# 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]
Impedances of the combination of DMC with end-ring capacitor C\({}_{\text{H}}\) and FS trap with C\({}_{\text{T}_{2}}\) at \(\omega_{\text{Na}}\) and \(\omega_{\text{H}}\) were \(-j\)19.75 \(\Omega\) and \(-j\)124.2 \(\Omega\), respectively. The FS trap-based circuit allows for the creation of two different capacitances. However, adjusting the component values during the practical construction of the RF coil proves complex. This complexity arises because the FS trap is designed not to resonate at any frequencies other than those designated for \(\omega_{\text{Na}}\) and \(\omega_{\text{H}}\), as previously mentioned. All circuit analyses were conducted under the ideal condition, with the assumption of negligible resistance of the inductance or capacitance. A dual-frequency trap was designed for both \(\omega_{\mathrm{H}}\) and \(\omega_{\mathrm{Na}}\), comprising RG-58 cable and two capacitors. As shown in Figure 4, two capacitors (\(\mathrm{C_{trap_{H}}}\) and \(\mathrm{C_{trap_{Na}}}\)) are connected in parallel to the inductor (\(\mathrm{L_{trap}}\)) with mutual inductance being used for both capacitors to enable operation at two frequencies. Figure 4: (**a**) Constructed DMBC RF coil; (**b**) comprehensive circuit diagram of DMBC RF coil, including the DMC and dual-frequency cable trap design. Figure 3: (**a**) Circuit diagram including DMC and \(\mathrm{C_{H^{\prime}}}\); the equivalent circuit of the DMC comprises an equivalent form of the parallel \(\mathrm{LC_{1}}\) circuit and \(\mathrm{C_{2}}\) (**b**) at \(\omega_{\mathrm{H}}\) and (**c**) \(\omega_{\mathrm{Na}}\); (**d**) circuit diagram of the FS trap based dual-tuning circuit; (**e**) calculated reactances (imaginary part of the impedance) and capacitances of the DMC with ideal values along the frequency range of 1–400 MHz. Reactances at \(\omega_{\mathrm{Na}}\) (red circle) and \(\omega_{\mathrm{H}}\) are \(-20.62\)\(\Omega\) and \(-5.8\)\(\mathrm{M}\Omega\) (almost infinity), respectively. Capacitances at \(\omega_{\mathrm{Na}}\) and \(\omega_{\mathrm{H}}\) depicted with blue circles are 97.7 pF and 0.00009 pF, respectively; (**f**) calculated reactances of the combination of DMC and \(\mathrm{C_{H}}\), and the FS trap along the frequency range of 1–400 MHz. The reactance at \(\omega_{\mathrm{Na}}\) pointed by the red triangle is \(-19.75\)\(\Omega\). Also, the reactance at \(\omega_{\mathrm{H}}\) pointed by the red circle is \(-124.2\)\(\Omega\). Each port of the \({}^{1}\)H/\({}^{23}\)Na DMBC RF coil was individually tuned to the homogeneous mode at \(\omega_{\text{Na}}\) and \(\omega_{\text{H}}\). At the end-ring, capacitors of 4.3 pF and 3.2 pF were connected to the upper end-ring and bottom end-ring, respectively, the same as the \({}^{1}\)H STBC RF coil to tune at 298 MHz. The equivalent capacitance of the DMC was added to each end-ring for tuning at 78.9 MHz. The DMC was constructed with the following components' values: L = 95 nH; C1 = 3 pF; C2 for upper DMC = 30.6 pF; C2 for bottom DMC = 24.2 pF. Capacitances of the upper and bottom DMC at \(\omega_{\text{Na}}\) were 97.8 pF and 51.8 pF, respectively. At \(\omega_{\text{H}}\), the DMC functioned as an open circuit, with the capacitance of less than 0.001 pF, which was sufficiently small to be considered immeasurable. Trimmer capacitors were attached at the end-ring of \({}^{1}\)H ports and C2 of DMC of \({}^{23}\)Na ports to fine-tune the RF coil. Bench tests at the \({}^{1}\)H and \({}^{23}\)Na frequency were performed as described in Section 2.2.