**4. Conclusions**

A simple technique for the designing and analysis of the birdcage RF coil based upon dominant resonance path and a novel analytical solution was demonstrated in this paper. We introduced a new concept of the dominant resonance path in the birdcage RF coil that identifies the specific closed current loop which is responsible of causing the dominant resonance frequency that is desired for NMR imaging. The concept is used to determine the numerical values of the lumped capacitances for the leg and/or end-ring segment of the birdcage RF coil using simple mathematical formulations. We also provided the analytical solutions for the birdcage RF coil in terms of its input impedance *Zin* by converting each of its section into a two-port network. The transmission parameters are used to determine the final analytical solution with the help of an equivalent *Te* matrix for the *N-*1 identical cascaded segments of the birdcage RF coil. Two separate analytical solution were developed by considering the feed port or external circuit location in the end-ring segment and leg segment. Both analytical solutions efficiently explain the characteristics of the birdcage RF coil, however the commonly used end-ring feed based analytical solution was mainly discussed in this paper. We implemented the birdcage RF coil in low pass, high pass and band pass configurations with FPCB etched conductor pattern technique for small volume NMR imaging applications at 1.5 T and 3.0 T MRI system. The numerical values of the lumped capacitance for the different configurations of the birdcage RF coil provided by the dominant resonance path method were found more accurate in comparison to those which were obtained through the conventional methods. The analysis of the implemented birdcage RF coils performed with the proposed analytical solution were also in complete agreement with the conventional methods.

**Author Contributions:** S.F.A. and H.D.K. conceived the idea, developed the analytical solution and carried out simulations. Y.C.K. designed the coils completed the experiments. B.-J.Y. developed the programming code for the implementation of the proposed analytical solution. H.D.K. analyzed the data and S.F.A. wrote the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was financially supported by the Ministry of Trade, Industry, and Energy (MOTIE), Korea, under the "Regional industry based organization support program" (reference number R0004072) and the BK21 Plus project funded by the Ministry of Education (21A20131600011).

**Conflicts of Interest:** The authors declare no conflict of interest.
