  We use Maxwell's equations to derive several models describing the
interaction of the multi-mode fundamental field and its second harmonic in a
ring microresonator with quadratic nonlinearity and quasi-phase-matching. We
demonstrate how multi-mode three-wave mixing sums entering nonlinear
polarisation response can be calculated via Fourier transforms of products of
the field envelopes. Quasi-phase-matching gratings with arbitrary profiles are
incorporated seamlessly into our models. We also introduce several levels of
approximations allowing to account for dispersion of nonlinear coefficients and
demonstrate how coupled-mode equations can be reduced to the envelope
Lugiato-Lefever-like equations with self-steepening terms. An estimate for the
$\chi^{(2)}$ induced cascaded Kerr nonlinearity, in the regime of imperfect
phase-matching, puts it above the intrinsic Kerr effect by several orders of
magnitude.
