## 2 Method  
To find a common expression for PEH and EMEH power output, we use the second-order mass-spring-damper system and describe the proof mass movement of both PEHs and EMEHs, using Equation (1). Proof mass displacement is expressed as \(x(t)\), \(m\) is the effective mass, \(C_{M}\) is the total mechanical damping factor, \(K_{M}\) is the mechanical spring constant, \(y(t)\) is the base displacement, and \(f\) is the damping force resulting from extracting electrical energy from the system. The corresponding free-body diagram is shown in Figure 1 (right). \[m\ddot{x}+C_{M}\dot{x}+K_{M}x+f=m\ddot{y} \tag{1}\]  
The electrical damping force, \(f\), for the PEH and EMEH is given by Equation (2). \[ \begin{array}{ll}\text{PEH}:&f=\theta v\\ \text{EMEH}:&f=\theta i\end{array}  \tag{2}\]  
Here, \(v\) and \(i\) are the instantaneously generated voltage and current for the respective VEH. \(\theta\) is a system-specific coupling factor and is a function of harvester dimensions and material properties, such as the piezoelectric coefficient or the magnetic remanence. The coupling factor relates displacement to current or voltage by:  
\[ \begin{array}{ll}\text{PEH}:&\theta\dot{x}=i\\ \text{EMEH}:&\theta\dot{x}=v\end{array}  \tag{3}\]  
Figure 1: Representation of PEH (**left**) and EMEH (**middle**), with a corresponding free-body diagram (**right**). \(C_{E}\) and \(C_{M}\) are the electrical and mechanical damping coefficients. \(K_{M}\) is the mechanical spring coefficient. \(m\) is the effective mass of the system. \(x(t)\) is the proof mass or tip displacement, and \(y(t)\) is the base displacement. Next, we use the lumped element model in the electrical domain to express the generated power as a function of \(f\). In Figure 2, we model the EMEH as a voltage source, \(v\), in series with an ideal inductor, \(L\), and winding resistance, \(R_{w}\). The PEH is modeled as a current source, \(i\), in parallel with an ideal capacitance, \(C_{P}\), and an electric resistance, \(R_{P}\), due to the resistivity of the PEH. We use [PERSON]'s first and second law to find the relationship between current and voltage in each case. \[ \begin{split}\text{PEH}:&\qquad i=C_{P}\dot{v}+ \frac{v}{R_{L}}+\frac{v}{R_{P}}\\ \text{EMEH}:&\qquad v=L\frac{di}{dt}+(R_{w}+R_{L})i \\ \end{split}  \tag{4}\]  
We can use Equation (4) together with Equations (2) and (3) to express proof mass displacement velocity as a function of the electrical damping, resulting in Equation (5). \[ \begin{split}\text{PEH}:&\qquad\dot{x}=\frac{1}{ \theta^{2}}\Big{(}C_{P}\dot{f}+\frac{f}{R_{L}}+\frac{f}{R_{P}}\Big{)}\\ \text{EMEH}:&\qquad\dot{x}=\frac{1}{\theta^{2}} \Big{(}L\dot{f}+(R_{w}+R_{L})f\Big{)}\\ \end{split}  \tag{5}\]  
To find a general governing equation, we perform the variable substitutions \(A=1/R_{T}\) and \(B=C_{P}\) for the PEH and \(A=R_{T}\) and \(B=L\) for the EMEH. \(R_{T}\) is the total output resistance and is given by Equation (6):  
\[ \begin{split}\text{PEH}:&\qquad R_{T}=\frac{R_{P}R_ {L}}{(R_{P}+R_{L})}\\ \text{EMEH}:&\qquad R_{T}=R_{w}+R_{L}\\ \end{split}  \tag{6}\]  
Applying the above-defined variables to Equation (5) yields a common expression for the proof mass velocity, as given by Equation (7). \[\dot{x}=\frac{1}{\theta^{2}}\Big{(}Af+B\dot{f}\Big{)} \tag{7}\]  
The total average power, generated over the resistive components, can be derived from the RMS voltage or current as:  
\[ \begin{split}\text{PEH}:&\qquad P_{T}=A\Big{(}|v|/ \sqrt{2}\Big{)}^{2}\\ \text{EMEH}:&\qquad P_{T}=A\Big{(}|i|/\sqrt{2} \Big{)}^{2}\\ \end{split}  \tag{8}\]  
Using Equations (2) and (8), we can arrive at a general expression for the average power:  
\[P_{T}=\frac{A\Big{|}\dot{f}\Big{|}^{2}}{2\theta^{2}} \tag{9}\]  
Figure 2: Lumped element model of electrical domain. **Left**: EMEH. \(L\), \(R_{w}\) and \(R_{L}\) are the coil inductance, coil resistance and load resistance. \(v\) is the generated voltage, and \(i\) is the resulting current. **Right**: PEH.