When a 3-phase cage induction motor is started on half winding, space harmonic m.m.f.＇s occur, and the machine usually fails to accelerate beyond the speed of the principal harmonic. A mathematical theory is presented in this paper which not only explains such behavior observed in the past, but also predicts whether and at what speed a motor will hang up or crawl on any particular arrangement of half winding. The theory also provides a basis for the design of harmonic-free half windings on which the machine will accelerate smoothly to full speed. This is illustrated by application to a 6-pole motor.