Nowadays, there is a major international motivation driving towards the electrification of the aviation industry to reduce its impact on climate change. One of the most promising possibilities for achieving this ambitious goal involves the use of superconducting electrical machines featuring a high power density. This thesis investigates the preliminary design of an Air-Core Superconducting Synchronous Machine for future aircraft applications. After introducing superconductivity from a macroscopic point of view, the state of the art of superconductors for electrical machine applications is presented. Then, some of the most commonly used modeling methods for the simulation of superconductors are analyzed. Two simple case studies are introduced to understand whether it is necessary to consider specific formulations for the analysis of superconductors in electrical machines.