  A derivation of the single particle model (SPM) is made from a porous
electrode theory model (or Newman model) of half-cell (dis)charge for an
electrode composed of uniformly sized spherical electrode particles of a single
chemistry. The derivation uses a formal asymptotic method based on the
disparity between the size of the thermal voltage and that of the
characteristic change in overpotential that occurs during (de)lithiation.
Comparison is made between solutions to the SPM and to the porous electrode
theory (PET) model for NMC, graphite and LFP. These are used to identify
regimes where the SPM gives accurate predictions. For most chemistries, even at
moderate (dis)charge rates, there are appreciable discrepancies between the PET
model and the SPM which can be attributed to spatial non-uniformities in the
electrolyte. This motivates us to calculate a correction term to the SPM. Once
this has been incorporated into the model its accuracy is significantly
improved. Generalised versions of the SPM, that can describe graded electrodes
containing multiple electrode particle sizes (or chemistries), are also
derived. The results of the generalised SPM, with the correction term, compare
favourably to the full PET model where the active electrode material is either
NMC or graphite.
