Let $k$ be a real abelian field such that the conductor of every nontrivial character belonging to $k$ agrees with the conductor of $k$. Note that real quadratic fields satisfy this condition. For a prime $p$, let $k_{infty}$ be the $mathbb{Z}_p$-extension of $k$. The aim of this paper is to produce a set of generators of the Tate cohomology group $hat{H}^{-1}$ of the circular units of $k_n$, the $nth$ layer of the $mathbb{Z}_p$-extension of $k$, where $p$ is an odd prime. This result generalizes some earlier works which treated the case when $k$ is real quadratic field and used them to study ${lambda}$-invariants of $k$.