transition.

It is standard practice in TES physics, to expand non-linear terms such as $P_B$, $P_J$ and $R(T,I)$ to first order in the small-signal limit around the steady state operating point $T_0$, $I_0$ and $R_0$, giving [@IrwinChapter] $$\frac{\mathrm{d}}{\mathrm{d}t}
\begin{pmatrix}
\delta I \\
\delta T
\end{pmatrix}
=-
\begin{pmatrix}
\frac{1}{\tau_{el}} & \frac{\alpha P_{J0}}{T_0I_0L} \\[0.7em]
-\frac{I_0R_0(2+\beta)}{C} & \frac{1}{\tau_I}
\end{pmatrix}
\begin{pmatrix}
\delta I \\
\delta T
\end{pmatrix}
+
\begin{pmatrix}
\frac{\delta V}{L} \\[0.5em]
0
\end{pmatrix},
\label{eq:Diff_Matrix}$$ where $\delta I = I-I_0$, $\delta T = T-T_0$, $P_{J0} = I_0^2R_0$, and $\delta V$ represents a small change in the applied bias voltage. The resistance-temperature and resistance-current sensitivities are given by $\alpha = (\partial \ln R/\partial \ln T)_I$ and $\beta = (\partial \ln R/\partial \ln I)_T$ respectively. The time constants $\tau_{el} = L/(R_L+R_0(1+\beta))$ and $\tau_I = \tau/(1-P_{J0}\alpha/(GT_0))$ represent electrical and thermal time constants. The natural thermal time constant in the absence of electrothermal feedback, $\alpha = 0$, and strictly $\beta = 0$, is given by $\tau = C/G$.

Adapting the approach of Lindeman[@LindemanThesis; @IrwinChapter], Eq. (\[eq:Diff\_Matrix\]) may be solved for the specific case of a small step in bias voltage, $\delta V$ at $t=0$, subsequently maintained over the course of a measurement, giving $$\delta I = \frac{\delta V}{L}\frac{1}{\lambda_+-\lambda_-}(A_-\mathrm{e}^{-\lambda_-t}-A_+\mathrm{e}^{-\lambda_+t}+B),
