The Archimedean spiral antenna is well-known for its broadband characteristics with circular polarization and has been investigated for several decades. Since their development in the late 1950's, establishing an analytical expression for the characteristics of spiral antenna has remained somewhat elusive. This has been studied qualitatively and evaluated using numerical and experimental techniques with some success, but many of these methods are not convenient in the design process since they do not impart any physical insight into the effect each design parameter has on the overall operation of the spiral antenna. This work examines the operation of spiral antennas and obtains a closed-form analytical solution by conformal mapping and transmission line model with high precision in a wide frequency band. Based on the analysis of spiral antenna, we propose two novel design processes for the stripline-fed Archimedean spiral antenna. This includes a stripline feed network integrated into one of the spiral arms and a broadband tapered impedance transformer that is conformal to the spiral topology for impedance matching the nominally-high input impedance of the spiral. A Dyson-style balun located at the center facilitates the transition between guided stripline and radiating spiral modes. Measured and simulated results for a probe-fed design operating from 2 GHz to over 20 GHz are in excellent agreements to illustrate the synthesis and performance of a demonstration antenna. The research in this work also provides the possibility to achieve conformal integration and planar structural multi-functionality for an Unmanned Air Vehicle (UAV) with band coverage across HF, UHF, and VHF. The proposed conformal mapping analysis can also be applied on periodic coplanar waveguides for integrated circuit applications.