## 3 Low-Power Electronic Converters
### Voltage Multipliers (Doublers and Quadruplers)
#### 3.2.1 Voltage Doublers  
The basic circuit of a voltage multiplier is shown in Figure 17[25; 46]. For this circuit, the voltage is multiplied by a factor of two. The circuit contains two diodes (\(D_{1}\) and \(D_{2}\)), which are most often Schottky due to the possibility of obtaining a smaller voltage drop in the forward mode of operation, and two capacitors (\(C_{1}\) and \(C_{2}\)). The input source is represented as a current source \(i_{p}\) with an internal resistance \(r_{p}\) and a capacitance \(C_{p}\). The frequency of the input signal is determined by the period, as \(T=1/f\) (\(f=\omega/2\pi\)). The magnitude of the open-circuit input voltage (in the condition that the internal resistance \(r_{p}\) is neglected) over the piezoelectric capacitance \(C_{p}\) yields [25]  
\[V_{pm,0}=\frac{I_{pm,0}}{\omega C_{p}}, \tag{16}\]  
where \(I_{pm,0}\) is the magnitude of the input current through the piezoelectric capacitance.  
The principle of operation of the circuit is implemented as follows: During the _first half-cycle_ (or the so-called _positive half-period_) for which the polarity of the input source is shown without brackets, the capacitor \(C_{1}\) is charged through the diode \(D_{1}\) to the amplitude of the input voltage--\(V_{\text{C1}}=V_{pm,0}\). For this polarity, the diode \(D_{1}\) is in the forward mode of operation. During the _next half-cycle_ (or _negative half-period_) for which the input voltage reverses polarity (signs in brackets), the diode \(D_{1}\) is turned off, and the sum of the capacitor voltage and the amplitude of the input voltage is applied to it. During this time interval, the diode \(D_{2}\) is turned on, and the capacitor \(C_{2}\) is charged, with the capacitor \(C_{1}\) transferring part of its electrical energy until a voltage equal to the sum of the input voltage and the voltage of \(C_{1}\) is reached, i.e., \(V_{\text{C2}}=V_{pm,0}+V_{\text{C1}}\approx 2V_{pm,0}\).  
For the circuit in Figure 17, the input and output voltages and currents as a function of time [14; 60] are given in Figure 18. As can be seen, the shape of the output voltage \(v_{o}\) is similar to the basic rectifier circuit (Figure 12), obtaining a similar ratio of the charge and discharge time intervals of the capacitors. With a larger value of the capacitors, an output voltage with smaller ripples can be obtained. Also, at a higher frequency \(\omega\) of the mechanical oscillations, a decrease in the ratio can occur.  
Figure 17: Circuit diagram of a rectifier with a voltage multiplication factor of two (voltage-doubler rectifier).  
The maximum value of average power \(\hat{P}_{PEH,\max}\) that can be extracted from a piezoelectric element for the circuit in Figure 17 is found at half the no-load rectified voltage [25]:  
\[\hat{P}_{PEH,\max}=\frac{C_{p}\omega V_{om,0}^{2}}{2\pi}\text{ at }V_{om,\infty}=V_{om,0}. \tag{17}\]  
The comparison of Formulas (15) and (17) shows that the average values of the maximum power that can be obtained from a piezoelectric generator are equal, but they are obtained at different values of the rectified voltage \(V_{om,\infty}\). For the doubler circuit, the voltage is higher and the current is smaller. A lower current flow for the voltage-doubler circuit also determines a lower power dissipation on the rectifier diodes. Furthermore, the voltage-doubler circuit uses two diodes instead of four as in the bridge rectifier circuit, which further reduces the total value of the power dissipation. An advantage of the bridge circuit compared to the voltage-doubler rectifier is the possibility of obtaining smaller ripples in the output voltage shape, which is determined by the specifics of the structure and the principle of operation.