## 2 Analytical Solution for Birdcage RF coil
### Equivalent Circuit Analysis  
Most of the analytical solutions for the birdcage RF coil which are commonly developed by using the basic circuit analysis, transmission line theory, numerical electromagnetics or any other mathematical technique are limited to the determination of the resonance frequency modes only. However, the analytical solutions proposed in this paper provide a comprehensive mathematical formulation for the input port impedance in the leg or end-ring segment of the birdcage RF coil (regardless of its configuration) as a function of frequency. This is used to compute the numerical value of the impedance at any desired position for any desired frequency. This solves the problem of external circuit interfacing with the birdcage coil without effecting its resonance characteristics. The proposed method is based upon the determination of a single equivalent transmission matrix (T\({}_{\text{e}}\)-matrix) for the total equivalent circuit of the birdcage RF coil. A conventional birdcage coil is an RF resonator which is composed of identical cascaded segments of inductive and capacitive elements. The conductor segments are the sources of inductance in the birdcage coil. The numerical value of the equivalent inductance (self and mutual) of the conductor segments with respect to its geometry (cylindrical wires or rectangular strips) which is determined by using mathematical equations is used as the lumped inductance [24]. While the lumped capacitors are the capacitive elements of the birdcage coil circuit. The block diagram of a unit segment of the birdcage RF coil consisting of the leg impedance \(Z_{l}\) and the end-ring impedance \(Z_{\sigma}\) is shown in Figure 5a. The two ports equivalent circuit of the unit segment which is required to compute the \(T\)-matrix of the single segment of birdcage RF coil is also shown in Figure 5b. The impedance of leg segments \(Z_{l}\) and the impedance of the end-ring segments \(Z_{cr}\) can be given as follows,  
\[Z_{l}=R_{l}+j\left(2\pi f\cdot L_{l}-\frac{1}{2\pi f\cdot C_{l}}\right) \tag{10}\]  
\[Z_{cr}=R_{cr}+j\left(2\pi f\cdot L_{cr}-\frac{1}{2\pi f\cdot C_{cr}}\right) \tag{11}\]  
The input and output voltage and currents of the two-port network of Figure 5b can be related to each other via following matrix equation. \[\left[ \begin{array}{c}V_{k}\\ I_{k}\end{array} \right]=T\left[ \begin{array}{c}V_{k+1}\\ I_{k}+1\end{array} \right] \tag{12}\]  
where \(T\) represents the transmission (ABCD) matrix of a unit segment of the birdcage RF coil that can be obtained by using the following matrix Equation (13). \[T=\left[ \begin{array}{cc}A&B\\ C&D\end{array} \right]=\left[ \begin{array}{cc}1+\frac{Z_{cr}}{Z_{l}}&2Z_{cr}+ \frac{Z_{cr}^{2}}{Z_{l}}\\ \frac{1}{Z_{l}}&1+\frac{Z_{cr}}{Z_{l}}\end{array} \right] \tag{13}\]  
In a conventional birdcage RF coil, \(N\) number of such identical segments as shown if Figure 5 are connected in cascade with each other. As the symmetry conditions prevail in this circuit, a single transmission matrix \(T_{e}\) can be defined for the equivalent circuit of birdcage RF coil which can represent its overall transmission characteristics. However, the equivalent transmission matrix \(T_{e}\) is a product of the \(N\)-\(I\) identical transmission matrices \(T\) because the \(N^{th}\) segment (which can be chosen arbitrarily) of the birdcage RF coil is used to establish an interface with the receiver port of the MRI apparatus. Thus, its transmission matrix is not included in the computation of \(T_{e}\). The single equivalent transmission matrix \(T_{e}\) can be expressed by Equation (14). \[T_{e}=T^{(N-1)}=\left[ \begin{array}{cc}A_{e}&B_{e}\\ C_{e}&D_{e}\end{array}