## 3 Parametric Analysis of RF Rectifier Circuits Efficiency
It can be observed that, in Figure 3b, only four different traces are appreciated, because N = 2 Dickson, N = 2 Greinacher, and the full wave variation rectifier efficiencies are degenerated in this representation. It can be observed that the more complex circuits (N = 2 and full wave variation) offer better efficiencies at higher input power (\(P_{in}>0\) dBm). However, for low and ultra-low input power (\(P_{in}\leq-30\) dBm), the best efficiencies correspond to the simple diode-capacitor rectifier, N = 1 Dickson-Greinacher rectifier, and four-bridge rectifier. In ultra-low power operation conditions (\(P_{in}\leq-30\) dBm), it can be observed that the best efficiency is around 12% and corresponds to the simple diode-capacitor rectifier. The second-best performance is obtained with the first order N = 1 Dickson-Greinacher rectifier (around 8%) and finally the four-diode rectifier with less than 4% efficiency. Figure 4 shows the intrinsic efficiency \(\eta_{rec}(P_{in},\ R_{L})\) for \(R_{L}=100\ \Omega\) (Figure 4a) and for \(R_{L}=100\ \mathrm{k\Omega}\) (Figure 4b). In both cases, the load is far from the optimum values  
Figure 3: Simulated rectifier efficiency for the six circuits in Table 1 as a function of \(P_{nom}\) in (**a**) and as a function of \(P_{in}\) in (**b**). Figure 2: Representation of the simulated intrinsic efficiency function \(\eta_{rec}\ (P_{in},\ R_{L})\) for N = 2 Dickson RF rectifier operating at 2.45 GHz. (around 10 k\(\Omega\)), and maximum efficiencies are lower than 50%. In the case of low load impedances (\(R_{L}\sim 100\)\(\Omega\)), the best efficiencies correspond to the four-diode bridge rectifier for \(P_{in}\geq 0\) dBm. However, under high load-resistance conditions (\(R_{L}\sim 100\) k\(\Omega\)), the best performance corresponds to both the second-order Dickson and Greinacher rectifiers and are achieved in the range of \(-20\) dBm \(\leq P_{in}\leq 15\) dBm. In both cases, the conversion efficiency under ultra-low power (\(P_{in}\leq-30\) dBm) is lower than 2% for \(R_{L}=100\)\(\Omega\) and 5% for \(R_{L}=100\) k\(\Omega\). Even in the extreme loads described in Figure 4, the simple diode-capacitor rectifier (Rect. Simple) efficiency is not below the other rectifiers. Results in Figures 3 and 4 discard the utilization of [PERSON] and Greinacher high-order rectifiers at ultra-low power. Figure 5 shows the simulated efficiency as a function of the load resistance for the three best rectifiers identified in Figures 4 and 5. As observed, the maximum efficiency of 12.8% corresponds to the simple diode-capacitor rectifier loaded with \(R_{L}=4.5\) k\(\Omega\). The second-best efficiency corresponds to the first-order Dickson-Greinacher rectifier, which is considerably lower, showing an efficiency of 7.8% when loaded with \(R_{L}=8.9\) k\(\Omega\). The four-diode bridge rectifier is the less efficient rectifier showing a maximum of 4.3% for \(R_{L}=5\) k\(\Omega\). The better performance of the diode-capacitor rectifier at ultra-low input power is attributable to its simpler structure that minimizes losses due to parasitic impedances in the constituent elements. Figure 5: ADS-harmonic balance simulated (blue, red, and black lines) and measured (blue, red, and black dots) efficiencies for simple diode-capacitor rectifier, four-diode Bridge rectifier, and the first-order Dickson–Greinacher rectifier circuits. \(P_{in}=-30\) dBm. The optimal efficiencies and \(R_{L}\) values for each rectifier are labelled. Simulated results in Figure 5 have been corroborated with experimental measurements using an experimental setup that reproduces the simulation setup Figure 1. A vector analyzer with zero span has been used as a 50 \(\Omega\) input source at 2.45 GHz, with \(P_{nom}=-30\) dBm. The same vector analyzer measures the \(S_{11}\) parameter needed to evaluate the power delivered to the rectifier (\(P_{in}\)) according to Equation (3). The rectifiers under test have been fabricated by minimizing connection paths using a FR4 PCB prototype board.