Let $\mathbb{F}_q$ be the finite field with $q$ elements and $\beta$ Salem series in $ \mathbb{F}_q((X^{-1}))$. It is proved in \cite{K} that, in this case, all elements in $\mathbb{F}_q(X,\beta)$ have purely periodic $\beta$-expansion. We characterize the formal power series $f$ in $\mathbb{F}_q(X,\beta)$ with purely periodic $\beta$-expansions by the conjugate vector $\widetilde{f}$ when $\beta$ is a cubic unit. No similar results exist in the real case.