## 2 Analytical Solution for Birdcage RF coil
### Dominant Resonance Path  
Basic principal behind the establishment of the birdcage coil was the development of a circular ladder structure which should be composed of the phase delay lines who can establish a unique phase pattern of the axial current on each coil leg [14]. The inductance (\(L\)) and capacitance (\(C\)) parameters of leg and end-ring segments are adjusted in such a manner that an ideal sinusoidal distribution of currents in terms of their intensities is realized on the legs of birdcage coil. An ideal sinusoidal intensity profile of currents along with their respective direction in different legs for the dominant resonance mode of an 8-leg birdcage RF coil is shown in Figure 3.  
This sinusoidal legs currents distribution is responsible of producing the homogeneous non-zero magnetic field everywhere inside the birdcage RF coil [24]. The resonance caused by this current distribution is known as the dominant resonance mode. The direction of currents in different legs of the birdcage RF coil as shown in Figure 3 provides the idea of the closed current loop which is  
Figure 3: Sinusoidal legs currents distribution on an 8-legs birdcage coil in dominant resonance mode. responsible for the dominant resonance mode. The total path length of such closed current loop in the birdcage RF coil can be given by the following Equation (2). \[P=2(\pi r+l) \tag{2}\]  
where \(r\) is radius of the coil and \(l\) is the length of its leg. The above path length equation provides the information about the segments of the birdcage RF coil which are involved in the creation of the closed current loop whose resonance frequency is used for NMR imaging operation. A dominant resonance closed current loop consists of two coil legs which are joined by N/2 consecutive end-ring segments on each end-ring as shown in Figure 4a. There exists total N/2 closed current loops of path length \(P\) in birdcage RF coil. Any leg of the birdcage RF coil which is involved in the establishment of a closed current loop dose not involved in the other simultaneously existing N/2-1 closed current loops under dominant resonance condition. The desired dominant resonance mode required in the birdcage coil is the resonance of this closed current loop which can be determined as  
\[f=\frac{1}{2\pi\sqrt{L_{T}\cdot C_{T}}} \tag{3}\]  
where \(L_{T}\) and \(C_{T}\) are the total inductance and capacitance of the dominant resonance current loop. The total inductance \(L_{T}\) is the sum of the inductances of two rungs and \(N\) end-ring segments which are connected in series as shown in Figure 4b. It can be given as follows. \[L_{T}=2L_{l}+N\cdot L_{\sigma} \tag{4}\]  
Here the \(L_{l}\) and \(L_{\sigma}\) are the total inductances of the legs and end-ring segments of a birdcage coil respectively. The numerical value of the self-inductance of a conductor with cylindrical or rectangular geometry can be determined by using the already established following relationships [23]. Figure 4: A closed current loop in the birdcage RF coil representing: (**a**) The dominant resonance path; (**b**) Total lumped inductance and capacitance of end-rings and legs in the dominant resonance path. For a cylindrical conductor of length \(s\) and radius \(r\),  
\[L=\frac{\mu_{0}\,s}{2\pi}\cdot\left[\ln\frac{2s}{r}-1\right] \tag{5}\]  
For a rectangular strip conductor of length \(s\) and width \(w\),  
\[L=\frac{\mu_{0}\,s}{2\pi}\cdot\left[\ln\frac{2s}{w}+\frac{1}{2}\right] \tag{6}\]  
The total inductance \(L_{T}\) of the dominant resonance loop for all three configurations of the birdcage RF coil which is determined by using Equation (4) remains unchanged. However, a single generalized equation cannot be established to determine the total capacitance \(C_{T}\) of the dominant resonance loop for all three configurations of the birdcage coil.