## Materials and Methods  
As explained in [12], the majority of published works on vibration energy harvesting using multiple PEC beams are difficult to transfer to industrial applications. An engineer wishing to design a broadband vibration harvester for a particular application usually has to rely on commercially available PEC beams, which are not available off the shelf with any geometry, and on simple circuit simulation software for predicting the performance of the entire electric circuit. During the design stage of a PE energy harvester, it is important to establish a particular analytical or numerical model to estimate the output power of the system [20]. A review of such models is given in [20]. In this paper, we shall rely on equivalent circuit (EC) modelling of PE energy harvesters combined with the power conversion circuit and the electrical load. This approach is similar to that described in [21]. The model is based on the Butterworth-Van Dyke EC of a single PE element, but can be easily generalized to incorporate the more complex results from the finite-element analysis [20]. Since the EC model is kept simple, the freeware SPICE (Simulation Program with Integrated Circuit Emphasis) simulator of electronic circuits LTspice (Linear Technology Corporation, Milpitas, CA, USA) can be employed. In order to be specific, the commercially available PE vibration energy harvester Mide(tm) PPA-1022 (Mide Technology Corporation, Medford, OR, USA) is used through the entire work [22]. It is a unimorph cantilever, where a single active PE layer (made of the PZT-5H material) is sandwiched between conductive and glass-reinforced epoxy laminate sheets [22]. By attaching an additional tip mass, the electromechanical resonance frequency of the circuit can be shifted below the resonance frequency of the free beam (nominal value is 408.4 Hz, \(m_{\text{tip}}=0\) g). The lumped model of a single PEC is shown in Figure 1 with the parameters of the linear EC model, tuned to the experimental resonance frequency of 40 Hz. Figure 2 compares the simulated (denoted as \"sim\") and measured (denoted as \"meas\") results for the electrical power dissipated in the load resistance of 275 k\(\Omega\) dependent upon the mechanical frequency \(f\) at a peak acceleration of 0.5 \(g_{\mathrm{n}}\), where \(g_{\mathrm{n}}\) is the standard gravity. A softening-spring behavior is observed. Obviously, the conventional simulation (\"con sim\") cannot completely describe the nonlinear behavior of the PEC beam circuit. The behavior is reproduced very well only in the vicinity of the resonance frequency. When simulating an array with several beams, this nonlinear asymmetry effect is known to introduce some deviations between measurements and simulations [23]. We propose a simple adjustment to the common equivalent circuit for obtaining better simulation results of the entire array. From the recorded curve the bandwidth limits (half-power frequencies) \(f_{1}\) and \(f_{2}\) have to be determined. These frequency points are defined by the value of half of the maximum power, which can be extracted with the optimum resistance load. To match the common simulation, a new resonance frequency \(f_{0}\)' has to be calculated and the curve is shown in Figure 2 (\"adj sim\"). This new frequency is the geometric mean value of \(f_{1}\) and \(f_{2}\):  
\[f_{0}^{\prime}=\sqrt{f_{1}f_{2}} \tag{1}\]  
Figure 2 shows that the resonance frequency of the adjusted simulation is slightly larger, but the half-power frequencies are reproduced correctly. This improvement leads to a much better similarity between the experimental and simulation results over a wider frequency band. The proposed correction changes to the EC parameters are small. The parameter adjustment works well since the quality factor of the resonance circuit is large (\(Q\approx 25\)). In the following, the lowest frequency limit of the array is determined by the lowest resonance frequency of a single (first) element, \(f_{0}\) = 40 Hz (\(m_{\mathrm{tip}}\) = 7.4 g).