46 electronic applications, usually the AC voltages generated by a micro-generator are
converted to usable DC voltage. In this conversion, diodes are normally used for simple full wave direct rectification, which need a minimum forward voltage of 300 mV for
each diode to operate. The minimum voltage was able to be reduced to 150 mV by replacing the diodes with active switches in a four stages voltage multiplier circuit as
studied by Saha
et al
[89]. A few tens of micro-watts of electrical power are needed for powering ultra low-power electronics, MEMS sensors and RF communications system.
As reported by Torah
et al
[65], 58 W of power is needed to power an accelerometer
based micro-system.
3.4 Theoretical Analysis of Multilayer Structures
Generally, the base excited harmonic motion is modelled as a spring-mass-damper system with the equation of motion [90],
3-1
where
y
denotes the displacement of the base and
x
the displacement of the mass from its static equilibrium position. The vibration body is assumed to have a harmonic
motion, 3-2
By defining the relative displacement
z
=
x
–
y
. The magnitude of the displacement and acceleration can be derived as,
3-3
3-4
and the phase difference is, 3-5
y x
y x
b x
M
t Y
t y
sin
Y
r r
r Z
2 2
2 2
2 1
Y
r r
Z
2 2
2
2 1
1
2
1
1 2
tan
r r
47 where
r
and ζ is the frequency and damping ratio respectively as,
3-6
3-7
3.4.1 Natural Frequency of a Unimorph Cantilever
From the Bernoulli-Euler equation derivation, a thin cantilever beam with one end clamped and the other end free, the natural transverse vibration can be written as,
3-8
where
v
i
is a coefficient related to boundary conditions,
h
is the total thickness of the cantilever beam,
l
b
is the length of the cantilever beam,
e
T
is the resultant elastic modulus and
is the density of the structure. However, for a more detail analysis on each layer of the structure, the Bernoulli-Euler equation can be derived in a term related
to bending modulus Appendix B as,
3-9
where
m
w
is the mass per unit area. The coefficient,
v
i
of the first three modes are:
3-10
The natural frequency of a multilayer cantilever consists of piezoelectric and electrode can be calculated accurately, if the thicknesses of the piezoelectric layer,
h
p
and electrode layer,
h
e
are known. Assume that the lengths of the piezoelectric and electrode are similar to the beam length,
l
b
and thickness of upper electrode and lower electrode are
h
e
. The mass per unit area of the cantilever for a unimorph as shown in Figure 3-2 is
n
r
n
M c
2
w b
i i
m D
l v
f
2 2
2
875104 .
1
1
v
694091 .
4
2
v
854757 .
7
3
v
T b
i i
e l
h v
f
2 2
2 2
