4.2. Mechanical and Electrical Performance
Since the electrical signal output from the energy harvester varies with excitation, an energy harvesting circuit is required to capture or store the electric energy generated by the structure [
35,
36]. While the energy harvesting circuit also has certain requirements for the output signal of the energy harvester, it generally requires the minimum output voltage of the energy harvester to be around 0.5~1.4 V. If the output voltage of the energy harvester is less than the minimum turn-on voltage, the circuit will be in an open circuit and will not be able to transmit or store electric energy. Therefore, the output voltage of the energy harvester is one important electrical characteristic.
As shown in
Figure 6a, the stress of the disk energy harvester was concentrated at the force transmission plate, and the peak stress was 0.45 MPa. As shown in
Figure 6b, the output voltage of the disk energy harvester had an obvious laminar distribution along the axial direction, and the maximum value was 0.62 V. The total electricity generated was calculated to be 1.09 × 10
−9 J.
As shown in
Figure 7a, the stresses in the cymbal energy harvester were concentrated at the top and bottom edges of metal caps with a peak stress of 3.62 MPa. As shown in
Figure 7b, the output voltage of the cymbal energy harvester was obviously distributed in layers along the axial direction, and the maximum value was 6.5 V. The total electricity generated by this energy harvester was calculated to be 9.37 × 10
−8 J.
Compared with the disk energy harvester, the output voltage and total output energy of the cymbal energy harvester were greatly increased. The output voltage was about 10 times higher than that of the disk energy harvester, and the total output energy was an order of magnitude higher than that of the disk energy harvester. However, the cymbal energy harvester showed a significant stress concentration at the change of geometric sizes, and the peak stress was about eight times higher than that of the disk energy harvester.
As shown in
Figure 8a, the stress of the spherical energy harvester was concentrated at the bottom edge of the spherical shell, but there was an obvious uniform gradual change of stress along the surface of the spherical shell, thus effectively reducing the peak stress, which was 1.25 MPa. As shown in
Figure 8b, the output voltage of the spherical energy harvester also showed obvious laminar distribution along the axial direction, and the maximum value was 6.7 V. The total electricity generated by the spherical energy harvester was calculated to be 1.17 × 10
−7 J.
Compared with the disk energy harvester, the total output energy of the spherical energy harvester was greatly improved, being two orders of magnitude higher than that of the disk energy harvester. And compared with the cymbal energy harvester, although the spherical energy harvester also showed some stress concentration at the bottom of the spherical shell, the peak stress was reduced by 65% compared with the cymbal one due to the good stress homogeneity of the spherical shell structure. The spherical energy harvester had both good mechanical properties as well as superior electrical properties. Its output voltage was 11 times more than that of the disk energy harvester, and the total output electric energy was 20% more than that of the cymbal energy harvester.
4.3. Load Match
In general, the piezoelectric energy harvester can be equivalent to an AC power supply with internal resistance, and part of the generated electric energy was consumed by internal resistance, while the other part was used for the load resistor to do work. Therefore, for the piezoelectric energy harvester, load resistance matching is particularly important, and a suitable external load is beneficial to improve the output performance of the energy harvester. For the piezoelectric energy harvester, it is generally regarded as a sinusoidal alternating voltage source with constant amplitude, while for the piezoelectric material itself, it has capacitive and resistive properties.
Figure 9 shows the equivalent circuit model of the piezoelectric energy harvester. Here, V is the voltage source;
R0 is the equivalent internal resistor of the energy harvester;
C0 is the equivalent capacitance; and
RL is the load resistor.
For the above equivalent circuit, the capacitance
C0 is calculated as Equation (1):
where
(
) represents the vacuum dielectric constant;
denotes the relative dielectric constant;
indicates the dielectric constant in F/m;
is the piezoelectric layer area; and
is the piezoelectric layer thickness.
The capacitive impedance
, resistance
, and the ratio of impedance to capacitive reactance is
. PZTs are capacitive elements, and the capacitive reactance is far greater than the impedance, so the impedance of PZT can generally be ignored and regarded as a pure capacitive element. Therefore, the general value of
is very small and negligible, and the load voltage is
Since the voltage source is a sinusoidal AC voltage, the load power is
From Equation (3), the output power is maximum when the external load resistance
is equal to the internal impedance
. The value of the load impedance at this time is called the matching impedance, that is,
Therefore, the matching impedance is not only related to the capacitance value of the piezoelectric energy harvester, but also related to the excitation frequency. The output power when the load is matched is
The load match model of disk energy harvester is established in COMSOL. In the global definition, the load resistance “
” is defined, and the resistor
is added with the resistance value “
”, that is, the load resistor is set between the output end of the piezoelectric energy harvester and the ground node of the circuit. After setting up, frequency domain analysis can be used to study the effect of external load resistance on the output power of piezoelectric energy harvester, and the output excitation frequency used was 200 Hz. The voltage on the load resistor varied with the resistance, as shown in
Figure 10a, and the power varied with the resistor, as shown in
Figure 10b.
