*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 (Te-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 *Zl* and the end-ring impedance *Zer* 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.

**Figure 5.** The block diagrams of: (**a**) The unit segment of a conventional birdcage RF coil; (**b**) Two port equivalent circuit of the unit segment.

The impedance of leg segments *Zl* and the impedance of the end-ring segments *Zer* can be given as follows,

$$Z\_{l} = R\_{l} + j \left( 2 \pi f \cdot L\_{l} - \frac{1}{2 \pi f \cdot \mathbf{C}\_{l}} \right) \tag{10}$$

$$Z\_{\text{eff}} = R\_{\text{eff}} + j \left( 2\pi f \cdot \text{L}\_{\text{er}} - \frac{1}{2\pi f \cdot \text{C}\_{\text{er}}} \right) \tag{11}$$

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.

$$
\begin{bmatrix} V\_k \\ I\_k \end{bmatrix} = T \begin{bmatrix} V\_{k+1} \\ I\_k + 1 \end{bmatrix} \tag{12}
$$

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).

$$T = \begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} 1 + \frac{Z\_{cr}}{Z\_l} & 2Z\_{cr} + \frac{Z\_{cr}}{Z\_l}^2 \\ \frac{1}{Z\_l} & 1 + \frac{Z\_{cr}}{Z\_l} \end{bmatrix} \tag{13}$$

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 *Te* can be defined for the equivalent circuit of birdcage RF coil which can represent its overall transmission characteristics. However, the equivalent transmission matrix *Te* is a product of the *N-1* identical transmission matrices *T* because the *Nth* 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 *Te*. The single equivalent transmission matrix *Te* can be expressed by Equation (14).

$$T\_{\mathfrak{c}} = T^{(N-1)} = \begin{bmatrix} A\_{\mathfrak{c}} & B\_{\mathfrak{c}} \\ C\_{\mathfrak{c}} & D\_{\mathfrak{c}} \end{bmatrix} \tag{14}$$

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 *Nth* segment is shown in Figure 6.

**Figure 6.** Equivalent circuit model of *N* segments birdcage RF coil by converting each segment into a two-port network.

The equivalent circuit impedances *Z ER* and *Z <sup>L</sup>* can be determined by using the elements of the equivalent transmission matrix *Te* with the help of following Equations (15) and (16) respectively.

$$Z\_L = \frac{1}{\overline{C}\_v} \tag{15}$$

$$Z\_{ER} = \frac{A\_{\varepsilon} - 1}{\mathcal{C}\_{\varepsilon}} \tag{16}$$

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 *Zer*/*2*, the general analytical solution can be developed by calculating the equivalent impedance of the circuit model shown in Figure 6 as follows.

$$Z\_4 = \frac{\left(Z\_{cr} + Z\_{ER}\right)}{2} \tag{17}$$

$$Z\_b = \frac{\left(Z\_{cr} + Z\_{ER}\right)}{4} \tag{18}$$

$$Z\_{\rm c} = \frac{(Z\_{\rm cr} + Z\_{\rm ER})}{4} + (Z\_{\rm l} + Z\_{\rm L}) \tag{19}$$

$$Z\_{\rm c\eta} = \frac{(Z\_1 \cdot Z\_3)}{(Z\_1 + Z\_3)} = \frac{(Z\_{\rm cr} + Z\_{\rm ER}) \cdot [4(Z\_{\rm l} + Z\_{\rm L}) + (Z\_{\rm cr} + Z\_{\rm ER})]}{8(Z\_{\rm l} + Z\_{\rm L}) + 6(Z\_{\rm cr} + Z\_{\rm ER})} \tag{20}$$

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).

$$Z\_{ER\\_p} = \frac{1}{2} \cdot \frac{Z\_{cr} \cdot \left(Z\_{ER} + 2Z\_{\text{eq}}\right)}{Z\_{cr} + Z\_{ER} + 2Z\_{\text{eq}}} \tag{21}$$

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).

$$Z\_{L\cdot p} = \frac{Z\_l \cdot (2Z\_L + Z\_{cr} + Z\_{ER})}{2(Z\_l + Z\_L) + (Z\_{cr} + Z\_{ER})} \tag{22}$$

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.
