#### 2.1. 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.

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.

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_{1}) and end-ring segment (

L_{2}), and the lumped capacitance in the leg (

C_{1}) 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 Hayes C.E. in 1985 with the aid of its equivalent circuit analysis is given in Equation (1) [

14].

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_{1} 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_{1} 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 Jiaming J. in 1991 who also included the mutual inductance effect in the calculations of

L_{1} and

L_{2} [

23].

#### 2.2. 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 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).

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

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.

Here the

L_{l} and

L_{er} 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].

For a cylindrical conductor of length

s and radius

r,

For a rectangular strip conductor of length

s and width

w,

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. As for the low pass coil, capacitors are present only in the legs, for high pass only in the end rings while for the band pass in both locations, so the relationship to determine the total capacitance in each case can be given as follows,

The lumped capacitors required in the legs and end-rings of the low pass and high pass birdcage RF coil can easily be determined using Equations (8) and (9) respectively for the given coil dimensions and required resonance frequency. However, for the band pass birdcage coil, the end-ring capacitor value is usually needed to be assumed while the leg capacitor value is calculated using Equation (7). Moreover, there exist different combinations of leg and end-ring capacitors for the band pass birdcage coil and the one which causes more homogeneous magnetic field distribution is selected.

#### 2.3. 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_{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_{er} 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_{er} can be given as follows,

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.

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).

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-1 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).

Since an arbitrary transmission matrix can be converted to a two-port network, the block diagram of the equivalent circuit of a birdcage RF coil containing two-port equivalent circuit model of the

N−1 segments connected in cascade with the two-port equivalent circuit model of the

N^{th} segment is shown in

Figure 6.

The equivalent circuit impedances

Z _{ER} and

Z _{L} can be determined by using the elements of the equivalent transmission matrix

T_{e} with the help of following Equations (15) and (16) respectively.

In a birdcage RF coil port can be established by interfacing the external circuit across any leg or end-ring segment but it exhibits the similar frequency characteristics. However, the port impedance as viewed from the external circuit would be different at both positions which results in different expressions for the general analytical solution. By considering the port connected across the end-ring segment

Z_{er}/2, the general analytical solution can be developed by calculating the equivalent impedance of the circuit model shown in

Figure 6 as follows.

The final expression of the general analytical solution for the case when the port is created in the end-ring segment is given by the Equation (21).

In a similar way the general analytical solution for the case when the port is created in the leg segment can be derived and the final expression is given by the Equation (22).

The proposed method allows the input impedance to be obtained directly from the leg or end-ring of the birdcage RF coil. Precisely speaking, unlike the existing solutions for analyzing the relationship between the components values and the resonant frequency, the input impedance can be easily obtained at any point of the birdcage coil. Therefore, the method is not only useful to determine the resonance frequencies of the birdcage RF coil, but also to provide the port impedance which is an essential parameter to be known for interfacing any external circuitry like impedance matching circuit, detuning circuit etc. to the birdcage RF coil.