The linear and nonlinear theories of large-orbit gyrotron traveling wave amplifiers ͑gyro-TWAs͒ have been developed based on the corresponding theories of small-orbit gyro-TWAs. The linear theory is in good agreement with the nonlinear theory in the small signal region of large-orbit gyro-TWAs. The phenomenon that most electrons move toward the axis of interaction circuit during the beam-wave interaction is observed and its potential effect on the design of large-orbit coaxial gyro-TWAs is emphasized.High power and broad bandwidth capability of the gyrotron traveling wave amplifier ͑gyro-TWA͒ makes it an attractive coherent radiation source in the millimeter and submillimeter wavelength ranges. Gyro-TWAs could be applied widely, especially in radar and communication systems. Major advances in gyro-TWA performances have been reported in the review papers. [1][2][3] Two types of electron beam have been employed in gyro-TWAs. One is the small-orbit beam consisting of many off-axis beamlets, which is usually produced by a magnetic injection gun, 4,5 and the other is the large-orbit beam with electrons encircling about the axis of the amplifier, which is usually produced by a cusp gun. 6,7 A large-orbit beam can be considered as a special small-orbit beam with its guiding center on the axis. So, beam radius of a small-orbit beam may be much larger than that of a large-orbit beam. Larger beam radius can mitigate the restriction of beam current caused by space charge effect, and hence is helpful for generating higher output power at short wavelength. However, compared with small-orbit gyrotrons, large-orbit gyrotrons have an important advantage of greatly reduced mode competition at higher cyclotron harmonics. Only the corotating transverse electric modes TE mn with azimuthal indices m equal to the resonant harmonic number s can be excited in large-orbit gyrotrons. 8 For small-orbit gyrotrons, the procedure of deriving beam-wave interaction equations is usually lengthy and tedious. For large-orbit gyrotrons, the derivation may be simplified by applying the well developed theory of small-orbit gyrotrons. In Ref. 8, it was shown that the existing nonlinear theory of small-orbit gyrotrons could be applied to largeorbit gyrotrons by setting the value of guiding center radius to zero. However, the theory presented in Ref. 8 is nonconsistent and limited to gyrotrons with resonance-cavity-type interaction structures. In this paper, the procedure of developing linear and nonlinear theories of large-orbit gyro-TWAs from the corresponding theories of small-orbit gyro-TWAs is presented in detail, and numerical investigations of the electron motion trajectories are carried out for the gyro-TWAs operated at the first and second harmonics of the electron cyclotron frequency.According to the linear theories developed in Refs. 9 and 10, for a small-orbit TE mn sth-harmonic gyro-TWA, the dispersion relation of its beam-wave interaction is D͑ ,where k = / c, k z is the axial wave number, k mn = x mn / r w is the cutoff wave number, and x...