## 2 Materials and Methods
### Equivalent Thermo-Electrical Circuit
Given the heating resistor's resistance of 15,950 \(\Omega\), these voltage settings resulted in the corresponding power values. These were combined with the four different power loss values (0 W, 3.3 W, 22.2 W, 59.4 W) generated by the DC current in the excitation coil circuit. The upper limit of the selected range was chosen to match the maximum current available for the excitation coil load at the highest frequency designed for AC magnetometry [12 mH (25 A) at 350 kHz]. The power losses corresponded to different temperature curves recorded at the coil carcass and were subsequently used as the excitation signal for the voltage source \(u_{m}(t)\). As a result, after discarding the combination that provided no meaningful contribution (0 mW, 0 W), a total of 19 calibration combinations were used. The proposed lumped model circuit (C2R5) and the equivalent lumped model corresponding to the corrected-slope method (C1R1) were created in the Simscape Simulink simulator (Figure 3). The heating resistor is represented by the current source unit pulse function \(i_{m}(t)\) and its gain k, while the temperature of the inner wall of the coil carcass is described by the continuous function \(u_{m}(t)\). The input excitation \(i_{m}(t)\) was a rectangular pulse with a duration of 3600 s, after which it was turned off. Once the pulse ended, the system entered the cooling phase, and the temperature was recorded for an additional 3600 s. Throughout both the heating and cooling phases, \(u_{m}(t)\) reflected the thermal response of the heated carcass. Figure 3: **(a)** Equivalent enhanced thermo-electrical circuit created in created in MATLAB Simscape Toolbox (Version 5.4) (C2R5). **(b)** Equivalent lumped model corresponding to the corrected-slope method (C1R1). The \(R_{1}\), \(C_{1}\), \(R_{2}\), \(R_{3}\), \(C_{3}\), \(R_{4}\), \(R_{5}\) are time-invariant parameters detailed in Section 2.2, while the functions \(u_{m}(t)\), \(i_{m}(t)\), \(v_{1,m}(t)\), \(v_{2,m}(t)\) are time-registered signals, as presented in Section 2.3. Several measurements were performed to identify the system RC parameters using two Osensa LUX 200+ temperature transmitters (OSENSA Innovations, Burnaby, BC, Canada) and three non-magnetic fiber-optic PRB G-40 probes, sampled at a frequency of 11 samples per second. As previously mentioned, the first probe measured the temperature of the 2 mL sample filled with water. The second and third probes recorded the temperature of the calorimeter's airspace and the coil carcass, respectively. All temperatures were measured according to the manufacturer's calibration note, with an accuracy of 0.1 \({}^{\circ}\)C and a precision of 0.02 \({}^{\circ}\)C. The transient temperature curves (Figure 4), measured by probes 1 and 2, correspond to the Simulink model's potentials, measured by the voltmeters at two circuit nodes, namely \(v_{1,m}(t)\) and \(v_{2,m}(t)\) (Figure 3). The transient curves of the recorded temperature in the water sample, the airspace inside the calorimeter, and the excitation coil's carcass, together with the input power pulse of the heating resistor, are presented in Figure 4.