## 3 Design
The presence of a considerable discontinuity in the SIR is discovered for an exceptionally big \(Z\), which results in a low insertion loss [1; 30]. According to Figure 3, there are two distinct forms of SIRs, which are referred to as symmetric (SSIR) structures and asymmetrical (ASIR) structures, respectively. Both of these structures are referred to as SIRs. It is a well-known fact that, among the microstrip sections that are utilized in SIR structures, the microstrip segment that has a narrower width possesses a higher impedance in contrast to the microstrip section that has a wider width. It is possible to observe symmetric SIR in situations where the value of Z1 is lower than Z2, and when the value of Z4 is higher than Z5. The values of d1 and d2 are the symbols that are used to represent electrical lengths in symmetrical SIR, while e1 and e2 are used in asymmetrical SIR. A symmetrical SIR shows that the lengths are the same, which is the reason for this difference. In general, symmetrical SIR features end-to-end impedance symmetry in addition to three-step impedance sections. These sections are also characterized by their electrical length. When it comes to the symmetrical SIR, the impedance ratio acts as a replacement for the dimensions condition (\(K\)) that is associated with the SIR. The SIR arrangement makes use of both odd and even excitation to ensure that it processes electrical signals correctly. When the value of Yin is equal to zero, it is possible to observe the resonances that are produced by odd-even excitations with a reasonable amount of ease [31]. It is possible to examine the SIR configuration and this can be accomplished with the assistance of odd and even excitations. It is possible to obtain the consequent resonant frequencies for odd and even excitations by using the condition Yin = 0, as stated in reference [31]. As a result of the condition, these frequencies have been extracted directly. (a) Odd-mode resonance condition:  
\[tan\ \theta_{0}\ tan\ \theta_{s}=\mathrm{R} \tag{2}\]  
(b) Even-mode resonance condition:  
\[\frac{1}{2R1}\ \bigg{[}1+\frac{G2}{R}\bigg{]}\bigg{[}\frac{G0}{R1}+\frac{G1}{R2} \bigg{]}+\bigg{[}G3+\frac{G4}{R}\bigg{]}\bigg{[}\frac{1}{R1}+\frac{G5}{R2} \bigg{]}=0 \tag{3}\]  
The following factors are taken into consideration when determining the parameters: \(\mathrm{R}=Z_{0}/Z_{8}\), \(\mathrm{R1}=Z_{1}/Z_{8}\), \(\mathrm{R2}=Z_{2}/Z_{8}\), \(\mathrm{G0}=tan\ \theta_{1}\), \(\mathrm{G1}=tan\ \theta_{2}\), \(\mathrm{G2}=tan\ \theta_{0}\ tan\ \theta_{1}\), \(\mathrm{G3}=tan\ \theta_{s}\), \(\mathrm{G4}=tan\ \theta_{0}\), \(\mathrm{G5}=tan\ \theta_{1}\ tan\ \theta_{2}\). Figure 4 and Equations (2) and (3) are used to predict the resonance (Fr) for odd mode and even mode, respectively, using Equations (4) and (5). \[\mathrm{Fr}\bigg{[}\mathrm{G4}+\frac{G3}{R}\bigg{]}-\frac{1}{\pi\ Zs\ R}+\frac{ G3\ G4}{\pi\ Zs\ R^{2}}=0 \tag{4}\](5)  
Figure 4: (a) Odd and (b) even resonance representations for SIR. Figure 5 depicts the lumped element circuit that corresponds to a resonant frequency [32], which can be used to analyze the SIRs. Figure 5: The lumped element circuit equivalent for SIR under operating resonant frequency. When it comes to planner layouts, meander line resonators are a representation of easily fabricable formats. The resonators in question are primarily utilized as compactness tools, and they are well-known for the great qualities and compact size that they possess. Figure 6 illustrates the outcomes of their implementation in an LC lumped element circuit, which has resulted in their extensive application [1]. Given the information presented in Figure 6b, the impedance of the LC series can be represented using the equation \(\mathrm{j}(\omega\mathrm{L}-1/\omega\mathrm{C})\). Therefore, at the terminal of the meander line resonator, the resistance load can be observed.