## 3 Low-Power Electronic Converters
### Single-Phase Rectifiers
#### 3.1.1 Half-Wave Rectifier  
The basic analog circuit that can be used to rectify an AC input voltage is shown in Figure 10[45; 46]. This circuit contains a rectifier diode \(D\) and a capacitor \(C\), with an external load \(R_{L}\) connected in parallel with the capacitor. According to the [PERSON]'s theorem, the input signal is represented as an input voltage source \(e_{p}\) and internal resistance \(r_{p}\). Provided that no load is included in the output of the circuit, the capacitor \(C\) is charged during the positive half-cycle to a peak value equal to \(V_{om,0}=V_{pm,0}-V_{D}=\sqrt{2}V_{p,rms,0}-V_{D}\), where \(V_{D}\) is the voltage in the forward direction of the diode, and for Si diodes, the typical values are from 0.6 to 0.7 V, while for Schottky diodes, the typical values are from 0.3 to 0.4 V. The voltage \(V_{pm,0}\) is the peak value of the input voltage, provided that no load is connected to the output because when an external load is connected, due to the influence of the internal resistance \(r_{i}\) of the input signal source, the voltage value can be reduced. For the negative half-wave of the input voltage, the peak value of the voltage across the diode is equal to \(V_{pm,0}\). Figure 9: Image of flexible piezoelectric energy harvesting with Ga-ZnO piezoelectric materials on a seed polymer layer (**left**), morphology of the nanostructured Ga-ZnO grown on the seed layer (**middle**), and increased piezoelectric voltage of the nanostructured piezoelectric film grown on a polymeric seed layer, as compared to metal electrode (**right**) [44]. Reprinted from _Microelectr. Eng._, [PERSON], Copyright (2020), with permission from Elsevier. When a load with resistance \(R_{L}\) is connected to the output of the rectifier, the value of the output voltage changes because the capacitor \(C\) will discharge while the diode is reverse-biased. Only during the time intervals when the input voltage \(V_{pm,0}\) is greater than the output voltage of value \(V_{D}\), the current flows from the source to the capacitor, and it is recharged. In this case, provided that the capacitor is of a sufficiently large value, the output voltage is determined by the following formula: \(V_{om,\infty}=V_{om,0}\Big{(}1-\sqrt{\frac{r_{p}}{R_{L}}}\Big{)}\)[46]. Therefore, the value of the output voltage can be significantly reduced when a load is turned on, especially when the internal resistance of the source is relatively large. The ratio between the DC component of the rectified current (the average value of the current \(I_{L,av}\)) and the output current \(i_{L}(t)\) with the load on is found as follows. Assuming that the diode is an ideal element, the curve of the current \(i_{D}(t)\) in the time intervals in which the diode is forward-biased has a sinusoidal waveform, and the repetitive peak current [47] is obtained as follows:  
\[I_{D,\max}=\frac{V_{om,0}}{\sqrt{r_{p}R_{L}}}. \tag{7}\]  
Also, based on the value of the output voltage, the average value of the output current is found by the formula \(I_{L,av}=\frac{V_{pm,\infty}}{R_{L}}\); then, the relation between the DC component of the rectified current and the repetitive peak current through the diode is \(\delta_{I_{L}}=\frac{I_{L,av}}{I_{D,\max}}=\Big{(}1-\sqrt{\frac{r_{p}}{R_{L}} }\Big{)}\sqrt{\frac{r_{p}}{R_{L}}}\). In this case, the average value of the rectified current is relatively small and depends on both the internal resistance of the source and the resistance of the load. Therefore, the relatively simple structure with a small number of elements should be defined as an advantage of the basic circuit. A disadvantage of the circuit is that due to the relatively small value of the ratio of the time intervals of charge and discharge of the capacitor, the output voltage can often be obtained with a small value, even for small values of the current through the load, which is essential for energy harvesting sources providing small input power of the input signal. A modification of the basic circuit is shown in Figure 11[47; 48].