The construction of self-dual codes over small fields such that their minimum distances are as large as possible is a long-standing challenging problem in the coding theory. In 2009, a family of binary self-dual cyclic codes with lengths $n_i$ and minimum distances $d_i \geq \frac{1}{2} \sqrt{n_i}$, $n_i$ goes to the infinity for $i=1,2, \ldots$, was constructed. In this paper, we construct a family of (repeated-root) binary self-dual cyclic codes with lengths $n$ and minimum distances at least $\sqrt{n}-2$. New families of lengths $n=q^m-1$, $m=3, 5, \ldots$, self-dual codes over ${\bf F}_q$, $q \equiv 1$ $mod$ $4$, with their minimum distances larger than or equal to $\sqrt{\frac{q}{2}}\sqrt{n}-q$ are also constructed.