# New Miniature Narrow Band Microstrip Diplexer for Recent Wireless Communications
## 3 Design Concept
The relationship between the frequency of the first spurious resonance and the frequency of SIR's fundamental resonance is as follows:  
\[Y_{in}=jY_{2}\frac{Y_{2}\tan\theta_{1}\text{. }\tan\theta_{2}-Y_{1}}{Y_{2}\tan \theta_{1}+Y_{1}\tan\theta_{2}} \tag{2}\]0  
\[Y_{in}=jY_{2}\frac{Y_{2}\tan\theta_{1}\text{. }\tan\theta_{2}-Y_{1}}{Y_{2}\tan \theta_{1}+Y_{1}\tan\theta_{2}} \tag{2}\]1  
Equation (7) stands for the quarter-wavelength SIR ratio in the case of \(\frac{f_{s}}{f_{0}}=3\) and \(R_{z}=1\). Equation (8) represents a ratio of the half-wavelength SIR when \(\frac{f_{s}}{f_{0}}=2\) and  
\[Y_{in}=jY_{2}\frac{Y_{2}\tan\theta_{1}\text{. }\tan\theta_{2}-Y_{1}}{Y_{2}\tan \theta_{1}+Y_{1}\tan\theta_{2}} \tag{2}\]2 For various impedance ratios, a depiction between a total electrical length and \(\theta_{1}\) for the resonant. \(R_{z}=1\). Both categories of resonators have their normalized spurious resonance frequencies illustrated in \[Y_{in}=jY_{2}\frac{Y_{2}\tan\theta_{1}\text{. }\tan\theta_{2}-Y_{1}}{Y_{2}\tan \theta_{1}+Y_{1}\tan\theta_{2}} \tag{2}\]3  
Step impedance filters are a great choice if you are looking for an alternative to stub-based filters. They are the least performing type of filter because they have the highest and lowest practicable impedance transmission lines. They are an excellent option for tasks that do not require precise cutting. Increased isolation and a broad and deep stopband can be achieved with microstrip filters and diplexers based on varying SIRs [15]. The schematic for the designed circuit is shown in \[Y_{in}=jY_{2}\frac{Y_{2}\tan\theta_{1}\text{. }\tan\theta_{2}-Y_{1}}{Y_{2}\tan \theta_{1}+Y_{1}\tan\theta_{2}} \tag{2}\]4 The entire circuit, including the substrate, takes up only 24 \(\times\) 18 mm\({}^{2}\) in area, making it incredibly small and straightforward. Two BPFs operating in separate frequency ranges must be created in the design method. It should be noted that adjusting the length of the depicted UIR and SIR elements can change the filter center frequencies and bandwidths. Because of the coupled junction's space-saving and diplexer design, it was chosen to connect the two filters. Because the required frequency response is compromised, combining the two filters in practice could be better. This necessitates thorough optimization of the entire circuit by an electromagnetic simulator. Furthermore, separating the SIR- and UIR-based open-loop resonators from the feed lines can alter the simulated outcomes. Since the new Rogers Kappa-438 substrate (\(\varepsilon\)r = 4.38, tan\(\delta\) = 0.005) and the FR4 substrate (\(\varepsilon\)r = 4.25, tan\(\delta\) = 0.015) share the same dielectric constant (\(\varepsilon\)r) and substrate height (h = 1.524 mm), they are envisioned as performance substitutes for one another. Microstrip devices' size and scattering parameters are all affected by the loss tangent and dielectric constant, assuming a constant substrate thickness. Kappa-438 excels over FR4 with a minor loss tangent. Kappa-438 has a smaller loss tangent (0.005) than FR4 (0.015). I/O feeders are all non-symmetrical SIR varieties, but the upper and lower sections include clearly symmetrical and non-symmetrical parts of SIR varieties. \[Y_{in}=jY_{2}\frac{Y_{2}\tan\theta_{1}\text{. }\tan\theta_{2}-Y_{1}}{Y_{2}\tan \theta_{1}+Y_{1}\tan\theta_{2}} \tag{2}\]5 Normalized spurious resonance frequency concerning impedance ratio.