We present an electrochemical model for lithium-ion batteries based on porous electrode theory that is derived by taking volume averages of microscale equations. Using a rigorous scaling analysis, we then show how the model can be systematically reduced to a pair of equivalent circuit models and, importantly, provide three simple conditions that are required for the reduction to be valid. Since the reduced model is derived from a detailed electrochemical model, it is based on physical parameters rather than empirical equivalent circuit parameters. The reduced model consists of two first-order ordinary differential equations for the solid-phase electric potentials and can be straightforwardly simulated using traditional fast methods. This makes the reduced model ideal for use in battery management systems and for parameter estimation and optimisation. Finally, we validate the reduced model against three sets of experimental data and large-scale numerical simulations and find excellent agreement in all cases.