## 2 Design of a Unit Cell for the Single-Layer Shared-Aperture Antenna
As shown in Figure 2d, the S-band element with the mutual complementary configuration including small gaps has a fractional bandwidth of 14.0%, which is 13.1% greater than that of the small size radiator, while the bore-sight gain for all cases are not significantly different. Figure 3 shows the reflection coefficient according to the patch widths. The bandwidths of the S- and X-band elements increase as S- and X-band patch widths \(w_{1}\) and \(w_{x}\) increase, respectively. The element widths are determined by considering the bandwidths and the average reflection coefficients. Figure 3: Reflection coefficient according to patch widths: (**a**) for the S-band element; (**b**) for the X-band elements. Figure 2: Comparison of antenna performances according to S-band element configurations: (**a**) without the mutual complementary configuration; (**b**) with the mutual complementary configuration; (**c**) with the mutual complementary configuration and gap; (**d**) bandwidths and bore-sight gains according to the S-band element configuration. To examine the operating principles of the S-band element, this element is modeled as an equivalent circuit. Figure 4 presents the impedance characteristics of the full EM simulation (CST) and the equivalent circuit model. The equivalent circuit model is obtained using a data-fitting method [22]. In the equivalent circuit model, the feeding patch and the parasitic patches are expressed as parallel circuits using resistances (\(R_{n}\)), inductances (\(L_{n}\)), and capacitances (\(C_{n}\)) [23; 24]. In addition, \(n\) is the element index, where \(n=1\) refers to the feeding patch, and \(n=2\), 3 refers to the parasitic patches. In addition, \(L_{f}\) is the feed inductance, and \(C_{mn}\) is the capacitive coupling between the feeding patch and the parasitic patches. The values of each lumped element are determined to minimize the difference between the impedance characteristics of the equivalent circuit and the EM simulation, as shown in Table 2. Figure 4b compares the real and imaginary characteristics of the antenna impedance of the equivalent circuit model and the full EM simulation model. In this result, averages of the real and imaginary impedance differences are 5.9 \(\Omega\) and 3.4 \(\Omega\), respectively. Thus, the tendency of the impedance characteristics of the equivalent circuit model is similar to the results of the full EM simulation model. Figure 4c,d show the antenna bandwidths of the equivalent circuit model and the full EM simulation model depending on the parameters \(C_{mn}\) and \(g\), respectively. As shown in Figure 4c, the bandwidth of the equivalent circuit model varies when the capacitance \(C_{mn}\) is adjusted. The bandwidth of the full EM simulation model is then changed by adjusting the gap \(g\), as shown in Figure 4d. This result is similar to that demonstrated in Figure 4c. Thus, these results demonstrate that the relation between \(C_{mn}\) and \(g\) is similar to the tendency of the capacitance according to the distance between conductors. Furthermore, the bandwidth of the S-band element is improved by adjusting the capacitive coupling between the feeding patch and parasitic patches. Figure 5 illustrates the normalized surface current distributions of the proposed antenna. Strong surface currents are observed in both the feeding patch and the parasitic patches, which indicate that the feeding patch and parasitic patch are electrically coupled. Figure 6 shows the antenna performance variation of the X-band element according to the substrate characteristics. As shown in Figure 6a, the isolation [25; 26] between the S- and X-band elements increases with the dielectric constant. However, radiation efficiency decreases as the dielectric constant increases [27]. Figure 6b represents the fractional bandwidth [25] and the front-to-back (F/B) radiation ratio of the X-band element according to the substrate thickness. The bandwidth of the X-band element increases with substrate thickness.