## 2 Method
\(C_{P}\), \(R_{P}\) and \(R_{L}\) are the capacitance, resistance due to PEH resistivity and load resistance. \(i\) is the generated current, and \(v\) is the resulting voltage. As our model assumes a purely resistive load, the average load power (Equation (10)) is found by applying a resistive division term to Equation (9). \[ \begin{array}{ll}\text{PEH}:&P=P_{T}\frac{R_{P}}{R_{P}+R_{L}}\\ \text{EMEH}:&P=P_{T}\frac{R_{L}}{R_{W}+R_{L}}\end{array}  \tag{10}\]  
As in the previous literature [7; 8], we normalize the power output by a reference power \(\Big{(}\big{(}\omega^{2}y_{0}\big{)}^{2}m^{2}\Big{)}/C_{M}\), where \(\omega\) is angular excitation frequency and \(y_{0}\) is the base displacement amplitude and use a set of dimensionless parameters according to Table 1. For a more concise power expression, we further parameterize with dimensionless parameters according to Table 2. The same variable substitution as described earlier, regarding A and B, is used in Tables 1 and 2 where applicable. Using Equations (1), (7), (9) and (10) and the dimensionless parameters described in Tables 1 and 2, we can derive a unified expression for dimensionless average load power under inertial load, \(\overline{P}_{IL}\) (see derivation in Supplementary Materials). \[P_{IL}\frac{C_{M}}{\big{(}\omega^{2}y_{0}\big{)}^{2}m^{2}}=\overline{P}_{IL}= \frac{\alpha\beta\xi_{C}}{2\Big{(}\big{(}1+\alpha\big{)}^{2}+\big{(}\beta\gamma \big{)}^{2}-2\varepsilon\alpha\beta\gamma+\varepsilon^{2}\Big{(}1+\big{(} \beta\gamma\big{)}^{2}\Big{)}\Big{)}} \tag{11}\]  
Equations (7), (9) and (10) can be used to derive the equivalent expression under prescribed displacement since \(x(t)=y(t)\). As the effect of electric damping on proof mass displacement is negated in this case, the load power under prescribed displacement (\(P_{PD}\)) can be derived in a more direct way by taking the square of the RMS voltage (for a PEH) or current (for an EMEH) acting on the load, multiplied or divided by the load resistance (see derivation in Supplementary Materials). We can express this in a unified manner using the same set of dimensionless parameters as before. For comparability with the case of inertial load, we can also extract the same reference power as mentioned previously, which results in Equation (12). \[\overline{P}_{PD}=2\zeta^{2}\frac{\alpha\beta\xi_{C}}{\gamma^{2}\Big{(}1+ \big{(}\beta\gamma\big{)}^{2}\Big{)}} \tag{12}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Expression** & **Name** \\ \hline \(Q\) & \(m\omega_{N}/C_{M}\) & Mechanical quality factor \(\big{(}\omega_{N}=\sqrt{K_{M}/m}\big{)}\) \\ \hline \(\zeta\) & \(1/(2Q)\) & Mechanical damping ratio \\ \hline \(\eta\) & \(\theta^{2}/K_{M}B\) & Effective electromechanical coupling coefficient \\ \hline \(\gamma\) & \(\omega/\omega_{N}\) & Normalized angular excitation frequency \\ \hline \(\xi_{C}\) & \(\begin{array}{l}PEH:\ 1/R_{L}\omega_{N}B\\ EMEH:\ R_{L}/\omega_{N}B\end{array}\) & Load coefficient \\ \hline \(\xi_{E}\) & \(\begin{array}{l}PEH:\ 1/R_{P}\omega_{N}B\\ EMEH:\ R_{W}/\omega_{N}B\end{array}\) & Resistive loss coefficient \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensionless parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Expression** \\ \hline \(\alpha\) & \(\theta^{2}/C_{M}A\) \\ \hline \(\beta\) & \(1/\big{(}\xi_{C}+\xi_{E}\big{)}\) \\ \hline \(\varepsilon\) & \(Q(\gamma^{2}-1)/\gamma\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensionless parameters used to simplify the power expression. The efficiency is defined as the average generated power by average input power [1]. For the case of inertial load, we use the input power as derived by [PERSON] et al. (see Equation (13))  
\[P_{In}=F\times\dot{y}=m\big{(}\ddot{x}+\ddot{y}\big{)}\dot{y} \tag{13}\]  
Given the derivations in [1], this results in the average input power given by Equation (14).