# Design of an Energy Harvester Based on a Rubber Bearing Floating Slab Track
## 4 Performance Comparison of Energy Harvesters
### 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; \(R_{0}\) is the equivalent internal resistor of the energy harvester; \(C_{0}\) is the equivalent capacitance; and \(R_{\text{L}}\) is the load resistor. For the above equivalent circuit, the capacitance \(C_{0}\) is calculated as Equation (1):  
\[C_{0}=\frac{\overline{x}_{33}\text{A}}{\text{H}} \tag{1}\]  
where \(\overline{x}_{33}=\epsilon_{r33}^{T}\times\varepsilon_{0}\) (\(\varepsilon_{0}=8.85\times 10^{-12}\text{ F/m}\)) represents the vacuum dielectric constant; \(\varepsilon_{r33}^{T}\) denotes the relative dielectric constant; \(\overline{x}_{33}\) indicates the dielectric constant in F/m; \(A\) is the piezoelectric layer area; and \(H\) is the piezoelectric layer thickness. The capacitive impedance \(R_{C_{0}}=\frac{1}{2\pi/C_{0}}\), resistance \(R_{0}=\frac{tan\delta}{2\pi/C_{0}}\), and the ratio of impedance to capacitive reactance is \(tan\delta=\frac{R_{0}}{R_{C_{0}}}\). 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 \(tan\delta\) is very small and negligible, and the load voltage is  
\[V_{L}=V\left|\frac{R_{L}}{R_{L}+R_{0}+R_{C_{0}}}\right|\approx V\left|\frac{R_ {L}}{R_{L}+R_{C_{0}}}\right| \tag{2}\]  
Since the voltage source is a sinusoidal AC voltage, the load power is  
\[P_{L}=\frac{V_{L}^{2}}{2R_{L}}\approx V^{2}\left|\frac{R_{L}}{2\left(R_{L}+R_ {C_{0}}\right)^{2}}\right| \tag{3}\]  
From Equation (3), the output power is maximum when the external load resistance \(R_{L}\) is equal to the internal impedance \(R_{C_{0}}\). The value of the load impedance at this time is called the matching impedance, that is,  
Figure 9: Equivalent circuit of the energy harvester. \[R_{match}=\left|R_{C_{0}}+R_{0}\right|\approx\left|R_{C_{0}}\right|=\frac{1}{ \omega C_{0}}=\frac{1}{2\pi f\cdot C_{0}} \tag{4}\]  
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  
\[R_{match}\approx\frac{V^{2}}{8R_{L}} \tag{5}\]  
The load match model of disk energy harvester is established in COMSOL. In the global definition, the load resistance \"\(R_{load}\)\" is defined, and the resistor \(R_{L}\) is added with the resistance value \"\(R_{load}\)\", 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 9(a), and the power varied with the resistor, as shown in Figure 9(b). 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.