The AC losses are calculated for superconducting filaments in the regime where the filaments are smaller than lambda ' and the amplitude of the oscillation is low. In a bias field analytic expressions can be obtained by assuming that the effect of pinning forces on the field distribution is small. In zero applied field the losses are still further decreased due to an increase in the effective value of B_c1. The results are in qualitative agreement with the experimental results of Sumiyoshi and co-workers (1986).