# GUI for Analysis of Parameters, Accurate Design and Optimization of Microstrip Filters
## 2 Mathematical Preamble to Microstrip Filters  
The basic configuration of a microstrip is depicted in Figure 1. It consists of a conductive strip (microstrip line) with a width \(W\) and length \(l\), placed on top of a dielectric substrate characterized by a relative dielectric constant \(\varepsilon_{r}\) and a thickness \(h\). The bottom side of the substrate features a conductive ground plane.  
The nomenclature of the variables corresponding to the design of microstrip filters is shown in Table 1.  
The relations used to compute the design parameters are the following. The effective dielectric constant, in terms of \(\varepsilon_{r}\), \(W\) and \(h\), is obtained through [24]  
\[\varepsilon_{eff}=\frac{\varepsilon_{r}+1}{2}+\frac{\varepsilon_{r}-1}{2} \bigg{(}1+\frac{10}{u}\bigg{)}^{-ab}, \tag{1}\]  
where \(u=W/h\), and  
\[a = 1+\frac{1}{49}\ln\Bigg{(}\frac{u^{4}+\Big{(}\frac{u}{52}\Big{)}^{ 2}}{u^{4}+0.432}\Bigg{)}+\frac{1}{18.7}\ln\bigg{[}1+\Big{(}\frac{u}{18.1} \Big{)}^{3}\bigg{]},\] \[b = 0.564\bigg{(}\frac{\varepsilon_{r}-0.9}{\varepsilon_{r}+3} \bigg{)}^{0.053}.\]  
The equation used to compute the characteristic impedance of a transmission line is  
\[Z_{c}=\frac{\eta}{2\pi\sqrt{\varepsilon_{eff}}}\ln\Bigg{(}\frac{F}{u}+\sqrt{ \Big{(}\frac{2}{u}\Big{)}^{2}}\Bigg{)}, \tag{2}\]  
where \(\eta=376.73\)\(\Omega\) is the wave impedance in free space, and  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Variable** & **Description** \\ \hline \(W\) & Width of the strip line. \\ \(\varepsilon_{r}\) & Relative dielectric constant of substrate \\ \(h\) & Thickness of the dielectric substrate \\ \(\mu\) & Permeability of the material \\ \(\varepsilon_{eff}\) & Effective dielectric constant \\ \(Z_{c}\) & Characteristic impedance of a strip line \\ \(\lambda_{g}\) & Guided wavelength of microstrip line \\ \(\beta\) & Associated propagation constant \\ \(l\) & Physical length of the strip line \\ \(f\) & Frequency in Hz \\ \(f_{c}\) & Cutoff frequency \\ \hline \hline \end{tabular}
\end{table}
Table 1: Nomenclature for variables of design.  
Figure 1: General microstrip structure.  
\[F=6+(2\pi-6)\exp\Bigl{[}-(30.666/u)^{0.7528}\Bigr{]}.\]  
The guided wavelength of a microstrip line is given, in millimeters, by  
\[\lambda_{g}=\frac{300}{f_{GHz}\sqrt{\varepsilon_{eff}}}\;\mathrm{mm}, \tag{3}\]  
where \(f_{GHz}\) is the frequency, given in GHz.  
The associated propagation constant is found through  
\[\beta=\frac{2\pi}{\lambda_{g}}=\frac{2\pi f\sqrt{\varepsilon_{eff}}}{c}, \tag{4}\]  
where \(c\) is the speed of light (\(c\approx 3\times 10^{8}\) m/s) in free space. Note that the frequency \(f\) in (4) is in Hertz.  
The electrical length for a microstrip line of physical length \(l\) is defined by  
\[\theta=\beta l. \tag{5}\]  
In some cases it will be useful to compute the width of the microstrip from its characteristic impedance \(Z_{c}\). In order to compute the width of a microstrip line, in terms of \(Z_{c}\), \(h\), and \(\varepsilon_{r}\), the following equation can be used  
\[W=h\frac{8\exp(A)}{\exp(2A)-2}, \tag{6}\]  
where  
\[A=\frac{Z_{c}}{60}\biggl{(}\frac{\varepsilon_{r}+1}{2}\biggr{)}^{0.5}+\frac{ \varepsilon_{r}-1}{\varepsilon_{r}+1}\biggl{(}0.23+\frac{0.11}{\varepsilon_{ r}}\biggr{)}.\]  
If \(W/h\geq 2\), then  
\[W=\frac{2h}{\pi}\biggl{\{}(B-1)-\ln(2B-1)\quad+\frac{\varepsilon_{r}-1}{2 \varepsilon_{r}}\biggl{[}\ln(B-1)+0.39-\frac{0.61}{\varepsilon_{r}}\biggr{]} \biggr{\}}, \tag{7}\]  
with  
\[B=\frac{60\pi^{2}}{Z_{c}\sqrt{\varepsilon_{r}}}.\]