handbook of modern sensors

68 3 Physical Principles of Sensing
Fig. 3.22. Piezoelectric sensor is formed by applying
electrodes to a poled crystalline material.
the crystal and the other pair of electrodes to pick up charge resulting from developed
strain. This method is used quite extensively in various piezoelectric transducers.
The magnitude of the piezoelectric effect in a simplified form can be represented
by the vector of polarization [4]:
P = P xx + P yy + P zz , (3.64)
where x, y, and z refer to a conventional ortogonal system related to the crystal axes.
In terms of axial stress, σ , we can write 6
P xx = d 11 σ xx + d 12 σ yy + d 13 σ zz ,
P yy = d 21 σ xx + d 22 σ yy + d 23 σ zz ,
P zz = d 31 σ xx + d 32 σ yy + d 33 σ zz ,
(3.65)
where constants d mn are the piezoelectric coefficients along the orthogonal axes of the
crystal cut. Dimensions of these coefficients are C/N (coulomb/newton) (i.e., charge
unit per unit force).
For the convenience of computation, two additional units have been introduced.
The first is a g coefficient which is defined by a division of corresponding d mn coefficients
by the absolute dielectric constant
g mn = d mn
ε 0 ε mn
. (3.66)
This coefficient represents a voltage gradient (electric field) generated by the crystal
per unit applied pressure; its dimension is
/
V N
m m 2 .
6 The complete set of coefficients also includes shear stress and the corresponding d-
coefficients.