# A Quad-Band RF Circuit for Enhancement of Energy Harvesting
## 4 Design and Simulation
### Single-Band Circuit  
The applied input power varied from \(-40\) dBm to 20 dBm. The operating frequency of the RF source was adjusted at each band of the mentioned frequency bands.  
L-type matching was inserted, as shown in Figure 4. This matching type has the advantages of simplicity and the limited dimensions the circuit can occupy. The Smith chart was used to determine the suitable lumped elements for matching. As illustrated in Figure 5, lumped elements' suitable values are determined at each band of the four interested bands. The optimal lumped elements values are as the following: L = 100.9 nH and C = 2.545 pF (for the GSM-900), L = 25.43 nH and C = 2.989 pF (GSM-1800), L = 18.58 nH and C = 2.819 pF (UMTS-2100), and L = 13.52 nH and C = 2.606 pF (Wi-Fi 2.45).  
The matching network elements (L and C) were determined by using the Smith chart. The matching between the antenna and the rectifier can be done using transmission line sections (TLs) and lumped elements. TLs are preferably used for the high-frequency bands of UMTS-2100 and Wi-Fi 2.45 in this work to avoid any losses. In contrast, lumped elements are sufficiently used for the GSM-900 and GSM-1800 low-frequency bands to reduce the circuit dimensions. Both the inductance (L) and capacitance (C) are realized using a short-circuited and open-circuited transmission line sections in the high-frequency bands' circuit.  
Figure 4: A matched single-band rectifier circuit.  
The lengths of the transmission line sections are found as follows [75]:  
\[Z_{in}=Z_{O}\frac{Z_{L}+jZ_{O}tan\beta\ell}{Z_{O}+jZ_{L}tan\beta\ell}, \tag{22}\]  
where \(Z_{in}\), \(Z_{O}\), \(Z_{L}\), and \(\beta\) are the transmission line's input impedance, the characteristic impedance, the load impedance, and the phase constant, respectively. The phase constant is calculated as follows:  
\[\beta=\frac{2\pi F}{v} \tag{23}\]  
Figure 5: Smith chart for single-bands. (**a**) GSM-900. (**b**) GSM-1800. (**c**) UMTS-2100. (**d**) Wi-Fi 2.45.  
where F and \(\
u\) are the wave frequency and the speed of light in air. The transmission line's input impedance is obtained by Equation (24) for the inductance element (L)  
\[Z_{in}=jZ_{O}tan\beta\ell_{1}, \tag{24}\]  
then, \(\ell_{1}\) can be calculated by Equations (25) and (26).  
\[j\omega L=jZ_{O}tan\beta\ell_{1} \tag{25}\] \[L=\frac{Z_{O}tan\beta\ell_{1}}{\omega}, \tag{26}\]  
where \(\omega\) and \(\ell_{1}\) are the angular frequency and the first transmission line's length. On the other hand, the transmission line's input impedance for the capacitance (C) is found by Equation (27):  
\[Z_{in}=Z_{O}\frac{1}{jtan\beta\ell_{2}}, \tag{27}\]  
therefore, the second transmission line's length (\(\ell_{2}\)) can be calculated by Equations (28) and (29):  
\[\frac{1}{j\omega C}=\frac{Z_{O}}{jtan\beta\ell_{2}} \tag{28}\] \[C=\frac{tan\beta\ell_{2}}{\omega Z_{O}}. \tag{29}\]  
Equations (30)-(33) are used to find the width of the transmission line (W) [76]:  
\[\frac{W}{h}=\frac{2}{\pi}\Big{\{}\{\frac{\varepsilon_{r}-1}{2 \varepsilon_{r}}\Big{[}ln(B-1)-\frac{0.61}{\varepsilon_{r}}+0.39\Big{]}+B-1 -ln(2B-1)\}\Big{\}},forA>1.52 \tag{30}\]  
\[\frac{W}{h}=\frac{8e^{A}}{e^{2A}-2}\qquad,forA<1.52 \tag{31}\]  
where h, and \(\varepsilon_{r}\) are substrate thickness and substrate relative permittivity, respectively.  
\[A=\frac{Z_{O}}{60}\sqrt{\{\frac{\varepsilon_{r}+1}{2}\}}^{1/2}+\frac{ \varepsilon_{r}-1}{\epsilon_{r}+1}\{\frac{0.11}{\varepsilon_{r}}+0.23\} \tag{32}\]  
In this case, A is calculated, and it is found to be <1.52; therefore, Equation (31) is used to find h. Both A and B are variables.  
\[B=\frac{529.18}{Z_{O}\sqrt{\varepsilon_{r}}}. \tag{33}\]  
Finally, the dimensions of transmission lines and microstrip lines are calculated and presented in Table 1.