Field-aligned electrons accelerated upward from the ionosphere to the magnetosphere are the principal charge carriers in the auroral downward field-aligned current region. Current densities, typically of the order of a few ¿Am-2, are sustained by potential drops of several 100 V up to a few kV. This paper uses a model presented in a separate paper in this special section (Cran-McGreehin and Wright, 2005) to obtain an analytical nonlinear current-voltage relationship for the downward current, which is complimentary to the well-known linear current-voltage relation for the upward current region (Knight, 1973). Exact and approximate current-voltage relations are given. These relations are compared with FAST observations and show good agreement.