (finally) —
| \( \kappa_2 \) | = piezoelectric coupling coefficient with the stack bolt |
| \( \kappa_1 \) | = piezoelectric coupling coefficient without the stack bolt |
| \( k_b \) | = bolt axial stiffness |
| \( k_c \) | = ceramic stiffness |
Berlincourt (3) (p. 269, endnote 9, referenced on p. 249) gives the same equation for a heavily mass-loaded transducer (one with centered ceramics whose ceramic volume is small compared to the two end masses), although with slightly different notation and some rearranging. (A heavily mass-loaded transducer is one with centered ceramics whose ceramic volume is small compared to the two end masses. Then the ceramics and stack bolt experience nearly uniform elongation, as was assumed here.)
If there is no bolt (so \( k_b = 0 \)) then \( \kappa_2 = \kappa_1 \), as expected. As the bolt stiffness \( k_b \) increases, \( \kappa_2 \) is progressively reduced below \( \kappa_1 \). If the bolt stiffness \( k_b \) becomes (theoretically) infinite then \( \kappa_2 \) is zero. This is because the stack is restrained by the bolt against any expansion regardless of the applied voltage. Thus, all of the applied voltage goes toward charging the ceramic and none results in expansion of the ceramic.
For each component of the ceramic stack, the stiffnesses can be expressed in terms of the component's modulus and dimensions —
\begin{align} \label{eq:10038a} k &= Y A / L \end{align}
where —
| \( Y \) | = Young's modulus |
| \( A \) | = cross-sectional area |
| \( L \) | = length (height) of component |
Thus, in terms of the actual ceramic and bolt parameters, equation \eqref{eq:10037a} can be written as —
\begin{align} \label{eq:10039a} {\kappa_2}^2 &= \frac{1}{ \left({\Large\frac{1}{{\kappa_1}^2}} -1 \right) \left({\Large\frac{Y_b \, A_b \, / \, L_b}{Y_c \, A_c \,/ \, L_c}} +1\right) +1} \end{align}
where —
| \( Y_c \) | = Young's modulus of ceramic |
| \( A_c \) | = cross-sectional area of ceramic (i.e., the area of the individual ceramic's flat face) |
| \( L_c \) | = length (height) of ceramic stack |
| \( Y_b \) | = Young's modulus of bolt |
| \( A_b \) | = cross-sectional area of bolt shank |
| \( L_b \) | = length of bolt |