handbook of modern sensors

3.6 Piezoelectric Effect 69
Another coefficient, designated h, is obtained by multiplying the g coefficients by the
corresponding Young’s moduli for the corresponding crystal axes. Dimension of the
h coefficient is
/
V m
m m .
Piezoelectric crystals are direct converters of mechanical energy into electrical. The
efficiency of the conversion can be determined from the so-called coupling coefficients
k mn :
k mn = √ d mn h mn . (3.67)
The k coefficient is an important characteristic for applications where energy efficiency
is of a prime importance, like in acoustics and ultrasonics.
The charge generated by the piezoelectric crystal is proportional to applied force,
for instance, in the x direction the charge is
Q x = d 11 F x . (3.68)
Because a crystal with deposited electrodes forms a capacitor having capacitance C,
the voltage, V , which develops across between the electrodes is
V = Q x
C = d 11
C F x. (3.69)
In turn, the capacitance can be represented [see Eq. 3.23] through the electrode surface
area, 7 a, and the crystal thickness, l:
C = κε 0
a
l . (3.70)
where ε 0 is permitivity constant and κ is dielectric constant. Then, the output voltage
is
V = d 11
C F x = d 11
Fx. (3.71)
κε 0 a
The manufacturing of ceramic PZT sensors begins with high purity metal oxides
(lead oxide, zirconium oxide, titanium oxide, etc.) in the form of fine powders having
various colors. The powders are milled to a specific fineness and mixed thoroughly
in chemically correct proportions. In a process called “calcining,” the mixtures are
then exposed to an elevated temperature, allowing the ingredients to react to form a
powder, each grain of which has a chemical composition close to the desired final
composition.At this stage, however, the grain does not yet have the desired crystalline
structure.
The next step is to mix the calcined powder with solid and/or liquid organic binders
(intended to burn out during firing) and mechanically form the mixture into a “cake”
which closely approximates a shape of the final sensing element. To form the “cakes”
of desired shapes, several methods can be used. Among them are pressing (under
force of a hydraulic powered piston), casting (pouring viscous liquid into molds and
7 Not the crystal area. Piezo-induced charge can be collected only over the area covered by
the electrode.