Page 242 - Mechanics of Microelectromechanical Systems

4. Microtransduction: actuation and sensing 229
The energy formulation is useful as it allows introducing an amount, the
piezoelectric coupling factor, which is defined as:

and which gives the measure of the degree of energy conversion efficiency.

Example 4.12
Determine the coupling factor for the case defined in Example 4.11
knowing that the electrical permittivity is

Solution:
For the particular problem of the previous example, the stress vector
reduces to the component. Similarly, the compliance matrix is single-
termed as it only contains the permittivity matrix reduces to its
component, and the electric field vector reduces to The piezoelectric
energy will be in this case:

The mechanical energy is:

and the electrical energy simplifies to:

By substituting Eqs. (4.116), (4,117) and (4.118) into the definition equation
– Eq. (4.115), the following equation is obtained for the coupling factor:

and it can be shown by analyzing the strain-stress relationship of Eq. (4.105)
– Example 4.11 – that (where E is the Young’s modulus about
direction 3 or z). As a consequence, Eq. (4.119) yields a coupling factor of
approximately 0.2.
The case of utilizing piezoelectric layers sandwiched with other
structural or active layers in bimorph/multimorph microcantilevers for