*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.

**Figure 1.** *Cont.*

**Figure 1.** *N-*segments birdcage RF coil: (**a**) Line diagram; (**b**) Block diagram of the equivalent circuit model.

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 *Zl* and identical end-rings with impedance *Zer* as shown in Figure 1b. The impedances *Zl* and *Zer* 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 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 (*L1*) and end-ring segment (*L2*), and the lumped capacitance in the leg (*C1*) and in the end-ring segment (*C2*). The first generalized analytic solution to determine the *m* possible resonance frequency modes (*fm*) 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].

$$f\_m = \frac{1}{2\pi} \sqrt{\frac{\frac{2}{C\_1} \sin^2 \frac{\pi m}{N} + \frac{1}{C\_2}}{L\_1 \sin^2 \frac{\pi m}{N} + L\_2}} \quad (m = 0, \ 1, \ 2, \ \dots, N/2) \tag{1}$$

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 *C1* and *C2* respectively. The numerical values of all possible resonance frequency modes of a birdcage coil obtained using Equation (1) are approximate as the terms *L1* and *L2* 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 *L1* and *L2* [23].
