Implementation of the ‘smart’ rotor concept - Wind Power Generation and Wind Turbine Design - page 478

The other symbols are the piezoelectric constants and they are defi ned as
follows:
e : Permittivity. It is the electric fi eld per unit applied electric displacement. e is
often related to the permitivity of vacuum: 8.85 10 -12 F/m

d : matrix of piezoelectric charge constants. It is the mechanical strain per
unit applied electric fi eld or the electrical polarization per unit mechanical
stress applied.
S : Elastic compliance constant which is defi ned as the amount of strain in the
material per unit applied stress.
There are many ways to rewrite these equations into other forms. The IEEE [62]
and Moulson and Herbert [66] present a good overview. These are chosen here
because eqn (1) is very useful in describing the behavior of piezo-electrics as actu-
ator and eqn (2) as sensor. Moheimani and Fleming [63] describes very well how
these equations can be applied to patches and Waanders [64] to stacks. Another
very important piezoelectric parameter is the effective coupling coeffi cient, k ,
which is a measure for the ability of the material to convert mechanical energy in
electrical energy, or vice versa:

×

converted energy
input energy

2

k = (3)

k can also be expressed in terms of piezoelectric constants described above:

d

ij

( 4)

2

k =

ij

ES
ij

s e

ij

However, the total effi ciency of a piece of piezoelectric material is not only defi ned
by k , but also by the way it is incorporated into a mechanical system. Giurgiutiu
and Rogers [68] defi ne r as the ratio between the (internal) stiffness of the piezo-
electric material and the (external) stiffness of the structure against which it acts,
and then derives that the total energy conversion coeffi cient equals:

⎛

2

⎞

1
4

k

( 5)

h =

⎜ ⎟ 2 (/

1 − k r r + 1 )

⎝ ⎠

Note that this equation holds for low frequent and quasi-static applications. Many
dynamic analyses [64, 66, 68] of piezoelectric materials in electro-mechanical
systems exist.

3.1.1 PZT
PZT, or lead zirconium titanate, is the most widely used piezoelectric material.
It is a very brittle ceramic. Below the Curry temperature the ceramic crystal
exhibits a lattice structure with a dipole because of tetragonal symmetry. Crys-
tals with adjoining dipoles are grouped in domains, called Weiss domains, which
are randomly orientated in the material. Therefore the material has no net dipole.