## 2 Analytical Solution for Birdcage RF coil  
### Configuration of the Birdcage RF coil  
The line diagram of a conventional \(N\)-segments birdcage RF coil structure and the block diagram for its equivalent circuit model are shown in the Figure 1.  
Figure 1: \(Cont.\)  
A conventional birdcage RF is a three-dimensional cylindrical structure which is composed of \(N\) (even) number of segments which are connected to each other and are precisely arranged in the axial and azimuthal plans as shown in Figure 1a. The axial element of a segment is known as leg (or rung) which is connected to the conductor segments known as end-ring on both sides in the azimuthal planes. Each segment of a birdcage RF coil contains identical legs with impedance \(Z_{l}\) and identical end-rings with impedance \(Z_{er}\) as shown in Figure 1b. The impedances \(Z_{l}\) and \(Z_{er}\) are composed of inductive and capacitive elements with negligible resistance (practically less than 1 ohm). The inductive elements of a segment can be the cylindrical wires or the rectangular conductors (of finite thickness or thin foils). These are arranged in the axial and azimuthal planes. The capacitive elements in a segment are the lumped capacitors (with non-magnetic characteristics) which can be inserted in its leg or/and end-rings. Based on the capacitor position in a segment, three configurations of the birdcage RF coil can be realized. These are known as low pass (LP), high pass (HP), and band pass (BP) [14]. A single segment of each configuration is shown in Figure 2.  
Figure 1: \(N\)-segments birdcage RF coil: (**a**) Line diagram; (**b**) Block diagram of the equivalent circuit model.  
Figure 2: Equivalent circuit diagrams of a single segment of the birdcage RF coil configurations: (**a**) Low pass (LP); (**b**) High pass (HP); (**c**) Band pass (BP).  
Due to circular ladder structure with multiple cascaded identical segments, there does exist multiple resonance frequencies (some time referred as resonance modes) simultaneously in a conventional birdcage RF coil. The total number of these resonance frequencies mainly depend upon the number of legs (\(N\)), the inductances of leg (\(L_{I}\)) and end-ring segment (\(L_{2}\)), and the lumped capacitance in the leg (\(C_{I}\)) and in the end-ring segment (\(C_{2}\)). The first generalized analytic solution to determine the \(m\) possible resonance frequency modes (\(f_{m}\)) of an \(N\) leg birdcage RF coil was developed by [PERSON] in 1985 with the aid of its equivalent circuit analysis is given in Equation (1) [14].  
\[f_{m}=\frac{1}{2\pi}\sqrt{\frac{\frac{2}{C_{1}}\sin^{2}\frac{mn}{N}+\frac{1}{C _{2}}}{L_{1}\sin^{2}\frac{mn}{N}+L_{2}}}\ \ \ \ (m=0,\ 1,\ 2,\ \ldots,N/2) \tag{1}\]  
The above equation is although for a band pass configuration which however can be converted for low pass and high pass configurations by removing the terms containing \(C_{I}\) and \(C_{2}\) respectively. The numerical values of all possible resonance frequency modes of a birdcage coil obtained using Equation (1) are approximate as the terms \(L_{I}\) and \(L_{2}\) in Equation (1) represents the self-inductances of leg and end-ring segments. A more accurate but rather complex solution was provided by [PERSON] in 1991 who also included the mutual inductance effect in the calculations of \(L_{I}\) and \(L_{2}\)[23].