## 8 Developing a Methodology to Evaluate the Effectiveness of Piezoelectric Energy Harvesters  
The effectiveness of PEHs can be defined as the ability to reach the maximum available power according to [PERSON] et al. (2018). The investigations showed how important the volume criterion is in piezoelectric energy harvesters. Hence, the effectiveness must be estimated based on the entire volume occupied by the proposed structure not only the volume of the transducer. The bandwidth of PEHs is first discussed. Then, a procedure on how to estimate the effectiveness is suggested. For broadband piezoelectric energy harvesters, the average power as a term is more convenient than the maximum power due to the variation in the power output at different excitation frequencies. For a linear transducer, the average power for a harmonic excitation frequency varying from \(\Omega_{\text{min}}\) to \(\Omega_{\text{max}}\) can be expressed according to [PERSON] et al. (2018) by Equation (15). \[P_{av}|_{linear}=\frac{1}{\Omega_{\text{max}}-\Omega_{\text{min}}}\int\limits _{\Omega_{\text{min}}}^{\Omega_{\text{max}}}P(\Omega)d\Omega\,, \tag{15}\]  
where \(\Omega_{\text{min}}\) and \(\Omega_{\text{max}}\) are the minimum and the maximum frequencies in the bandwidth of the PEH, respectively. The bandwidth of vibration energy harvesters can be defined as the band of frequencies over which a specific, steady level of power can be supplied to a certain value/range of the applied load resistance. This means that, for the same energy harvester, there are infinite values of bandwidth and a corresponding average power for each bandwidth. It can be seen in Figure 13 that the average power over the bandwidth BW1 is less than the peak power but larger than the average power over the bandwidth BW2. Multiplying the bandwidth BW1 by its corresponding average power gives a different result than multiplying BW3 by its corresponding average power, although they are of the same PEH. Hence, multiplying the power by the bandwidth in one formula will not give an accurate estimation of the effectiveness. The power output and the bandwidth are two different factors. All bandwidths and their corresponding average power from the same energy harvester are options for the user. It depends firstly on the conditions of the working environment, such as the band within which the excitation frequency varies, and secondly, on the power requirements for the application; for example, if the power requirement is very low, an average power sufficient for the application can be generated over a very wide bandwidth. When proposing a novel PEH, it is important to calculate the average power and clearly state the band of frequencies over which this power is generated. The evaluation of the proposed PEHs is based on how effective they are in generating the maximum power when compared to a single piezoelectric energy harvester of the entire occupied volume (see Figure 14). It is suggested that the average power over a band of frequencies (the expected bandwidth) be compared to the average power output of a solid, single PEH of the same entire volume and over the same band of frequencies (see Equation (16)). \[e_{eff}=\frac{P_{av}|_{\Omega_{\max}-\Omega_{\min}}}{P_{ref\_av}|_{\Omega_{ \max}-\Omega_{\min}}}, \tag{16}\]  
where \(e_{eff}\) is the effectiveness of the proposed PEH, \(P_{av}\) is the average power output of the proposed PEH over a range of frequencies from \(\Omega_{\min}\) to \(\Omega_{\max}\), and \(P_{ref\_av}\) is the average power output of the solid, single PEH over the same range of frequencies. The entire volume of the proposed PEH (referred to as \(V_{PEH}\)) must be estimated. The entire volume occupied by the optimized rectangular-shaped PEH is referred to as \(V_{nf}\)and is expressed by Equation (17). \[V_{ref}=(l_{b}+l_{mt})\Big{(}h_{b}+2x_{ref}\Big{)}b_{b}, \tag{17}\]  
Figure 14: Broadband (BD) energy harvester versus a single reference harvester. Figure 13: Different bandwidths at different power levels.