## 4 Circuit Model of the Proposed Meta-Atom  
The proposed meta-atom structure contains the inductive-capacitive circuit, so the resonance frequency (_f_),  
\[f=\frac{1}{2\pi\sqrt{L_{T}C_{T}}}, \tag{6}\]  
The total inductance (\(L_{T}\)) and capacitance (\(C_{T}\)) for the proposed structure can be obtained from [27]. Therefore, the total inductance (\(L_{T}\)) can be obtained from,  
\[L_{T}=\tau\times\mu_{0}t\left\{\frac{2(10d+h)^{2}}{(2w+g)^{2}}+\frac{\sqrt{ \left(2w+g\right)^{2}+l^{2}}}{(10d+h)}\right\}, \tag{7}\]  
However, the total capacitance (\(C_{T}\)) can be calculated by:  
\[C_{T}=\varepsilon_{0}\left[\frac{(2w+g)}{\pi(10d+g+h)^{2}}ln\left\{\frac{2(10d +g+h)}{(a-l)}\right\}\right], \tag{8}\]  
where the free-space permeability is \(\mu_{0}=4\pi\times 10^{-7}\) H/m, the permittivity is \(\varepsilon_{0}=8.854\times 10^{-12}\) F/m and the parametric constant is \(\tau=0.0001\). For left-handed characteristics, the series and shunt branch are formed by the inductive and capacitive effect. The capacitive effect is maintained by gaps or splits, which are symbolized by C\({}_{1}\), C\({}_{2}\), C\({}_{3}\), C\({}_{4}\) and C\({}_{5}\) in the circuit shown in Figure 6. Conversely, the metal strips are accountable for the inductive effect and denoted by L\({}_{1}\), L\({}_{2}\), L\({}_{3}\), L\({}_{4}\), L\({}_{5}\) and L\({}_{6}\). When the width of the metal strip is increased, the inductive effect is also increased. As a result, owing to the inductive effect, the resonance frequency is decreased. Similarly, by decreasing the width and increasing the number of splits, the capacitive effect can be raised for the proposed unit cell, which swings the resonance frequency downwards [28]. However, the addition of further splits and metal arms in the meta-atom structure produces a small phase delay and increases the total inductive and capacitive effect. Figure 5: Experimental setup for measuring the \(S_{11}\) and \(S_{21}\) parameters: (a) horn antenna method; (b) waveguide (WG) method. In the above equivalent circuit, the current I (from the source) is divided into I\({}_{1}\), I\({}_{2}\), I\({}_{3}\), I\({}_{4}\), I\({}_{5}\) and I\({}_{6}\), where I = I\({}_{1}\) + I\({}_{2}\) + I\({}_{3}\) + I\({}_{4}\) + I\({}_{5}\) + I\({}_{6}\) ([PERSON]'s first law). If the applied voltage is \(V\) and the charge in the capacitor is \(Q\), then according to [PERSON]'s second law, the differential equations for the loops are [29],  
From the loop ABEA in the circuit model,  
\[\mathrm{IR}+\frac{Q}{C_{1}}+L_{1}\times\frac{dI_{1}}{dt}=\mathrm{V},\]  
After the Laplace transformation,  
\[\mathrm{IR}+\frac{I_{1}}{SC_{1}}+L_{1}SI_{1}=\mathrm{V}, \tag{9}\]  
From the loop BEDB in the circuit model,  
\[\frac{Q}{C_{1}}+L_{1}\times\frac{dI_{1}}{dt}-\frac{Q}{C_{4}}-L_{4}\times\frac {dI_{4}}{dt}-\frac{Q}{C_{3}}=0,\]  
After the Laplace transformation,  
\[\frac{I_{1}}{SC_{1}}+L_{1}SI_{1}-\frac{I_{4}}{SC_{4}}-L_{4}SI_{4}-\frac{I_{3}} {SC_{3}}=0,\]  
\[\frac{1}{S}\bigg{(}\frac{I_{1}}{C_{1}}-\frac{I_{4}}{C_{4}}-\frac{I_{3}}{C_{3}} \bigg{)}+S(L_{1}I_{1}-L_{4}I_{4})=0, \tag{10}\]  
From the loop CEFC in the circuit model,  
\[L_{T}=\tau\times\mu_{0}t\left\{\frac{2(10d+h)^{2}}{(2w+g)^{2}}+\frac{\sqrt{ \left(2w+g\right)^{2}+l^{2}}}{(10d+h)}\right\}, \tag{7}\]0  
After the Laplace transformation,  
\[L_{T}=\tau\times\mu_{0}t\left\{\frac{2(10d+h)^{2}}{(2w+g)^{2}}+\frac{\sqrt{ \left(2w+g\right)^{2}+l^{2}}}{(10d+h)}\right\}, \tag{7}\]1  
\[L_{T}=\tau\times\mu_{0}t\left\{\frac{2(10d+h)^{2}}{(2w+g)^{2}}+\frac{\sqrt{ \left(2w+g\right)^{2}+l^{2}}}{(10d+h)}\right\}, \tag{7}\]2  
\[L_{T}=\tau\times\mu_{0}t\left\{\frac{2(10d+h)^{2}}{(2w+g)^{2}}+\frac{\sqrt{ \left(2w+g\right)^{2}+l^{2}}}{(10d+h)}\right\}, \tag{7}\]3 Equivalent circuit model of the proposed meta-atom structure.