# Predicting Electromagnetic Performance Under Wrinkling in Thin-Film Phased Arrays
## 4 Discussion  
### Dual-Domain Displacement Patch Analysis  
Firstly, we performed the analysis of Equation (17) under the condition \(c_{fz1}=c_{fz2}\), thereby predicting the effect of local deformation of the patch antenna on its electrical performance. The impact of local film deformation (\(c_{fz1}\)) on antenna performance was analyzed separately via S-parameter (S11) curves and gain radiation patterns. As shown in Figure 12, the S11 curves depict how the antenna's reflection coefficient varies with frequency for different \(c_{fz1}\) values. Figure 12 shows the antenna gain versus \(\theta\) at 1.4 GHz under the same \(c_{fz1}\) conditions. Figure 11: Effect of film translation on S11 and gain (\(d_{zm}\)). Figure 12: Effect of local deformation on S11 and gain (\(c_{fz1}\)). In Figure 12, changing \(c_{fz1}\) causes clear shifts in both the resonant frequency and matching depth: for certain \(c_{fz1}\) values, the resonance moves to higher frequencies, and the S11 minimum degrades from about \(-\)12 dB to \(-\)6.7 dB. This indicates that local deformation alters the patch's effective electrical length and impedance environment, worsening matching or shifting the resonance. At \(c_{fz1}\) = 0.003, no S11 response appears between 1 GHz and 2 GHz, implying the resonance has moved outside this band or become unmatched. The gain patterns in Figure 12 illustrate the far-field behavior at 1.4 GHz for each \(c_{fz1}\). The blue curve (\(c_{fz1}\) = 0.003) shows an unreasonably low main lobe gain and uneven side-lobe distribution, indicating that this deformation severely distorts the current distribution and polarization, effectively degrading radiation performance. Other cases (e.g., the green and orange curves) maintain smoother patterns, though with a reduced peak gain. The phenomenon that the gain decreases first and then increases can be qualitatively analyzed using the following equations. When the surface distortion increases to \(c_{fz1}\) = 0.003, due to the geometric changes in the patch-ground gap and the current path, the equivalent capacitance and inductance of the antenna increase simultaneously, causing the resonant frequency to shift out of the original design frequency band in the low-frequency direction; at the same time, the distortion enhances the excitation of high-order modes and the patch-film coupling, significantly reducing the quality factor, making the original S\({}_{11}\) matching valley shallow and wide, and making it difficult to distinguish within a given scanning range. This mechanism can qualitatively explain why no obvious matching point can be seen at 0.003. Based on the previous section's analysis, among the three offset errors, the \(c_{fz1}\) error has the most significant impact on resonant frequency drift. As the \(c_{fz1}\) offset increases, the resonant point shifts the furthest and may even fall outside the matching band, preventing the antenna from operating normally at its design frequency and undermining system stability. It also inflicts the greatest direct damage to gain--distorting the pattern and even exceeding the ideal peak--indicating that the form of local deformation represented by \(c_{fz1}\) is the most severe. In high-precision applications (e.g., satellite communications or phased arrays), such frequency drift can cause element-to-element phase mismatches, thereby degrading overall directivity and gain balance. In general, local deformation of the film leads to shifts in the resonant frequency and matching characteristics and causes varying degrees of distortion or sidelobe increase in the gain pattern. For a phased array antenna, if such deformation varies significantly between elements, it can degrade beam-forming precision and compromise pattern consistency. Therefore, during design and fabrication, the film's local deformation must be strictly controlled to ensure each element's matching and gain remain stable. In contrast, the \(d_{pch}\) error ranks second in severity.