## 2 Materials and Methods
### Methods of Magnetic Measurements and Losses Segregation  
In this study, the ring method was employed for the measurement of printed material magnetic properties. The measurements were conducted at frequencies of 25 mHz (quasi-static), 1, 10 and 50 Hz, in the magnetization range of 0.9-1.6 T. The measurements were completed in accordance with the European standard EN 60404-6 [22]. The quasi-static measurement methods agree with the EN 60404-4 standard [23]: employing continuous recording method with the toroid first demagnetized with gradually reducing the magnetic field and completing the magnetization cycle time in the range of 30-60 s. The drift correction of the measured induced voltage for DC measurements was performed similar to [24]. The schematic of the magnetic measurements setup is shown in Figure 5a. Dewetron DEWE2-M data acquisition system at 10 kHz sampling frequency was used for logging the induced voltage on the sensing coil and the voltage drop over the shunt precision resistor of 75 mV/15A in the primary coil circuit. In all experiments, sinusoidal excitation current was used. Signal Amplifer Omicron CMS 356 was used for amplifying the generated signal from the waveform generator and maintaining its sinusoidal waveform throughout the experiment. The magnetic field strength H was calculated by (1) and the average flux density by (2):  
\[H=\frac{N_{1}i}{l_{t}} \tag{1}\]  
\[B=\frac{1}{N_{2}S}\int\limits_{l}e(t)dt \tag{2}\]  
where \(N_{1}\)--is number of turns of the primary coil, \(N_{2}\)--the number of turns of the secondary coil, \(i\)--the instantaneous current value, \(e\)--the instantaneous induced electromotive force, \(l_{t}\)--the toroid average length and \(S\)--toroid cross-sectional area. After constructing the hysteresis curves from the measured data, the hysteresis curves were halved into two sets of data corresponding to the direction of the magnetizing force. Both halves were then fitted with cubic fitting splines for noise and data points reduction for simplifying the integration of the curves. The specific iron losses (\(P_{s}\)) were then found as a subtraction of the curve areas and accounting for the average density (\(\rho\)) and period (\(T\)) of each hysteresis cycle, as expressed in (3).  
\[P_{s}=\frac{1}{T\rho}\left(\int_{0}^{T}H_{1}dB_{1}-\int_{0}^{T}H_{2}dB_{2}\right) \tag{3}\]  
Calculated total iron losses were then segregated into hysteresis, classical and excess eddy current losses for more accurate description of the printed material behavior. Hysteresis losses were calculated from the quasi-static measurements. For the estimation of eddy current losses in the printed material, FEM (finite element method) analysis was conducted, as the analytical means are insufficient for the calculation of generated eddy currents in the nonlaminated printed toroids. Eddy current losses were simulated in a commercial FE software Infolytica Magnet (Version v2020_1, Montreal, Canada). The model geometry imitated the experimental measurements of the toroid. Due to the physical nature of the eddy currents, three-dimensional transient analysis was conducted. The axial symmetry of the toroid favored the simulation of only a small part of the toroid in 3D (length 0.5 mm), with symmetric boundary conditions segregating the sector. The geometry and mesh of the model are shown in Figure 5b. The setup was modeled with a single coil, since the software enables the calculation of the flux linkage through it and a separate sensing coil is unnecessary. The model was excited with an undistorted AC current with RMS value altered until the desired maximum flux density \(B_{max}\) was achieved. Transient step size of a two hundredths of the excitation current period was used for simulating two current periods of each simulation. The magnetization curves of the FEM models were determined from the centerlines of the experimentally measured quasi-static hysteresis curves and the resistivity from the previously experimentally measured average value of 98 u\(\Omega\)-cm.