## 2 Micro-Strip Patch Antenna and DR Process  
### Micro-Strip Patch Antenna  
An MSP antenna is composed of a patch, a feed line, a dielectric substrate, and a ground, as depicted in Figure 2a. Owing to the effect of the inductance and the capacitance, intrinsic resonance frequency (R\({}_{\text{f}}\)) is produced where inductive and capacitive reactance offset each other, which results in a transfer of signal and power. The R\({}_{\text{f}}\) fluctuates as the antenna geometry is changed, which directly indicates how much the antenna is distorted. The antenna employed in a strain gauge normally adheres to the fragile surface of a structure, and the antenna's shape will be changed in which the structure is deformed. R\({}_{\text{f}}\) is then promptly changed in accordance with the structural deformation of the antenna. Therefore, it is ineludible to monitor the R\({}_{\text{f}}\) variation of the antenna to observe the status of the structured deformities [4]. As shown in Figure 2b, the R\({}_{\text{f}}\) can be appraised based on the geometric information as follows:  
\[\text{R}_{\text{f}}=\frac{1}{2(2\Delta L_{\text{f}}+L_{\text{e}})\sqrt{\varepsilon _{\text{r}}}}c_{l}, \tag{1}\]where \(\Delta L_{\text{\emph{L}}}\), \(L_{\text{\emph{e}}}\), \(\varepsilon_{\text{\emph{re}}}\), and \(C_{\text{\emph{l}}}\) indicate variation of the electrical length, the phase length, the effective dielectric constant, and the velocity of the light, respectively. Here, the phase length stands for the geometric dimension of the patch along the direction of antenna's line extension and radiation mode. The effective dielectric constant is rephrased by  
\[\varepsilon_{\text{\emph{e}}dc}=\frac{\varepsilon_{r}-1}{2\sqrt{(1+12H/W_{e})} }+\frac{\varepsilon_{r}+1}{2}, \tag{2}\]  
where \(W_{e}\), \(H\), and \(\varepsilon_{r}\) represent the patch's electrical width, constant thickness, and substrate's dielectric constant, respectively. On the basis of Equation (2), the phase length, \(\Delta L_{\text{\emph{L}}e}\), is denoted by the dimensions:  
\[\Delta L_{\text{\emph{L}}e}=\frac{(W_{e}/H+0.264)(\varepsilon_{\text{\emph{re }}}+0.3)H}{2.427(W_{e}/H+0.813)(\varepsilon_{\text{\emph{e}}dc}-0.258)}. \tag{3}\]  
The \(\text{R}_{\text{\emph{f}}}\) fluctuation depends on the path of the tensile strain (\(\varepsilon_{L}\)). While an observed \(\text{R}_{\text{\emph{f}}}\) that is higher than the initial \(\text{R}_{\text{\emph{f}}}\) implies the strain was employed in the electrical width direction (\(L_{\text{\emph{e}}}\)), the value of the \(\text{R}_{\text{\emph{f}}}\) should be less under the strain that is applied along electrical length direction (\(W_{e}\)). The frequency oscillation is caused by [PERSON]'s effect by  
\[W_{e}=\left(1-v_{p}\varepsilon_{L}\right)W_{o}\quad\text{and}\quad H=(1-v_{s} \varepsilon_{L})H_{o}, \tag{4}\]  
where \(v_{p}\), \(W_{o}\), \(v_{s}\), and \(H_{o}\) denote the Poisson's ratios of the patch, the original width, the Poisson's ratios of the substrate, and the thickness, respectively. By a variation of the tensile strain, (\(W_{o}\)/\(H\)) is elucidated as independent in the case where the proportion of width to thickness is less than one. Here, the Poisson's ratios, \(v_{p}\), and \(v_{s}\), will be the same, leading to the same \(\varepsilon_{\text{\emph{re}}}\) and \(\varepsilon_{r}\). Hence, the \(\text{R}_{\text{\emph{f}}}\) when a strain, \(\varepsilon\), is applied along the vertical direction will be denoted by  
\[R_{f}=\frac{c_{\text{\emph{l}}}}{2\sqrt{\varepsilon_{r}}}\frac{1}{(1- \varepsilon)(L_{\text{\emph{e}}}+2\Delta L_{\text{\emph{L}}e})}\approx\frac{ R_{o\text{\emph{f}}}}{1-\varepsilon}\approx R_{o\text{\emph{f}}}(1+ \varepsilon), \tag{5}\]  
where \(R_{o\text{\emph{f}}}\) indicates the \(\text{R}_{\text{\emph{f}}}\) that is estimated by \(W_{o}\) and \(H_{o}\). In this study, an example of the tensile strain applied along the width direction is demonstrated. The patch and substrate are able to derive significant effects on the antenna performance.