# Predicting Electromagnetic Performance Under Wrinkling in Thin-Film Phased Arrays
## 5 Conclusions  
This study proposes a wrinkle-aware electromechanical coupling model for space-deployable membrane antennas with copper-coated patch elements. By incorporating angular distortion, displacement variation, and stiffness contrast into a unified framework, this model enables the performance prediction of phased array antennas under structural deformation.  
A dual-domain displacement model was constructed to account for non-uniform stiffness across the membrane surface. The concept of the stiffness ratio--defined as the bending stiffness ratio between the reinforced and base regions--was introduced as a key parameter linking structural reinforcement to electromagnetic behavior. Based on this model, a series of simulations were performed to evaluate the influence of structural design parameters \(c_{fz1}\) and \(c_{fz2}\) on antenna performance.  
The key findings are summarized as follows:  
1. The proposed model accurately captures wrinkling-induced deformations and their effects on far-field radiation and impedance. It enables rapid evaluation of antenna behavior under varying \(c_{fz1}\) and \(c_{fz2}\) values.
2. Variations in \(c_{fz1}\) and \(c_{fz2}\) lead to significant changes in membrane surface accuracy, which, in turn, affect the peak gain and sidelobe structure. Excessive structural modification in either parameter can degrade beam symmetry and increase sidelobe interference.
3. As \(c_{fz2}\) increases (e.g., reaching 0.30), the main lobe gain decreases, and sidelobes intensity, indicating potential degradation in directional performance and an increased risk of interference.
4. S-parameter analysis reveals that increasing \(c_{fz1}\) and \(c_{fz2}\) causes resonant frequency shifts and impedance mismatches. Notably, at \(c_{fz2}\) = 0.30, the return loss significantly worsens, potentially compromising the antenna's operational efficiency at 1.3 GHz.
5. Phased array simulations show that while the antenna maintains high gain at broad-side (\(\theta\approx 0^{\circ}\)), its radiation pattern becomes increasingly distorted at larger \(c_{fz2}\) values. In particular, gain symmetry is preserved at \(\varphi\) = \(0^{\circ}\), but deformation is more pronounced at \(\varphi\) = \(90^{\circ}\), demonstrating directional sensitivity to the structural design.  
In summary, this work provides a structural-electromagnetic co-simulation framework based on \(c_{fz1}\) and \(c_{fz2}\) parameterization, enabling rapid assessment of how structural design influences wrinkling, impedance, and radiation pattern. These findings offer valuable theoretical and simulation guidance for future design and optimization of thin-film phased array antennas.  
**Author Contributions:** Conceptualization, X.Z. and X.M.; methodology, X.Z.; software, L.Z.; validation, H.L., Y.F. and X.J.; formal analysis, J.Y.; investigation, X.Z.; resources, X.M.; data curation, J.Y.; writing--original draft preparation, Y.F.; writing--review and editing, Y.X.; visualization, L.Z.; supervision, X.M.; project administration, X.M.; funding acquisition, X.M. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the National Natural Science Foundation of China, grant number 12494564, and the National Key R&D Program of China, grant number: 2023 YFB3407104. This research was funded by the CSC Program, grant number 202204980008, and the Pre-research Project of Civil Aerospace Technology, Grant No. D030207. The APC was funded by the Xi'an Institute of Space Ratio Technology.  
**Data Availability Statement:** All data used for the numerical analyses that support the findings of this study are available from the corresponding author upon reasonable request.  
**Conflicts of Interest:** The authors declare no conflicts of interest.