# Parallel-Coupled-Line Bandpass Filter with Notch for Ultra-Wideband (UWB) Applications
## 2 UWB Filter Design and Analysis  
The proposed configuration of the microstrip wideband bandpass filter with a narrow notch using a microstrip line is illustrated in Figure 1. The wideband bandpass microstrip filter was designed using a single section of parallel-coupled lines with an electrical length (\(\theta_{c}\)) of a quarter-wavelength at the center frequency of the desired UWB passband. At the end of the central line of the coupled lines, a short-circuited stub was connected. The electrical length (\(\theta_{s}\)) of the short-circuited stub is a quarter-wavelength at the midband frequency of the desired passband (6.9 GHz). In addition, a stepped-impedance resonator was placed close to the three parallel lines section. The stepped-impedance resonator was designed to have a length of a half-wavelength at the mid-band frequency of the notch. A general six-port three parallel microstrip line configuration is shown in Figure 2.  
The relationship of the port currents and voltages of the symmetrical three parallel-coupled lines can be described as follows:  
\[\frac{\mathrm{d}[V]}{\mathrm{d}x}=-[z][I] \tag{1}\]  
where \([I]\) and \([V]\) are the three-dimensional column vectors and \([z]\) is the 3 \(\times\) 3 impedance matrix, which are defined by  
\[[z]= \begin{bmatrix}z_{11}&z_{12}&z_{13}\\ z_{12}&z_{22}&z_{12}\\ z_{13}&z_{12}&z_{11}\end{bmatrix} ,\ \ \ [V]= \begin{bmatrix}V_{1}\\ V_{2}\\ V_{3}\end{bmatrix} ,\ \ \ [I]= \begin{bmatrix}I_{1}\\ I_{2}\\ I_{3}\end{bmatrix}  \tag{2}\]  
The eigenvalue equation for \([z]\) can be written as follows:  
\[([z]-\lambda[E])[x]=0 \tag{3}\]  
where \([\lambda]=\lambda[E]\) is a matrix whose diagonal elements are the eigenvalues of \([z]\), \([E]\) is the unity matrix, and \([x]\) is a matrix with columns denoting the eigenvalues of \([z]\). More details of the design equations of the parallel line are explained in [18; 19]. The circuit model of the presented microstrip filter without a notch is depicted in Figure 3. From the six-port network expression in Figure 2, the circuit model in Figure 3 can be considered as a two-port network with the following termination conditions:  
\[I_{1}=I_{2}=I_{6}=0 \tag{4}\]  
\[I_{i}=I_{3} \tag{5}\]  
\[I_{o}=I_{4} \tag{6}\]  
\[V_{i}=V_{3} \tag{7}\]  
Figure 2: Schematic of six-port three parallel-coupled microstrip lines. \[V_{o}=V_{4} \tag{8}\]  
\[V_{s}=I_{5}(jZ_{s}tam\theta_{s}) \tag{9}\]  
Taking these conditions into account, the scattering parameters can be extracted as described in [20]. Following the procedure in [18; 19; 20], the characteristic impedances, admittances, and the length of the coupled three lines of equal width and separation using the defined microstrip substrate can be computed for the three normal modes taking into account the conditions in (4) to (9). The proposed microstrip filter was developed to attain a passband with a fractional bandwidth of 110% at a midband frequency of around 6.9 GHz to meet the FCC requirements for the UWB spectrum. It was developed and implemented on a low-cost microstrip substrate with a relative dielectric constant of 10.8. The thickness of the substrate 'was chosen to be 1.27 mm. The microstrip configuration for the bandpass filter is presented in Figure 4. The microstrip physical length and width for the parallel lines are represented by \(L\) and \(W\), respectively. The spacing between the coupled three lines is represented by \(g\). The microstrip physical length and width for the short-circuited stub are represented by \(L_{s}\) and \(W_{s}\), respectively. In order to attain a passband with a FBW of more than 110% at 6.9 GHz midband frequency, the physical sizes for the coupled line section were selected to be as follows: \(W=0.1\) mm, \(L=3.9\) mm, \(g=0.05\) mm. The width of the 50 \(\Omega\) feedlines for the input and output ports was found to be 1.1 mm.