P_{t}$ and $\delta P_{a}$ are power excitations into the TES and absorber respectively. This model can also be seen diagrammatically in Fig. \[fig:Det\_2DModel\]. Variable definitions and estimated sizes can be found in Table \[tab:PhononDet\]. One relatively unique feature of the CRESST design is the capability to directly heat the TES electronic system through an additional heater circuit ($Q$). When held constant, this is effectively equivalent to being able to easily vary the bath temperature $T_b$ on a detector by detector basis.

![[]{data-label="fig:Det_2DModel"}](figs/Gen/2Dthermal_diag.pdf){width="2.0in"}

Taylor expanding to first order around the operating point, $[I_{to},T_{to},T_{ao}]$ and then Fourier transforming, these equations simplify to

$$\left[ \!
            \begin{array}{c c c}
                 j\omega + \frac{R_{to} (1+\beta) +R_{l}}{L}    & \frac{I_{to} R_{to} \alpha}{L T_{to} }                                                    & 0 \\
                 -\frac{I_{to} R_{to}(2+\beta)}{C_{t}}          & j \omega +\frac{\frac{I_{to}^{2}R_{to} \alpha}{T_{to}}- G_{ta}-G_{tb}}{C_{t}} &   -\frac{G_{ta,a}}{C_{t}} \\
                 0                                                                          &  -\frac{G_{ta}}{C_{a}}                                                    &   j \omega + \frac{G_{ta,a}+ G_{ab}}{C_{a}}
            \end{array}
            \! \right]
            \left[\!
            \begin{array}{c}
                \Delta I _{t}(\omega)\\
                \Delta T_{t}(\omega) \\
                \Delta T_{a}(\omega)
            \end{array}
            \!\right]
            =
            \left[\!
            \begin{array}{c}
                \frac{\delta V(\omega)}{L} \\
                \frac{\delta P_{t}(\omega)}{C_{t}} \\
                \frac{\delta P_{a}(\omega)}{C_a}
            \end{array}
            \!\right]               
\label{eq:3D_ODE_Fourier}$$

which in the pertinent limit of $L \rightarrow 0$ can be simplified to only thermal DOF $$\left[ \!
            \begin{array}{c c}
                   j\omega + ({\mathcal{L}_{2D}}\!+\!1) \frac{G_{ta} +G_{tb}}{C_{t}}   &   \frac{-G_{ta,a}}{C_{t}} \\
                   \frac{-G_{ta}}{C_{a}}                                                &j \omega  + 