# Electronic Beam Steering Metamaterial Antenna with Dual-Tuned Mode of Liquid Crystal Material
## 3 Design, Simulation and Analysis of Dual-Tuned CRLH Metamaterial Antenna Based on LC
### Theoretical Analysis of CRLH Structure with Dual-Tuned Mode
The phase constants corresponding to the four extreme states of the dual-tuned mode can be expressed as follows:  
\[\beta_{00}(\omega)=\omega\sqrt{L_{R00}C_{R00}}-\frac{1}{\omega\sqrt{L_{L00}C_ {L00}}} \tag{7}\]  
\[\beta_{V0}(\omega)=\omega\sqrt{L_{RV0}C_{RV0}}-\frac{1}{\omega\sqrt{L_{LV0}C_ {LV0}}} \tag{8}\]  
\[\beta_{0V}(\omega)=\omega\sqrt{L_{R0V}C_{R0V}}-\frac{1}{\omega\sqrt{L_{L0V}C_ {L0V}}} \tag{9}\]  
\[\beta_{VV}(\omega)=\omega\sqrt{L_{RVV}C_{RVV}}-\frac{1}{\omega\sqrt{L_{RVV}C_ {RVV}}} \tag{10}\]where \(\beta_{00}\), \(L_{R00}\), \(C_{R00}\), \(L_{L00}\) and \(C_{L00}\) are the phase constant, the right-handed inductance, the right-handed capacitance, the left-handed inductance and the left-handed capacitance of the CRLH unit cell when two LC layers are both under the alignment state; \(\beta_{0V}\), \(L_{R0V}\), \(C_{R0V}\), \(L_{L0V}\), \(C_{R0V}\), \(L_{L0V}\) and \(C_{L0V}\) are the phase constant, the right-handed capacitance, the left-handed capacitance, the left-handed capacitance and the left-handed capacitance, the left-handed capacitance of the CRLH unit cell when the LC layer is loaded with saturated bias voltage; \(\beta_{V0}\), \(L_{R0V}\), \(C_{R0V}\), \(L_{L00}\) and \(C_{L00}\) are the phase constant, the right-handed capacitance, the right-handed capacitance, the right-handed capacitance, the left-handed capacitance and the left-handed capacitance of the CRLH unit cell when the two LC layers are both loaded with saturated bias voltage. Then, under the two extreme states (two LC layers under alignment state and loaded with saturated bias voltage), Equation (6) can be expressed as follows:  
\[ \begin{array}{c}k_{0}(\sin\theta_{VV}-\sin\theta_{00})=\omega\Big{(}\sqrt{ L_{RVV}(C_{R1 VV}+C_{R2 VV})}-\sqrt{L_{R00}(C_{R100}+C_{R200})}\Big{)}\\ +\frac{\sqrt{L_{LVV}C_{LVV}}-\sqrt{L_{L00}C_{L00}}}{\omega\sqrt{L_{LVV}}C_{LVV }L_{L00}C_{L00}}\end{array}  \tag{11}\]  
where \(\theta_{00}\) is the main lobe direction of antenna; \(C_{R100}\) and \(C_{R200}\) are the right-hand capacitance formed by LC1 and the right-handed capacitance formed by LC2 when both layers of LC are in alignment; \(\theta_{VV}\) is the main lobe direction; and \(C_{R1 VV}\) and \(C_{R2 VV}\) are the right-handed capacitance formed by LC1 and the right-handed capacitance formed by LC2 when both layers of LC are in the state of saturated bias voltage. Since the right-handed inductance is invariable with the tuning of the LC state, \(L_{R}\) can be used to replace the right-handed inductance in any state, and the above formula is simplified to obtain the following formula:  
\[ \begin{array}{c}k_{0}(\sin\theta_{VV}-\sin\theta_{00})=\\ \\ \omega\sqrt{L_{R}}\big{(}\sqrt{C_{R1 VV}+C_{R2 VV}}-\sqrt{C_{R100}+C_{R200}})+ \frac{1-\sqrt{\frac{L_{L00}C_{L00}}{L_{LV}C_{LV}}}}{\omega\sqrt{L_{L00}C_{L00 }}}\end{array}  \tag{12}\]  
Employing the same simplified method to process the expression of the single-tuned CRLH LC metamaterial antenna, Equation (12) can be rewritten as follows:  
\[ \begin{array}{c}k_{0}(\sin\theta_{V}-\sin\theta_{0})=\omega\sqrt{L_{R}} \Big{(}\sqrt{C_{RV}}-\sqrt{C_{R0}}\Big{)}+\frac{1-\sqrt{\frac{L_{L0}C_{L0}}{ L_{LV}C_{LV}}}}{\omega\sqrt{L_{L0}C_{L0}}}\end{array}  \tag{13}\]  
From Equations (12) and (13), for the dual-tuned LC CRLH metamaterial antenna, the maximum beam scanning range is \(\Delta\theta^{\prime}=\theta_{VV}-\theta_{00}\), and the maximum beam scanning range of the single-tuned CRLH LC metamaterial antenna is \(\Delta\theta=\theta_{V}-\theta_{0}\). Compared with the beam scanning range expression of a single-tuned CRLH LC metamaterial antenna, in a dual-tuned LC metamaterial antenna, the change rate of the right-hand capacitance is enhanced due to the addition of the second LC layer. Thus, the phase constant of the right-handed transmission line improves accordingly, meaning that the first item on the right of the equal sign is enlarged. Therefore, the beam scanning range of the metamaterial antenna is improved.