Coupling Depends on Geometry AND Material Properties
For a particular piezoelectric transducer, the coupling coefficients, Kf and Ki (which
are identically valued), and the mechanical spring coefficient, ka, depend both on
inherent material properties
the specific geometry of the particular transducer.
Let's cover the spring coefficient for the actuator (ka) first:
where Y is Young's modulus of elasticity [N/m^2], A is area [m^2], and t is thickness [m]
NOTE: blue highlights the GEOMETRIC dependences throughout.
In the electrical domain, the capacitance of the actuator (C) has a similar depence on geometry:
where Eo is the permittivity of free space, and E=Er*Eo (that is, Er tells how many times greatly that the permittivity of free space the permittivity of the piezoelectric material (E) is. (You just want the
permittivity of the PZT material, but it's common practice to list permittivities in 'units of free space permittivity'...)
To get the coupling coefficients, we will use our newly-calculated mechanical spring constant (ka) and capacitance (C), in addition to using the piezoelectric strain constant 'd' given for the material. This constant is a material property which does not depend on geometry. Our calculations here seem a bit strenuous because we cannot simply
be told a generic 'coupling coefficient' for an arbitrary transducer made of this material. The coupling
coefficient for a specific actuator-sensor element must incorporate information about both geometry and material properties (which unfortunately makes this whole process complicated).
To obtain Kf, we solve the following quadratic:
After algebraic gymnastics (assuming I haven't made errors...), we obtain the following positive-valued solution for Kf:
On this page, I am simply restating/summarizing Prof. Crandall's solutions to Problem 4 from the
2.004 final exam (spring 2000). Please refer to them to make sure you understand the more-detailed analysis!
[Here are links for
pdf
and
ps versions of the solutions, if you missed them from other pages.]
