# A Vibrational Energy Harvesting Sensor Based on Linear and Rotational Electromechanical Effects
## 6 Empirical and Model Analysis of the VEH Sensor
### Harvesting from Low-Level Non-Stationary Signals
The similarity between the Figures 5 and 8 is evident in the bandwidth (130\(-\)150 Hz). In this region, the sensor has an average rms harvesting efficiency of 6.3%, with a peak value of 9.8%  
Figure 8: VEH sensor harvesting efficiency defined in terms of the ratio of the powers: \(\eta=P_{\text{out}}/P_{\text{in}}\). Here, \(P_{\text{out}}\) is defined as the output voltage power spectrum per Hz across the 50 \(\Omega\) load resister (the red line of Figure 7). \(P_{\text{in}}\) represents the mechanical power spectrum per Hz of the combined mass of the coil and magnets (the black line of Figure 7) as a function of frequency over the 130\(-\)150 Hz region of interest. The rms harvesting efficiency for this region is \(\eta_{\text{rms}}=6.3\%\), with a peak harvesting efficiency of \(\eta_{\text{pk}}=9.7\%\) at 138 Hz. at 138 Hz. Here, the impedance match between the coil resistance, 60 \(\Omega\), and the load resistance, 50 \(\Omega\), supports that the approximation is close to optimal in terms of impedance matching with the load resistance. In a third set of measurements, the harvesting performance of the VEH prototype is recorded while connected to the EH300 harvesting circuit. The harvested voltage across the 50 \(\Omega\) load resistor is connected to the input buffering circuit of the EH300. The oscilloscope is now used to record the voltage gain across the circuit's storage capacitor, and as before, the sensor is exposed to the same band limited random acceleration signal described previously. The total harvested voltage from the VEH sensor in the EH300 is recorded for approximately 7 min. The sampling rate for the oscilloscope was set at 0.2 ms, which resulted in a data record of approximately 20 k samples. This is the largest window that could be held in the oscilloscope's window before the first samples are lost. The sampling rate ensures a Nyquist limit of 5 kHz, which is a factor of 25 times larger than the highest frequency associated with the noise spectrum. Figure 9 shows a linear fit of the data results in a voltage harvesting rate of approximately 10 mV h\({}^{-1}\) for random acceleration noise \(\hat{y}_{\text{rms}}\sim 10^{-2}\) g. The spikes in the temporal voltage signal may be generated in part by electronic noise present in the unshielded circuit. Another consideration is that the random acceleration magnitudes may be introducing the fluctuations in the voltage in the storage capacitor. Finally, it is possible to estimate the VEH sensor's power density by normalizing the output power with the volume, \(\text{Vol}_{\text{mag}}\), of the effective moving elements (i.e., the volume of the two coil and magnet assemblies [29]). This can be calculated as \(P=\left|V_{\text{out}}\right|^{2}/\left(R_{l}\Delta f\right)/\text{Vol}_{ \text{mag}}\), with a resulting power density of 0.2 \(\upmu\text{W}/\text{cm}^{3}\). With this information, the prototype sensor can be compared to existing results published in the literature [30]. Figure 9: VEH sensor charging performance over a seven\(-\)minute charging window using the EH300 interface circuit. Blue line: Raw DC voltage across the output terminals from the EH300. Red line: Fitted data using least squares approximation. The measured harvesting performance is approximately 10 mV h\({}^{-1}\), as harvested from the random acceleration input acceleration power spectra shown in Figure 7.