We argue that many aspects of improper ferroelectric (FE) activity in orthorhombic manganites can be rationalized by considering the limit of infinite intra-atomic splitting between majority-and minority-spin states (or the double-exchange limit), which reduces the problem to the analysis of a spinless double-exchange (DE) Hamiltonian. We apply this strategy to the low-energy model, derived from the first-principles electronic-structure calculations, and combine it with the Berry-phase theory of electric polarization. We start with the analysis of the simplest two-orbital model, describing the behavior of the e g bands, and apply it to the E-type antiferromagnetic (AFM) phase, which in the DE limit effectively breaks up into one-dimensional zigzag chains. We derive an analytical expression for the electronic polarization (P el ) and explain how it depends on the orbital ordering and the energy splitting between e g states. Then, we evaluate parameters of this model for the series of manganites. For these purposes, we start from a more general five-orbital model for all Mn 3d bands and construct a new downfolded model for the e g bands. From the analysis of these parameters, we conclude that the behavior of P el in realistic manganites always corresponds to the limit of large . This property holds for all considered compounds even in the local-density approximation, which typically underestimates . We further utilize this property in order to derive an analytical expression for P el in a general twofold periodic magnetic texture, based on the five-orbital model and the perturbation-theory expansion for the Wannier functions in the first order of 1/ . This expression explains the functional dependence of P el on the relative directions of spins. Furthermore, it suggests that P el is related to the asymmetry of transfer integrals, which should have simultaneously symmetric and antisymmetric components in the crystal-field representation. The main contribution to this asymmetry comes from the antiferro-orbital ordering in the ab plane. Finally, we explain how the FE polarization can be switched between orthorhombic a and c directions by inverting the zigzag AFM texture in every second ab plane. We argue that this property is generic and can be realized even in the twofold periodic texture.