Normal-mode helical antennas (NMHA) are well known for achieving high antenna efficiencies in very small sizes when antenna structures are tuned to self-resonant situations. The design equation determining self-resonant structures were developed in free space conditions. However, in practically, NMHA mostly used dielectric or magnetic materials as cores. The design equation for these cases is not studied yet. In this paper, fundamental self-resonant structures are studied for NMHA placed in dielectric or magnetic material. In order to obtain correct calculation results, calculation models of an commercial electromagnetic simulator (FEKO) is studied. Many data for structural change from free space condition to inside materials are achieved. Moreover, input resistance and antenna efficiency changes are also obtained.