\frac{G_{ta,a}\! + \! G_{ab}}{C_{a}}
            \end{array}
            \! \right]
            \left[\!
            \begin{array}{c}
                \Delta T_{t} \\
                \Delta T_{a}
            \end{array}
            \!\right]
        =   
            \left[\!
            \begin{array}{c}
                \frac{\delta P_{t}}{C_{t}} + \frac{ I_{to}R_{to} (2+\beta) \delta V  }{C_{t} \left(R_{l} + R_{to}\left[1+\beta \right] \right)} \\
                \frac{\delta P_{a}}{C_a}
            \end{array}
            \!\right]               
\label{eq:2D_ODE}$$

where $$\delta I_{t} = \frac{-I_{to} R_{to} \alpha}{\left( R_{l} + R_{to}\left[1+\beta\right]\right) T_{to}} \delta T_{t} + \frac{1}{R_{l} + R_{o}\left[1+\beta\right]} \delta V
\label{eq;dI_2D}$$ As is standard, $\beta$ or ${\frac{I_{to}}{R_{to}} \frac{\partial R_{t}}{\partial I_{t}}(T_{to},I_{to}) |_{T_{t}}}$ characterizes the unwanted dependence of $R_{t}$ on the current. Also note that we have generalized Irwin’s loop gain parameter $\mathcal{L}_{I}$ to $${\mathcal{L}_{2D}}= \alpha \frac{R_{to}-R_{l}}{R_{to}(1+\beta)+R_{l}}  \frac{I_{to}^{2}R_{to}}{\left(G_{tb}+G_{ta}\right)T_{to}}
\label{eq:LoopGain}$$ to account for the 2 thermal DOF as well as the fact that CRESST has purposely chosen to use a load resistor ($R_{l}$) that is of similar magnitude to $R_{to}$ to suppress electro-thermal feedback. Finally, due to the possibility that $T_{to}$ and $T_{ao}$ could be macroscopically different, the thermal conductances $\frac{\partial P_{ta}}{\partial T_{t}}$ and $\frac{\partial P_{ta}}{\partial T_{a}}$ could have different values and consequently we define $$\begin{split}
G_{ta,a} &= \frac{\partial P_{ta}}{\partial T_{a}}\\
          &= \frac{\partial P_{ta}}{\partial T_{t}} \left(\frac{T_{a}}{T_{t}}\right)^{n_{ep}-1}\\
          &= G_{ta} \left(\frac{T_{a}}{T_{t}}\right)^{n_{ep}-1}
\end{split}$$ Inversion of the generalized impedance matrix, $M_{2D}$, found in Eq. \[eq:2D\_ODE\] leads to the current transfer functions for the thermal power response for direct heating 