###### Abstract
performs a similar analysis to that conducted by [PERSON] et al. [8] (although theirs includes the aspect of harvesting efficiency) but assumes an effective coupling coefficient under the critical value, and thus the anti-resonant point is lost in the analysis. Contrary to [8], [PERSON] et al. neglects the intrinsic resistance for the PEH/EMEH (i.e., coil and dielectric resistance), which is common practice for PEHs but is not generally applicable for an EMEH and results in incorrect optimal load resistance [6; 14]. [PERSON] et al. [15] derive separate dimensionless expressions for the average load power generated from a PEH and EMEH. Optimization here is primarily performed and analyzed relative to the electrical damping ratio and normalized excitation frequency. The case of nonnegligible intrinsic resistance is only considered for the EMEH and treated only numerically for two discrete values of the ratio of load resistance to intrinsic resistance. Their analysis deviates from the common approach in that they mainly consider the case of constant base displacement amplitude. The critical effective electromechanical coupling coefficient, as defined in [3], is only treated for the case of a PEH and only discussed regarding the optimal load resistance. In [16], [PERSON] et al. derive a general expression for the PEH power output using the equivalent circuit method, in which the entire system is expressed and solved in the electrical domain. The same theoretical power limit as mentioned previously is derived here and applies regardless of the interface circuit. A significant contribution of their analysis is an increased understanding of the critical electromechanical coupling coefficient and how it is affected by various interface circuits. The intrinsic resistance, and thus resistive loss coefficient, is not included in the analysis, and the characteristics of the resonant and anti-resonant solutions are not discussed. [PERSON] et al. and others [1; 7; 17] arrive at the same expression for PEH efficiency. [PERSON] [18] notes that previous models of PEH efficiency are lacking in a common approach to calculation. According to [PERSON], the theoretical efficiency over-estimates the real scenario as the damping from air viscosity is not considered. Although the analysis by [PERSON] et al. [7] includes efficiency, in the context of both PEHs and EMEHs, it does so under the assumption of the negligible effect from intrinsic resistance. In this paper, we use the lumped second-order mass-spring-damper system for the mechanical domain, and the lumped element system for the electrical domain, to derive a common dimensionless expression for generated power and efficiency, valid for both PEHs and EMEHs. In order to capture all the essential VEH characteristics, our model consists of a mechanical quality factor, an effective electromechanical coupling coefficient, an excitation frequency normalized by the natural frequency and two dimensionless coefficients accounting for intrinsic resistance and load resistance, defined as the resistive loss coefficient and load coefficient. Using the same set of parameters, we derive expressions for output power and efficiency based on system excitation from both inertial load and prescribed displacement. The comparison between these two types of excitations has not been performed in the previous literature. To show the effect of all input parameters, in a relevant and clear way, we perform a numerical investigation in the domain of the mechanical quality factor and resistive loss coefficient, under the condition of optimal excitation frequency and load resistance. The effective electromechanical coupling coefficient is also varied to examine its dependence. As there are local power optima at both resonance and anti-resonance, we characterize the model at both points. Based on typical systems parameters, we compare the EMEH and PEH in the above defined space.