The partial breakdown (PBD) and complete breakdown (BD) phenomena in a composite insulation system of glass fiber reinforced plastic (GFRP) and liquid nitrogen are investigated to find the PBD and BD characteristics in high temperature superconducting (HTS) coils at quench. The electrode system used is made from a coaxial spiral coil-to-cylindrical electrode with an insulation barrier and spacers, and is immersed in liquid nitrogen. A heater is mounted inside the coil electrode to generate boiling which occurs on quenched superconducting coils. The experimental results show that (1) the polarity of PBD initiation under 60 Hz ac applied voltage depends on the width lc of the cooling channel, (2) PBD appears through a gas bubble locked between the coil and concentric insulation barrier due to the electrical gradient and the spacer, (3) breakdown voltages are affected severely by the risetime of the applied voltage and the current amplitude associated with PBD, (4) two kinds of BD mechanisms are found depending on the shape of the spacer, length of cooling channel and heater power.