## 3 Low-Power Electronic Converters
### Switching DC/DC Regulators
The FoM parameter is defined as the ratio of the power \(P_{o}\) that can be delivered from the storage element (rechargeable battery or ultra-capacitor) to the maximum average power \(\hat{P}_{PEH,\text{max}}\) that can be obtained from the piezoelectric element, provided that there is a lossless rectifier (\(V_{D}\approx 0V\)) [7]:  
\[FoM=\frac{P_{o}}{\hat{P}_{PEH,\text{max}}}=\frac{P_{o}}{C_{p}fV_{p,\text{max}} ^{2}}. \tag{22}\]  
Although miniaturized single piezoelectric generators can provide a higher voltage (>10 V) from mechanical vibrations compared to electrostatic and electromagnetic generators, the current that can be obtained is of relatively small value (<0.1 mA), and as a result, the electrical power \(\hat{P}_{PEH}\) will also be of small value (with typical values from 10 to 100 \(\upmu\)W). Then, a value greater than unity can be obtained for the \(FoM\) parameter if a rechargeable battery or a supercapacitor is connected to the output port of the DC-DC regulator. Thus, the instantaneous value of the electrical power that can be obtained from a charged battery or supercapacitor can be significantly greater than the output power of  
Figure 21: Block diagram of a PWM controller: (**a**) circuit variant using output voltage as a feedback signal; (**b**) circuit variant using piezoelectric input current as a feedback signal. Figure 20: Functional diagram of a rectifier with DC-DC regulator (or PSD), intended for extracting electrical energy from piezoelectric generators. the same DC-DC converter even with minimal power dissipation in the bridge rectifier or switching circuit. The electrical energy that spreads from the input to the output of the power source with an energy storage element (supercapacitor) connected to the output is determined as follows [7; 8]:  
\[\Delta E_{o}=\frac{1}{2}C_{STORAGE}(v_{F}^{2}-v_{o}^{2})=\frac{1}{2}C_{STORAGE}v_ {o}^{2}\bigg{[}f(C_{p},C_{STORAGE},r_{p},R_{L}^{2})\bigg{(}\frac{v_{p}}{v_{o}} \bigg{)}-1\bigg{]}, \tag{23}\]  
where \(C_{STORAGE}\) is the capacitance of the energy storage element, \(f^{2}(C_{p},r_{p},R_{L}C_{STORAGE})\) is a function depending on the type of voltage rectifier selected and its circuit parameters, and \(v_{F}\) is the output voltage of the rectifier over the filter capacitor \(C\) (see Figures 15 and 17) (voltage fluctuations typically do not exceed 5% of the nominal dc value). Then, the charging efficiency can be calculated as the ratio of the energy that is transferred to the output to the energy that is generated by the vibrations [7; 8]:  
\[\eta_{ST}=\frac{\Delta E_{o}}{E_{PEH}}=\frac{C_{STORAGE}}{C_{p}}\bigg{(}\frac{ v_{o}}{v_{p}}\bigg{)}^{2}\bigg{[}f(C_{p},C_{STORAGE},r_{p},R_{L}^{2})\bigg{(} \frac{v_{p}}{v_{o}}\bigg{)}-1\bigg{]}, \tag{24}\]  
Based on the functional diagram (Figure 20) in Figure 21a,b, block diagrams of PWM controllers are presented depending on the way of obtaining the feedback signal. In cases where the output DC voltage \(V_{o}\) is used as the input signal to the feedback network for the PWM controller, the block diagram [46; 60] in Figure 21a is obtained. In it, the output voltage is initially subtracted from the reference voltage \(V_{ref}\), with \(V_{o}\) previously being multiplied by a factor \(K_{V}\), which is less than unity. The output voltage \(V_{R}\) of the subtractor is applied to one input terminal of a PWM, while a voltage \(V_{SO}\) from a sawtooth oscillator is applied to the other input terminal. As a result of the comparison of the two voltages at the output of the PWM through the transistor gate driver, a control signal with a rectangular shape and a variable duty cycle \(\delta=\frac{t_{pulse}}{T}=\frac{V_{R}}{V_{SO}}\) is obtained, depending on how much the voltage \(V_{R}\) is greater than the triangular voltage \(V_{SO}\), produced by the sawtooth oscillator (Figure 22). The time during which the switch is closed is as long as the time interval \(t_{pulse}\).