## 2 Materials and Methods
From Equation (6), is evident that the equivalent input resistance of the converter that drives an output load \(R_{L}\) is not constant, since  
\[R_{DC/DC}=\frac{V_{\text{TEG}}}{I_{\text{TEG}}}=\frac{V_{out}(1-D)}{\frac{I_{ out}}{1-D}}=\frac{V_{out}}{I_{out}}(1-D)^{2}=R_{L}(1-D)^{2} \tag{7}\]  
From Equation (7) it can be stated that the equivalent input resistance of the DC/DC converter depends on the output load and the switching signal duty cycle. The same dependence of the output voltage and current, thus the equivalent DC/DC input resistance, from the duty cycle and the output load occurs when the converter operates in discontinuous mode, for which the current flowing through the inductor goes to zero during a single operation cycle  
\[V_{out}=V_{\text{TEG}}+\frac{V_{\text{TEG}}^{2}D^{2}T}{2 LI_{out}} \tag{8}\] \[I_{out}=\frac{V_{\text{TEG}}^{2}D^{2}T}{2L(V_{out}-V_{\text{TEG}})}\]  
where \(T\) is the time period of the switching signal. According to the well-known maximum power transfer theorem, to extract the maximum power from the TEG source, the equivalent input resistance \(R_{\text{DC/DC}}\) of the converter should be equal to the source resistance \(R_{\text{TEG}}\). Therefore, a control algorithm for the DC/DC is needed, in order to dynamically adjust the converter input impedance to guarantee the maximum power transfer. For this reason, many MPPT algorithms have been developed and described in the literature. They differ for complexity, number of sensors or required measured data, They differ for complexity, number of  
Figure 4: An example of a complete energy harvesting power management system with cascaded maximum power point track (MPPT) and fixed voltage regulated DC/DC converters. sensors or required measured data, efficiency, and hardware, and can be classified into two major categories. The first continuously track voltage and current, in order to dynamically adjust the power operation point, without the need for any empirical information. Algorithms like perturb and observe (P&O), incremental conductance (IC), or hill climbing (HC) belongs to this category [47; 48; 49] and are characterized by high hardware, complexity, continuous functionality, and high efficiency, especially if the behavior of the energy source is unknown. Other MPPT techniques like fractional open-circuit voltage (FOCV) or fractional short-circuit current (FSCC) requires less complexity and measured data, without the need of continuously track the input voltage and/or current since they rely on a priori information about the source [50; 51]. These algorithms represent the second category of MPPT techniques. The drawback of such type of algorithms is the conversion efficiency, which is affected and lowered if the knowledge and description of the source are not accurate. The identification of the TEG cell SPICE model enables the usage of a priori MPPT techniques for thermal energy harvesting, which are better in terms of occupied hardware area, cost, and complexity. For the purpose of this work, a FOCV technique has been selected for a design example. The technique expects that the source output voltage corresponding to the maximum power point is linearly proportional to the source open-circuit (OC) voltage with a constant \(K\) coefficient. Therefore, the goal of this algorithm is to measure the open-circuit voltage and modulate the required source current, in order to guarantee the output voltage is a constant fraction \(K\) of the OC measured voltage. The adopted scheme is depicted in Figure 5.  
In this scheme, the MPPT algorithm is divided into two working steps, that are repeated over time. \(\bullet\) Step 1:  
In the first step, the switch \(S1\) is open while \(S2\) and \(S3\) are closed. The capacitor \(C\) is then charged at a voltage \(V_{A}\) as  
\[V_{A}=V_{TEG,\ OC}\frac{R_{2}}{R_{1}+R_{2}}=K\cdot V_{TEG,\ OC} \tag{9}\]  
where \(V_{TEG,\ OC}\) is the open-circuit output voltage of the TEG cells, since \(S1\) disconnects the source from the DC/DC, thus no load is attached.