# Design of a Compact Ultra-Wideband Microstrip Bandpass Filter
## 2 Filter Design  
Figure 1 shows the laminated structure and design patterns of each layer, which consists of three layers of Rogers RO4350 dielectric substrate and four layers of RO4450F prepreg. Rogers RO4350 dielectric substrate has a relative dielectric constant of 3.66 and a loss tangent of 0.009. The thicknesses of the three dielectric substrates are 0.1016 mm, 0.254 mm and 0.508 mm respectively. Thermosetting resin bonding material is selected according to the processing requirements when processing MPCB structure. In this paper, RO4450F thermosetting resin bonding sheet compatible with R04350B is selected, with a thickness of 0.1016 mm, a dielectric constant of 3.52 and a loss tangent of 0.004. Two pieces of RO4450F need to be filled between every two substrates. The thickness of the copper skin of the top layer and the grounding layer is 0.035 mm, and the thickness of the copper skin of the middle two layers related to the lamination of RO4450F prepreg is set to 0.018 mm. The volume of the filter is 20.8 \(\times\) 20 \(\times\) 1.34 mm\({}^{3}\). The upper surfaces of three dielectric substrates are respectively coated with copper according to different design patterns, and four pieces of RO4450F with the same thickness are used for bonding them. Two blind holes are drilled in the wider part of the stepped impedance resonator (SIR) from the bottom ground plate through the third dielectric substrate to its upper surface. A DGS consisting of three rectangular slots is etched on the metal ground layer. When processing, the multilayer PCB to be pressed filled with RO4450F is clamped first, and then the clamping is positioned. After that, the temperature-programmed operation is carried out. With the gradual increase of laminating temperature, the prepreg resin will flow with it, and it will be filled between the copper circuit patterns with the help of uniform pressure attached to the multilayer PCB to be pressed, so that there will be no gap between the substrates. The whole design process includes four steps of optimization and improvement. Besides some marked design parameters being adjusted according to the actual processing conditions, the main optimization direction is to change the design structure of the top coupling patch and introduce the DGS composed of rectangular slots. Figure 2 shows the optimization process of the overall design of the filter. The curves of their corresponding transmission characteristics and reflection characteristics are shown in Figure 3.  
Figure 1: (a) Side view of filter structure; (b) Front view of filter structure; (c) Structure of top coupling patch, Copper_1; (d) Structure of second coupling patch, Copper_2; (e) SIR structure of the third layer, Copper_3; (f) Ground layer of DGS, Copper_4. Figure 3. Two pairs of terminal-open microstrip stubs are used to enhance out-of-band attenuation and roll-off factor from the perspective of impedance analysis. The transmission line input impedance is defined as:  
\[Z_{in}=Z_{0}\frac{Z_{L}+jZ_{0}\tan(\beta l)}{Z_{0}+jZ_{L}\tan(\beta l)} \tag{1}\]  
where \(Z_{0}\) is the transmission line of characteristic impedance. When the transmission line terminal is open-circuit, \(Z_{L}\rightarrow\infty\). By substituting \(Z_{L}\) into (1), \(Z_{in}\) is expressed as follows:  
\[Z_{in}=-jZ_{0}ctg(\beta l) \tag{2}\]  
Here, \(\beta=2\pi/\lambda\). When the length of the stub line is a quarter-wavelength, the input impedance is the following:  
\[Z_{in}=0 \tag{3}\]  
After a quarter-wavelength, the open circuit of the stub becomes a short circuit. Therefore, the quarter wavelength stub lines implement grounding. Because the terminal-open stubs are quarter wavelengths, two transmission zeros outside the passband will be generated, which correspond to 9.4 GHz and 16.8 GHz. At the same time, considering the coupling between two quarter-wavelength terminal-open stubs, the two groups of stubs are set at an appropriate distance to prevent the interference of short-range coupling on the transmission signal.