# Self-Isolated Dual-Mode High-Pass Birdcage RF Coil for Proton and Sodium MR Imaging at 7 T MRI
## 2 Materials and Methods
### 1H/23 Na Dual-Mode Birdage (DMBC) RF Coil [42]  
The DMBC RF coil was designed with same size and structure as the STBC RF coil described in Section 2.2, including the coil frame. Meanwhile, the DMC was employed at each end-ring to tune the RF coil at 23 Na resonant frequency. The DMC comprises two capacitors (C\({}_{1}\) and C\({}_{2}\)) and one inductor (L), as shown in Figure 3. In this configuration, the parallel LC\({}_{1}\) circuit operates as an open circuit, specially designed to resonate at 1H resonant frequency, and its impedance goes infinity. On the other hand, the LC\({}_{1}\) circuit acts as an inductive circuit (L\({}_{\text{eq}}\)) at the 23 Na resonant frequency. The series L\({}_{\text{eq}}\) and C\({}_{2}\) work as a capacitive circuit (C\({}_{\text{eq}}\)), so the addition of the equivalent capacitance facilitates the tuning of the RF coil at 23 Na frequency. Thus, the DMC functions as additional capacitance for tuning at 23 Na resonant frequency without affecting the end-ring capacitor (C\({}_{\text{H}}\)) for tuning the 1H resonant frequency [42]. In other words, the combined capacitance of the C\({}_{\text{eq}}\) and C\({}_{\text{H}}\) functions equivalently to the end-ring capacitor in the 23 Na STBC RF coil. The impedance of the DMC at 1H and 23 Na resonant frequency (Z\({}_{\text{H}}\) and Z\({}_{\text{Na}}\)) and the C\({}_{\text{eq}}\) of the DMC at 23 Na resonant frequency can be described as follows:  
\[Z_{\text{H}}(\omega)=\frac{1}{\frac{1}{j\omega_{\text{H}}\text{L}}+j\omega_{ \text{H}}\text{C}_{1}}+\frac{1}{j\omega_{\text{H}}\text{C}_{2}} \tag{1}\]  
\[Z_{\text{Na}}(\omega)=j\omega_{\text{Na}}\text{Leq}+\frac{1}{j\omega_{\text{ Na}}\text{C}_{2}} \tag{2}\]  
\[\omega_{\text{H}}=\frac{1}{\sqrt{\text{LC}_{1}}} \tag{3}\]  
\[\text{C}_{\text{eq}}=\frac{\text{C}_{2}}{1-\omega_{\text{Na}}^{2}\text{C}_{2 }\text{Leq}} \tag{4}\]  
where \(\omega_{\text{H}}\) and \(\omega_{\text{Na}}\) are the resonant frequency of 1H and 23 Na at 7 T, respectively. The impedance and capacitance of the DMC along 1 to 400 MHz were calculated with the following values of each component: L = 95 nH; C\({}_{1}\) = 3 pF; C\({}_{2}\) = 28.24 pF; C\({}_{\text{H}}\) = 4.3 pF. The capacitance of the C\({}_{\text{H}}\) is chosen for the same value with the upper end-ring capacitor of the 1H STBC RF coil, and L and C\({}_{1}\) are chosen to the practical values of the DMC in the DMBC RF coil. The capacitance of the C\({}_{2}\) was selected to operate the combination of DMC and C\({}_{\text{H}}\) as the capacitance of 102 pF at \(\omega_{\text{Na}}\) which was used for upper end-ring capacitor of the 23 Na STBC RF coil. Calculated impedances of the DMC at \(\omega_{\text{Na}}\) and \(\omega_{\text{H}}\) were \(-j\)20.62 \(\Omega\) and \(-j\)5.8 M\(\Omega\), respectively. Capacitances computed from the impedance of the DMC at \(\omega_{\text{Na}}\) and \(\omega_{\text{H}}\) were 97.7 pF and 0.00009 pF, respectively. At the \(\omega_{H}\), the impedance of the DMC is extremely high, and the capacitance is small enough to reject the current, so the DMC operates as close to open state. To compare with the FS trap based dual-tuning method [28; 29], the impedance of the FS trap was calculated along the frequency range of 1 to 400 MHz. The circuit with FS trap comprises the same components as DMC (L\({}_{\text{T}}\): inductor of FS trap, C\({}_{\text{T}_{1}}\): capacitor of FS trap, C\({}_{\text{T}_{2}}\): additional capacitor for dual-tuning), but not resonate at the 1H frequency. In this case, there are infinite combinations of component values available. Therefore, to ensure that the FS trap does not resonate at 298 MHz, we selected the same inductance value for the L\({}_{\text{T}}\) as used in the DMC and an arbitrary C\({}_{\text{T}_{1}}\) of 8.2 pF. The C\({}_{\text{T}_{2}}\) of 25.8 pF was chosen to operate the circuit as the capacitance of 102 pF and 4.3 pF at \(\omega_{\text{Na}}\) and \(\omega_{\text{H}}\), respectively.