## 2 Electromagnetic Analysis for Proposed Antenna Design  
Figure 1 shows the proposed frequency reconfigurable kirigami-inspired antenna, designed using the ANSYS high-frequency structure simulator (HFSS). The antenna operates in circular disk mode when the SMA spring is pulled circular disk sector mode when the SMA spring is released. The circular antenna radius was determined from the cavity model [35; 36]:  
\[a=\frac{K_{mm}c}{2\pi f_{r1}\sqrt{\epsilon_{r}}}=\frac{K_{11}c}{2\pi f_{r1} \sqrt{\epsilon_{r}}} \tag{1}\]  
where \(f_{r1}\) is the resonant frequency of circular patch antenna, \(c\) is the speed of light, \(\epsilon_{r}\) is the dielectric constant, \(K_{mm}\) is the \(m\)th zero of derivative of the Bessel function of order \(v\), and \(K_{11}\) is the lowest value in circular patch antenna.  
Here is the first Bessel's function of order \(v\):  
\[J_{v}(K_{mm}a)=0\bigg{\{}\begin{array}{cl}v=n,&if\ \theta=2\pi\\ v=\frac{n\pi}{\theta},&otherwise\end{array} \tag{2}\]  
where \(a\) is the circular patch antenna radius, \(\theta\) is the ratio of the angle (expressed in radians), and \(n=\) 0,1,2,3.....  
From (1), it is possible to calculate the effective radius \(a_{e}\):  
\[a_{e}=\bigg{[}a^{2}+\bigg{(}\frac{2 ha}{\pi\epsilon_{r}}\bigg{)}\bigg{(}ln\frac {a}{2h}+(1.41\epsilon_{r}+1.77)+\frac{h}{a}(0.268\epsilon_{r}+1.65)\bigg{)} \bigg{]}^{1/2} \tag{3}\]where \(h\) is the substrate height.  
Finally, the expected resonant frequency of the circular sector antenna equation is calculated with the following:  
\[f_{r2}=\frac{K_{vm}c}{2\pi a_{e}\sqrt{e_{e}}}=\frac{K_{61}c}{2\pi a_{e}\sqrt{e_{e}}} \tag{4}\]  
where \(f_{r2}\) is the resonant frequency of circular patch antenna, \(e_{e}\) is the effective dielectric constant and \(K_{61}\) is the lowest value when \(\theta\) = 300\({}^{\circ}\) circular sector patch antenna.  
We set the proposed antenna radius \(a\) = 20.5 mm, and length from the center (\(d_{1}\)) was cut to implement the proposed kirigami-inspired structure. The circular patch was realized on an Ecoflex substrate with size \(w\)\(\times\)\(l\) = 60 mm \(\times\) 60 mm, thickness \(h\) = 5 mm, dielectric constant = 2.8, and tangential loss = 0.045. Both \(w\) and \(l\) correspond to 0.8\(\lambda_{\mathrm{g}}\) at 2.4 GHz. Figure 1c,d show side views for each antenna mode with the SMA spring for mode transformation. The proposed antenna was fed by a coaxial probe and excited at distance \(d_{2}\) = 11 mm from the proposed antenna center for 50 \(\Omega\) impedance matching. We allowed space \(r_{1}\) to insert the SMA spring, and the length was within the substrate height when the SMA was pulled.  
Figure 2 shows that simulated real and imaginary input impedances for the circular disk and circular sector modes change according to the mode conversion. Figure 3 shows the simulation results of the proposed kirigami-inspired antenna. As shown in Figure 3a, when the radius (C\({}_{1}\) = 16 to 21 mm) of circular disk mode is larger, the resonant frequency is decreased. As shown in Figure 2b, when the fixed radius (C\({}_{1}\) = 20.5 mm) with a longer arc length (larger \(\theta\) = 220\({}^{\circ}\), 250\({}^{\circ}\), 280\({}^{\circ}\), and 320\({}^{\circ}\)) of the major sector, the resonant frequency is decreased, as well. Figure 4a,b show electric field magnitude for the circular disk and circular sector modes at mode at 2.41 and 3.25 GHz, respectively. The circular disk mode exhibited a large electric field at the antenna edges (Figure 4a), whereas the circular sector mode exhibited a reduced electric field in regions where the electric field gathers due to  
Figure 1: Proposed frequency reconfigurable kirigami-inspired antenna: perspective view in (**a**) circular disk and (**b**) circular sector modes; side view in (**c**) circular disk and (**d**) circular sector modes.