  In this paper, we construct a family of non-weight modules over the untwisted
$N=2$ superconformal algebras. Those modules when regarded as modules over the
Cartan subalgebra (modulo the center) are free of rank $2$. We give a
classification of isomorphism classes of such modules. Moreover, all submodules
of such modules are precisely determined. In particular, those modules are not
simple. The corresponding simple quotient modules are classified. Furthermore,
these simple modules when restricted as modules over $N=1$ superconformal
algebras coincide with those modules constructed in [H. Yang, Y. Yao, L. Xia, A
family of non-weight modules over the super-Virasoro algebras, J. Algebra 547
(2020), 538-555].
