In this paper we exploit a relationship between certain pairwise balanced designs with v points and periodic Golay pairs of length v, to classify periodic Golay pairs of length less than 40. In particular, we construct all pairwise balanced designs with v points under specific block conditions having an assumed cyclic automorphism group, and using isomorph rejection which is compatible with equivalence of corresponding periodic Golay pairs, we complete a classification up to equivalence. This is done using the theory of orbit matrices and some compression techniques which apply to complementary sequences. We use similar tools to construct new periodic Golay pairs of lengths greater than 40 where classifications remain incomplete and demonstrate that under some extra conditions on its automorphism group, a periodic Golay pair of length 90 will not exist. Length 90 remains the smallest length for which existence of a periodic Golay pair is undecided. Some quasi-cyclic self-orthogonal codes are constructed as an added application.