## 2 Characteristic Theory-Based Antenna Methodology
Mode 4 had a narrow bandwidth from 24 to 25 GHz, and Mode 5 attained significance from 29 to 44 GHz, as depicted in Figure 1b. The phase difference between these modes could be analyzed with a characteristic angle in Figure 1c. It is worth noting that the structure has an inherent inductance and capacitance nature, which is reflected through its various modes when an electric field is incident. Modes 1 and 2 generated an opposing EMF for an incident field (\(E^{i}\)), leading to inductance from 20 to 30 and 20 to 37 GHz. A similar effect was observed for Mode 3 from 30 to 44 GHz. However, Mode 1 changed its character from inductance to capacitance from 30 to 44 GHz, and Mode 4 was largely capacitance. The zero-phase lag that was a CA of \(180^{\circ}\) was observed at 22 GHz, where Modes 1, 2, and 3 had almost zero phase differences. Consequently, this may have resulted in a good impedance match. Also, at 30 GHz, Modes 1 and 3 had zero-phase lag (i.e., at CA = \(180^{\circ}\)); however, Modes2, 4, and 5, which had a huge phase lag, created an impedance mismatch, impacting the antenna performance. Further, these modes' behavior was improved by adding another circular ring inside the outer ring to the antenna structure in Figure 2a. In this case, all the modes appeared significant from 24 to 34 GHz, and Modes 1, 3, 4, and 5 were prominent from 34 to 42 GHz, as shown in Figure 2b. However, a phase lag of 15\({}^{\circ}\) (i.e., 180\({}^{\circ}\)\(\pm\) 15\({}^{\circ}\)) was seen from 23 to 28.5 GHz for Modes 1 and 2 with a capacitance effect and Modes 3 to 5 with an inductance effect, as depicted in Figure 2c. Moreover, these modes exhibited non-linear behavior for changes in frequency. The non-linearities in the modes were corrected by adding a circular patch inside the inner ring, which terminated the feed line. Also, the conductive area of the ground plane was reduced, excluding the vertical edge portion, as illustrated in Figure 3a. The resultant MS in Figure 3b revealed that Modes 1, 2, 3, and 5 were dominant from 20 to 23 GHz; beyond this frequency, Modes 2 and 3 remained significant until 27 GHz. From 27 GHz onwards, a linear response of Modes 1 to 4 could be observed until 44 GHz. Further investigation through phase angle difference indicated that the antenna structure had three convergence points with a fundamental frequency at 23 GHz, as depicted in Figure 3c. The second and third convergence occurrences were noticed at the second and third harmonic frequencies of 33 and 43 GHz. The phase difference between the modes at these convergence points was close to zero, which was a CA \(\approx\) 180\({}^{\circ}\). It is also worth noting that Modes 1 and 3 and Modes 2 and 4 had similar current characteristics where their maximum equal phase difference of 40\({}^{\circ}\) (i.e., 180\({}^{\circ}\)\(\pm\) 40\({}^{\circ}\)) was reached between 23 and 33 GHz. This linear phase change from 23 to 33 GHz was expected to generate good impedance matching with a wide bandwidth. A similar impedance matching could be  
Figure 1: (**a**) The antenna structure of a simple ring (Ant1) and its results: (**b**) modal significance difference and (**c**) characteristic angle difference. expected from 33 to 43 GHz with a linear maximum equiphase difference of 15\({}^{\circ}\). However, the situation from 33 to 43 GHz was quite different. The surface current \(J_{5}\) (destructive mode) was lagging by 60\({}^{\circ}\) in response to the incident field due to the slow accumulation and discharge of the electric field. This large phase difference impacted the antenna with poor impedance matching from 33 to 43 GHz, so wide bandwidth could not be expected between these frequencies. Further, in the proposed antenna structure, four equilateral triangle-shaped stubs were embedded in the radiator at an angle of 90\({}^{\circ}\), as illustrated in Figure 4a. This modification improved the Mode 5 response from 33 to 43 GHz compared to the earlier antenna structure, as depicted in the MS graph in Figure 4b.