# A Quad-Band RF Circuit for Enhancement of Energy Harvesting
## 3 System Modeling
The maximum received voltage (\(V_{Max}\)) from the antenna, which is considered the input voltage to the voltage doubler circuit (\(V_{Max}=V_{input}\)) can be calculated as in Equation (1):  
$\text{$1  
Then, the equivalent input impedance of the rectifier (\(Z_{input}\)) is a function of the circuit impedance at the two different operation states based on diodes biasing as in Equation (2). Hence, the two impedance can be calculated as in Equations (3) and (4). $\text{$2  
$\text{$3  
$\text{$4  
where \(Z_{D,R}\) and \(Z_{D,F}\) are the diode impedance at the reverse biasing and forward biasing states, respectively, to find them, a sequence of equations must be followed as expressed in Equations (5)-(12) regarding the HSMS-2852 Schottky diode datasheet. $\text{$5  
$\text{$6  
$\text{$7  
$\text{$8 (**a**) Equivalent circuit of a single-stage voltage doubler circuit. (**b**) A single cycle of the applied RF wave to the voltage doubler. $\text{$9  
$}}$0  
$}}$1  
$}}$2  
$}}$3  
where \(I_{s}\), n, \(C_{JO}\), \(V_{I}\), m, \(R_{s}\), \(V_{T}\) and T are the saturation current, the ideality factor, the diode junction capacitance at the zero bias case, the diode junction voltage, the grading coefficient, the series resistance, the thermal voltage, and the temperature in \({}^{\circ}\)K, respectively. \(I_{b,F}\), \(V_{a,F}\), \(R_{J,F}\), and \(C_{J,F}\) are the externally applied bias current, the applied voltage on the diode, the diode junction resistance, and the diode junction capacitance, respectively when the diode is forward biased. On the other hand, \(I_{b,R}\), \(V_{a,R}\), \(R_{J,R}\) and \(C_{J,R}\) are the externally applied bias current, the applied voltage on the diode, the diode junction resistance, and the diode junction capacitance respectively, when the diode is in the reverse case. Using the power analysis mode for the showed full-cycle in Figure 1, it can be analyzed as follows [73]:  
$}}$4  
where T is the periodic time, considering the diode operation theory based on the applied periods of the RF wave, the power integral will be as the following:  
$}}$5  
The second integral in Equation (14) must be from \(t_{1}\) to \(t_{2}\), and \(t_{2}\) is approximately T/4, i.e., \(t_{2}\approx\) T/4 (Figure 1, that is why the approximation sign is inserted in Equation (14). $}}$6  
$}}$7  
Since  
$}}$8  
then,  
$}}$9  
Equation (16) is substituted in Equations (18) and (19)$}$0  
$}$1  
\[ \begin{array}{ll}Z_{input}\approx&2\Bigg{[}\Big{(}\frac{1}{Z_{in,R}}+\frac{1 }{Z_{in,F}}\Big{)}+\frac{1}{\pi}\Big{(}\frac{1}{Z_{in,R}}-\frac{1}{Z_{in,F}} \Big{)}*\\ &\left.\left[2 sin^{-1}\Big{(}\frac{2V_{th}+V_{out}}{2V_{Max}}\Big{)}-sin\Big{(} 2 sin^{-1}\Big{(}\frac{2V_{th}+V_{out}}{2V_{Max}}\Big{)}\Big{)}\right]\right] \right]^{-1}.\end{array}  \tag{21}\]  
Based on the analyzed model's extracted formula, the input impedance of the rectifier vs. the frequency at different input power levels is presented in Figure 2. The rectifier input impedance was found under \(-20\) dBm, \(-10\) dBm, \(0\) dBm, and \(+10\) dBm input powers. Figure 2: Modelling results of input impedance of the rectifier vs. frequency at different input power levels. (**a**) Real part. (**b**) Imaginary part.