  We produce explicit generators of the classical W-algebras associated with
the principal nilpotents in the simple Lie algebras of all classical types and
in the exceptional Lie algebra of type $G_2$. The generators are given by
determinant formulas in the context of the Poisson vertex algebras. We also
show that the images of the W-algebra generators under the Chevalley-type
isomorphism coincide with the elements defined via the corresponding Miura
transformations.
