  A method is proposed and experimentally explored for in-situ calibration of
complex transmission data for superconducting microwave resonators. This
cryogenic calibration method accounts for the instrumental transmission
response between the vector network analyzer reference plane and the device
calibration plane. Once calibrated, the observed resonator response is analyzed
in detail by two approaches. The first, a phenomenological model based on
physically realizable rational functions, enables the extraction of multiple
resonance frequencies and widths for coupled resonators without explicit
specification of the circuit network. In the second, an ABCD-matrix
representation for the distributed transmission line circuit is used to model
the observed response from the characteristic impedance and propagation
constant. When used in conjunction with electromagnetic simulations, the
kinetic inductance fraction can be determined with this method with an accuracy
of 2%. Datasets for superconducting microstrip and coplanar-waveguide resonator
devices were investigated and a recovery within 1% of the observed complex
transmission amplitude was achieved with both analysis approaches. The
experimental configuration used in microwave characterization of the devices
and self-consistent constraints for the electromagnetic constitutive relations
for parameter extraction are also presented.
