In this work, we focus on two locally active device models and present, firstly, the simplification of a 3D modified Poole-Frenkel conduction based model on a mathematical basis, which leads to a compact and numerically efficient form of the conductance expression. Then, we present a transformation of the equations of a thermally induced phase transition based locally active device model such that the resulting set of equations is numerically stable and requires less simulation time. The new versions of both models share a similar form, though their state variables have different physical meanings. As an important aspect of their S-shaped DC I-V curves, we examine the DC characteristics of these equivalent models and derive analytical expressions for the peak and valley points in terms of the physically meaningful model parameters. Theoretical results are verified through numerical simulations while our findings may support device manufacturers to adopt a systematical approach to tailor the fundamental characteristics of locally active devices and help circuit designers to execute low cost and robust simulations of large scale systems utilizing them.