  We show that the electrocaloric (EC) effect -- e.g., the temperature change
experienced by an insulator upon application of an electric bias -- lends
itself to a straightforward interpretation when expressed as a Taylor series in
the external field. Our formalism explains in a unified and simple way the most
notable small-field effects reported the literature -- e.g., the so-called {\em
normal} (increase of temperature under bias, as usually found in
ferroelectrics) and {\em inverse} (decrease of temperature, as e.g. in
antiferroelectrics) EC responses -- and clarifies their physical
interpretation. We also discuss in detail second-principles simulations for
prototype ferroelectric PbTiO$_{3}$, explicitly evaluating subtle predictions
of the theory, such as the occurrence of competing contributions to the EC
response.
