## 2 Proposed UWB Antenna with Quadruple Band Rejections
### Input Impedance and Equivalent Circuit Model Analysis  
Figure 6a gives the impedance curve of the UWB antenna without the notch-band structures. The real part of the impedance is around 50 \(\Omega\) and the imaginary part is around 0 \(\Omega\). The impedance curve of the UWB antenna with notch-band structures is given in Figure 6b. At 3.5 GHz, 5.2 GHz, 5.8 GHz, and 7.6 GHz, the real part of the impedance is approximately zero and the slope of the imaginary part is positive, showing series resonance characteristics. A low impedance at the feed point results in a mismatch and a band-notch characteristic is observed.  
For a series RLC circuit, the bandwidth is defined as the frequency range where the resistance value is less than 1/0.707 times the minimum resistance value. The resistance can also be derived from the simulation results in Figure 6. The lumped element value can be obtained by the following equation and is listed in Table 1:  
\[\mathrm{Q}_{0}=\frac{f_{0}}{\mathrm{BW}} \tag{3}\]  
\[\mathrm{Q}_{0}=\frac{1}{2\pi f_{0}\mathrm{RC}} \tag{4}\]  
\[f_{0}=\frac{1}{2\pi\sqrt{\mathrm{LC}}} \tag{5}\]  
Figure 7 shows the equivalent circuit model of the proposed antenna. According to Figure 7, S11 of the equivalent circuit model can be obtained from the Equations (6)-(8). Figure 8a shows the S11 curve of the circuit model obtained through ADS simulation, which agrees with the S11 parameters calculated by the formula. The circuit parameters are analyzed. Taking the curled C-ring as an example, Figure 8b illustrates that when \(\mathrm{L}_{1}\) and \(\mathrm{C}_{1}\) are unchanged, the notch frequency and band-stop width of the antenna do not change. Figure 8b shows that both \(\mathrm{R}_{1}\) and \(\mathrm{L}_{1}\) remain unchanged; the notch-band frequency decreases with the increase of \(\mathrm{C}_{1}\). Figure 8d depicts that when \(\mathrm{R}_{1}\) and \(\mathrm{C}_{1}\) remain unchanged,  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Circuit.** & **BW (MHz)** & **R (f)** & **L (nH)** & **C (fF)** & \(\mathbf{Q_{0}}\) \\ \hline Curled C-shaped slot & 40 & 1.95 & 7.7 & 266 & 87.5 \\ Interdigital inductance slots 1 & 40 & 9.49 & 37.7 & 24.9 & 129.8 \\ Interdigital inductance slots 2 & 50 & 9.23 & 29.4 & 25.6 & 116 \\ Curled split-ring resonator & 110 & 11.51 & 16.6 & 26.2 & 69.09 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Calculated values of lumped elements.  
Figure 6: Simulated impedance characteristics of the proposed antennas. (**a**) Without band notch structures. (**b**) With band notch structures.  
the frequency of the notch band and notch width decrease with the increase of L\({}_{1}\). From the above rule, it can be concluded that when the notch frequency needs to be changed, only the capacitance of the structure needs to be changed. When the notch width needs to be changed, it is necessary to change the L of the structure first to obtain a suitable notch width, then change C to move the notch to the appropriate frequency.  
\[Z_{i}=\text{R}_{i}+j\omega\text{L}_{i}+\frac{1}{j\omega\text{C}_{i}} \tag{6}\]  
\[\frac{1}{Z_{in}}=\sum_{i=1}^{5}\frac{1}{Z_{i}} \tag{7}\]  
\[\text{S11}=\frac{Z_{in}-Z_{0}}{Z_{in}+Z_{0}} \tag{8}\]  
Figure 8: Simulated S11 of equivalent circuit model under different parameters. (**a**) Comparison between simulation and calculation. (**b**) Equivalent resistance R\({}_{1}\). (**c**) Equivalent capacitance C\({}_{1}\). (**d**) Equivalent inductance L\({}_{1}\).  
Figure 7: Equivalent circuit model of the proposed band-notched UWB antenna.