# A Compact Thevenin Model for a Rectenna and Its Application to an RF Harvester with MPPT
## 2 Rectenna and Its Thevenin Model
Thus, apart  
Figure 3: (**left**) Equivalent input circuit of the antenna and L-matching network considering the parasitic effects of the coil and diode and (**right**) its Thévenin equivalent circuit. from boosting the voltage, the matching network ideally acts as an input band-pass filter that prevents any of the DC current and harmonics to flow through the antenna resistance and dissipate power. This leads to an ideal rectenna efficiency of 100%, assuming no losses in the circuit components and in the diode [34]. Contrariwise, when no matching network is present, maximum rectenna efficiency decreases to 46%, due to the additional losses at \(R_{\mathrm{a}}\) originated by the current harmonics generated by the diode pulsed current, as demonstrated in [35]. Appendix B confirms these results via simulations. Finally, the value of \(C_{\mathrm{o}}\) has to be much higher than the diode junction capacitance (\(C_{\mathrm{j}}\)), as explained in Appendix A, to keep \(V_{\mathrm{o}}\) nearly constant, that is, with a low voltage ripple (\(\Delta V_{\mathrm{o}}\)). This second condition leads to:  
\[C_{\mathrm{o}}>\frac{I_{\mathrm{o}}}{\Delta V_{\mathrm{o}}f_{\mathrm{o}}}. \tag{6}\]  
The left-hand circuit of Figure 4 leads to the equivalent Thevenin circuit of the rectenna, represented by the right-hand circuit of Figure 4, by linking their output voltage-current relationship. For the left-hand circuit, we have:  
\[V_{\mathrm{o}}=V_{\mathrm{inp}}-V_{\mathrm{\gamma}}, \tag{7}\]  
assuming a fixed forward voltage drop \(V_{\mathrm{\gamma}}\) at the diode and:  
\[V_{\mathrm{inp}}=V_{\mathrm{eep}}-R_{\mathrm{eq}}I_{\mathrm{inp}}, \tag{8}\]  
where \(V_{\mathrm{eep}}\) and \(I_{\mathrm{inp}}\) are the amplitudes of \(v_{\mathrm{eq}}\) and \(i_{\mathrm{in}}\), respectively. Substituting (8) into (7) provides:  
\[V_{\mathrm{o}}=V_{\mathrm{eep}}-V_{\mathrm{\gamma}}-R_{\mathrm{eq}}I_{\mathrm{inp }}. \tag{9}\]  
On the other hand, for the right-hand circuit we have:  
\[V_{\mathrm{o}}=V_{\mathrm{oc}}-R_{\mathrm{T}}I_{\mathrm{o}}. \tag{10}\]  
Then, by equating powers, we obtain:  
\[P_{\mathrm{in}}=P_{\mathrm{o}}+P_{\mathrm{d}}, \tag{11}\]  
where  
\[ \begin{array}{c}P_{\mathrm{in}}=\frac{V_{\mathrm{inp}}I_{\mathrm{inp}}}{P_{ \mathrm{o}}},\\ P_{\mathrm{d}}=V_{\mathrm{\gamma}}I_{\mathrm{o}},\end{array}  \tag{12}\]  
and \(P_{\mathrm{d}}\) is the average power dissipated across the diode. Thus, replacing (12) into (11) and using (7), we arrive at the following:  
\[I_{\mathrm{inp}}=2I_{\mathrm{o}}. \tag{13}\]  
Figure 4: (**left**) Equivalent circuit of the rectenna using the right-hand circuit of Figure 3 and (**right**) its Thevenin equivalent. Finally, using (13) in (9) and equating (9) and (10), we obtain the parameters of the Thevenin model:  
\[ \begin{array}{c}V_{\rm{oc}}=V_{\rm{eep}}-V_{\gamma},\\ R_{\rm{T}}=2R_{\rm{eq}},\end{array}  \tag{14}\]  
where \(V_{\rm{eep}}\) can be derived from \(v_{\rm{eq}}\) in (5), using \(V_{\rm{ap}}\) instead of \(v_{\rm{a}}\), resulting in:  
\[ \begin{array}{c}V_{\rm{eep}}=2G_{\rm{t}}V_{\rm{ap}}\frac{R_{\rm{p}}}{4G_{\rm {t}}^{2}R_{\rm{a}}+R_{\rm{p}}}.\end{array}  \tag{15}\]  
Then, using (15) and \(R_{\rm{eq}}\) of (5) in (14), we have:  
\[ \begin{array}{c}V_{\rm{oc}}=2G_{\rm{t}}V_{\rm{ap}}\frac{R_{\rm{p}}}{4G_{\rm {t}}^{2}R_{\rm{a}}+R_{\rm{p}}}-V_{\gamma},\\ R_{\rm{T}}=2\big{[}\big{(}4G_{\rm{t}}^{2}R_{\rm{a}}\big{)}\|R_{\rm{p}}\big{]}. \end{array}  \tag{16}\]  
Therefore, from (16), with an increasing \(P_{\rm{av}}\) and thus \(V_{\rm{ap}}\), \(V_{\rm{oc}}\) increases whereas \(R_{\rm{T}}\) holds constant. Next, from (10), we can express \(I_{\rm{o}}\) as:  
\[ \begin{array}{c}I_{\rm{o}}=(V_{\rm{oc}}-V_{\rm{o}})/R_{\rm{T}},\end{array}  \tag{17}\]  
and the output power \(P_{\rm{o}}\) over a load resistor \(R_{\rm{o}}\) can be simply calculated as:  
\[ \begin{array}{c}P_{\rm{o}}=V_{\rm{o}}I_{\rm{o}}=\frac{V_{\rm{oc}}V_{\rm{o}} -V_{\rm{o}}^{2}}{R_{\rm{T}}},\end{array}  \tag{18}\]  
being the power efficiency of the rectenna as:  
\[