## 4 Applications
### Differential Lines and Balanced Filters with Common-Mode Suppression  
It was already pointed out in Section 3.1 that differential lines loaded with CSRRs suppress the common mode whilst the differential mode is kept unaltered. A differential line loaded with CSRRs \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(d^{\prime}\)**(mm)** & \(L\)**(nH)** & \(C\)**(pF)** & \(C_{r}\)**(pF)** & \(L_{r}\)**(nH)** & \(M\)**(nH)** & \(M^{\prime}\)**(nH)** \\ \hline
0.105 & 1.82 & 1.58 & 0.44 & 6.85 & 0.82 & 1.74 \\
0.305 & 1.86 & 1.58 & 0.44 & 6.85 & 0.82 & 1.29 \\
0.505 & 1.84 & 1.57 & 0.43 & 6.85 & 0.81 & 1.02 \\
0.755 & 1.85 & 1.55 & 0.43 & 6.85 & 0.80 & 0.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Extracted circuit parameters (symmetric case) for different values of \(d^{\prime}\). Figure 16: Variation of the notch frequencies as a function of \(d^{\prime}\) for the asymmetric SRR-loaded CPW (**a**), and detail of the fabricated sample (bottom face) for \(d^{\prime}\) = 0.105 mm (**b**). From [26]; reprinted with permission IEEE 2015. Figure 15: Frequency response (symmetric case) for different values of \(d^{\prime}\)**(**a**), and detail of one of the fabricated samples (bottom) where the ground planes have been connected through vias and strips to avoid the slot mode (**b**). From [26]; reprinted with permission from IEEE 2015.  
optimized to enhance the common-mode rejection bandwidth is shown in Figure 17[36]. To widen the rejection bandwidth for the common mode, the CSRRs have been designed with wider rings and inter-ring space [27,36]. The target was to achieve at least 20-dB common-mode rejection in the frequency range between 1.2 GHz and 1.8 GHz. To this end, three tightly-coupled square-shaped CSRRs suffice to achieve the target specifications. Figure 18 shows the simulated and measured differential- and common-mode insertion loss of the structure. The dimensions of the active region of the structure are 32.8 mm \(\times\) 10.8 mm, that is 0.43 \(\lambda\times\) 0.14 \(\lambda\). It is remarkable that the measured insertion loss for the differential signal is smaller than 0.5 dB in the considered range. Other common-mode suppressed differential microstrip lines implemented by loading the line with double-slit CSRRs (DS-CSRRs) are reported in [27]. The DS-CSRR is useful to broaden the rejection bandwidth for the common mode, since this particle exhibits a lower inductance than the CSRR, as required for that purpose. For instance, DS-CSRRs have been demonstrated to be useful for the efficient suppression of the common mode in balanced filters [27]. By merely cascading a pair of differential lines, each one loaded with a single DS-CSRR, the suppression of the common mode in differential filters with intrinsic common-mode suppression can be significantly enhanced. This is  
Figure 17: Photograph of the **(a)** top view and **(b)** bottom view of the differential line with wideband common-mode rejection. Dimensions are: \(c\) = 1.2 mm, \(d\) = 0.8 mm, \(W\) = 1 mm, \(S\) = 2.5 mm, and CSRR side length of 10.8 mm. The substrate is _Rogers RO3010_ with \(h\) = 1.27 mm, \(\varepsilon_{r}\) = 10.2, and loss tangent tan\(\delta\) = 0.0023. Figure 18: Common-mode and differential-mode insertion loss \(|S_{cc21}|\) and \(|S_{dd21}|\) for the structure of Figure 17. From [36]; reprinted with permission from IEEE 2011.  
illustrated in Figure 19, where a balanced filter consisting of a pair of coupled stepped impedance resonators (SIRs) fed by a differential line, is depicted. The filter by itself rejects the common mode due to the symmetry of the structure, since the symmetry plane of the resonator exhibits an electric wall at the first SIR resonance. Therefore, such resonators cannot be excited by means of common-mode signals, and the even mode is reflected back to the source due to the presence of the slots between the pair of SIRs.