An exact equivalent circuit including terminal parts, which takes account of electrical and chemical control parameters in a unified way, is derived for a cell with a mixed conductor (or electrolyte) without internal sources or sinks. In one-dimensional problems electrochemical kinetics can be mapped by two-dimensional circuits exhibiting the spatial and the thermodynamic displacement as two independent coordinates. One main advantage of the exact circuits with respect to the underlying differential equations is the ability to simplify the description according to specific situations. As we show in several examples in the second part of the paper, it is straightforward to select the elements relevant for the particular experimental conditions and so to make appropriate approximations. This is most helpful for the description of electrochemical systems, such as fuel cells, membranes, pumps and batteries.