## 3 Parametric Analysis of RF Rectifier Circuits Efficiency  
As it has been commented, the choice of an architecture for the RF rectification is barely justified in many examples that can be found in the literature. In the case of the design of a rectifier for ambient RF that is destinated to operate in ultra-low power conditions, the most important parameter is the efficiency. To determine the best rectifier for this specific conditions, Table 1 rectifiers have been simulated using ADS-harmonic balance analyses to evaluate and compare the efficiencies. Each rectifier circuit has a particular input impedance that reflects a fraction of the injected power (\(P_{nom}\)). The intrinsic efficiency defined in Equation (2) is referred to the effective power delivered to the load (\(P_{in}\)). Therefore, it is a magnitude independent from the input impedance that can be compared between different rectifiers. Figure 1 shows the harmonic-balance simulation setup used to evaluate the intrinsic rectifier efficiency of different rectifiers. The different rectifier circuits have been simulated using identical elements. The SMS-7630 Schottky diode model has been chosen because of its low forward voltage (ranging from 60 to 120 mV at 0.1 mA [33]). The rectifier output capacitors have been fixed to the standard value of 4.7 \(\mu\)F in all cases for the simulations. The analysis of the efficiency as a bidimensional function on \(P_{in}\) and \(R_{L}\) points out the existence of different combinations that optimize the efficiency. Figure 2 shows, as example, the simulated \(\eta_{rec}(P_{in},\ R_{L})\) for the N = 2 Dickson rectifier circuit. As it can be observed, there are multiple \(P_{in}\) and \(R_{L}\) combinations producing the maximum efficiency (close to 70%). It is also observed that for the expected ambient RF power (\(P_{in}\leq-30\) dBm), the maximum DC conversion efficiencies are lower than 30%, which are higher than 10% only in the range of 1 k\(\Omega<R_{L}<\) 300 k\(\Omega\). The efficiency mapping in Figure 2 offers a good overview of the efficiency general behavior but renders the comparison between rectifiers difficult. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Frequential**} & \multicolumn{3}{c}{**Half-Wave Rectifier (\(f_{true}\))**} & \multicolumn{3}{c}{**Full-Wave Rectifier (\(f_{true}\))**} \\ \cline{2-7}  & **Fourier** & **Output Power** & **Input Power** & **Fourier** & **Output Power** & **Input Power** \\  & **Coefficient** & **Coefficient** & **Percentage** & **Coefficient** & **Coefficient** & **Percentage** \\ \hline DC & \(\dfrac{A}{\pi}\) & \(\dfrac{1}{\pi}=0.32\) & & & \(\dfrac{2A}{\pi}\) & \(\dfrac{2}{\pi}=0.64\) & \\ \hline \(f_{o}\) & \(\dfrac{A}{2}\) & \(\dfrac{1}{\sqrt{2}\pi}=0.35\) & & - & - & 64\% \\ \hline \(2f_{o}\) & \(\dfrac{2A}{3\pi}\) & \(\dfrac{2}{3\sqrt{2}\pi}=0.15\) & 15\% & \(\dfrac{4A}{3\pi}\) & \(\dfrac{4}{3\sqrt{2}\pi}=0.30\) & 30\% \\ \hline \(3f_{o}\) & - & - & - & - & - & - \\ \hline \(4f_{o}\) & \(\dfrac{2A}{15\pi}\) & \(\dfrac{2}{15\sqrt{2}\pi}=0.03\) & 3\% & \(\dfrac{4A}{15\pi}\) & \(\dfrac{4}{15\sqrt{2}\pi}=0.0\) & 6\% \\ \hline \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Harmonic coefficients of the full- and half-rectifier waves and the associate power. Figure 1: Simulation setup to evaluate the intrinsic efficiency \(\eta_{rec}\) of the different rectifiers included in Table 1, using harmonic balance analysis with a 50 \(\Omega\) single-tone source operating at 2.45 GHz. Figure 3a shows intrinsic efficiencies as a function of the nominal power (\(P_{nom}\)) for \(R_{L}=10\) k\(\Omega\), and a curve for each different rectifier can be observed. However, since the input impedance of each rectifier is different, the curves in Figure 3a are not comparable. Figure 3b shows the same data plotted as a function of \(P_{in}\) enabling the comparison.