The design of a 220GHz high power coaxial gyrotron is presented. The device operates in the fundamental harmonic TE04circular electric mode. The operation parameters for cavity were 50kV, 10A, and 8.4 Tesla which were optimized by self-consistent nonlinear theory. The electron beam is produced by a coaxial triode magnetron injection gun (MIG) with an average perpendicular-to-parallel velocity ratio 1.5 and perpendicular velocity spread of 5.2%. Particle-in-cell (PIC) simulations have been performed for the whole gyrotron. Both the results from theoretical analysis and PIC simulations show that the designed gyrotron can produce more than 200kW output power with power conversion efficiency of more than 40%.