# Shape and Weighting Optimization of a Subarray for an mm-Wave Phased Array Antenna
## 2 Weighting Optimization for Individual Subarrays by Quadratic Programming
### Array Antenna Design and Calculation of an Array Pattern Considering Mutual Coupling  
Using the designed single antenna, a 24 \(\times\) 24 array antenna was designed as shown in Figure 10. In addition, the AP was calculated by applying the optimized weighting scheme in Figure 3 to the designed array antenna. In order to calculate the AP considering mutual coupling, the calculation applied an active element pattern (AEP). The AEP is defined as the radiation pattern by the center element when radiating elements other than the center radiating element of a large array antenna are matched in terms of the impedance. In [27], the characteristics of the AEP according to the location of the radiating element were explained as follows: in a large array, most radiating elements are in a similar coupling environment (except for the edge radiating elements). Therefore, there is a similar radiation pattern for each radiating element. However, the edge radiating elements have somewhat different radiation patterns. In a typical phased array antenna, the amplitude of the edge radiating elements is 10-15 dB lower than the amplitude of the central elements. For this reason, the AEP is used for the radiation pattern of the central radiating element. Using the AEP, the array pattern of a finite array in which the effect of mutual coupling between array elements is considered can be calculated through Equation (24) [27]. In order to derive the AEP, only the central element (port 300) was fed while the other radiating elements were impedance matched (termination 50-\(\Omega\)).  
\[AP=AEP\times AF \tag{24}\]  
Figure 10: A 24 \(\times\) 24 rectangular lattice array antenna: (**a**) cavity-type array antenna structure, (**b**) port definition from a front view (z-axis).  
Figure 9: Simulated \(S_{11}\) and 3-D radiation pattern of a cavity-backed patch antenna: (**a**) \(S_{11}\), (**b**) 3-D radiation pattern.  
Figure 11 shows the radiation pattern according to the AP calculation to which the AEP is applied when the beam direction is the boresight direction. It was confirmed that the SLL was \(-26.18\) dB through AP calculations considering the presence of mutual coupling. Figure 4 shows the radiation pattern without considering mutual coupling. It was confirmed that the radiation pattern is slightly different from AP considering the influence of mutual coupling and AP when ignoring the mutual coupling effect. Figures 11-13 show the radiation pattern applied with the AEP when the beam is scanned at the azimuth and elevation. Each \(5^{\circ}\) beam was scanned at an azimuth elevation, and through AP calculations, it was confirmed that the SLLs were \(-22.70\) dB and \(-24.44\) dB. In addition, it was confirmed that the radiation pattern derived through the AP calculations and the radiation pattern obtained using the EM full-wave simulation are practically equal. Table 3 quantitatively summarizes the beam characteristics shown in Figures 11-13. As the beam scanning angle increases, it can be seen that the SLL and HPBW both increase.