The design of a class of short period superconducting undulators is presented. We begin with a parameter-based analysis that provides insight into potential device performance as a function of the properties of superconducting materials. We present data on candidate low-temperature superconducting materials and the motivation to consider low-copper fraction, high Jc materials. Measured data on recent Nb3Sn conductors is provided, together with wire and cable design issues that are tailored for undulator applications. Key design concerns are then addressed, in particular the quench protection system limitations and the system performance. Progress on the construction and testing of a prototype 30mm period device is described.