## 2 TES Physics and Models
### TES Electro-Thermal Equations
For the sake of simplicity, in the electrical circuit equations Equation (18b), we have omitted the terms related to the superconducting transformer drawn in \[V_{TES}=\frac{\hbar}{2e}\frac{\partial\varphi}{\partial t},\ \ I_{I}(t)=I_{c}(T)\sin \varphi(t), \tag{14}\]1 These can be easily derived by a standard electrical circuit analysis. In the last two thermal equations, \(P_{X}\) is the input power generated by an absorbed photon, while \(P_{bath}\) refers to the power flowing to the thermal bath, given by  
\[V_{TES}=\frac{\hbar}{2e}\frac{\partial\varphi}{\partial t},\ \ I_{I}(t)=I_{c}(T)\sin \varphi(t), \tag{14}\]2  
where \(k=G_{bath}/n(T^{u-1})\), with \(G_{bath}\) the differential thermal conductance to the thermal bath, \(n\) the thermal conductance exponent, and \(T_{bath}\) the bath temperature [14]. The heat capacity \(C\) in Equation (18) is typically the sum of the TES bilayer and the absorber heat capacity, since the two structures are generally thermally well coupled, while \(C_{2}\) is a potential decoupled heat capacitance which could have different physical sources, like a fraction of the TES bilayer, the supporting \(Si_{x}N_{y}\) membrane, or the leads [50; 51; 52; 53; 54]. The terms \(e_{int}\) and \(e_{ext}\) indicate the internal and external voltage noise sources, respectively, while \(p_{bath}\) and \(p_{12}\), are the power noise sources generated by the finite thermal conductance to the bath (\(G_{bath}\) and \(G_{2}\)) and between the different internal thermal bodies (\(G_{12}\)). The term \(p_{ext}\) accounts for potential parasitic external power sources like, for example, stray light. All these noise sources will be extensively discussed in Section 2.3. The full equations for a dc-biased TES are retrieved by setting \(Q(t)/C_{F}=0\) in Equation (18b). In this case, \(L_{f}\) and \(r_{sl}\) are, respectively, the Nyquist inductor and the shunt resistor typically used in a TDM read-out. The ac voltage term \(V_{ac}(t)\) is absent in normal bias conditions. However, it can be used to calculate the TES response to a small, \(|V_{ac}|<<V_{dc}\), ac excitation, or to evaluate the impact on the weak-link TES of electro-magnetic interferences (EMI) coupled to the bias line, by assuming, for example, \(V_{ac}(t)=V_{pk}\cos(\omega_{EMI}t)\). Interesting enough, by adding to the bias line a controlled small signal at a fixed MHz frequency, Shapiro steps can be observed in the TES IV characteristics, according to Equation (16), and the position of the steps can be used to accurately calibrate the TES voltage [44; 45; 46]. The system of coupled equations given in Equation (18) is typically solved in the small signal regime and in the linear approximation using standard linear matrix algebra. In this case, the RSJ equation Equation (18a) is generally ignored and all the terms including \(\partial\varphi/\partial t\) are replaced by \(I(t)R\)[14; 17; 59; 60]. The Equations (18a) and (18b) can be solved analytically, for all the equilibrium values \(I_{0}\), \(R_{0}\), and \(T_{0}\) along the TES transition, by simultaneously impose that, at each value of \(I(t)\), Equation (19) is satisfied. This has been shown in [61] for a TES bolometer ac voltage biased, at resonance, at \(f_{bias}=1.4\) and \(2.4\,\mathrm{MHz}\), with \(V_{ac}(t)=V_{pk}\cos(2\pi f_{bias})\). The RSJ model was extended to calculate the stationary non-linear response of a TES to a large ac bias current, following the approached described by [PERSON] et al. in [62; 63]. A clear signature of the ac Josephson effect in a TES was observed in the quadrature component of the current. Using the analytic expression for the non-linear admittance of a weak-link, changing in accordance with the power balance variation through the resistive transition, they could well reproduce the measured TES impedance as a function of bias voltage and frequency, at the operating temperature.