# Predicting Electromagnetic Performance Under Wrinkling in Thin-Film Phased Arrays
## 3 Structural-Electromagnetic Coupling Model  
### Electromechanical Coupling Model  
The coordinate system used for analyzing the far-field electrical performance of the phased array antenna is given in Figure 8. According to the theory of antenna design and analysis, the formula for calculating the field strength at point P is shown in the following equation [18]. \[E(r,\theta,\phi)=\frac{-j\omega\mu_{0}}{4\pi r}{\int_{dS}{I^{\prime}}_{n}(x,y) e^{-jk\cdot}\cos\theta dS} \tag{18}\]  
Figure 7: Deformation plots of the equivalent structure under separate dual-domain adjustments (20 th-order fitting results). In the equation, \(J_{n}(n=1,2,\ldots,\mathrm{N})\) denotes the current density of the nth patch, \(\omega\) is the angular frequency, \(\mu_{0}\) is the vacuum permeability, \(\theta,\phi\) is the observation direction at point P in space, \(k\) is the propagation constant (wavenumber), and \(\widehat{r}=\frac{\tau}{r}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\) is the unit vector in the radial (r) direction. According to the antenna design theory, the change in patch area displacement affects the spacing between the radiating patch and ground plane of the thin-film antenna, thus affecting the electrical performance of the antenna. And the effect of fold deformation in the film region represents the deformation in the medium region. In the equation, this deformation has an effect on antenna performance, but the effect of this deformation on the electrical performance has not been analyzed in previous studies. The formula for calculating the nth patch antenna's electrical performance, accounting for deformation, is as follows:  
\[E(r,\theta,\phi)=\frac{-j\omega\mu_{0}}{4\pi(r+\Delta r)}\underset{dS}{\int}J _{n}(x,y)e^{-j\mathcal{H}\cdot\varphi}e^{j\Delta\varphi(x,y)}\cos\theta dS \tag{19}\]  
In the formula, \(\Delta r\) and \(\Delta\varphi\), respectively, denote the effects introduced by the structural deformation. Here, \(\Delta r\) and \(\Delta\varphi\) represent the deformation errors in displacement and rotation, respectively. \(\Delta r\) and \(\Delta\varphi\) are, respectively, extracted from the overall antenna displacement function. \(\Delta r\) and \(\Delta\varphi\) can be expressed as  
\[\left\{ \begin{array}{l}\Delta r=\frac{1}{S_{\mathrm{patch}}} \underset{S_{\mathrm{patch}}}{\iint}f_{patch}(x,y)dS\qquad f_{patch}(x,y)=c _{fz1}\underset{i}{\sum}x^{i}y^{i}\\ \Delta\varphi(x,y)=\arccos[\hat{n}(x,y)\cdot\hat{r}_{0}]=\arccos \left(\frac{1}{\sqrt{1+\left(\partial_{x}f\right)^{2}+\left(\partial_{y}f \right)^{2}}}\right)\approx\frac{\left(\partial_{x}f\right)^{2}+\left( \partial_{y}f\right)^{2}}{2}\end{array} \right. \tag{20}\]  
Equation (20) follows directly from the small-slope geometry sketched in Figure 8. Denoting the local surface profile by _f(x,y)_, the height change of each patch, \(\Delta r\), is obtained by averaging the patch deflection over the patch area \(S_{patch}\). The phase shift \(\Delta\varphi\) is then the angle between the deformed normal \(\hat{n}_{i}\) and the reference normal \(\hat{n}_{0}\), where the final approximation is valid for \(\partial f\ll 1\). When the electrical performance of a single patch unit is obtained, the electrical performance of the entire antenna can be obtained through Equation (21). \[E_{Ant}=\underset{n,m}{\iint}E(n,m)e^{j(k_{x}x_{mn}+k_{y}y_{mn})} \tag{21}\]  
In this equation, \(n\) and \(m\) represents the number of rows and columns. The sum represents the vector sum. Figure 8: Geometric relationships of far-field electrical performance. In this paper, the effects of two typical deformations on the electrical properties of the patch antenna unit are investigated separately by varying the parameters of the two dual-domain displacement patterns. Since the film scale of thin-film antennas is particularly small compared to the area, direct modeling in simulation software results in a huge amount of computation.