The rate equations for integrated erbium-doped waveguide amplifiers (EDWA′s) pumped at 980 nm wavelength are analyzed using overlapping factors methods. An analytical implicit gain solution of EDWA′s is derived. Based on which an analytical threshold expression of pumping power is obtained, and overlapping factors between the erbium concentration and the pumping and signal light field intensity distribution inside the erbium-doped planar optical waveguide amplifiers are calculated. The effect of erbium-doped concentration on pump threshold power and the effect of pump power on gain are discussed.