## 2 Piezolectric Harvesters (PEHs)  
### Piezolectric Generators--Equivalent Circuits and Basic Parameters  
Piezoelectric generators are electro-mechanical elements for which the energy of mechanical vibrations is converted into electrical energy. The piezoelectric materials consist of asymmetrical unit cells within their crystal lattice structures. These asymmetrical unit cells include atoms or molecules with a center of positive and negative charges within the cell that do not coincide, thus creating a permanent dipole moment. This internal arrangement allows piezoelectric materials to respond to external mechanical and electrical stimuli by generating electric polarization [2; 23]. In order to use the piezoelectric elements, the most widely spread structure, due to its compatibility with the silicon microfabrication technology, is the cantilever. This is a flexible beam coated with piezoelectric material so that mechanical vibrations are produced when driven, and an electrical AC voltage is generated between both sides of the element (Figure 1), to which electrodes are mounted [24; 25; 26; 27]. Depending on the structure of the harvesters, this electrical voltage is proportional basically to the amplitude and frequency \(f=\omega/2\pi\) of the mechanical vibrations (Figure 2). When an external force is applied, resulting in vibrations of sinusoidal waveform \(u=U_{m}\cos\omega t\) and constant amplitude (maximum deflection of the beam) \(U_{m}\), the static charge equation is [25]  
\[Q=\Theta u+C_{P}v_{p} \tag{1}\]  
where \(u\) is the tip deflection of the cantilever, \(v_{p}\) is the voltage, generated between the two electrodes of the piezoelectric element, \(\Theta\) is the coupling coefficient of the beam geometry, and \(C_{P}\) is the internal electrode capacitance of the PEH. If the terminals of the electrodes of the piezoelectric element are connected to a load with certain parameters in the network flow, there is a current whose value is given by the following formula:  
\[i_{o}(t)=\frac{dQ}{dt}=\Theta\frac{du}{dt}+C_{P}\frac{dv_{p}}{dt}=\underbrace{ \Theta\omega U_{m}\sin\omega t}_{i_{p}(t)}+\underbrace{C_{P}\frac{dv_{p}}{dt}}_ {i_{C_{p}}(t)}. \tag{2}\]  
As can be seen from Formula (2), the piezoelectric element can be represented as a current generator and a capacitor \(C_{P}\). Moreover, for sinusoidal excitation of the cantilever, the current is also sinusoidal in shape \(i_{p}=I_{P}\sin\omega t\), and the amplitude has the form \(I_{P}=\Theta\omega U_{m}\). Then, the amplitude of the current is proportional to the geometry of the cantilever expressed by the value of the coefficient \(\Theta\), and it is also proportional to the amplitude \(U_{m}\) and frequency \(\omega\) of the mechanical vibrations. The electrical energy that can be obtained at the maximum (peak) value of the voltage \(V_{p,\max}\) can be given as  
\[E_{PEH}=\frac{1}{2}C_{P}V_{p,\max}^{2}. \tag{3}\]  
Then, based on the above value, the maximum average power (or _raw power_) \(\hat{P}_{PEH,\max}\) that can be obtained from a piezoelectric element is  
\[\hat{P}_{PEH,\max}=\left(\frac{dE_{PEH}}{dt}\right)_{\max}=\frac{C_{P}\omega V _{p,\max}^{2}}{2\pi}=C_{P}fV_{p,\max}^{2}. \tag{4}\]  
Based on the structures given in Figures 1 and 2, Figure 3 shows the equivalent circuit diagram of a piezoelectric element. Figure 3a shows an equivalent circuit according to [PERSON]'s theorem, showing a current source and a capacitor connected in parallel with it, and Figure 3b shows an equivalent circuit according to [PERSON]'s theorem, in which a current source is converted into a voltage source and a capacitor is connected in series with  
Figure 1: Simplified diagram of a piezoelectric element fixed to a flexible beam (cantilever). Figure 2: Piezoelectric element connected to a load. it. For the equivalent circuit in Figure 3b, the voltage of the piezoelectric element is found, \(e_{p}(t)=\frac{1}{C_{p}}\int i_{p}(t)dt\). Impedance matching is a necessity in systems where the load is required to use the maximum power generated by the source.