## 2 Time Exponential Rate P&O Algorithm  
Considering the low output voltage of TEG (<400 mV), a boost DC-DC converter is used to step up the voltage, as shown in Figure 3. According to the volt-second theory of inductors, the relationship between the voltage and on-time of a boost converter is  
\[t_{MN}\mathrm{V_{TEG}}=t_{MP}(\mathrm{V_{OUT}}-\mathrm{V_{TEG}}) \tag{2}\]  
where \(\mathrm{V_{OUT}}\) is the output voltage of the boost converter, and \(t_{MN}\) and \(t_{MP}\) are the on-time of the power transistors MN and MP, respectively. Because of the high conversion rate for this implementation, the inductor current IL declines much faster than it increases and the boost converter works in discontinuous conduction mode. Accordingly, the average current through the inductance can be expressed as  
\[\overline{\mathrm{L}}=\frac{t_{MN}(t_{MN}+t_{MP})\mathrm{V_{TEG}}}{2\mathrm{ TL}} \tag{3}\]  
where L is the inductance and T is the cycle time of the converter. The input power of the converter can be expressed as  
\[\mathrm{P_{IN}}=\mathrm{V_{TEG}}\cdot\overline{\mathrm{L}}=\frac{\mathrm{V_{ TEG}}^{2}\cdot t_{MN}(t_{MN}+t_{MP})}{2\mathrm{TL}} \tag{4}\]  
By substituting Equation (2) into Equation (4) and considering the high conversion rate (\(\mathrm{V_{OUT}}\gg\mathrm{V_{TEG}}\)), \(\mathrm{P_{IN}}\) can be approximated to  
\[ \begin{array}{ll}\mathrm{P_{IN}}&=\frac{\mathrm{V_{TEG}}(t_{MN}+t_{MP})\cdot \mathrm{V_{TEG}}t_{MN}}{2\mathrm{TL}}\\ &=\frac{\mathrm{V_{OUT}}t_{MP}\cdot(\mathrm{V_{OUT}}-\mathrm{V_{TEG}}t_{MP}}{2 \mathrm{TL}}\approx\frac{\mathrm{V_{OUT}}^{2}\cdot t_{MP}}{2\mathrm{TL}}\end{array}  \tag{5}\]  
Observing Equation (5), we find that the on-time \(\mathrm{t_{MP}}\) reflects the \(\mathrm{P_{IN}}\) when the converter works in pulse width modulation (PWM) mode and the output \(\mathrm{V_{OUT}}\) changes slightly. As a function of the equivalent resistance of the boost converter (\(R_{eq}\)), \(\mathrm{P_{IN}}\) can be expressed as  
\[\mathrm{P_{IN}}=(\frac{\mathrm{E_{TEG}}}{\mathrm{R_{TEG}}+R_{eq}})^{2}\cdot R _{eq} \tag{6}\]  
The \(\mathrm{P_{IN}}\) reaches the maximum when \(R_{eq}\) is equal to \(\mathrm{R_{TEG}}\). According to Equation (6), tracking efficiency, which is the ratio of input power to ideal maximum input power, can be expressed as  
\[\eta_{\mathrm{track}}=\frac{\mathrm{P_{IN}}}{\mathrm{P_{IN(MAX)}}}=\frac{4 \mathrm{R_{TEG}}R_{eq}}{(\mathrm{R_{TEG}}+R_{eq})^{2}}>\varepsilon,0< \varepsilon<1 \tag{7}\]  
Figure 3: Proposed boost DC–DC converter: (**a**) topology architecture; (**b**) inductor current and control signals. where \(\varepsilon\) is any constant in the range of 0 to 1. By rearranging Equation (7), we can obtain  
\[\frac{\text{R}_{\text{TEG}}}{\beta}<R_{\alpha q}<\text{R}_{\text{TEG}}\cdot\beta, \ \beta=\frac{1+\sqrt{1-\varepsilon}}{1-\sqrt{1-\varepsilon}} \tag{8}\]  
By analyzing Equation (7) and Equation (8), we find that if \(R_{\alpha q}\) is in the range of \(\text{R}_{\text{TEG}}/\beta\) to \(\beta\text{R}_{\text{TEG}}\), the tracking efficiency of the MPPT system will remain higher than \(\varepsilon\). When \(R_{\alpha q}(n)\) is a geometric sequence with a common ratio \(\beta^{1/2}\), there are at least three \(R_{\alpha q}(n)\) in the range of \(\text{R}_{\text{TEG}}/\beta\) to \(\beta\text{R}_{\text{TEG}}\) for arbitrary \(\text{R}_{\text{TEG}}\), as shown in Figure 4.  
Figure 5 graphically represents the entire process of the time exponential rate P&O algorithm. In each MPPT cycle, the present \(\text{t}_{\text{MP}}\) is sampled and compared with the \(\text{t}_{\text{MP}}\) sampled in the previous MPPT cycle. The comparison reveals the input-power changes of two consecutive MPPT cycles based on Equation (5), as \(\text{V}_{\text{OUT}}\) shows little change between the short sampling time intervals. If the \(\text{t}_{\text{MP}}\) of an MPPT cycle is larger than the previous cycle, it means that input power has increased in the cycle.