## 2 Theoretical Infrastructure
### UCPW Transmission Lines and Key Feature  
Coplanar waveguide transmission lines typically consist of a signal line flanked by ground planes on the same layer, with some configurations including an additional ground plane beneath the signal line. Ungrounded coplanar waveguide transmission lines differ by omitting this lower ground plane, resulting in distinct electromagnetic characteristics. This simplified design is particularly advantageous for compact and cost-sensitive high-frequency circuits. Figure 1 illustrates the UCPW configuration, where key parametersinclude the line width (\(w\)), spacing between the signal line and ground planes (\(s\)), copper thickness (\(t\)), dielectric height (\(h\)), and dielectric constant (\(\varepsilon_{r}\)).  
Maintaining consistent impedance along transmission lines is crucial for preserving signal integrity. Impedance mismatches can cause signal reflections, losses, and parasitic interactions that degrade overall performance. The reflection coefficient used to evaluate these mismatches is calculated by Equation (1) [15].  
\[\rho=\frac{V_{\text{reflected}}}{V_{\text{incident}}}=\frac{Z_{s}-Z_{in}}{Z_{s}+Z _{in}}, \tag{1}\]  
where \(V_{reflected}\) and \(V_{incident}\) are the reflected and incident voltages, \(Z_{in}\) is the instantaneous impedance at the signal entry point, and \(Z_{s}\) represents the initial impedance.  
When stacking layers in a PCB design, precise control over the distance between layers is essential. Parameters such as the dielectric material type, its constant \(\varepsilon_{r}\), and the spacing between the ground and signal layers all play critical roles in determining transmission line impedance. The stack design must consider material choices, layer thicknesses, and insulating properties, which collectively impact circuit performance and signal integrity [26; 27; 28].  
In UCPW transmission lines, the impedance calculation follows a structured series of steps outlined by [PERSON] [16].  
\[\text{a}=\text{w}, \tag{2}\]  
\[\text{b}=\text{w}+2\text{s}, \tag{3}\]  
\[\text{a}_{t}=\text{a}+\frac{1.25\text{t}}{\pi}\bigg{(}1+\ln\bigg{(}\frac{4\pi \text{a}}{\text{t}}\bigg{)}\bigg{)}, \tag{4}\]  
\[\text{b}_{t}=\text{b}-\frac{1.25\text{t}}{\pi}\bigg{(}1+\ln\bigg{(}\frac{4\pi \text{a}}{\text{t}}\bigg{)}\bigg{)}, \tag{5}\]  
Figure 1: The structure of UCPW transmission line.  
The resulting parameter \(k\) is derived from \(a\) and \(b\), and \(k_{t}\) is similarly computed using \(a_{t}\) and \(b_{t}\):  
\[k=\frac{a}{b}, \tag{6}\]  
\[k_{t}=\frac{a}{b_{t}}, \tag{7}\]  
\[k_{t}^{\prime}=\sqrt{1-k_{t}^{2}}, \tag{8}\]  
An additional variable \(k_{1}\) is calculated based on hyperbolic sine functions of at and \(b_{t}\) relative to dielectric height \(h\):  
\[k_{1}=\frac{\sin h\left(\frac{\pi n_{\text{H}}}{4h}\right)}{\sin h\left(\frac {\pi n_{\text{H}}}{4h}\right)}, \tag{9}\]  
\[k_{1}^{\prime}=\sqrt{1-k_{1}^{2}}, \tag{10}\]  
With the calculated \(k\) values, the elliptic integral \(K(k)\) is determined as follows:  
\[K(k)=\int_{0}^{\frac{\pi}{2}}\frac{d\Phi}{\sqrt{1-k^{2}\sin^{2}\Phi}}, \tag{11}\]  
Using \(K(k)\) and dielectric constant \(\varepsilon_{r}\), the effective dielectric constant \(\varepsilon_{eff}\) is given by Equation (12).  
\[\varepsilon_{\text{e}_{H}}=1+\frac{\varepsilon_{r}-1}{2}\frac{K(k^{\prime})} {K(k)}\frac{K(k_{1})}{K(k_{1}^{\prime})}, \tag{12}\]  
Finally, after accounting for copper thickness, the modified effective dielectric constant \(\varepsilon_{eff,t}\) and characteristic impedance \(Z_{0}\) are calculated with Equations (13) and (14).  
\[\varepsilon_{\text{e}_{H,z}}=\varepsilon_{\text{eff}}-\frac{\varepsilon_{ \text{e}_{H}}-1}{\frac{(b-a)/2}{0.7}\frac{K(k)}{K^{\prime}(k)}+1}, \tag{13}\]  
\[z_{0}=\frac{30\pi}{\sqrt{\varepsilon_{\text{eff},t}}}\frac{K(k_{t}^{\prime}) }{K(k_{t})}, \tag{14}\]