## 3 Design and Analysis of the Birdcage RF coil  
### Design Based on Analytical Solution  
Birdcage RF coils are designed in various sizes with respect to their length and diameter in according with the specified applications. The main tasks in the designing of a birdcage RF coil is the choice of coil configurations, number of legs, dimensions of legs and end-ring segments, lumped capacitance, and port impedance matching at desired resonance frequency. Legs and end-ring segments conductors are the sources of inductance in the birdcage coil. Their count and dimensions are the controlling factors of the magnetic field homogeneity as well as SAR inside the coil. While on the other hand, the lumped capacitors are responsible for producing the resonance at required radio frequency without effecting the magnetic field distribution. A balance between the conductor size and lumped capacitance value is quite necessary for an appropriate design of the birdcage RF coil. In order to validate the proposed analytical technique, we designed the low pass, high pass and band pass configurations of the linearly polarized birdcage RF coil for \({}^{1}\)H NMR imaging in small volume applications at 1.5 T and 3 T MRI systems as shown in Figure 7.  
Figure 7: Configurations of the birdcage RF coil for the reception of \({}^{1}\)H NMR signal at 1.5 T and 3 T MRI systems: (**a**) Low pass; (**b**) high pass; (**c**) band pass. (All dimensions are in mm). The designed birdcage RF coils are linearly polarized having 8 legs and 2 end-rings which are made of 35 \(\upmu\)m thick and 5 mm wide rectangular copper strips etched on a flexible FR4 substrate (\(\varepsilon_{\tau}\) = 4.4) of 0.2 mm thickness. A 2 mm gap is created in the respective conductor segments of the legs and/or end-rings for SMT lumped capacitors. Total length of a single leg of the birdcage RF coil including capacitor gap is 162 mm. In order to keep the permanent 3D symmetry, the birdcage conductor pattern etched flexible FR4 was further attached on a solid acryl (\(\varepsilon_{\tau}\) = 3.2) cylinder of 5 mm thickness and 40 mm internal diameter. The length and diameter of the coil along with the dimension of the legs and end-rings segments are assumed to be constant for all three configurations of the 1.5 T and 3 T birdcage RF coils. The self-inductance (referred as calculated inductance \(L_{cal}\) in this paper) of the leg and end-rings segments which is constant for all designed coils is determined using Equation (6). However, in the actual scenario there also exist very strong mutual inductance between the parallel conductor segments of the birdcage RF coil which affects the self-inductances of the coil. The self-inductance of the leg and end-ring segment which includes the effect of mutual-inductance was computed using the methods described by the birdcage builder [25]. This self-inductance is referred as effective inductance \(L_{\mathit{eff}}\) in this paper. For the given dimensions of the birdcage RF coil the inductance \(L_{cal}\) and \(L_{\mathit{eff}}\) for the leg and end-ring segment respectively are given in the Table 1 below. These calculated and effective inductance values were further used to compute the lumped capacitances required in the legs and/or end-rings for all three configurations of the birdcage RF coil at the desired frequencies of 63.8 MHz and 127.7 MHz (for 1.5 T and 3 T MRI systems respectively) by using the Equations (7)-(9). In order to prove the validity of the proposed method, we also computed the lumped capacitances for all three configurations by using the conventional method described by Equation (1), 2D circuit simulation and 3D electromagnetic simulation. The actual values of the lumped capacitance used in the implementation of the prototypes of low pass, high pass and band pass configurations of 1.5 T and 3 T birdcage RF coils were also measured. A comparison among lumped capacitor values determined through different techniques for 1.5 T and 3 T birdcage coils are given below in the Tables 2 and 3 respectively.