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Mathematical proof of the validity of reciprocity in one-dimensional bipolar transistors with arbitrary base parameters

We consider the solution of the general continuity equation for the minority electrons in the base of a one-dimensional n-p-n bipolar transistor. The solution of the continuity equation can be expressed as the superposition of two linearly independent expressions. The collector current is derived as a function of these two expressions. Two cases are considered: one junction is injecting at bias VFand the other is collecting at zero bias and vice-versa. The two collector current expressions are found to be identical which confirms the reciprocity theory for the very general one-dimensional low-level injection case.