## 2 TES Physics and Models
### TES Electro-Thermal Equations
In Figure 4a, the measured TES resistance and reactance of a \(\mathrm{Ti}/\mathrm{Au}\)\(50\times 50\,\mathrm{\SIUnitSymbolMicro m}^{2}\) TES bolometer biased at \(2.4\,\mathrm{MHz}\) is show as a function of the TES voltage. The reactance has a periodic dependency on the voltage as predicted by the RSJ model (red line in the graph) [61]. Using a similar approach, but then solving the equations numerically, the peculiar structures observed in the TES reactance of a low resistance (\(R_{n}=8\,\mathrm{m}\Omega\)), Mo/Au TES microcalorimeters with noise mitigation structures developed at NASA-GSFC, have been explained as well [64]. An example of the results presented in [64] is reproduced in figure Figure 4b, where the TES electrical impedance is shown as a function of bias voltage. The sharp jumps in the TES reactance \(X_{\omega}\), are related to the Josephson effects in combination with a large TES bias current (detector with high saturation power) and low \(R_{n}\) as predicted in [65]. Numerical simulations of the resistive transition of a dc biased TES, based on the RSJ model, have been performed by [PERSON] et al. [34]. The work includes an extensive study on the effect of (self-)magnetic field perpendicular to the TES surface. The coupled differential equations in Equation (18) have been fully solved numerically, for an ac biased pixel, in the simplified single-thermal body case [66]. In the numerical simulations, the TES impedance is the results of the solution of the RSJ equation and the power balance, as for the analytical solution described in [61]. Aside from the electric circuit and thermal bath parameters, and the TES normal resistance \(R_{n}\), the most important input for the calculation is the \(I_{c}(T)\) curve, which can be evaluated experimentally. The equilibrium value of the simulated TES current has been shown to be in excellent agreement with the experimental IV curve of the modeled pixel. The strength of this numerical time-domain model is that, beside predicting the steady state of the TES, it can also be used to simulate the TES response to the incoming photons in the non-linear and large signal regime. As a matter of fact, this TES simulator will become part of the end-to-end (e2e) simulator of the X-IFU on Athena [67; 68]. In Figure 5, we give an example of the simulation of high normal resistance TES microcalorimeters developed at NASA-GSFC for the Athena/X-IFU and characterized at SRON under MHz bias. The devices consist of Mo/Au \(100\times 100\upmu\)m\({}^{2}\) bare TESs, with a normal resistance \(R_{n}=41\,\mathrm{m}\Omega\), a \(G_{bath}=83\,\mathrm{p}\mathrm{W}/\mathrm{K}\) and a saturation power \(P_{sat}\sim 2\,\mathrm{p}\mathrm{W}\) at \(T_{bath}=55\,\mathrm{m}\mathrm{K}\). They have been characterized under ac-bias at several bias frequencies (from 1.1 up to 4.7 MHz). The color maps in Figure 5a,b show, respectively, the TES resistance \(R(I,T)\) and reactance \(X(I,T)\) as a function of the current \(I\) and temperature for a TES biased at \(f_{bias}=3.081\,\mathrm{MHz}\). The red curve gives the bias trajectory as the solution of the power balance equation. The TES impedance has been calculated analytically as in [61; 62]. A set of \(R(T)\) curves calculated at a constant TES current are given in Figure 5c. The oscillating structures observed experimentally with this type of TESs are clearly visible on each \(R-T\) line as well as on the resulting bias curve in red. The oscillation are generally much smaller when the TES is biased at lower frequency, following the prediction from the Josephson equations. \[V_{TES}=\frac{\hbar}{2e}\frac{\partial\varphi}{\partial t},\ \ I_{I}(t)=I_{c}(T)\sin \varphi(t), \tag{14}\]3 TES resistance \(R_{TES}\) and reactance \(X_{TES}\) as a function of the TES voltage for (**a**) a \(50\times 50\upmu\)m\({}^{2}\) bare Ti/Au TES bolometer [61] and (**b**) a \(120\times 120\upmu\)m\({}^{2}\) NASA-GSFC TES microcalorimeters. The detectors are biased, respectively, at 2.4 and 1.7 MHz. The red lines show the prediction from the RSJ model.