## 4 Simulation Results  
The proposed MPPT circuit using the time exponential rate P&O algorithm for TEG was implemented and simulated with a 110 nm standard complementary metal-oxide  
Figure 9: Time exponential rate power modulator (a) schematic and (b) input control signal.  
semiconductor (CMOS) process. The total silicon chip occupied an area of 1.5 mm\({}^{2}\) (with pads), as shown in Figure 10.  
A voltage source E\({}_{\text{TEG}}\) and a resistance R\({}_{\text{TEG}}\) were connected in series to mimic the TEG. Figure 11 displays the simulation waveform of the TEG output voltage V\({}_{\text{TEG}}\) when E\({}_{\text{TEG}}\) changed from 50 mV to 400 mV and R\({}_{\text{TEG}}\) was 6 \(\Omega\). There is an enlarged view of the simulation results at the bottom of the figure. As described in Section 2, V\({}_{\text{TEG}}\) oscillated between the three states closest to E\({}_{\text{TEG}}\)/2, which is the maximum power point.  
The input power of the DC-DC converter was obtained by the calculator module embedded in Spectre. The input power was divided by the ideal maximum power output of the TEG to obtain the tracking efficiency of the proposed MPPT circuit. The simulation result of the tracking efficiencies was between 98.9% and 99.5% for E\({}_{\text{TEG}}\) from 50 mV to 400 mV and R\({}_{\text{TEG}}\) from 1 \(\Omega\) to 12 \(\Omega\), as shown in Figure 12, indicating that the MPPT circuit maintained extremely high tracking efficiency in a wide TEG range.  
Figure 11: The waveform of V\({}_{\text{TEG}}\), when E\({}_{\text{TEG}}\) changes from 50 mV to 400 mV and R\({}_{\text{TEG}}\) is 6 \(\Omega\).  
Figure 10: Layout view of the proposed MPPT circuit.  
For comparison, Figure 13 illustrates the tracking efficiencies of the compared MPPT circuit, which realizes P&O technology by fixed time step length perturbing. With the increase in \(\text{R}_{\text{TEG}}\), the perturbation of \(t_{MN}\) should be smaller to maintain the same tracking efficiency. Therefore, the \(\eta_{\text{track}}\) of the long-time step circuit gradually decreases from 98.7% to 92.3%. However, when the fixed-time step length is set to short, the change in power is too small to be observed correctly at low \(\text{R}_{\text{TEG}}\). Therefore, the circuit will oscillate before reaching the MPP. Then, the \(\eta_{\text{track}}\) maintains a high level at high \(\text{R}_{\text{TEG}}\) and decreases significantly at low \(\text{R}_{\text{TEG}}\), as shown in Figure 13. Comparing with the fixed-time step P&O scheme, the proposed time exponential rate P&O scheme maintains high tracking efficiency in a wider \(\text{R}_{\text{TEG}}\) range.  
Table 1 shows a comparison of this work with other state-of-the-art MPPT techniques for TEG. A previous MPPT circuit with the FOCV technique by [PERSON] [20] achieved peak tracking efficiencies of 96%, but only TEGs with fixed internal resistance were utilized and the influence of internal resistances on tracking efficiency was not considered. [PERSON] [21] tested different internal resistances, and the tracking efficiency decreased by 7% when the internal resistance varied from 5 to 10 \(\Omega\). Moreover, the FOCV technique must frequently stop the boost circuit for a while to sample the open-circuit voltage, which reduces the system efficiency. Comparing with previous works using the P&O technique [22; 23], the proposed MPPT circuit using the time exponential rate P&O technique maintains tracking efficiency at a high level from 98.9 to 99.5%, and the applicable range of the TEG resistance is from 1 to 12 \(\Omega\), which reflects an enhancement of at least 2.2 times.  
Figure 12: Tracking efficiencies of the proposed circuit at different conditions.  
Figure 13: Tracking efficiencies of the compared MPPT circuit with a fixed-time step P&O scheme.