From the results, it can be seen that as the circuit load resistance increased, the output voltage increased monotonically and the rate of increase (i.e., the slope of the curve) decreased. When the load resistance increased to a near open circuit, the output voltage was 0.61 V, which approached the open circuit voltage of 0.62 V, and the curve tended to be flat. While the output power was low when the load resistance was very small or very large, it first increased and then decreased with the increase in the load resistance, and there was a maximum value. In addition, as can be seen from the shaded part under the curve, when the load resistance was in a certain range (50~200), the output power of the disk energy harvester was about 1.7 × 10−3 mW, which was at a low level. And the output voltage was less than 0.5 V in this resistance range, making it difficult to achieve the minimum input requirements of the energy capture circuit, so the electrical output performance of the disk energy harvester was found to be relatively ordinary. In this range, the load resistance that made the output power optimal was the “matching impedance”. Therefore, when , the maximum output power was 2.03 × 10−3 mW, and the internal resistance of the piezoelectric energy harvester was calculated to be 90.
The load match model of the cymbal energy harvester was established, and the effect of external load resistance on the output power of the cymbal energy harvester was studied by frequency domain analysis. The results of voltage variation with resistance on the load resistor are shown in
Figure 11a, and the results of power variation with resistance are shown in
Figure 11b.
According to the results, as the circuit load resistance increased, the voltage across the resistor increased monotonically and at a decreasing rate. When the load resistance increased to a near open circuit, the output voltage was 5.2 V, which approached the open circuit voltage of 6.5 V, and the curve tended to flatten out. While the output power was low when the load resistance was very small or very large, it first increased and then decreased with the increase in the load resistance, and there was a maximum value in the process. When the load resistance was within a certain range (50~200), the output voltage of the cymbal energy harvester was able to reach the input requirement of the energy harvesting circuit, which was greater than 1.4 V. And the output power reached a high level, up to 0.13 mW. Meanwhile, when , the maximum output power was 0.15 mW, and the internal resistance of the cymbal piezoelectric energy harvester was calculated to be 90.
Compared with the disk energy harvester, the cymbal energy harvester had the same matching impedance of 90. But the output power of the cymbal energy harvester was two orders of magnitude higher than that of the disk type at a better load resistance, and it was up to the microwatt level.
The load match model of the spherical energy harvester was established, and the effect of external load resistance on the output power of the energy harvester was studied through frequency domain analysis. The voltage on the load resistor varied with the resistance, as shown in
Figure 12a, and the power varied with the resistance, as shown in
Figure 12b.
As can be seen from
Figure 12, with the increase in circuit load resistance, the voltage across the resistor increased monotonously, and the growth rate decreased. When the load resistance increased to a near open circuit, the output voltage was 6.53 V, which approached the open circuit voltage of 6.7 V, and the curve tended to level off at this time. The output power was lower when the load resistance was very small or very large, and the law was consistent with other energy harvesters, that is, the power first increased and then decreased with the increase in load resistance, and there was a maximum value in the process.
In addition, it can be seen from the shaded part under the curve that when the load resistance was within a certain range (50~200), the output voltage of the energy harvester was greater than 1.4 V, being able to meet the input requirements of the energy harvesting circuit. And the output power was able to reach a high level, up to 0.2 mW. When , the maximum output power was 0.23 mW, and the internal resistance of the spherical piezoelectric energy harvester was calculated to be 100 .
Compared with the disk and cymbal energy harvester, the matching impedance of the spherical energy harvester was slightly larger than the other two, at 100 . At the better load resistance, the output power of the spherical energy harvester was two orders of magnitude higher than that of the disk one and 53% higher than that of the cymbal energy harvester, which was also able to reach the microwatt level.
The output electrical properties of the three energy harvesting structures was as shown in
Table 3.
The same load of 0.3 MPa was applied to the disk, cymbal, and spherical energy harvesting structures, and the power output efficiencies were 2.03 × 10−3 mW, 0.15 mW, and 0.23 mW, respectively.
In terms of the three types of energy harvesting structures, the disk energy harvester is simple in structure, easy to make, and long in service, but its energy harvesting efficiency is low. The energy harvesting efficiency of the cymbal type is higher than that of the disk type, but due to its complex structure, it can easily produce stress concentration, which is not conducive to long-term service. Compared with the disk energy harvesting structure, the output of the spherical energy harvesting structure is greatly improved. Compared with the cymbal energy harvesting structure, although the spherical energy harvesting structure also has a certain stress concentration phenomenon at the bottom of the spherical shell, the stress concentration phenomenon is weakened due to the good stress equalization of the spherical shell structure. Therefore, the spherical energy harvesting structure not only has good mechanical properties, but also has better electrical properties.