Wind Energy Harvesting System - Energy Harvesting Autonomous Sensor Systems - page 75

where L , w , and t are the bimorph piezoelectric generator length, width,
and thickness, respectively.

33 is the dielectric constant of the piezoelectric
material under a free condition; d 31 is the transverse piezoelectric coefficient,
and k 31 is the transverse piezoelectric coupling coefficient. When only an
external tip force F is acting on the tip of the bimorph piezoelectric generator
( x

L ), the generated electric charge in the bimorph defined as Equation
2.25 by Wang et al. [80], becomes

=

3 d 31 L 2
2 t 2

Q = F (2.26)

For a bimorph piezoelectric generator, two pieces of piezoelectric material
are mechanically bonded together as a whole; hence, the overall dielectric
constant of the bimorph piezoelectric generator is smaller than the free dielec-
tric constant of only the piezoelectric material. As such, the overall dielectric
constant

b is given as

33 1

4

X

k 31 /

b = − (2.27)

The total capacitance of the bimorph piezoelectric generator can be obtained
by

( overall dielectric constant )( electrode surface area )
thickness separating the electrodes

C = (2.28)

33 Lw 1

4

X

k 31 /

−

C = (2.29)

t

Based on Equations 2.26 and 2.28 , the open circuit electric voltage V oc gen-
erated by the bimorph piezoelectric generator when an external tip force is
applied is expressed as

Q
C

3 d 31 L

33 wt 1

4

V oc = = F (2.30)

X

−

k 31 /

2

X

The dielectric constant

33 , piezoelectric constant d 31 , and coupling coefficient
k 31 of the piezoelectric material and the dimensions of the bimorph are given
by the vendor in the technical datasheet. The properties of the piezoelectric
material are given in Table 2.2 .
Alternatively, the open circuit electric voltage V oc generated by the bimorph
piezoelectric generator can also be a function of the tip deflection y L . Rear-
ranging Equation 2.24 , it can be seen that the external tip force F is related to
the tip deflection of the piezoelectric generator by

3 E y Iy L
L 3

F = (2.31)

Since the Young modulus E y of the piezoelectric material is known in
Table 2.2 and the moment of inertia I of the cantilever piezoelectric generator
can be calculated using Equation 2.16 , the open circuit electric voltage V oc