  We examine the transmission through nonideal microwave resonant circuits. The
general analytical resonance line shape is derived for both inductive and
capacitive coupling with mismatched input and output transmission impedances,
and it is found that for certain non-ideal conditions the line shape is
asymmetric. We describe an analysis method for extracting an accurate internal
quality factor ($Q_i$), the Diameter Correction Method (DCM), and compare it to
the conventional method used for millikelvin resonator measurements, the $\phi$
Rotation Method ($\phi$RM). We analytically find that the $\phi$RM
deterministically overestimates $Q_i$ when the asymmetry of the resonance line
shape is high, and that this error is eliminated with the DCM. A consistent
discrepancy between the two methods is observed when they are used to analyze
both simulations from a numerical linear solver and data from asymmetric
coplanar superconducting thin-film resonators.
