# Flexible Phased Antenna Arrays: A Review
## 4 Major Challenges
### Mechanical Deformation and Its Impact on Performance  
FPAs are often deployed on either curved surfaces or dynamically deforming surfaces, consequently, introducing phase distortions, impedance mismatches, and degradation of beamforming accuracy. Arrays mounted on deformable surfaces suffer from phase variation as a result of geometric distortions. The array factor describes how the geometrical combination of multiple antenna elements affects the overall radiation pattern. It represents the impact of the geometry of the array, amplitude and phase of the excitation signals applied to each radiating element. Generally, in an FPA, the radiating elements are properly arranged on a curved surface. So, the array factor is given by  
\[AF(\theta,\phi)=\sum_{n=1}^{N}a_{n}e^{\mathrm{jk}[x_{n}(u-u_{n})+y_{n}(v-v_{n} )+z_{n}(w-u_{n})]} \tag{1}\]  
with  
\[ \begin{split} u&=\sin\theta\cos\phi\\ u_{s}&=\sin\theta_{s}\cos\phi_{s}\\ v&=\sin\theta\sin\phi\\ v_{s}&=\sin\theta_{s}\sin\phi_{s}\\ w&=\cos\theta\\ w_{s}&=\cos\theta_{s}\end{split}  \tag{2}\]  
In (1)-(2), \(a_{n}\) is the amplitude of the \(n\)th radiating element, \(k=2\pi/\lambda\) is the wave number at the frequency \(f\), \(x_{n},y_{n}\), and \(z_{n}\) are the position coordinates of the \(n\)th element relative to the center oh the global coordinate system, \(\theta\) and \(\phi\) are the elevation and azimuth angles pertaining an arbitrary observation point, \(\theta_{s}\) and \(\phi_{s}\) are the scanning angles, \(u_{s}\), \(v_{s}\), and \(w_{s}\) are the scanning directions in the \(u\)-\(v\)-\(w\) space. Unlike the planar array, in the FPA, the \(z\)-axis coordinate of each radiating element changes according to its position over a 3D surface. As a result, the radiation pattern of FPA differs from that of the planar one even with the same number, shape, and spacing of the radiating elements. Phase of the signal is determined by its position relative to a reference point, so the variation due to the mechanical deformation affects the array \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Materials Used** & **Mechanical Properties** & **Electromagnetic Properties** & **Reason for Use** \\ \hline SNP ink & Flexible, Printable & Conductive, low resistivity & Printability, thermal stability \\ \hline Taconic@FastFilmTM & Durable, Lightweight & Dielectric constant \(\pm\) 3,55, Loss & \\  & & tangent = 0.0027 & Low loss, flexibility \\ \hline PP & Lightweight, Robust & Dielectric constant = 2 2–34, Loss & \\  & & tangent = 0.0002–0.001 & \\ \hline PET & Flexible for small thickness, Durable & Dielectric constant = 3.0–3.4, Loss & \\  & & tangent = 0.02 & Cost-effectiveness \\ \hline PDMS & Flexible, Conformable & Dielectric constant = 2.65, Loss & \\  & & tangent = 0.031 & Flexibility and thermal stability \\ \hline PEI fiber & Flexible, Durable & Dielectric constant = 3.0, Loss & \\  & & tangent = 0.0012 & Thermal stability \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._factor. This variation can alter the conditions for constructive and destructive interference leading to the distortion in main lobe and side lobe level (SLL). This issue can be dealt with compensation mechanisms and calibration techniques for the impact of mechanical deformation on electromagnetic performance. For example, two studies utilized mutual coupling measurements to sense shape distortions and compensate for phase and one of them implements a self-calibration mechanism [69; 70]. Similarly, another study addressed this challenge by integrating flexible resistive sensor and analog circuitry to monitor surface deformations and adjust phase shifters for real-time radiation pattern correction in dynamically changing conditions [36]. The problem of shape reconstruction was dealt with, in a different way with the introduction of Frequency Shift Key (FSK) radar approach to reconstruct the geometry of FPAs by estimating inter-element spacing [62]. For the preservation of the radiation pattern, one approach was implemented using phase in 2022 [71].