146 Figure 6-21 shows a comparison of the maximum output power and resonant frequency
for sample D3, C2 and IDa1. Sample D3, with thinner non-active PZT protective layer compared to the other samples, was expected to have the lowest resonant frequency at
875 Hz. Although samples C2 and IDa1 were printed with similar numbers of layers of films, their resonant frequencies are slightly different, at 1155 Hz and 960 Hz
respectively. The difference maybe because the effective thickness of an IDE sample is less than the plated sample and therefore resonance occurs at a lower frequency.
Figure 6-19: Output power of sample IDa1 as a function of resistive load at an acceleration of 0.05 g and 0.5 g.
147
Figure 6-20: Frequency response comparison for sample D3, C2 and IDa1 at an acceleration of 0.1 g
and with resistive load of 30 kΩ.
Figure 6-21: Comparison of output power and natural frequency for sample D3, C2 and IDa1 at 0.05 g.
Resonant Freq.
Output Power
148
6.6 Energy Conversion Efficiency
The efficiency of energy conversion from mechanical to electrical energy is given by [115],
6-10
where
k
is the actual coupling factor k
31
of the piezoelectric free-standing structure after taking into account its dielectric and mechanical losses, which can be measured
from experiment when the optimum electrical resistive load,
R
opt
is known. The value is derived from [12] as,
6-11
where
s0
is the angular resonant frequency,
C
p
is the capacitance of the material and
T
is the total damping ratio. The damping ratio is relatively small 0.05 for the ceramic structure, and can be determined experimentally by measuring the
Q
-factor as shown in equation 6-8.
The optimum electrical resistive load,
R
opt
can be derived from equation 3-35 by differentiating the output power with load resistance. At optimum output power,
dP dR
= 0, this gives,
6-12
Figure 6-22 shows the relationship between coupling factor,
k
with the optimum resistive load,
R
opt
and total damping ratio,
T
. The coupling coefficient is equal to zero when the load resistance is equal to the inverse of the natural frequency multiplied by
the capacitance of the material
p r
opt
C R
1
. For the case of
p r
opt
C R
1
, the coupling coefficient is almost uniform with changing resistive load. The coupling
coefficient changes critically in the range of
p r
opt p
r
C R
C
1 1
5 2
, therefore the
2 2
2 2
1 2
1 k
k Q
k k
E
T eff
4 1
2 2
4 2
T p
s opt
T
C R
k
4 2
4 2
1
k C
R
T T
p r
o p t
149 coupling coefficient can be improved with adjusting the optimum resistive load in this
region, with the assumption that the mechanical damping can be modified.
Figure 6-22: Coupling coefficient as a function of optimum resistive load for different damping ratio for sample D5, with resonant frequency at 505.5 Hz and capacitance of 6.82 nF.
Figure 6-23 shows the relationships between the efficiency of energy conversion for a piezoelectric cantilever with the coupling factor and
Q
-factor. The efficiency can be improved by increasing the coupling factor and the
Q
-factor of the cantilever structure.
Figure 6-23: Efficiency of energy conversion as a function of coupling coefficient and
Q
-factor.
= 0.25 = 0.1
= 0.01 = 0.001
