Electrically small loops and monopoles are modeled over infinite lossy ground planes using a modification to the Sommerfeld approach, in order to assess the performance of YBCO antennas connected to typical metallic ground planes. Calculated results suggest that for antennas with low radiation resistances, significant efficiency improvements can be obtained with YBCO elements even if the ground planes remain lossy, and that little is to be gained by making the ground planes superconducting if the antenna elements remain lossy.