# Simulation and Design of Three 5G Antennas
## 4 Metal Frame Mobile Terminal Dual-Element \"Loop-Groove\" MIMO Antenna Design
### The Decoupling Principle of the \"Ring-Slot\" Structure  
In MIMO antenna design, the characteristic mode theory is often used to address the issue of mutual coupling between antenna elements. However, when an antenna has  
Figure 37: The impact of the chip inductor on the transmission coefficient of the “ring-slot” structure. Figure 36: The influence of ground slot length (L) on the reflection coefficient of the slot antenna. multiple resonant modes, many of these modes may be closely spaced or adjacent within the specified frequency band. In such cases, it becomes impossible to excite only a single resonant mode in the MIMO antenna. Furthermore, regardless of how antenna elements are arranged, multiple resonant modes are often excited. The characteristic mode theory can be employed to analyze scenarios where two elements excite the same resonant modes. Typically, there are two situations:  
* Both antenna elements excite completely different resonant modes. When this occurs, an analysis of the overall structure is needed to find the most suitable placement for the antenna elements. However, this approach has limitations in practical antenna design. * Both antenna elements can excite partially similar or completely identical resonant modes. In this section, for a mobile terminal antenna with added metal frame, operating in the 3.4-3.6 GHz frequency range, the ring antenna element and the slot antenna element can excite multiple identical resonant modes. In a dual-element \"ring-slot\" MIMO antenna system, the radiated electric fields from the ring antenna and slot antenna can be represented as a superposition of characteristic electric fields. \[E_{Ant1}=\alpha_{1,1}E_{1}+\alpha_{M,1}E_{M}+\alpha_{C_{1,1}}E_{C_{1}}+\cdots+ \alpha_{C_{N1,1}}E_{C_{N1}},\] (29) \[E_{Ant2}=\alpha_{1,2}E_{1}+\alpha_{M,2}E_{M}+\alpha_{D_{1,2}}E_{D_{1}}+\cdots+ \alpha_{D_{N2}}2E_{D_{N2}},\] (30)  
In the equations \(\alpha_{n,1}\) and \(\alpha_{n,2}\) represent the mode weighting coefficients for the ring antenna and slot antenna, respectively. M represents the number of identical modes that the ring antenna and slot antenna can excite and which are the different resonant modes excited by the ring antenna and slot antenna. The envelope correlation coefficient in a MIMO system can be expressed as in Equation (31). \[\rho_{c}(i,j)=\frac{\left|\sum_{n=1}^{N}\alpha_{n,i}\alpha_{n,j}\right|^{2}}{ \left(\sum_{n=1}^{N}\left|\alpha_{n,i}\right|^{2}\right)\left(\sum_{n=1}^{N} \left|\alpha_{n,j}\right|^{2}\right)}, \tag{31}\]  
Substituting Equation (29) and Equation (30) into Equation (31), we obtain Equation (32). \[\left|\sum_{n=1}^{M+N1+N2}\alpha_{n,j}\alpha_{n,j}\right|^{2}=\left|\alpha_{1, 1}\alpha_{1,2}\right.^{\ast}+\alpha_{2,1}\alpha_{1,2}\right.^{\ast}+\cdots+ \alpha_{M,1}\alpha_{M,2}\right|^{2}, \tag{32}\]  
From Equation (32), it is evident that the envelope correlation coefficient is solely dependent on the mode weighting coefficients associated with the M identical modes excited. Identical modes can be categorized into two types: in-phase modes and out-of-phase modes, with a phase difference of 180 degrees. Considering the influence of different phase factors in the mode weighting coefficients, Equation (32) can be reformulated as Equation (33). \[\left|\sum_{n=1}^{P_{+}Q}\alpha_{n,i}\alpha_{n,j}\right|^{2} =\left|(\left|\alpha_{E1,1}\alpha_{E1,2}\right|+\cdots+\left|\alpha _{EP,1}\alpha_{EP,2}\right|)\right. \tag{33}\] \[\left.-\left(\left|\alpha_{A1,1}\alpha_{A1,2}\right|+\cdots+\left| \alpha_{AQ,1}\alpha_{AQ,2}\right|\right)\right|^{2}=\left|EC-AC\right|^{2},\]  
In the equation, among the M identical modes, assuming that there are P co-polar modes excited, \(\alpha_{\text{En,1}}\) and \(\alpha_{\text{En,2}}\) represent the mode weighting coefficients for the nth co-polar mode. Similarly, assuming that there are Q cross-polar modes excited, \(\alpha_{\text{An,1}}\) and \(\alpha_{\text{An,2}}\) represent the mode weighting coefficients for the nth cross-polar mode.