**3. 2<sup>K</sup> Factorial**

Altering the ten structural parameters helps in reducing EMI, whereas changing the operating frequency affects its area of application [30]. Table 1 shows an example of the different structural parameters that can affect the return loss, bandwidth, and maximum EMI. As full factorial design requires 2<sup>10</sup> experiments, which is too excessive, fractional 2K factorial was employed [31] with a <sup>1</sup> 64 fraction, which means we can identify the important structural parameters from 32 simulation runs. The maximum and minimum values taken for each parameter are 50% and 150% of their nominal value (shown in Figure 1b). Table 2 provides respective P values for three different responses for which fractional 2<sup>K</sup> factorials were performed. The P value helps in identifying the importance of the parameters [32].

**Table 1.** Examples of the changes in the antenna's structural parameters on its performance.


**Table 2.** List of significant parameters for optimization. None of the interaction terms other than these parameters were found to have P value less than 0.05, hence they are not statistically significant and are not shown here.


#### **4. Optimization Results and Discussion**

After the key structural parameters were identified from the factorial design, GA was employed for optimization. The GA optimizer setup included maximum number of generations (10,000), number of individuals for parents (30), number of mating pools per individual (30), number of individuals for children (30), number of survivors (10), and selection pressure for the next generation (10). For the reproduction setup, a uniform distribution mutation type was utilized with 0.1 uniform mutation probability, 0.5 individual mutation probability, 0.2 variable mutation probability, and 0.05 standard deviation [33]. These GA set up values were taken from [33], which also employed GA to optimize antenna design parameters with highly accurate results.

The optimized values of the parameters are shown in Table 3. Figure 5 shows the optimal design of the antenna in comparison with the initial antenna design.


**Table 3.** Initial and optimized values of tl, tw, ga, and gb obtained from the genetic algorithm (GA).

**Figure 5.** (**a**) Initial antenna and (**b**) optimized antenna, to show the difference in the design.

Figure 6 shows the EMI results for the nominal initial antenna design and the optimized design. We can see that in the frequency range covered by the antenna (shown in Figure 7 where the return loss is less than −10 dB), EMI was reduced by about 1 V/m. The initial antenna design covers a frequency range from 7.84 to 16.44 GHz, while the optimized antenna covers a frequency range from 7.72 to 17.25 GHz In Figure 7, the major change can be seen in the frequency at which the return loss is at a minimum.

**Figure 6.** *Cont*.

**Figure 6.** EMI for the 1 m spherical environment from the antenna.

(**a**) Simulated return loss for initial antenna.

(**b**) Simulated return loss for optimized antenna.

**Figure 7.** Return loss of antenna.

The antenna was excited using a wave port excitation at 10.3 GHz frequency and it was observed that the maximum surface current distribution concentrated at the excited port, and this is depicted in Figure 8. Surface current distribution was reduced for the same excitation in the optimized antenna, by almost half of that of the initial antenna. Lower current density points towards lower power consumption of the antenna.

**Figure 8.** Surface current distribution.

The 2-D and 3-D radiation patterns obtained in the E-fields and H-fields of the initial and optimized antennas at 10 GHz (operating frequency) was simulated and depicted in Figure 9. The antenna was simulated in the xz-plane orientation, where red and violet color patterns in Figure 9a and b represent radiation patterns for phi angles of 0◦ and 90◦, respectively, and the theta angle is 0◦. 3-D radiation patterns help to study the spread from the antenna into the environment. It is noticed that both the initial and optimized antennas achieved omnidirectional radiation patterns at a central frequency.

(**a**) 2-D radiation pattern for the initial antenna.

(**b**) 2-D radiation pattern for the optimized antenna.

**Figure 9.** *Cont*.

(**c**) 3-D radiation pattern for the initial antenna.

(**d**) 3-D radiation pattern for the optimized antenna.

**Figure 9.** Radiation patterns for the initial and optimized antennas: (**a**,**b**) show 2-D radiation patterns and (**c**,**d**) show 3-D radiation patterns.

Peak gain was plotted with respect to the operating frequency of the antenna as shown in Figure 10. An increase in peak gain was observed with the increase in operating frequency: 0.7 dB to 4.95 dB was the average peak gain of the initial antenna design, whereas 0 dB to 5 dB was the average peak gain for the optimized antenna for the given frequency range (where return loss was less than −10 dB). The advantage of stubs is that it helps to achieve small peak gains at lower operating frequencies [34]. A small variation of 0.05 dB found in the peak gain of an antenna is because of the reduced ground plane [34].

**Figure 10.** Peak gain in dB for initial and optimized antenna designs.

Mutual coupling between ports for the initial (found to be < −15 dB) and optimized antennas (found to be < −12 dB) are shown in Figure 11. Higher isolation was observed for ports 1 to 2 and 3 to 4 due to the identical structure of elements. The peak values of the isolation (within the obtained band) were −22 dB and −21.24 dB for the initial and optimized antennas, respectively, at 10.3 GHz. Thus, the isolation was sufficiently good with the optimized antenna. Table 4 represent the difference between initial and optimized antenna parameters.

**Figure 11.** S-parameters of the MIMO antenna for mutual coupling between ports for (**a**) the initial antenna and (**b**) the optimized antenna.


**Table 4.** Antenna parameter values for the initial and optimized antennas.

#### **5. Conclusions**

In this paper, a miniaturized single element of a T-shaped MIMO antenna was designed, optimized, and simulated. The 2K factorial and genetic algorithm optimization techniques were used for the optimization of the antenna. It was found that gb, ga, tl, and tw, as defined in Figure 1, were the major dimensional parameters affecting the return loss, peak gain, bandwidth, radiation pattern, surface current distribution, and EMI of the antenna. The optimized antenna had a wider frequency band that ranged from 7.72 to 17.25 GHz. This corresponded to a 1 GHz improvement in the bandwidth due to the improved return loss that provides new opportunities for the antenna's utilization in different

applications. The optimized antenna had lower current distribution that gives lower power dissipation and its 1 m sphere EMI was also reduced.

**Author Contributions:** Conceptualization, C.M.T.; methodology, D.K.; software, D.K., V.S., V.P., N.T.; validation, D.K.; formal analysis, D.K., V.S., V.P.; investigation, D.K.; resources, C.M.T.; data curation, D.K., V.S., V.P.; writing—original draft preparation, D.K.; writing—review and editing, C.M.T.; visualization, D.K., N.T.; supervision, C.M.T.; project administration, C.M.T.; funding acquisition, C.M.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the Chang Gung University research Grants QZRPD123 and CIRPD2F0024.

**Conflicts of Interest:** The authors declare no conflict of interest.
