# A Compact Thevenin Model for a Rectenna and Its Application to an RF Harvester with MPPT
## 5 Experimental Results and Discussion  
As for the proposed rectenna, Figure 7 shows the measured values (in dots) of \(I_{\rm o}\) (blue circles) and \(\eta_{\rm rect}\) (red squares) as a function of \(V_{\rm o}\) at different values of \(P_{\rm av}\). A least-squares fitting of (17) to the experimental data of \(I_{\rm o}\) was performed (blue continuous line) to obtain the Thevenin parameters (\(V_{\rm oc}\) and \(R_{\rm T}\)) at each power level, which are shown in Table 1. Calculated values of \(V_{\rm ap}\), from (1), and of \(V_{\rm eqp}\), from (15), are also included in Table 1. This fitting differs from that performed in [32], where the efficiency data (\(\eta_{\rm rect}\)) were used instead, which leads to slight differences in the Thevenin parameters. The new fitting procedure was considered more convenient as both \(V_{\rm oc}\) and \(R_{\rm T}\) can be more easily inferred from the fitting curve. As can be seen, the fitting curves match well the experimental data, and more at the highest power of \(-10\) dBm, which confirms that the rectenna can be well approximated by a Thevenin equivalent circuit. Then, \(V_{\rm oc}\) and \(R_{\rm T}\) were used to obtain \(\eta_{\rm rect}\) using (19), and the resulting curves are also represented in Figure 7 (red continuous line). The match with the experimental data is good, and again better at \(-10\) dBm. With \(C_{\rm m}\) = 0.5 pF, \(R_{\rm a}\) = 50 \(\Omega\), and \(f_{\rm o}\) = 808 MHz, \(Q\) = 7.88 results from (3), and \(G_{\rm t}\) = 3.97 from (2). Then, from (16) and assuming the value of \(R_{\rm p}\) = 9.21 k\(\Omega\) derived in Appendix A, \(R_{\rm T}\) = 4.7 k\(\Omega\) is obtained, which is within the range of values found in Table 1. The inferred values of \(R_{\rm T}\) moderately change with \(P_{\rm av}\) due to the relative low value of \(Q\), which limits the accuracy of the rectenna model proposed in Section 2. However, a higher value of \(Q\), which could be obtained using a lower value of \(C_{\rm m}\) and appropriately readjusting \(L_{\rm m}\), does not lead to the optimum gain \(G_{\rm t}\)[33], thus decreasing \begin{table}
\begin{tabular}{c c c c c} \hline \(P_{\rm av}\) (dBm) & \(V_{\rm ap}\) (mV) & \(V_{\rm oc}\) (mV) & \(R_{\rm T}\) (k\(\Omega\)) & \(V_{\rm eqp}\) (mV) \\ \hline \(-10\) & 200 mV & 937 & 3.56 & 1183 \\ \hline \(-20\) & 63.2 mV & 268 & 4.29 & 374 \\ \hline \(-30\) & 20.0 mV & 56.6 & 5.51 & 118.3 \\ \hline \end{tabular}
\end{table} 
Table 1: Inferred values of \(V_{\rm oc}\) and \(R_{\rm T}\) and calculated values of \(V_{\rm eqp}\) at different values of \(P_{\rm av}\). Figure 7: Measured values (dots) and least-squares fittings (continuous lines) of \(I_{\rm o}\) and \(\eta_{\rm rect}\) versus \(V_{\rm o}\) for the rectenna at \(P_{\rm av}\) equal to (**a**) \(-10\) dBm, (**b**) \(-20\) dBm, and (**c**) \(-30\) dBm. the power efficiency. On the other hand, \(V_{\rm{oc}}\) in Table 1 increases with increasing \(P_{\rm{av}}\) and thus \(V_{\rm{ap}}\), which agrees with (16). The values of \(V_{\rm{oc}}\) can be estimated in advance, when necessary, from (16) by calculating \(V_{\rm{ep}}\) from (15), shown in Table 1, and inferring a value of \(V_{\rm{\gamma}}\) from the manufacturer data or from simulations. From the measured data of \(\eta_{\rm{rect}}\) (red squares in Figure 7), Table 2 shows the achieved \(\eta_{\rm{rect,max}}\) and its corresponding voltage (\(V_{\rm{MPP,exp}}\)), as well as the experimental open circuit voltage (\(V_{\rm{oc,exp}}\)) of the rectenna. In Figure 7, \(\eta_{\rm{rect,max}}\), \(V_{\rm{MPP,exp}}\), and \(V_{\rm{oc,exp}}\) are also marked for \(P_{\rm{av}}=-20\) dBm. As can be seen, \(\eta_{\rm{rect,max}}\) increases with increasing \(P_{\rm{av}}\), ranging from 13.6% at \(-30\) dBm to 60.3% at \(-10\) dBm, which agrees with (21). The values of \(\eta_{\rm{rect,max}}\) can be estimated in advance, when necessary, from (21) and inferring a value of \(V_{\rm{\gamma}}\) from the manufacturer data or from simulations.