Cold field electron emission (CFE) from metals is described by a large family of approximate equations called Fowler-Nordheim-type (FN-type) equations. This article discusses FN-type equations that give emission current density in terms of local work function and barrier field. Starting from the widely used standard FN-type equation [E. L. Murphy and R. H. Good, Phys. Rev. 102, 1464 (1956)], which is a hybrid physical/mathematical equation applicable to CFE from flat free-electron-metal surfaces at 0 K, this article builds up a generalized "physical FN-type equation" in which physical correction factors are used to represent effects not included in standard CFE theory. The derivation starts by separating mathematical and physical aspects of standard theory, making use of a new independent physical variable (the scaled barrier field) and a physical "barrier-shape correction factor" that applies to any well-behaved barrier. Correction of long-standing error in applying Jeffries-Wentzel-Kramers-Brillouin-type approximations to CFE theory introduces a tunneling prefactor and a related small correction factor associated with integration over occupied electron states. Further correction factors represent effects due to non-free-electron band structures and emitter temperature. The outcome is an equation that provides a physical framework for FN-type equations that apply to CFE from a single metal-like conduction band. This equation is conceptually more complete than the equations normally used. Estimates are presented for the sizes of correction factors. (c) 2008 American Vacuum Society.