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The steady state drift-diffusion model for the flow of electrons and holes in semiconductors is simplified by perturbation techniques. The simplifications amount to assuming zero space charge and low injection. The limiting problems are solved and explicit formulas for the voltage-current characteristics of bipolar devices can be obtained. As examples, the p«-diode, the bipolar transistor and the thyristor are discussed. While the classical results of a one-dimensional analysis are confirmed indoi:10.1090/qam/1134751 fatcat:twouqbvxmbfcpplv47tmlz4oma