**1. Introduction**

NMR imaging is highly in demand for the medical diagnostics and various other nonclinical and research applications due to its ability of providing very high-quality anatomical images non-invasively [1–3]. A highly sensitive and sophisticated radio frequency (RF) coil (or resonator) is responsible for obtaining these high-quality images only in a highly uniform magnetic field environment of an MRI system [4,5]. One such resonator is known as birdcage RF coil which is a popular choice for the nuclear magnetic resonance (NMR) based volume imaging for medical diagnostics [6]. Having the features like excellent magnetic field homogeneity, high signal-to-noise ratio, design flexibility and the ability to be designed for multi-resonance operation, the birdcage coil is a widely used RF resonator for whole volume NMR imaging applications at high as well as ultra-high field MRI systems [7–13].

The birdcage coil is a closed ladder network which is composed of identical cascaded segments of inductive and capacitive elements [14]. There exist multiple closed current loops in a birdcage coil which are responsible for the simultaneous existence of a definite number of resonance frequency signals. In general, a birdcage coil with *N* number of cascaded segments can be used to support maximum *N*/2 + 2 resonance frequency modes [15]. Several analytical and numerical solutions have been presented which explain the working of birdcage RF coil quite efficiently and provide the information about its resonance frequency spectrum [15–20]. Most of the existing solutions either involve some tedious techniques with complex mathematical formulation or limited to the determination of the resonance characteristics of the birdcage coil. However, in practical applications, it is required to connect the external circuits like impedance matching circuit (with the input/output ports) and detuning circuits (in the end-ring and/or leg segments) with the birdcage RF coil. This process can seriously affect its resonance frequency spectrum [9]. The existing methods however do not address this fundamental issue of external circuit interfacing with the birdcage coil.

In this paper, we have introduced a new concept of the dominant resonance path for the birdcage RF coil which is a simple and intuitive designing technique. The method works by identifying the closed current loop in the birdcage RF coil who is responsible of causing the dominant resonance frequency which is used for NMR imaging applications [21]. The proposed strategy provides rather convenient way to determine the lumped capacitances for the leg and end-ring segments of the birdcage RF coil for its proper operation at desired dominant resonance frequency. We have also devised a simple analytical solution for the birdcage RF coil by establishing it equivalent circuit model which is based upon the transmission matrix theory [21,22]. The proposed analytical solution provides the mathematical expression for the input impedance *Zin* of the birdcage RF coil as seen from the port in terms of the impedances of its leg and end-ring segments. Along with explaining the resonance frequency characteristics of the birdcage RF coil, the proposed analytical solution also provides the impedance at any desired position in the coil for the desired resonance frequency. This methodology solves the problems of external circuit interfacing with birdcage coil without disturbing its resonance characteristics.

The paper is divided into two main sections. The first section is about the *analytical solution for the birdcage RF coil* which explains the concept of dominant resonance path in the birdcage RF coil. It is then followed by a novel analytical solution which is derived using simple equivalent circuit modeling-and T-matrix theory. Second section is the *design and analysis of the birdcage RF coil* which contains the results (analytical, simulated and measured) of the successful implementation of the low pass, high pass and band pass configurations of the birdcage RF coil for small volume NMR imaging applications at 1.5 T and 3 T MRI systems by using the proposed analytical solution.

## **2. Analytical Solution for Birdcage RF Coil**
