## 3 Design and Characterisation of CDNG-MTM  
The basic DNG-MTM consists of a ring with two stub lines [30]. A modification of this basic structure is proposed by addding two extra stubs on the existing stub lines. Even though the stub length is increased by this modification, the overall size and the DNG behaviour remain unaltered. At the same time, this modification enables wider frequency tuning. Thus, the modified DNG-MTM consists of a metallic ring having two L-shaped stub lines with a gap. The distributed inductance and capacitance [31; 32] effect on this ring, stub and the gap produce an LC resonator behaviour with a high-quality factor (\(Q\)). However, complementary MTM structures are more suitable for inducing MTM effect on patch antennas. On the basis of [PERSON]'s principle and the duality concept, a DNG-MTM has a complementary structure (CDNG-MTM) that exhibits similar operational behaviour and properties. By altering the various physical parameters--such as ring radius, stub length, etc.--tuning of the resonant frequency of the CDNG-MTM can be done.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Size (mm)** & **Parameters** & **Size (mm)** \\ \hline L\({}_{1}\) & 20 & W\({}_{1}\) & 1 \\ L\({}_{2}\) & 20 & W\({}_{2}\) & 0.6 \\ L\({}_{3}\) & 3.1 & G\({}_{1}\) & 0.35 \\ L\({}_{4}\) & 7.3 & G\({}_{2}\) & 0.5 \\ L\({}_{5}\) & 2.8 & R\({}_{1}\) & 6.5 \\ L\({}_{6}\) & 0.8 & R\({}_{2}\) & 7.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design parameters of the proposed antenna.  
Figure 2: Simulated and measured return loss characteristics of the circular patch antenna.  
The material parameters of the CDNG-MTM can be retrieved using S-parameter retrieval method [30; 33; 34; 35; 36; 37]. A two-port waveguide set up is used to extract S-parameters of the CDNG-MTM. The boundary conditions perfect magnetic conductor (PMC), perfect electric conductor (PEC) and waveguide for the extraction of \(S_{11}\) and \(S_{21}\) are shown in Figure 3. Material properties such as permittivity, permeability, refractive index, and impedance can be retrieved using S-parameters as per the following equations implemented in MATLAB(tm).  
\[z=\pm\sqrt{\frac{\left(1+S_{11}\right)^{2}-S_{21}^{2}}{\left(1-S_{11}\right)^ {2}-S_{21}^{2}}} \tag{1}\]  
\[e^{ink_{0}d}=\frac{S_{21}}{1-S_{11}\frac{z-1}{z+1}} \tag{2}\]  
\[n=\frac{1}{k_{0}d}\bigg{\{}\Big{[}\big{[}ln\ \Big{(}e^{ink_{0}d}\Big{)}\ \Big{]}^{\alpha}+2m\tau\Big{]}-i\big{[}ln\ \Big{(}e^{ink_{0}d}\Big{)}\ \Big{]}^{\prime}\bigg{\}} \tag{3}\]  
where '\(k_{0}\)'is the wavenumber in free space and '\(m\)' represents branch index of refractive index, which is an integer. Relative permittivity '\(\epsilon\)' and relative permeability '\(\mu\)' can be calculated using the relations (4) and (5).  
\[\epsilon=n/z \tag{4}\]  
\[\mu=nz \tag{5}\]  
Figure 4a,b show \(S_{11}\) and \(S_{21}\) magnitude and phase plots of the CDNG structure, obtained from the waveguide setup. Figure 4a shows resonance at 2.4 GHz and 5.2 GHz, whereas, Figure 4b shows phase change at the same frequencies. Since the CDNG structure shows resonance and phase change at these frequencies, the structure exhibits metamaterial behaviour in these frequency regimes. These S-parameter values are used to extract material properties. Figure 5a,b exhibit the extracted values of relative permittivity \(\epsilon\), relative permeability \(\mu\), refractive index \(n\), and impedance \(z\). It can be observed in Figure 5a that, both permittivity and permeability show negative values over a wide range in C-band. In Figure 5b the DNG property is exhibited where \(n\) is negative. A passive material shows DNG behaviour when its \(n\) shows negative values and \(z\) shows positive values.  
Figure 3: S-parameters retrieval waveguide setup.