High-temperature superconducting (HTS) coils wound from coated conductors without turn-to-turn insulation (non-insulated (NI) coils) have been proven with excellent electrical and thermal performances. However, the slow charging of NI coils has been a long-lasting problem. In this work, we explore using a transformer-rectifier HTS flux pump to charge an NI coil and a metal-insulated coil. The charging performance comparison is made between different coils. Comprehensive study is done to thoroughly understand the electrical-magnetic transience in charging these coils. We will show that the low-voltage high-current flux pump is especially suitable for charging NI coils with very low characteristic resistance.