55 where
A
p
is the area ratio of the PZT layer to the total cross-sectional area of the composite beam. Solving equation 3-31 for a beam attached with proof mass, we get,
3-33
Substituting equation 3-5, 3-18, 3-28 and 3-32 into 3-33, we get,
3-34
3.4.5 Estimated Output Voltage
The output voltage for a piezoelectric cantilever can be estimated with equation 2-15 deduced from the Roundy’s dynamic model [12] Appendix C. Although the model is
oversimplified, it does give a reasonably good approximation of the amount of voltage generated.
At resonant frequency, equation 2-15 can be simplified as,
3-35
where
e
T
is the resultant elastic modulus as defined in equation 3-32 and
d
is the distance from the centroid of the layer of PZT to the neutral axis of the structure as
defined in equation 3-22.
unimor ph T
b eff
I e
l z
y M
z
3
3
2 2
2 2
1 2
1 4
2 1
1 1
1 1
3 236
. 236
. r
r D
h h
h h
n y
l w
M h
h h
l z
unimor ph p
e e
p ep
b b
p p
e e
e b
p r
T r
r T
b in
p T
RC k
j l
da h
d je
V
2 4
3
2 31
2 2
2 31
56
3.5 Analysis and Discussion on Calculation Results
A cantilever as shown in Figure 3-6 with standard dimensions as summarised in Table 3-2 is used to verify the model as derived in section 3.4. The standard parameters for the
calculation are also incorporated into the same table.
Figure 3-6: Diagram of a multimorph cantilever structure. The theoretical model was based on a unimorph sandwiched structure of PZT, lower and upper electrodes.
Table 3-2: Standard dimensions of a cantilever used to verify theoretical model.
The calc
ulat ion
take s
into acc
oun t of
the effe
ct of
cant Dimension
Unit Value
Length
l
b
mm 18
Width
w
b
mm 9
PZT thickness
h
p
µm 80
Lower electrode thickness
h
e1
µm 15
Upper electrode thickness
h
e2
µm 20
PZT density kgm
3
7400 Electrode density
kgm
3
10900 Base excitation
A
in
ms
2
10 PZT elasticity
e
e
GPa 116
Electrode elasticity
e
p
GPa 60
Piezoelectric charge constant magnitude
d
31
pCN
1
50 Dielectric permittivity
nFm
1
4
Potential Free-Standing
Structure Length, l
b
Width, w
b
Solder Pad Sacrificial
Layer
