Limiting currents in annular unneutralized relativistic electron beams are calculated and compared with experiment. The limiting factor for short beams is found to be the space-charge depression of the potential due to the unneutralized beam; in longer beams self-magnetic and electric fields impose a power limit. Typical beam impedances are found to be of the order of 15 Ω with a 350 keV beam. The role of shear in the axial velocity of the beam is discussed.