The second-generation high temperature superconducting tapes (2G-HTS) such as the REBCO coated conductors (CC) are known for their strong mechanical properties and high current carrying capacity with low AC losses compared with first-generation HTS tapes as well as low temperature superconductors. Therefore, they seem to be good candidates to assemble coils for Superconducting Rotating Electric Machines (SuREMs). In this paper, to investigate influence of a Soft Magnetic Composite (SMC) material on the DC and AC HTS coil performances at cryogenic temperatures, GdBCO-and YBCO-based commercial tapes from different manufactures were used to assemble 10 HTS coils in doublepancake configuration. These tapes were wound using two different core structures, i.e. fixed and movable cores, having the same circular shape but different diameters as well as materials, i.e. fiberglass and SMC. Critical currents and AC transport current losses of these coils were measured using the electrical method. To evaluate the parallel and perpendicular components of the profile of magnetic flux density for each assembled HTS coil, a finite element model has been developed using Maxwell 2D. Eventually, the performance of different coils for SuREM application was discussed.