The acceleration of electrons into discrete auroras by parallel electric fields is intimately connected with the auroral current-voltage relation. Kinetic and fluid models are in this study combined to derive a relation between auroral currents and field-aligned electric fields. The derivation is based on the assumption that the plasma density along the flux tube is a parameter, which is independent of the field-aligned potential drop. On the other hand, recent observations suggest that the density in the acceleration region is an important parameter, which strongly influences the field-aligned electric field. Knight's model, leading to the widely accepted linear current-voltage relation, predicts an increase of the density in the acceleration region in proportion to the square root of the voltage, which seems incompatible with these observations. However, the observed correlation between large potential drops and deep auroral cavities is consistent with the results of this study, which shows that the voltage is inversely proportional to the square of the density in the acceleration region. Our model also predicts that the voltage is proportional to the square of the current.