A common approximation used in the analysis of power systems is the neglect of the dynamic saliency in synchronous machines. In this paper, it is shown that eliminating the error associated with neglecting dynamic saliency can be accomplished with the addition of a singular perturbation(s) into the machine model. By considering the elimination of error in such a way, singular-perturbation-based model-order-reduction techniques are used to derive detailed- and reduced-order models of synchronous machines where dynamic saliency is eliminated with zero error and no added numerical cost.