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An analysis of the relativistic electron motion at electron cyclotron resonance (ECR) condition in a magnetic field which is gradually growing in time is accomplished. The results show the existence of a strong mechanism (basically nonrelativistic) which automatically maintains the phase stability during the gyro-resonant acceleration. The conditions in which the gyro-resonant acceleration (up to arbitrarily high energies) is possible are established. The conceptual design of a gyro-resonant accelerator (Gyrac) is proposed. The parameters of possible Gyrac devices for the energies of 5 GeV, 1 GeV, 100 MeV, and 10 MeV with peak powers of accelerated electron bunches (e-vortex) correspondingly 2.5 TW, 0.5 TW, 25 GW, and 0.4 GW are given.