# New Miniature Narrow Band Microstrip Diplexer for Recent Wireless Communications
## 3 Design Concept  
A transmission line resonator with stepped impedances consists of two or more lines with a varied characteristic impedance that operates in TEM or quasi-TEM mode. Figure 2 depicts the two most common types of SIR, a short-circuited \(\lambda\)g/4 resonator and an open-circuited \(\lambda\)g/2 resonator. According to Figure 2, the quarter wavelength SIR comprises a short-ended line with characteristic impedance \(Z_{1}\) and electrical length \(\theta_{1}\) coupled to an open-ended line with characteristic impedance \(Z_{2}\) and electrical length \(\theta_{2}\). Two such elements connected through short-circuited ends, with this connection in place of grounding, might be thought of as the essential constituent of SIR and half-wavelength resonators [16; 17]. The definition of this element allows us to see \(\lambda\)g/4- and \(\lambda\)g/2-type SIR as composed of one and two elements, respectively. The SIR is defined in terms of electrical parameters by the ratio of the impedances \(Z_{1}\) and \(Z_{2}\) of the transmission lines, as shown in the following equation [17]:  
\[R_{z}=\frac{Z_{2}}{Z_{1}} \tag{1}\]  
The input admittance of \(\lambda_{\text{g}}/4\) SIR is equal to:  
\[Y_{in}=jY_{2}\frac{Y_{2}\tan\theta_{1}\text{. }\tan\theta_{2}-Y_{1}}{Y_{2}\tan \theta_{1}+Y_{1}\tan\theta_{2}} \tag{2}\]The \(\lambda\)g/4 resonator with a short circuit behaves like a parallel resonant circuit. For a quarter-wavelength SIR, the parallel resonance under condition \(Y_{in}\) = 0 is:  
\[\tan\theta_{1}.\tan\theta_{2}=Y_{1}/Y_{2}=Z_{2}/Z_{1}=R_{z} \tag{3}\]  
As shown in Equation (3), the resonant state of SIR can be determined by \(\lambda\)g/2 transmission line and the impedance ratio \(R_{z}\) relative to typical uniform-impedance resonators (UIR). The electrical length is \(2\theta\), and the characteristic impedance is \(Z_{1}\). This is depicted for approximately UIR in Figure 3. Half-wavelength UIR is used to determine the angular resonance frequency for this resonator [17]. As a rule, dielectric substrate materials for UIR applications should have a small loss-tangent, a high permittivity, and a stable temperature. Comparing the two types of resonators again, we can conclude that the electrical length is the primary determinant of the resonance condition for UIR. SIR has additional flexibility that can be employed in new designs. Referring to Figure 4, where \(\theta_{T}\) represents the resonant electrical length of the resonator, we have [17]:  
\[\theta_{T}=\theta_{1}+\theta_{2}=\theta_{1}+\tan^{-1}\biggl{(}\frac{R_{z}}{ \tan\theta_{1}}\biggr{)} \tag{4}\]  
Figure 3: Half-wavelength UIR. Figure 2: SIR varieties: (**a**) quarter-wavelength type; (**b**) half-wavelength type. For various values of the impedance ratio \(R_{z}\), the overall electrical length for SIR is depicted based on \(\theta_{T}\), as illustrated in Figure 4.  
As it can be comprehended, the resonator's overall electrical length has the highest magnitude at \(R_{z}\geq 1\) and the lowest magnitude when \(R_{z}\leq 1\). The condition for these highest and lowest values can be derived as:  
\[\theta_{1}=\theta_{2}=\tan^{-1}\sqrt{R_{z}} \tag{5}\]  
The condition \(\theta_{1}=\theta_{2}\) stands for a specific condition that provides the highest and lowest lengths for SIR, as expressed below:  
\[\theta_{T\ min}=\tan^{-1}\biggl{(}\frac{2\sqrt{R_{z}}}{1-R_{z}}\biggr{)} \tag{6}\]  
Equation (6) has a minimum value for \(\theta_{T}\) when \(0<R_{z}<1\) and \(0<\theta_{T}<\pi/2\). When comparing SIR to UIR, the critical difference is that the length of the resonators can be tuned by altering the impedance ratio \(R_{z}\). This can be utilized for the same fundamental resonance frequency to design SIRs that are physically smaller than their UIR equivalents. SIRs regulate the passband of a bandpass filter's first spurious mode, which can be employed in developing dual-band BPFs.