Novel structures and methods for forming useful 
high temperature superconducting devices, most particularly resonators, are provided. Structures resulting in reduced 
peak current densities relative to known structures achieve numerous desirable benefits, especially including the reduced 
intermodulation effects of earlier resonators. In one aspect of this invention, a spiral in, spiral out 
resonator is provided, characterized in that it has an odd number of long runs, at least equal to five long runs, where the long runs are connected by turns, and wherein there are at least two sequential turns of the same handedness, followed by at least two turns of the opposite handedness. In yet another aspect of this invention, it has been discovered that reducing the size of the input and output pads of HTS resonators increases the relative 
inductance compared to the 
capacitance. Yet another 
resonator structure is a spiral snake 
resonator having a terminal end disposed within the resonator. A wide in the middle structure and wide at 
peak current density resonator structures utilize enlarged width portions of the resonator in those areas where 
current density is largest. In yet another aspect of this invention, operation of resonators in high 
modes, above the fundamental mode, reduce 
peak current densities. Resonators operated in 
modes in which current in adjacent long runs are in the same direction further serve to reduce current densities, and 
intermodulation effects. Symmetric current structures and 
modes of operation are particularly advantageous where far field effects are compensated for.