$L$ & Inductance (SQUID+ parasitic) & & 350 nH [@CRESST_2007_electronics]\
$R_{l}$ & Load resistor (shunt+ parasitic) & & 40 m$\Omega$ [@CRESST_2007_electronics]\
$T_{l}$ & Temperature of $R_{l}$ & & 10mK [@CRESST_2007_electronics]\
$S_{I SQUID}$& SQUID Current Noise & & 1.2 pA/$\sqrt{\text{hz}}$[@CRESST_2007_electronics]\

![[]{data-label="fig:CaWO4_Qdiag"}](figs/CRESST_PDet/CRESST_PDet.pdf){width="3.3in"}

Modeling of complex calorimeters with multiple coupled thermal degrees of freedom (DOF) like the CRESST phonon detector has been a very active area of research, particularly within the x-ray and gamma-ray TES communities [@Tali_Thesis; @TaliComplexuCal; @Smith_Hydra; @Irwin_GammaTES]. Thus, we need only apply mature strategies, being careful to follow the conventions of [@Irwin_TES]. This is made even easier by the fact that CRESST has already modeled the dynamics of their detectors [@CRESST_1995_ProbstModel; @CRESSTII_2009_SplitDetector; @Emilija_2008_thesis] and even attempted to simulate noise [@Emilija_2008_thesis]; so we will concentrate in particular on developing analytical simplifications and physical intuition, neither of which has yet been done.

Dynamics
--------

The original (non-composite) CRESST-II phonon detector is a large 300 g cylindrical CaWO$_{4}$ crystal with a W TES (T$_{c} \approx$ 10 mK) and all of its required accessories (bonding pads, Al bias rails) fabricated directly upon its surface, as shown in Fig. \[fig:CaWO4\_Qdiag\] [@Emilija_2008_thesis]. Unfortunately, even after such painstaking fabrication efforts, the thermal coupling between the TES and 