# High-Efficiency Triple-Band RF-to-DC Rectifier Primary Design for RF Energy-Harvesting Systems
## 3 Results and Discussion
\begin{table}
\begin{tabular}{l r r r r} \hline \hline  & **Rectifier’s Input Impedance (Ohm)** & **Frequency** & **\% Deviation** \\ \cline{2-4}  & **Real Part** & **Imaginary Part** & **(MHz)** & **from 50 Ohm** \\ \hline \multicolumn{4}{c}{**without impedance matching**} \\ \hline  & \(11.42\) & \(-66.36\) & \(866\) & \(77.16\) \\  & \(27.1\) & \(-65.4\) & \(937\) & \(45.8\) \\ \hline \multicolumn{4}{c}{**with impedance matching**} \\ \hline
**dual-band** & \(44.3\) & \(6.1\) & \(866\) & \(11.4\) \\
**(gradient optimizer)** & \(55\) & \(8.5\) & \(937\) & \(10\) \\
**triple-band** & \(53.48\) & \(0.44\) & \(866\) & \(6.96\) \\
**(genetic optimizer)** & \(48.81\) & \(9.66\) & \(948\) & \(2.38\) \\  & \(48.19\) & \(-1.6\) & \(2423\) & \(3.62\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Input impedance values of the dual- and triple-band rectifier, with and without the impedance-matching network. The overall system RF-to-DC efficiency \(n\) is computed as follows:  
\[n=\frac{P_{DC}}{P_{in}} \tag{1}\]  
\[P_{DC}=\frac{V_{out}^{2}}{R_{L}} \tag{2}\]  
where \(P_{DC}\) is the total output \(DC\) power, calculated by the sum of each RF power \(P_{DC,f1}\), \(P_{DC,f2}\), and \(P_{DC,f3}\) corresponding to an RF input power at a frequency of \(f_{1}\) = 866 MHz, \(f_{2}\) = 948 MHz, and \(f_{3}\) = 2423 MHz, respectively, \(P_{in}\) is the RF input power, \(V_{out}\) is the output \(DC\) voltage, and \(R_{L}\) is the load resistance. We should point out that for an input signal of two (dual-band rectifier) or three tones (triple-band rectifier), the total power of the two or three tones is counted as the input power \(P_{in}\), respectively [33]. Figure 9 displays the RF-to-DC efficiency versus the \(R_{L}\) output load of the dual-band (Figure 9a: two-tone signal) and triple-band (Figure 9b: three-tone signal) rectifying circuit. From the presented curves of the graph, we can easily derive that the maximum efficiency of the proposed dual-band rectifier for an input power of 2 dBm is 85.9%, whereas the optimum output load is equal to 13 kOhm. At \(-\)20 dBm, \(-\)15 dBm, \(-\)10 dBm, and \(-\)5 dBm, the obtained conversion efficiencies are 38%, 55%, 69%, and 78%, respectively. For the  
Figure 8: Obtained S11 parameter (reflection coefficient) of the proposed Schottky diode-based dual- and triple-band rectifier for \(R_{L}\) = 13 kOhm (blue solid line: dual-band S11 parameter, green solid line: triple-band S11 parameter, red dash line: \(-\)10 dB limit). Figure 9: RF-to-DC efficiency vs. \(R_{L}\) output load of the demonstrated (**a**) dual-band (for a two-tone signal: 869 MHz and 940 MHz) and (**b**) triple-band (for a three-tone signal: 866 MHz, 948 MHz, and 2.423 GHz) rectifying circuit for various RF input power values (\(P_{in}\)). triple-band rectifier, the maximum power conversion efficiency is 81.9%. Additionally, at \(-20\) dBm, \(-15\) dBm, \(-10\) dBm, and \(-5\) dBm, the obtained conversion efficiencies for an output load of 13 kOhm are 36%, 53%, 67%, and 76%. Therefore, we can conclude that the given topology of the rectifier design has the features of stability and durability if we consider the efficiency values at different input power values. Finally, Figure 10a,b portrays the RF-to-DC efficiency and the DC output voltage \(V_{out}\) versus the RF input power \(P_{in}\), respectively. From Figure 10a, we can observe that the efficiency of the dual-band rectifier is 85% for an input power level of 0 dBm (two-tone signal), whereas in the triple-band system, it is 81% (three-tone signal). For an input power equal to 2 dBm, the maximum power conversion efficiency of the dual- and triple-band circuits is 85.9% and 81.9%, respectively. One may notice that there is a reduction in the PCE of the triple-band system compared to the PCE of the dual-band system; however, this variation is indiscernible.