Fast passage dynamic nuclear polarization on rotating solids

► We present experimental data of the dependence of dynamic nuclear polarization with MAS spinning frequency. ► We model a rotating {electron-nucleus} and {electron-electron-nucleus} spin system. ► We use Liouville equation and powder averaging, to compare with experiments. ► We explain the basics of solid effect and cross effect's mechanisms in MAS-DNP. ► We study the effect of relaxation on the steady state enhancement in both cases.