  We study a family of separable potentials with and without added contact
interactions by solving the associated Lippmann-Schwinger equation with two
coupled partial waves. The matching of the resulting amplitude matrix with the
effective-range expansion is studied in detail. When a counterterm is included
in the potential we also carefully discuss its renormalization. Next, we use
the matrix $N/D$ method and study whether the amplitude matrices from the
potentials considered admit an $N/D$ representation in matrix form. As a novel
result we show that it is typically not possible to find such matrix
representation for the coupled partial-wave case. However, a separate $N/D$
representation for each coupled partial wave, a valid option known in the
literature, is explicitly implemented and numerically solved in cases where the
matrix $N/D$ method is unavailable.
