  Understanding of the intimate cross-coupling between electric and magnetic
degrees of freedom in solids usually requires sophisticated models and
time-consuming calculation methods. Instead of macroscopic symmetry analysis,
we present a simple but general approach to explore the microscopic mechanism
of magnetoelectric (ME) effects in magnetic ordered materials based on local
spin and lattice symmetry analysis. Our methods are successfully applied to
Cr$_2$O$_3$ and orthorhombic RMnO$_3$ with linear and quadratic ME effects,
respectively. We revealed all the possible microscopic origins of every
non-zero ME coefficients that cannot be easily fulfilled by other theoretical
methods. Moreover, the contribution from each mechanism can be located down to
specific spins or spin pairs. Our theoretical approach is capable of providing
a detailed guide to explore rich spin-induced magnetoelectrics with strong ME
effects.
