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A new 3-dimensional theory of short-primary linear induction machines is presented. The theory almost completely incorporates both longitudinal and transverse end effects, especially including ferromagnetic end effects caused by the geometry of the primary laminated iron. The theory is based on the idea of considering two different regions of the primary iron and the airspace around it as one fictitious region having an inhomogeneous permeability, with step variations only at the four iron/air boundaries if the primary iron is assumed nonconducting. The 5-layer boundary-value problem including the fictitious layer is solved neatly in a mathematical sense using the new transfer-matrix method developed here. The theory is verified from test data on the double-sided linear induction machine at General Electric Co. The theory can very accurately predict the machine behaviour, including the vertical force, at low computational cost.