  Superconducting resonators provide a convenient way to measure loss tangents
of various dielectrics at low temperature. For the purpose of examining the
microscopic loss mechanisms in dielectrics, precise measurements of the
internal quality factor at different values of energy stored in the resonators
are required. Here, we present a consistent method to analyze a LC
superconducting resonator coupled to a transmission line. We first derive an
approximate expression for the transmission S-parameter $S_{21}$ based on a
complete circuit model. In the weak coupling limit, we show that the internal
quality factor is reliably determined by fitting the approximate form of
$S_{21}$. Since the voltage $V$ of the capacitor of the LC circuit is required
to determine the energy stored in the resonator, we next calculate the relation
between $V$ and the forward propagating wave voltage $V_{in}^+$. Due to the
dependence of the quality factor on voltage, $V$ is not simply proportional to
$V_{in}^+$. We find a self-consistent way to determine the relation between $V$
and $V_{in}^+$, which employs only the fitting parameters for $S_{21}$ and a
linear scaling factor. We then examine the resonator transmission in the cases
of port reflection and impedance mismatch. We find that resonator transmission
asymmetry is primarily due to the reflection from discontinuity in transmission
lines. We show that our analysis method to extract the internal quality factor
is robust in the non-ideal cases above. Finally, we show that the analysis
method on LC resonator can be generalize to arbitrary weakly coupled lumped and
distributed resonators. The generalization uses a systematic approximation on
the response function based on the pole and zero which are the closest to the
resonance frequency. This Closest Pole and Zero Method (CPZM) is a valuable
tool for analyzing physical measurements of high-Q resonators.
