A physical-based and analytical drain current model of amorphous indium gallium zinc oxide (a-IGZO) thin-film transistors (TFTs) is proposed. The model considers the combined contribution of both trapped and free charges that move through the a-IGZO film by multiple-trapping-and-release and percolation in conduction band. The model is compared with both measurements of TFTs fabricated on a flexible substrate and numerical simulations. It is accurate in the whole range of a-IGZO TFTs operation. The model requires only physical and geometrical device parameters. The resulting mathematical expressions are suitable for computer-aided design implementation and yield the material physical parameters that are essential for process characterization.