# Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network
## 2 Stochastic Differential Equations and Energy Harvesting System Modeling
Here, \(m\) is the inertial mass, \(x\) represents the displacement of the mass from the rest position (dots denote derivation with respect to time), \(\gamma\) is the internal friction constant, \(U(x)\) is the elastic potential of the beam (symbol \({}^{\prime}\) denotes derivation with respect to the argument), \(\alpha\) is the electro-mechanical coupling constant (in N/V or As/m), \(e\) is the output voltage of the piezoelectric transducer, and \(f_{\mathrm{ext}}(t)\) is the external force due to the vibrating support that, as discussed above, will be modeled as white Gaussian noise. Equation (2b) describes the piezoelectric transducer, where \(C_{\mathrm{p}\dot{x}}\) is the electrical capacitance of the transducer, and \(I_{\mathrm{L}}\) is the current through the electrical load. This model was experimentally validated by several groups; see, for instance, [14; 16; 26; 29; 44]. Finally, we consider the circuit description of the load. A matched load is composed by an electrical element that absorbs average power, such as a sensor, an actuator or a battery, modeled by a resistor (Figure 2b) and by a matching network (Figure 2a) made of reactive elements (inductors and/or capacitors) that do not absorb average power, designed to reduce the impedance mismatch between the load and the power source. Perfect matching can be achieved only at one specific frequency, even if wide-band matching networks can be designed to achieve a partial matching over a relatively wide frequency interval. In this work, we consider a low-pass L-matching network, such as the one shown in Figure 2a. The network is composed by an inductor and a shunt capacitor, connected to form an L-shaped structure. The network behavior is low-pass, as at very low frequency, the inductor and the capacitor behave as a short circuit and an open circuit, respectively, and thus \(v_{o}(t)=e(t)\). Conversely, at very high frequency, the asymptotic behavior of the inductor and of the capacitor are reversed so that \(v_{o}(t)=0\) V. The low-pass behavior is the most convenient for mechanical vibrations scavenging, as the energy of ambient vibrations is mostly concentrated at low frequencies. Application of Kirchhoff current law and of the capacitor characteristic relationship gives  
\[-I_{L}+C_{P}\dot{v}_{o}+G_{L}v_{o}=0 \tag{3}\]  
Similarly, application of Kirchhoff voltage law and using the inductor characteristic relationship provides  
\[L_{S}\dot{I}_{L}+v_{o}-e=0 \tag{4}\]  
Combining (2), (3) and (4), and rewriting as an SDEs system, finally yields  
\[dZ_{1}= Z_{2}\,dt \tag{5a}\] \[dZ_{2}= \frac{1}{m}\big{(}-U^{\prime}(Z_{1})-\gamma\,Z_{2}-\alpha\,Z_{3} \big{)}\,dt+\frac{\varepsilon}{m}dW_{t}\] (5b) \[dZ_{3}= \frac{1}{C_{\mathrm{P}z}}(\alpha Z_{2}-Z_{4})\,dt\] (5c) \[dZ_{4}= \frac{1}{L_{5}}(Z_{3}-Z_{5})\,dt\] (5d) \[d\mathbf{Z}_{t}=\mathbf{a}(\mathbf{Z}_{t})dt+\mathbf{B}(\mathbf{Z}_{t})dW_{t} \tag{1}\]0  
where \(\mathbf{Z}_{t}=[Z_{1},\dots,Z_{5}]^{T}=[x,\dot{x},e,I_{L},v_{o}]^{T}\) is the state vector of electro-mechanical variables, and \(\varepsilon\,dW_{t}\) is the external forcing that models ambient vibrations as a white Gaussian noise process with intensity \(\varepsilon\). To assess the performance of the matching network, we compare the output average power and the power efficiency of the matched load with those of a simple resistive load, as that shown in Figure 2b, which is used as a benchmark. For the resistive load, [PERSON]'s law gives \(I_{L}=G_{L}v_{o}\) that, combined with (2) and rewritten as an SDEs system, yields  
\[d\mathbf{Z}_{t}=\mathbf{a}(\mathbf{Z}_{t})dt+\mathbf{B}(\mathbf{Z}_{t})dW_{t} \tag{1}\]1 \[d\mathbf{Z}_{t}=\mathbf{a}(\mathbf{Z}_{t})dt+\mathbf{B}(\mathbf{Z}_{t})dW_{t} \tag{1}\]2 (6b) \[d\mathbf{Z}_{t}=\mathbf{a}(\mathbf{Z}_{t})dt+\mathbf{B}(\mathbf{Z}_{t})dW_{t} \tag{1}\]3  
where \(\mathbf{Z}_{t}=[Z_{1},Z_{2},Z_{3}]^{T}=[x,\dot{x},e]^{T}\) is the state vector of electro-mechanical variables.