## 2 Derivation of 3D Array Factor  
### Improvement of Pattern Degradation of Conformal Arrays through Phase Compensation  
To investigate the change in electrical characteristics according to the platform shape of the conformal arrays, we designed a patch antenna operating at 10 GHz (Figure 1). Taconic's RF-35 (dielectric constant = 3.5) was used as a substrate for the patch antenna with a thickness of 1.52 mm. Coaxial feeding was employed, and the broadside gain of a single radiation element was about 6.2 dB. In the patch antenna, the radiating element has the size of x-direction element spacing, dx = 0.5\(\lambda\), and y-direction element spacing, dy = 0.5\(\lambda\) (15 mm). We compared the pattern performance of a \(12\times 24\) planar array using the designed patch antenna as a single radiation element and the radiation pattern of a \(12\times 24\) conformal array. Figure 1: Single element information used for conformal-array characteristic analysis: (**a**) element type, (**b**) S-parameters information. The conformal array is cylindrical, and the variable \(r\) is the radius of the cylindrical array platform, as shown in Figure 2. When \(r\) for the \(12\times 24\) planar and the conformal arrays is \(r=200\), \(400\), and \(600\) mm, the 2D radiation patterns under uniform amplitude and in-phase feeding are as shown in Figure 3.  
The pattern of the conformal array significantly differs from that of the planar array according to the change in the platform (change in \(r\)) even with the same radiating element shape, number of radiating elements, and radiating element spacing. In the presented case, as \(r\) becomes shorter (as the curvature increases), the element location situation changes, and it can be seen that the gain is decreased and the beam is widely distributed due to the phase difference up to the same-phase wave front. The cause of performance degradation in uniform feeding was analyzed from a phase perspective by schematically illustrating a cylindrical situation bent along the x-axis. Figure 2: Description of array type and related variables: (**a**) \(12\times 24\) planar array, (**b**) \(12\times 24\) conformal array, (**c**) description of variable \(r\). Figure 4a shows the position of each radiation element in a planar array condition. Unlike the planar array (Figure 4a), the z-axis coordinates for the conformal array changes for each radiation element as the element position changes over the curve (Figure 4b). To obtain the phase length to the wavefront in Figure 4b, the array factor of the conformal array can be calculated by considering the z-axis coordinates or the phase compensation for the phase delay, which is the distance from the actual element position to the x-axis. In other words, for uniform amplitude feeding, conformal arrays require the modification of the array factor (Equation (1)) used for planar arrays. \[AF_{planar}=\sum_{k=1}^{M\cdot N}e^{i[(\frac{2\pi}{\lambda}x_{k}\sin(\theta) \cos(\phi)+\frac{2\pi}{\lambda}y_{k}\sin(\theta)\sin(\phi))-(\frac{2\pi}{ \lambda}x_{k}\sin(\theta_{0})\cos(\phi_{0})+\frac{2\pi}{\lambda}y_{k}\sin( \phi_{0})\sin(\phi_{0}))]} \tag{1}\]  
When the positions of the elements in Figure 4a correspond to the XZ plane, as shown in Figure 4b, it can be represented as shown in Figure 5. The element coordinate of the planar array \(P_{k}\) is moved to that in the form of a conformal array. To calculate the phase on the x-axis, we consider the phase difference as much as \(\textit{phasedelay}_{k}\), which is the length between the corresponding positions \(P_{k}^{\prime}\) and \(P_{k}^{\prime\prime}\) on the x-axis. If Equation (1) is solved considering the phase difference of \(\textit{phasedelay}_{k}\) for the conformal array, the array factor in Figure 4b can be obtained. The phase difference \(\textit{phasedelay}_{k}\) can be expressed as follows. \[\textit{phasedelay}_{k}=\frac{|z_{k}|}{\cos(\theta_{0})} \tag{2}\]  
Figure 4: Arrangement shape on the XZ plane: (**a**) linear array (planar array) shape and (**b**) curved array (conformal array) shape.