## 1 Introduction  
With the flourishing of wearable devices and implantable medical devices, self-powered applications are preferred in order to avoid regular battery replacement or recharging. Therefore, clean and environmentally friendly energy harvesting technology, which scavenges ambient energy, is receiving more and more attention due to its long service life [1, 2, 3, 4, 5, 6]. The thermoelectric generator (TEG), which converts thermal energy into electricity based on the Seebeck effect, stands out among many energy harvesters due to its simple structure, small size, rich thermal energy, and the absence of pollution and noise [7, 8, 9, 10, 11, 12].  
A TEG is composed of a series of thermocouples (TC) sandwiched between two high-thermal-conductivity substrates. It can be modeled as a DC voltage source E\({}_{\text{TEG}}\) and an internal resistance R\({}_{\text{TEG}}\) connected in series, as shown in Figure 1. According to the Seebeck effect, the ETEG can be expressed as  
\[\text{E}_{\text{TEG}}=N\alpha\Delta T \tag{1}\]where \(N\) is the number of thermocouples, \(\Delta T\) is the temperature difference between the two sides of the TEG, and \(\alpha\) is the Seebeck coefficient in V/K. \(\alpha\) is material-related and not relevant to the size of the thermocouples. The R\({}_{\text{TEG}}\) is mainly determined by the size and the electrical resistivity of the semiconductor materials. Limited by the device area and temperature differences, the E\({}_{\text{TEG}}\) fluctuates within the range of 50-400 mV and the R\({}_{\text{TEG}}\) varies from 1 \(\Omega\) to 12 \(\Omega\) in most wearable applications [13; 14]. Therefore, extracting as much power as possible is crucial for the TEG converter.  
Because of the restricted space condition, maximum power point tracking (MPPT) methods for wearable devices usually adopt fractional open-circuit voltage (FOCV) [15; 16] or perturbation and observation (P&O) [17; 18] techniques, rather than complex algorithms that must collect and manage large amounts of data [19].  
According to the TEG electrical model in Figure 1, the maximum output power occurs when the TEG output voltage (V\({}_{\text{TEG}}\)) is half of its present open-circuit voltage (E\({}_{\text{TEG}}\)). A conventional MPPT circuit using the FOCV technique is illustrated in Figure 2. The circuit periodically turns on S1 and turns off the connection of the TEG and boost circuit to sample the open-circuit voltage on the TEG. During the boost circuit operation, the control signals of the boost circuit are automatically tuned by a hysteresis comparator and an MPPT controller to keep V\({}_{\text{TEG}}\) around half of V\({}_{\text{OC}}\)[20; 21]. Although the FOCV technique is simple to implement, it must frequently stop the boost circuit for a while to sample the V\({}_{\text{TEG}}\), which reduces the system efficiency. To overcome the drawback of the FOCV technique, the P&O technique is proposed for TEG by [PERSON] [22] and [PERSON] [23]. However, the tracking efficiencies fluctuate wildly, especially for a wide R\({}_{\text{TEG}}\) range.  
The remainder of the paper is organized as follows. By analyzing the influence of TEG internal resistances on tracking efficiency, a time exponential rate P&O technology is proposed in Section 2. Using the time exponential rate P&O, the MPPT circuit observes the power by comparing a positive-channel metal-oxide semiconductor (PMOS) on-time and perturbs the power by adjusting a negative-channel metal-oxide semiconductor (NMOS) on-time exponentially. The detailed circuit modules including zero current detection and power modulator are demonstrated respectively in Section 3. Section 4 presents the simulation results of the proposed MPPT circuit, which shows an enhanced tracking efficiency for a wide R\({}_{\text{TEG}}\) range. Conclusions and discussion are included in Section 5.  
Figure 1: Thermoelectric generator and its equivalent electrical model.  
Figure 2: Conventional MPPT circuit using the fractional open circuit technique.