9 The anisotropic piezoelectric properties of the ceramic are defined by a system of
symbols and notations as shown in Figure 2-2. This is related to the orientation of the ceramic and the direction of measurements and applied stressesforces.
Figure 2-2: Notation of piezoelectric axes.
A cantilever piezoelectric can be designed to operate in either
d
31
or
d
33
modes of vibration depending on the arrangement of the electrodes [18].
d
31
is a thickness mode polarisation of plated electrode on the piezoelectric materials, with stress applied
orthogonal to the poling direction, as shown in Figure 2-3 a.
d
33
mode on the other hand, can be implemented by fabricating interdigitated IDT electrodes on piezoelectric
materials for in-plane polarisation where stress can be applied to the poling direction, as shown in Figure 2-3 b.
Figure 2-3: Cross-sectional view of piezoelectric configuration mode, a
d
31
and b
d
33
.
X Y
6
2 5
1 4
Direction of Polarisation
Z 3
E T
x
T
x
Poling distance
a b
T
x
T
x
E
Poling distance
10
2.2.2 Piezoelectric Material Measurement Methods
Generally there are two categories of piezoelectric material measurements methods: static and dynamic measurement methods. The static method is implemented by either
directly applying force to the material and observing the charge generation or by applying constant electric field and observing the dimension change, while the dynamic
method is using alternating electrical signal at high frequency to observe the frequency responses of the material [19].
Static piezoelectric measurement can be made by either direct or indirect methods. The direct method, also known as Berlincourt method, is conducted by applying a known
force to a piezoceramic sample and the charge generated is measured. The relationship between the generated charge and applied force is the piezoelectric charge coefficient as
given in equation 2-2.
2-2
2-3
The subscripts
i
and
j
are the notations for poling direction and the applied stress direction respectively as according to the Figure 2-2. A poled piezoelectric material
produces a voltage of the same polarity as the poling direction for compressive force and on the other hand, voltage in the opposite direction is produced when tensile force is
applied. This method is the simplest way to measure the
d
33
coefficient by using standard laboratory equipment [20]. The indirect method or converse method is an
opposite technique, where voltage is applied to generate deformation to the piezoceramic dimensions without changing the material volume. The relation of
applied field and developed strain is given in equation 2-3. When a voltage of opposite polarity is applied to the piezoceramic, the material will be compressed and voltage of
the same polarity will induce an expansion along the poling axis.
Resonant frequency measurement is one of the dynamic methods used to determine the piezoelectric and elastic properties of the ceramics. Since frequencies are very easily
N j
i d
ij
C direction
in stress
Applied direction
in density
charge circuit
Short
V i
j d
ij
m direction
in field
Applied direction
in developed
Strain
11 and accurately measured, this method provides a good basis for measuring the
properties of piezoelectrics [19]. This method involves the measurement of the resonant,
f
r
and antiresonant,
f
a
frequencies which are influenced by the dimensions of the material and the clamping condition. When excited at the resonant frequency, the
ceramic will resonate with greater amplitude which corresponds to the lowest impedance and follow by an antiresonant frequency, where the amplitude of the
oscillation become minimum, which corresponds to the highest impedance in the circuit as shown in Figure 2-4.
Figure 2-4: Impedance of a piezoelectric ceramic at resonance.
As the thickness of the samples was many times smaller than their widths and lengths
h w
50 and h
l
100, this method is suitable for measuring the piezoelectric constants related to transverse modes, where the direction of polarisation is perpendicular to the
direction of the applied stress. The transverse piezoelectric charge coefficient is given as [19]
2-4
This is related to the resonant frequency,
f
r
, the difference between resonant and antiresonant frequencies,
Δf, the length of piezoelectric material,
l
b
, the density, , and
the permittivity of the piezoelectric materials.
33 T
is the permittivity of the material, and usually compared with the permittivity of vacuum,
8.85
10
-12
Fm and described in a form of relative dielectric constant at constant stress,
K
33 T
. This value is related to
f
r
f
a
2 8
1 2
1
2 33
31
f f
f l
d
r T
r b
