## 3 Modeling of the TENG System  
### Buck Converter  
In this research, a buck converter (a type of DC-DC converter) is used to control the TENG's output voltage. In Figure 3, the buck converter circuit is connected to the rectifier circuit to obtain a steady output voltage. The buck converter can produce a voltage lower than the input voltage. The DC-DC converter operates in two states: ON and OFF. The switch state is controlled by a pulse-width modulation (PWM) signal. The duty cycle, defined as the ratio of ON time to the PWM period, is given in Equation (4). Here, \(T\) is the pulse period, and \(T_{ON}\) is the pulse width. \[\hat{d}=\frac{T_{ON}}{T} \tag{4}\]  
The parameters for the buck converter are as follows: \(V_{in}\) represents the input voltage, and its unit is V. \(V_{out}\) is the output voltage, and its unit is V. \(i_{L}\) denotes the inductor current, and its unit is A. The load resistance is denoted by \(R_{load}\), and its unit is \(\Omega\). \(L\) is the inductance, and its unit is H. Additionally, \(C_{out}\) refers to the output capacitor, and its unit is F.  
Figure 3: The TENG system with the buck converter. Figure 2: The equivalent circuit of the TENG. In this research, the widely applicable state-space averaging method (WASAM) is used for modeling, as it accommodates both CCM and DCM [28]. As it is first necessary to obtain a model using the state-space averaging method, the modeling process begins with this approach. For DC-DC converters, the system is modeled separately for each state, where the switch is either ON or OFF and weighted based on the PWM duty cycle (Figure 4). By superimposing these weighted models, an averaged model of the DC-DC converter can be obtained. In Figure 5, the circuit equation for the equivalent circuit in the ON state is expressed by Equation (5):  
\[ \begin{cases}\dfrac{di_{L}}{dt}=-\dfrac{1}{L}V_{out}+\dfrac{1}{L}V_{in}\\ \dfrac{dV_{out}}{dt}=\dfrac{1}{C_{out}}i_{L}-\dfrac{1}{R_{load}C_{out}}V_{out} \end{cases}  \tag{5}\]  
Similarly, the circuit equation for the equivalent circuit in the OFF state is expressed by Equation (6):  
\[ \begin{cases}\dfrac{di_{L}}{dt}=-\dfrac{1}{L}V_{out}\\ \dfrac{dV_{out}}{dt}=\dfrac{1}{C_{out}}i_{L}-\dfrac{1}{R_{load}C_{out}}V_{out} \end{cases}  \tag{6}\]  
By weighting and superimposing the circuit equations for the ON and OFF states, expressed in Equations (5) and (6), respectively, with the duty cycle \(d\), the model equation for the DC-DC converter is derived. The obtained model equation is presented in Equation (7):  
\[ \begin{cases}\dfrac{di_{L}}{dt}=-\dfrac{1}{L}V_{out}+\dfrac{\hat{d}}{L}V_{in}\\ \dfrac{dV_{out}}{dt}=\dfrac{1}{C_{out}}i_{L}-\dfrac{1}{R_{load}C_{out}}V_{out} \end{cases}  \tag{7}\]  
For CCM, modeling using the above method is sufficient; however, for DCM, additional characteristics must be taken into consideration. Figure 6 illustrates the characteristics in DCM, where the inductor current consists of three intervals: charging (\(\hat{d}_{1}\)), diode conducting (\(\hat{d}_{2}\)), and discontinuous (\(\hat{d}_{3}\)) intervals. The duty cycle of the charging interval is \(\hat{d}_{1}\), which corresponds to the on time within a PWM period. The duty cycles of the diode conducting and discontinuous intervals are \(\hat{d}_{2}\) and \(\hat{d}_{3}\), respectively, satisfying the relationship \(\hat{d}_{1}+\hat{d}_{2}+\hat{d}_{3}=1\). Figure 4: The pulse-width modulation (PWM) illustration with ON time (\(T_{ON}\)) and PWM period (\(T\)). The circuit configuration varies across different intervals. During the charging interval \(\hat{d}_{1}\), the switch is turned on, and the current through the inductor increases linearly. The circuit configuration at this time is shown in Figure 5a. In the diode conducting interval \(\hat{d}_{2}\), the switch is off, while the diode is on, and the current decreases linearly until it reaches zero. This state's circuit configuration is depicted in Figure 5b.