HTS transformer is one of the most promising devices to supply enough electric energy for quick increase consumption. However, for practical design of the HTS transformer, it is necessary to establish the research on breakdown, V-t characteristics, degradation, and so on. In this paper, we discussed breakdown characteristics and V-t characteristics of polyimide/L$N_2$ and glass fiber reinforced plastic/$LN_2$ composite insulations. These composite insulations have been used as turn-to-turn and layer-to-layer insulations for HTS transformer respectively, Moreover, we investigated the degradation of these insulation samples after breakdown using microscope and SEM photograph.