Resonant Measurement Piezoelectric Materials
114
Figure 5-4: Comparison of sample D and C series for the value of capacitance over the ratio of areathickness with ± 5 error.
The resonant and antiresonant frequencies that correspond to the minimum and maximum impedances of the materials are important variables to determine the
piezoelectric constants of the materials. The frequency response of the samples was measured by using NetworkSpectrum Analyser HP 4195A between 100 kHz to 500
MHz. The resonant and antiresonant frequencies for sample D series can be identified by the magnitude of the impedance as shown in Figure 5-5.
There are a few possible modes of vibration in the range of 120 kHz to 280 kHz: lateral, longitudinal and thickness modes. For all the samples of series C and D, the thickness
vibration mode is not significant compared to the lateral and longitudinal modes. This is due to the fact that the length and the width of the samples are more than 50 times
bigger than their thickness.
The lateral vibration mode was observed for samples D1 and D2 which is about 180 kHz, however, the lateral mode diminishes as the length of the sample increases which
can be see in sample D3 – D5 as shown in Figure 5-5. The resonant frequency of the
longitudinal mode for sample D1 is about 240 kHz and reduced to about 185 kHz for sample D5. From equation 2-4, the average value of
d
31
for sample D series is about 33.9 pCN.
115
Figure 5-5: Frequency response for sample D1, D2, D3, D4 and D5, corresponds to their impedance.
Sample D1
Sample D2
Sample D3
Sample D4
Sample D5
116 Similar to the case of sample D series, Figure 5-6 shows the frequency response for
sample C series. Sample C2, which has a square dimension displays two significant vibration modes. One of which is the lateral mode, at a resonant frequency of 165 kHz
and the other one is the longitudinal mode, which happens at around 235 kHz. For sample C1, with its length smaller than its width, the lateral mode occurs at the resonant
frequency similar to sample C2, at 165 kHz, due to the fact that their dimensions are almost similar which results in poor output from longitudinal vibration mode. As the
length of the sample increases and becomes larger than its width sample C3, the longitudinal vibration mode becomes prominent which happens at a resonant frequency
of 178 kHz, while the lateral mode diminishes as the length of the sample increases. The resonant frequency is inversely proportional to the length of the material, as shown in
Figure 5-7, which is in good agreement with equation 2-4. The average value of
d
31
for sample C series is 24 pCN, which is slightly smaller than sample D series.
Figure 5-6: Frequency response for sample C series.
The longitudinal resonant frequencies of sample D and C series are inversely proportional to the length of the structure as indicated in Figure 5-7, which is consistent
with equation 2-7. The elastic compliances at constant electric field,
s
11 E
for sample D ranges from 5.48 × 10
-12
m
2
N to 12.9 × 10
-12
m
2
N and ranges from 5.85 × 10
-12
m
2
N to 13.4 × 10
-12
m
2
N for sample C series, with the assumption that, the density of PZT type-5H is 7400 kgm
3
[31].
100 200
300 400
500
120 140
160 180
200 220
240 260
280
Z
m
O h
m
Excited Frequency kHz
C1 C2
C3
117
Figure 5-7: Resonant frequency as a function of inverse of cantilever length.
The coupling factor for the material can be estimated by substituting the measured values of
d
31
, and
into equation 2-6. Figure 5-8 shows that the coupling factor increases with length of the materials. For example, the coupling factor for sample D
series increases from 0.127 at a length of 6.75 mm to 0.216 at a length of 18 mm, while sample C series has a slight reduced coupling factor of 0.12 at a length of 6.75 mm and
increases to 0.192 at a length of 13.5 mm.
Figure 5-8: Coupling factor of sample D and C series as a factor of material length.
From equation 2-8, the constant displacement elastic compliance,
s
11 D
for sample D series ranges from 5.1 × 10
-12
m
2
N to 12.6 × 10
-12
m
2
N, while that for sample C series
118 ranges from 5.63 × 10
-12
m
2
N to 13.3 × 10
-21
m
2
N. The piezoelectric charge coefficients calculated from equation 2-9 for samples D and C series range from 9.38
× 10
-3
VmN to 11.4 × 10
-3
VmN and 8.3 × 10
-3
VmN to 9.1 × 10
-3
VmN, respectively.
As expected, the impedance reduces as the length of the material increases as shown in Figure 5-9. The minimum impedance impedance at resonant frequency is proportional
to the ratio of thickness to the area of the material. The impedances at resonant frequency were measured for evaluating the mechanical quality factor,
Q
m
of the materials according to equation 2-10. The mechanical quality factor for the samples
was calculated and plotted in Figure 5-10. On average both samples have a Q-factor,
Q
m
of the order of 120. The experimental results obtained by the resonant measurement method for all the piezoelectric properties are summarised in Table 5-2.
Figure 5-9: The impedance at resonance is proportional to the ratio of thickness to the area of the material.
119
Figure 5-10: Mechanical quality factor,
Q
m
for sample C and D series.
Table 5-2: Summary of measurement results from resonant measurement method for sample C and D series.
Piezoelectric Constant C
D C1
C2 C3
D2 D3
D4 D5
Constant electric field
elastic compliance
s
11 E
10
-12
m
2
N 13.4
10.2 5.9
12.9 9.9
7.6 5.5
Constant displacement
elastic compliance
s
11 D
10
-12
m
2
N 13.3
10.0 5.6
12.6 9.5
7.2 5.1
Permittivity 10
-9
Fm 2.9
2.6 2.6
3.6 3.3
3.1 3.0
Relative dielectric
constant K
33 T
dimensionless 325
295 295
4.8 372
347 336
Coupling factor
k
31
dimensionless 0.12
0.15 0.19
0.16 0.19
0.22 0.27
Piezoelectric charge
coefficient d
31
10
-12
CN -29
-26 -21
-39 -32
-25 -22
Piezoelectric voltage
coefficient g
31
10
-3
VmN -8.3
-9.1 -9.1
-9.4 -10.3
-11.0 -11.4
Impedance at resonance
Z
m
Ω 205
188 90
162 103
88 75
Mechanical quality factor
Q
m
dimensionless 99
89 125
100 103
138 130
120