## 2 Antenna Design and Theoretical Analysis
The specific performance is the gap between the four cambered strips; the gap between the strips and the circular patch is both capacitive and employed to reduce the inductance introduced by the coaxial probe. Compared with the simulated result of Ant. 1, the improved Ant. 2 also has three resonance modes, which are the same as Ant. 1. The specific performance of the reflection coefficient amplitudes of Ant. 2 at the three frequency points of 2.14 GHz, 3.64 GHz, and 4.43 GHz are \(-\)15.18 dB, \(-\)16.34 dB, and \(-\)32.08 dB,  
Figure 3: The effect of different parameters on simulated S\({}_{11}\) of Ant. 1. (**a**) d; (**b**) R4. Figure 2: The evolution of the proposed tri-band patch antenna: (**a**) Ant. 1; (**b**) Ant. 2; (**c**) Ant. 3.  
respectively, all of which satisfy the requirement of less than \(-15\) dB. On the basis of Ant. 2, Substrate 2, printed with a parasitic patch, is placed directly above Ant. 2, and an air layer with thickness of \(0.5\) mm is introduced between the two substrates to form Ant. 3. Attributed to the coupling effect between the circular patch and the parasitic patch, the surface current distribution on the circular patch changes, and the simulated results of \(\text{S}_{11}\) are given in Figure 4. The simulated results show that the reflection coefficients of Ant. 3 in the three frequency bands of \(1.86\)-\(1.89\) GHz, \(3.77\)-\(4.09\) GHz, and \(5.14\)-\(5.59\) GHz are all lower than \(-10\) dB. According to the location of the frequency band, three resonant frequency bands of Ant. 3 are named \(f_{1}\), \(f_{2}\), and \(f_{3}\) (from low to high). Compared with the previous two antennas, Ant. 3 achieves the characteristics of extended bandwidth while maintaining three resonance modes. In order to explain the working mechanism of the multi-mode and the expanded bandwidth more clearly, a parameter analysis is carried out for the two key parameters of R5 and \(H2\). The simulated results are shown in Figure 5. As shown in Figure 5a, as the radius R5 of the parasitic patch increases, \(f_{1}\) gradually shifts to a lower frequency, accompanied by a gradually deteriorating reflection coefficient. Similarly, \(f_{2}\), which has two resonance frequency points, also shows a tendency to move to a low frequency. However, the resonant point \(f_{22}\) at the high frequency within \(f_{2}\) remains basically unchanged, while the resonant point \(f_{21}\) at the low frequency gradually moves to an even lower frequency, resulting in the gradual division of \(f_{2}\) into two adjacent frequency bands. As for the high-frequency band \(f_{3}\), Figure 5b shows the detailed simulated results. When the radius R5 of the parasitic patch is \(29.8\) mm, Ant. 3 exhibits dual-frequency \(f_{31}\) and \(f_{32}\) resonance characteristics at \(5.24\) GHz and \(5.72\) GHz. When the radius R5 increases to \(31.8\) mm in steps of \(1\) mm, both \(f_{31}\) and \(f_{32}\) move to a lower frequency, and the distance between the two frequency points gradually decreases. As shown by the solid black line in Figure 5b, adjacent \(f_{31}\) and \(f_{32}\) form a wide band with an acceptable reflection coefficient, and the additional resonance point \(f_{33}\) further extends \(f_{3}\). However, as R5 continues to increase, \(f_{31}\) and \(f_{32}\) finally merge into the resonance frequency of \(5.31\) GHz, and the frequency \(f_{33}\) is maintained at \(5.57\) GHz. It is worth noting that the reflection coefficient within \(f_{3}\) gradually deteriorates as R5 increases from \(31.8\) mm to \(33.8\) mm, and finally, the advantage of broadband is lost. Figure 4: Comparison of simulated \(\text{S}_{11}\) for the three antennas. Figure 5c,d show the curve between the reflection coefficient of Ant. 3 and the air-layer thickness \(H2\). It can be concluded from Figure 5c that, as \(H2\) increases,\(f_{1}\) gradually moves to a higher frequency, and both\(f_{21}\) and \(f_{22}\) also show a tendency to move to a higher frequency.