## 3 Proposed MPPT Control Method
### Analyzing the Thermoelectric Power Generation System  
This section reveals that the output of the TM can estimate by using the linear analyzing result of the system. This paper uses the state space averaging method [7] for this analyzing. To reduce the power loss of the winding resistance of \(L\), \(L\) is low inductance. On the other hand, \(R_{O}\) is the high impedance. As a result, the boost chopper acts in discontinuous current mode (DCM). There are three modes in DCM. Figure 4 illustrates the relationship between switching condition and \(i_{L}\). \(D_{1}\) is the duty ratio during G-on period, \(D_{2}\) is the duty ratio during G-off and D-on period, \(D_{3}\) is the duty ratio during G-off and D-off period. Thus, we analyze the three modes. Figure 5 shows the equivalent circuits of Figure 3. The state equations of Figure 5 set up where the state variable \(x\) is \(\left(v_{T}\,i_{L}\,v_{O}\right)^{T}\). Figure 3: System configuration of the thermoelectric power generation system using a boost chopper. Figure 4: Relationship between the switching condition and \(i_{L}\). Figure 5: Equivalent circuits of the thermoelectric power generation system. (**a**) MODE 1; (**b**) MODE 2; (**c**) MODE 3. During the MODE 1, the switch G is ON, and \(i_{L}\) increases from 0 to the peak value \(i_{LP}\). Figure 5a illustrates the equivalent circuit in MODE 1. \(r_{S}\) is the ON-resistance of the MOSFET and \(r_{L}\) is the winding resistance of \(L\). From Figure 5a, the state equation of the MODE 1 is given as follows  
\[\frac{dv_{T}}{dt}= -\frac{v_{T}}{R_{M}C_{I}}+\frac{v_{M}}{R_{M}C_{I}}-\frac{1}{C_{I}}i_ {T} \tag{6}\] \[\frac{di_{T}}{dt}=\frac{v_{T}}{L}-\frac{r_{I}+r_{s}}{L}i_{T}\] (7) \[\frac{dv_{O}}{dt}= -\frac{v_{O}}{R_{O}C_{O}} \tag{8}\]  
From these equations, the matrix is given by  
\[\frac{d\mathbf{x}}{dt}=\mathbf{A_{1}x+b_{1}}V_{M} \tag{9}\]  
where \(\mathbf{A_{1}}\) and \(\mathbf{b_{1}}\) are given by  
\[\mathbf{A_{1}}= \begin{pmatrix}-\frac{1}{R_{M}C_{I}}&-\frac{1}{C_{I}}&0\\ \frac{1}{L}&-\frac{r_{L}+r_{S}}{L}&0\\ 0&0&-\frac{1}{R_{O}C_{O}}\end{pmatrix} ,\mathbf{b_{1}}= \begin{pmatrix}\frac{1} {R_{M}C_{I}}\\ 0\\ 0\\ \end{pmatrix} \]  
The switching time is very short. So, the right side of Equations (6), (7) and (8) are assumed the constant value. As a result, we approximate these equations as follows  
\[ \begin{split} v_{T}\big{(}D_{1}T_{s}\big{)}-v_{T}\big{(}0\big{)}& =D_{1}T_{s}\,\frac{dv_{T}}{dt}_{(t=0)}\\ &=-D_{1}T_{s}\,\frac{v_{T}\big{(}0\big{)}}{R_{M}C_{I}}+D_{1}T_{s} \,\frac{v_{M}\big{(}0\big{)}}{R_{M}C_{I}}-D_{1}T_{s}\,\frac{i_{T}\big{(}0\big{)} }{C_{I}}\\ i_{T}\big{(}D_{1}T_{s}\big{)}-i_{T}\big{(}0\big{)}&=D_{1}T_{s} \,\frac{di_{T}}{dt}_{(t=0)}\\ &=D_{1}T_{s}\,\frac{v_{T}\big{(}0\big{)}}{L}-D_{1}T_{s}\,\frac{r_{L}+r _{s}}{L}\,i_{T}\big{(}0\big{)}\\ & v_{O}\big{(}D_{1}T_{s}\big{)}-v_{O}\big{(}0\big{)}&=D_{1}T_{s} \,\frac{dv_{O}}{dt}_{(t=0)}\\ &=-D_{1}T_{s}\,\frac{v_{O}\big{(}0\big{)}}{R_{O}C_{O}}\end{split}  \tag{11}\]  
During the MODE 2, the switch G is OFF, and the current flows into D. Figure 5b illustrates the equivalent circuit in MODE 2. \(r_{D}\) is the conduction resistance of D. From Figure 5b, the state equation of the MODE 2 is given as follows  
\[\frac{dv_{T}}{dt}=-\frac{v_{T}}{R_{M}C_{I}}+\frac{v_{M}}{R_{M}C_{I}}-\frac{1} {C_{I}}i_{T} \tag{13}\]\[\frac{di_{T}}{dt}=\frac{v_{T}}{L}-\frac{r_{L}+r_{D}}{L}i_{T}-\frac{1}{L}v_{O} \tag{14}\]  
\[\frac{dv_{O}}{dt}=-\frac{i_{T}}{C_{O}}+\frac{v_{O}}{R_{O}C_{O}} \tag{15}\]  
From these equations, the matrix is given by  
\[\frac{d\textbf{x}}{dt}=\textbf{A}_{2}\textbf{x}+\textbf{b}_{2}V_{M} \tag{16}\]  
where \(\textbf{A_{2}}\) and \(\textbf{b_{2}}\) are given by  
\[\textbf{A}_{2}= \begin{pmatrix}-\frac{1}{k_{M}c_{I}}&-\frac{1}{C_{I}}&0\\ \frac{1}{L}&-\frac{r_{L}+r_{D}}{L}&-\frac{1}{L}\\ 0&-\frac{1}{C_{O}}&-\frac{1}{R_{O}C_{O}}\end{pmatrix} ,\textbf{b}_{2}= \begin{pmatrix}\frac{1}{R_{M}C_{I}}\\ 0\\ 0\end{pmatrix} \]  
The right side of Equations (13), (14) and (15) are assumed the constant value.