# A Multi-Frequency Omnidirectional Antenna Based on a Ring-Shaped Structure
## 2 Theory and Design  
The geometry of the antenna simulation model is shown in Figure 1. The antenna is printed on a 1.6 mm thick Taconic RF-60A substrate with a relative dielectric constant of 6.15 and a loss angle of 0.0028. The printed metal is copper with a thickness of 0.036 mm. Three open-loop resonator structures are loaded on the front in the form of radiation patches, as shown in Figure 1b, and circular defect ground structures with cross-shaped cross notches and rectangular ground structures are distributed in corresponding positions on the back, as shown in Figure 1c. The entire open ring resonator radiation patch structure is connected to the feeder end via a microstrip line with an impedance set to 50 \(\Omega\). The parameters of the proposed antenna are given in Table 1.  
The radiation patch on the front of the antenna structure consists of three split-ring resonator structures. When an electromagnetic wave is incident on an open resonant ring, the initially stable magnetic field induced by the incident wave changes and the magnetic field induces an induced current in the metal ring. The induced current flows over the metal ring such that the metal ring has an equivalent inductance. Moreover, due to the accumulation of charges between the metal rings, the metal rings generate equivalent inductance, and \(LC\) resonances occur in the metal rings [41]. According to the equivalence \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Parameter** & \(L\) & \(W\) & \(h\) & \(R_{1}\) & \(R_{2}\) & \(R_{3}\) & \(u\) & \(r\) & \(g\) & \(c\) & \(m\) & \(t\) \\ \hline Value/mm & 60 & 42 & 1.6 & 7.5 & 13.5 & 19.5 & 3.17 & 1.5 & 1.5 & 1.5 & 1 & 0.036 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design size of the multi-frequency antenna. principle of the open resonant ring, its resonant frequency can be expressed in the following formula [42]. \[f_{0}=\left[\sqrt{\left(C_{SRRs}R_{SRRs}^{2}\right)/L_{SRRs}-1}\right](2\pi C_{SRRs }R_{SRRs}) \tag{1}\]  
Here, the \(L_{SRR}\) is the overall inductance of the SRRs, which is formed by the metal ring inductors in series with the current direction. \(C_{SRR}\) is the overall capacitance of the SRRs equivalent circuit. \(R_{SRR}\)2 is the equivalent conductor loss, the shunt resistance. Its size is related to ring width g, metal thickness \(t\), and metal material \(\rho\). The capacitance of the open resonant ring SRRs is:  
\[C_{SRRs}=(n-1)[2R_{1}-(2n-1)(g+s)]/C_{0}/2 \tag{2}\]  
\[R_{SRRs}=\rho[R_{1}-(n-1)(g+s)]/(gt) \tag{3}\]  
\[L_{SRRs}=4\mu_{0}[R_{1}-(n-1)(g+s)][ln(0.98/\rho)+1.84\rho] \tag{4}\]  
where \(n=3\) is a ring number. In the open resonant ring structure, the resonance effect is mainly generated by the outer ring. The inner ring is used to strengthen the coupling between the inner and outer rings so that the total capacitance of the open resonant ring structure is increased to achieve a shift of the resonant frequency to a lower frequency. Figure 1: Antenna structure diagram. (**a**) Three-dimensional image of the antenna; (**b**) The front structure of the antenna; (**c**) The back structure of the antenna. The inner split-ring resonator radius is \(R_{1}\), the middle split-ring resonator radius is \(R_{2}\), and the outer split-ring resonator radius is \(R_{3}\) The inner, middle, and outer radii of the proposed radiation patch are proposed from [35]. \[R_{3}=\frac{29\times 10^{6}}{f_{0}\sqrt{\varepsilon_{reff}}} \tag{5}\]  
\[R_{2}=0.69R_{3} \tag{6}\]  
\[R_{1}=0.38R_{3} \tag{7}\]  
where \(f_{0}\) is the lowest resonant frequency of the proposed antenna, and \(\varepsilon_{reff}=\sqrt{(\varepsilon_{r}+1)/2}\) is the effective dielectric constant of the substrate. To explain the motives for developing the multi-frequency antenna, we show the process of antenna design more precisely. It should be emphasized that it is a common method to realize multi-frequency antennas by using multi-loop structures. Therefore, we compare the results of several multi-loop antennas here.