\label{eq:RtMod}$$ where $\lambda_\pm = \tau_\pm^{-1}$ are eigenvalues of the matrix in Eq. \[eq:Diff\_Matrix\], $A_\pm = 1-\lambda_\pm^{-1}\tau_I^{-1}$ and $B=\tau_I^{-1}(\lambda_-^{-1}-\lambda_+^{-1})$. For low inductance, $\tau_+ \ll \tau_-$, such that $$\begin{aligned}
\tau_+ & \to \tau_{el} \\
\tau_- & \to \tau_{eff},\end{aligned}$$ where $\tau_{eff}$ is the effective thermal time constant, given by $$\tau_{eff}  = \tau\frac{1+\beta+R_L/R_0}{1+\beta+R_L/R_0+(1-R_L/R_0)P_{J0}\alpha/(GT_0)}$$ $$\label{eq:tauEff_Approx}
\approx \frac{\tau}{1+\frac{\alpha}{n}(1-  T_B^n  / T_0^n )}.$$ Equation \[eq:tauEff\_Approx\] is a simplified form following the assumptions that $\beta$ and $R_{L}/R_{0}$ are small such that $R(T,I)\approx R(T)$, and the device is driven from a near-perfect voltage source [@irwin1998thermal; @IrwinThesis]. The empirical expression $P_{J0} = K(T_0^n-T_B^n)$ was used here, Eq. (\[eq:ThermalFlux\]), which is standard in the TES community.

The time constant $\tau_{eff}$ governs the rate at which the current stabilises after a voltage step has been applied, through the dominant exponential term in Eq. \[eq:RtMod\]. This response time is significantly shortened from its natural value $\tau = C/G$ due to negative electrothermal feedback when the TES is voltage-biased in its transition, where $\alpha \gg 0$. Since $\tau_{eff}$ is approximately proportional to $C/G$, a TES with reduced $G$ is expected to have a larger $\tau_{eff}$, which can be tested experimentally by fitting Eq. \[eq:RtMod\] to $\delta I(t)$. Measurements of $\tau_{eff}$ therefore provide an independent, 