A comprehensive Auxiliary Functions of Generalized Scattering Matrix (AFGSM) method is presented on the bandgap analysis of periodic structures encountered in many engineering applications. Proposed method accurately determines the bandgaps of periodic structures with symmetric and asymmetric unit cells. In order to test the feasibility of the proposed method, numerical results are compared with Eigenvalue method (EIV) and HFSS/CST frequency domain simulations. The validity and applicability of the presented method are demonstrated by analysing the bandgap characteristics of periodically dielectric loaded rectangular waveguides, photonic crystals, helix Slow Wave Structures (SWSs) of Traveling Wave Tubes (TWTs) and excellent agreements on the simulation results are obtained. Bandgap analysis and design of different periodic structure problems can be effectively achieved using the proposed AFGSM method.