  We study the connection between $N=2$ supersymmetry and a topological bound
in a two-Higgs-doublet system with an $SU(2)\times U(1)_Y\times
U(1)_{Y^{\prime}}$ gauge group. We derive the Bogomol'nyi equations from
supersymmetry considerations showing that they hold provided certain conditions
on the coupling constants, which are a consequence of the huge symmetry of the
theory, are satisfied. Their solutions, which can be interpreted as electroweak
cosmic strings breaking one half of the supersymmetries of the theory, are
studied. Certain interesting limiting cases of our model which have recently
been considered in the literature are finally analyzed.
