## 2 Antenna Design and Theoretical Analysis  
As shown in Figure 1, the geometric diagram of the proposed tri-band patch antenna and the optimal size of the corresponding configuration are given. It can be seen that the antenna is composed of two circular substrates and an air layer with a height of \(H2\). Substrate 1, which has a permittivity of 2.2, a tangent loss of 0.0001, and a height of \(H1\), is located in the lower layer of the antenna. A circular patch with a radius of R4 and a ground plane with a radius of R1 are both concentric with Substrate 1, and are printed on the upper and lower surfaces of Substrate 1. An off-center probe is introduced to change the surface current on the circular patch, which is similar to the principle of etching a slot on the patch [10]. Meanwhile, four cambered strips surrounding the outer side of the circular patch are also printed on the upper surface of Substrate 1. Substrate 2, which has the same electrical parameters as Substrate 1, is located at the uppermost layer of the proposed antenna, but the thickness is selected as \(H3\) to reduce the profile height of the overall antenna. A parasitic patch with a radius of R5 is printed on the upper surface of Substrate 2. The parasitic patch is excited by the coupling effect with the lower circular patch. Finally, the two substrates and a foam board with a thickness of 0.5 mm in the middle are fixed together with two nylon screws. In order to clearly explain the working mechanism of the designed antenna for multi-frequency, widened bandwidth, and diverse radiation patterns, the three antennas that appear in sequence during the antenna design process are shown in Figure 2. Ant. 1 is a traditional single-layer circular patch antenna with an off-center probe. Compared with the center-fed circular patch antenna in [1; 2; 6], the off-center feeding proposed in this paper introduces more resonance modes. As for Ant. 1, the distance between the feeding point and the center of the circular patch is set to d, and the radius of the circular patch is set to R4. In terms of the reflection coefficient, a detailed parameter analysis is carried out for these two parameters, and the corresponding simulated results are shown in Figure 3. It can be observed from Figure 3 that when d = 0--that is, when the center-fed probe is employed--the antenna has only one resonance point at 4.53 GHz. As d gradually increases, Ant. 1 exhibits the characteristics of multi-frequency resonance. In detail, when d = 9.8 mm, Ant. 1 exhibits a reflection coefficient of less than \(-\)10 dB at 2.18 GHz, 3.62 GHz, and 4.48 GHz, respectively. On the basis that d is determined to be 9.8 mm, we further study the influence of R4 on antenna resonance. It can be seen from Figure 3  
Figure 1: Geometry of the proposed patch antenna: (**a**) top view of Substrate 1; (**b**) top view of Substrate 2; (**c**) side view of the proposed antenna. R1 = 48, R2 = 32, R3 = 27, R4 = 26.25, R5 = 33, R6 = 9.8, 1g = 4, \(H1\) = 1.5, \(H2\) = 0.5, \(H3\) = 1.1 (all dimensions in mm). the circular patch continues to increase, the three resonant frequency points of Ant. 1 all move to lower frequencies. Based on the above discussion, it can be concluded that, for Ant. 1, the position of the feeding point determines the resonance mode of the antenna, and the position of the resonance frequency point is mainly influenced by the size of the circular patch. Since the optimal reflection coefficients of Ant. 1 at the frequency points of 2.18 GHz and 3.62 GHz are greater than \(-\)10.56 dB, which fails to meet the requirements of engineering applications, some improvements are implemented in Ant. 1. As shown in Figure 2a, four cambered strips surrounding the outside of the circular patch are introduced to form Ant. 2, and the simulated result of S11 is given in Figure 4. It is worth mentioning that metal cambered strips surrounding the circular patch are introduced to improve impedance matching.