## 2 Design Equations
### Even and Odd Mode Analysis without DGS Elements
By comparing Equations (5) and (6) with respect to their real and complex parts, Equations (7) and (8) are obtained:  
\[Y_{in}^{e}=Y_{1}\Bigg{(}\frac{\frac{Y_{0}}{2}+jY_{1}\tan\theta_{1}}{Y_{1}+j \frac{Y_{0}}{2}\tan\theta_{1}}\Bigg{)}=Y_{1}\frac{\frac{\left(Y_{1}Y_{0} \right)}{2}+\frac{\left(Y_{1}Y_{0}\right)}{2}\tan^{2}\theta_{1}}{Y_{1}^{2}+ \left(\frac{Y_{0}}{2}\tan\theta_{1}\right)^{2}} \tag{1}\]0  
\[Y_{in}^{e}=Y_{1}\Bigg{(}\frac{\frac{Y_{0}}{2}+jY_{1}\tan\theta_{1}}{Y_{1}+j \frac{Y_{0}}{2}\tan\theta_{1}}\Bigg{)}=Y_{1}\frac{\frac{\left(Y_{1}Y_{0} \right)}{2}+\frac{\left(Y_{1}Y_{0}\right)}{2}\tan^{2}\theta_{1}}{Y_{1}^{2}+ \left(\frac{Y_{0}}{2}\tan\theta_{1}\right)^{2}} \tag{1}\]1  
By solving the Equation (3), the electrical length of the admittance line \(Y_{1}\) is obtained:  
\[Y_{in}^{e}=Y_{1}\Bigg{(}\frac{\frac{Y_{0}}{2}+jY_{1}\tan\theta_{1}}{Y_{1}+j \frac{Y_{0}}{2}\tan\theta_{1}}\Bigg{)}=Y_{1}\frac{\frac{\left(Y_{1}Y_{0} \right)}{2}+\frac{\left(Y_{1}Y_{0}\right)}{2}\tan^{2}\theta_{1}}{Y_{1}^{2}+ \left(\frac{Y_{0}}{2}\tan\theta_{1}\right)^{2}} \tag{1}\]2  
where  
\[Y_{in}^{e}=Y_{1}\Bigg{(}\frac{\frac{Y_{0}}{2}+jY_{1}\tan\theta_{1}}{Y_{1}+j \frac{Y_{0}}{2}\tan\theta_{1}}\Bigg{)}=Y_{1}\frac{\frac{\left(Y_{1}Y_{0} \right)}{2}+\frac{\left(Y_{1}Y_{0}\right)}{2}\tan^{2}\theta_{1}}{Y_{1}^{2}+ \left(\frac{Y_{0}}{2}\tan\theta_{1}\right)^{2}} \tag{1}\]3  
By evaluating Equation (7), the electrical length \(\theta_{2}\) of the transmission line (TL2) is achieved. The electrical length of the admittance line \(Y_{2}\) is then:  
\[Y_{in}^{e}=Y_{1}\Bigg{(}\frac{\frac{Y_{0}}{2}+jY_{1}\tan\theta_{1}}{Y_{1}+j \frac{Y_{0}}{2}\tan\theta_{1}}\Bigg{)}=Y_{1}\frac{\frac{\left(Y_{1}Y_{0} \right)}{2}+\frac{\left(Y_{1}Y_{0}\right)}{2}\tan^{2}\theta_{1}}{Y_{1}^{2}+ \left(\frac{Y_{0}}{2}\tan\theta_{1}\right)^{2}} \tag{1}\]4  
where  
\[Y_{in}^{e}=Y_{1}\Bigg{(}\frac{\frac{Y_{0}}{2}+jY_{1}\tan\theta_{1}}{Y_{1}+j \frac{Y_{0}}{2}\tan\theta_{1}}\Bigg{)}=Y_{1}\frac{\frac{\left(Y_{1}Y_{0} \right)}{2}+\frac{\left(Y_{1}Y_{0}\right)}{2}\tan^{2}\theta_{1}}{Y_{1}^{2}+ \left(\frac{Y_{0}}{2}\tan\theta_{1}\right)^{2}} \tag{1}\]5  
Furthermore, subtracting Equation (4) from Equation (8) yields:  
\[Y_{in}^{e}=Y_{1}\Bigg{(}\frac{\frac{Y_{0}}{2}+jY_{1}\tan\theta_{1}}{Y_{1}+j \frac{Y_{0}}{2}\tan\theta_{1}}\Bigg{)}=Y_{1}\frac{\frac{\left(Y_{1}Y_{0} \right)}{2}+\frac{\left(Y_{1}Y_{0}\right)}{2}\tan^{2}\theta_{1}}{Y_{1}^{2}+ \left(\frac{Y_{0}}{2}\tan\theta_{1}\right)^{2}} \tag{1}\]6  
Equation (13) is rearranged as:  
\[Y_{in}^{e}=Y_{1}\Bigg{(}\frac{\frac{Y_{0}}{2}+jY_{1}\tan\theta_{1}}{Y_{1}+j \frac{Y_{0}}{2}\tan\theta_{1}}\Bigg{)}=Y_{1}\frac{\frac{\left(Y_{1}Y_{0} \right)}{2}+\frac{\left(Y_{1}Y_{0}\right)}{2}\tan^{2}\theta_{1}}{Y_{1}^{2}+ \left(\frac{Y_{0}}{2}\tan\theta_{1}\right)^{2}} \tag{1}\]7  
Thus, by inserting \(T_{2}\) from Equation (11) into Equation (13) and rearranging it, Equation (14) is obtained:  
\[Y_{in}^{e}=Y_{1}\Bigg{(}\frac{\frac{Y_{0}}{2}+jY_{1}\tan\theta_{1}}{Y_{1}+j \frac{Y_{0}}{2}\tan\theta_{1}}\Bigg{)}=Y_{1}\frac{\frac{\left(Y_{1}Y_{0} \right)}{2}+\frac{\left(Y_{1}Y_{0}\right)}{2}\tan^{2}\theta_{1}}{Y_{1}^{2}+ \left(\frac{Y_{0}}{2}\tan\theta_{1}\right)^{2}} \tag{1}\]8where  
\[Y_{in}^{e}=Y_{1}\Bigg{(}\frac{\frac{Y_{0}}{2}+jY_{1}\tan\theta_{1}}{Y_{1}+j \frac{Y_{0}}{2}\tan\theta_{1}}\Bigg{)}=Y_{1}\frac{\frac{\left(Y_{1}Y_{0} \right)}{2}+\frac{\left(Y_{1}Y_{0}\right)}{2}\tan^{2}\theta_{1}}{Y_{1}^{2}+ \left(\frac{Y_{0}}{2}\tan\theta_{1}\right)^{2}} \tag{1}\]9  
As Equation (14) is a function of \(Y_{0}\), \(Y_{1}\), \(Y_{2}\) and R, Equations (9), (11) and (14) can be plugged into a script or spreadsheet utilizing Matlab or Microsoft Excel, in order to calculate the values of \(\theta_{1}\), \(\theta_{2}\) and R.