## 2 Circuit Topology of RF-Energy-Harvesting Integrated Circuit
### RF-DC Rectifier  
For an N-stage rectifier, a pair of a metal-oxide-semiconductor field-effect transistor [MOSFET (M\({}_{\text{P}}\))] and capacitor (C\({}_{\text{P}}\)) can be considered as a rectifier with a small ripple voltage across C\({}_{\text{P}}\). An averaged output voltage of the \((p+1)^{\text{th}}\) stage can be expressed as follows:  
\[V_{O(p+1)}(V)=V_{O(p)}(V)+V_{\text{boost}}(V), \tag{2}\]  
where \(V_{O(p)}\), \(V_{O(p+1)}\), and \(V_{boost}\) are the output voltage of the \(p^{\text{th}}\) stage (present voltage), output voltage of the \((p+1)^{\text{th}}\) stage (next voltage), and incremental voltage of each stage [22], respectively. Approximation in charge computation was used to provide the incremental voltage \(V_{boost}\) as follows:  
\[V_{boost}(V)=V_{in}^{\prime}-V_{tn}-\left(\frac{15\pi}{8}\times\frac{I_{Ocf} ^{\prime}\sqrt{2V_{in}^{\prime}}}{\mu_{n}C_{ox}(W/L)}\right)^{2/5}, \tag{3}\]  
where \(V_{in}^{\prime}\), \(V_{tn}\), \(I_{\text{Oeff}}\), \(\mu_{n}\), and \(C_{ox}\) are an effective voltage amplitude, the threshold voltage of an NMOS transistor, effective loading current, electron mobility, and a gate oxide capacitor, respectively. \(W\) and \(L\) are the width and length of the MOSFET, respectively. \(I_{Off}^{\prime}\) and \(V_{in}\) can be given as follows:  
\[I_{Oeff}^{\prime}=I_{0}+\frac{I_{S0}}{\pi}\Big{(}\frac{W}{L}\Big{)}\Big{(}1- e^{-V_{in}^{\prime}/V_{T}}\Big{)}\Big{(}1+\lambda_{sub}V_{in}^{\prime}\Big{)}, \tag{4}\]  
\[V_{in}^{\prime}=\frac{C_{t}}{C_{t}+C_{par}} \tag{5}\]  
where \(I_{0}\), \(I_{S0}\), \(V_{T}\), \(\lambda_{sub}\), \(C_{t}\), \(C_{par}\), and \(V_{in}\) are the initial loading current, saturation current, thermal voltage, channel length modulation coefficient, total capacitance value of all capacitors, parasitic capacitance at each stage, and peak amplitude of an input signal of a voltage rectifier, respectively. If the body effect is ignored, the output voltage of the N-stage rectifier is as follows:  
\[V_{ON}(V)=N\times V_{boost}(V)=N\times\left(V_{in}^{\prime}-V_{tn}-\left( \frac{15\pi}{8}\times\frac{I_{Oeff}^{\prime}\sqrt{2V_{in}^{\prime}}}{\mu_{n}C_ {ox}(W/L)}\right)^{2/5}\right), \tag{6}\]According to the simulation results, many rectifiers are used for achieving the maximum efficiency. For example, 2-, 4-, 6-, 8-, 10-, and 16-stages are designed with respect to different loading currents or peak voltages [22]. For a rapid charging mechanism and stable power source at an operating frequency of 917 MHz, this study adopted a 6-stage voltage rectifier with a small threshold voltage of 0.45 V.  
Figure 3 shows a single-ended 6-stage Dickson voltage multiplier, which was published in [22]. It includes 6 diode-connected MOSFETs (M\({}_{1}\)-M\({}_{6}\)) and 6 capacitors (C\({}_{1}\)-C\({}_{5}\) and C\({}_{\rm L}\)). All transistors and capacitors are identical. A bottom plate, which is marked using a bold line, exhibited a large parasitic capacitance. It is grounded to reduce loss or is connected to the input terminal RF\({}_{\rm in}\), which is fed from the matching network. If the input signal of the voltage rectifier was sinusoidal with \(RF_{in}=\rm V_{in}\rm cos\ 2\pi\rm f\), where V\({}_{\rm in}\) and \(f\) are respectively the peak amplitude and operating frequency, the DC output voltage V\({}_{\rm dc}\) could be obtained for a charge transfer with a load capacitor C\({}_{\rm L}\). The capacitor is sufficiently large to store the transferred charge and to reduce the output ripple voltage [22]. If the voltage multiplier was in a steady state, RF\({}_{\rm in}\) was greater than or equal to zero for \(0\leq\rm t\leq\pi\)/4 and \(3\pi\rm/4\leq t\leq\pi\), and RF\({}_{\rm in}\) was less than or equal to zero for \(\pi\rm/4\leq t\leq 3\pi\rm/4\) in the first time cycle T (= 1/\(f\)). RF\({}_{\rm in}\) charged the capacitor C\({}_{\rm 1}\) through the MOSFET M\({}_{\rm 1}\) for RF\({}_{\rm in}\leq 0\), and a steady voltage of \(V_{in}-V_{TH}\) was then stored in C\({}_{\rm 1}\) by reducing a threshold voltage \(V_{TH}\).