  We present a framework for modeling superconducting circuits that integrates
classical microwave analysis with circuit quantization. Our framework enables
the calculation of the lossy eigenmodes of superconducting circuits, and we
demonstrate the method by analyzing several circuits relevant to multiplexed,
Purcell filtered transmon readout architectures. We show that the transmon
relaxation times obtained by our method agree with the established
approximation $T_1 \approx C/\text{Re}[Y(i\omega_q)]$ away from environmental
resonances and do not vanish on resonance. We also show that the hybridization
of the modes in the readout circuit is highly sensitive to the bandwidth of the
Purcell filter.
