Resonators in superconducting circuits are studied to extract the relative permittivity and dielectric loss of aluminium nitride in parallel-plate capacitors used in travelling wave parametric amplifiers. One approach relies on a lumped circuit with a parallel-plate capacitor and a spiral inductor and the other approach relies on comparing coplanar waveguides with and without a capacitive load. Resonance frequencies are estimated through analytical methods, to select the right parameters of physical implementations. Circuit simulations and 3D field simulations are also performed, with ADS and Comsol respectively. The resonators are fabricated on chips with the notch configuration for measurements. The complex transmission spectrum S_21(f) is measured for frequencies around 3GHz to 8GHz at a temperature around 10 mK.