Experimental observation of the vertical displacement between heating and levitation regions in an electromagnetic levitation coil

Scientific progress in the relevant fields of science and technology requires the production of crystals with quality beyond the current state of the art. Electro-magnetic levitation (EML) is a prospective method for the growth of high-purity crystals, allowing for avoidance of any contact between the crystal-melt and the crucible. Contactless crystal growth reduces the number of crystal defects commonly abundant in conventional crystal growth methods. The EML method also allows crystal growth of materials with very high melting points. In this article, we report detailed measurements of the EML method. The induction coil used in this study has three turns and one counterturn. We subject different metal material (Al, Cu, Sn, and Ni) samples to the induction coil’s electromagnetic field. For each sample, we measure the induced lift force, Joule heating, and components of magnetic induction as a function of position inside the coil. The results show that the maximum heating in an EML coil is emitted in the area below the levitation zone, a discrepancy not reported earlier. Our findings suggest that this shift should be considered in coil design to avoid instability of the levitated material. We hope this study will serve as a stepping stone for developing EML techniques. The experimental results we provide will be used to evaluate the accuracy of current and future theoretical models of EML coils. This, in turn, will facilitate progress in the application of EML to the growth of larger crystals of higher quality.

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The methods of Czochralski [13] and Bridgman [14] are the most widely used crystal growth techniques, which use a crucible to contain the melt of the material to be crystallized.These methods allow large monocrystals, however, the use of a crucible introduces several problems, including contamination, unwanted crystallization centers from the walls of the crucible, and the inability to contain molten materials with extremely high melting points.The floating zone [15] method avoids the use of a crucible by melting and recrystallising a rod of material locally by electromagnetic (EM) induction and/or laser irradiation.However, physical strains induced by heat gradients can still introduce defects in the resulting crystal.
Growth under levitation conditions can alleviate physical stresses, reducing the number of defects in the resulting crystals.Electromagnetic levitation (EML) relies on the Lorentz force between the coil and the conductive sample to counterbalance the force of gravity.A high-frequency alternating current passing through the coil generates a magnetic field that induces eddy currents in the conductive sample.The eddy currents generate an opposing magnetic field from the sample.Many EML coils use a 'counter-turn' setup, where the lower part of the coil generates the lift force, and the upper part of the coil is wound in the opposite direction to provide stability for the sample [16].Because eddy currents cause Joule heating in the sample, EML melting can often be performed without additional heat sources.However, the coupling between the lift force and the heating makes it difficult to work with some materials because of their physical properties (conductivity, permittivity, magnetic permeability, density, melting point, etc) because the sample material should be at or below the melting point to crystallize while satisfying the conditions for levitation.It should be noted that materials require low electrical resistivity to levitate in a magnetic field.For semiconductors, this means heating to significant temperatures or even melting.
The magnetic field generated by different coil configurations is commonly estimated using 2D models in combination with empirical data [17].Husley [18] studied the efficiency of heating along with the magnetic field in EML coils.Fromm and Jehn [19] developed a model for calculating lift force and power absorption for EML coils.Adapting and optimising the design of EML coils for specific applications is not a trivial task [20,21], but methods are being developed to help with the process [22].Despite that EML is based on old ideas and classical electrodynamics, it remains an active research field and is still poorly understood.New approaches to EML modeling are continuously being developed [23].Recently, truly 3D models of EML coils have been shown to have quantitative differences compared to simpler 2D models [24,25].EML coils have been used to grow monocrystals by melting and recrystallizing small amounts of various materials in levitation conditions [25][26][27][28][29]. EML coils have also been used to develop non-standard techniques for growing monocrystals, such as the Levitation-Assisted Self-Seeding Crystal Growth Method [30].
To the best of our knowledge, theoretical modeling of EML coils currently does not consider the distribution of the heating of the sample in detail and simply assumes that the coil is the hottest at the point where the intensity of the magnetic field is highest, and experimental measurements have only determined the maximum temperature for levitating samples [18].Since conductive samples interact with the magnetic field from the EML coil, calculating the distribution of the Joule heating becomes a non-trivial task.Information about the geometry of the magnetic field as well as the generated lift force is extremely important for the design of EML coils and for the development of new techniques for growing high-quality crystals.
Despite recent progress in developing levitation coils for crystal growth, a detailed understanding of the physical processes at play is still lacking.This hinders the design of suitable coils for growing crystals with exceptional purity through EML.Therefore, in this work, we aim to provide a better understanding of the interaction between the coil geometry, the generated EM field, the induced heating power, and the lift force on the metal sample.We do this by experiments in which we simultaneously measure the lift force, magnetic field, and Joule heating as a function of position along the vertical direction within an EML coil.By examining the data obtained with higher precision than before, we hope to observe evidence of previously unconsidered or undiscovered effects relating to samples of non-negligible size.We also hope that the measurement data of this study will serve as a reference for new EML coil simulations, models, and studies.
The coil was based on a patent by Priede et al [16], consisting of 2 sets of oppositely directed windings, which were connected to the same power source.An advantage of such a design is the ability to stabilize the levitated material by trapping it in between the 2 sets of windings.A particular design of the coil was chosen by manually adjusting the position and distance for several different sets of windings and selecting the coil which could levitate the largest amount of molten aluminum without it spilling out of the coil, with the power supply we had available.The physical properties (electrical conductivity, density, etc) of the materials change with heating and melting, but a molten sample would damage the measurement apparatus.Therefore, we decided to experiment with a variety of materials in low-power, short-pulse mode to avoid melting and obtain data for a wide range of physical properties of materials.It should be noted that the force acting on a conductive sample within an EML coil was previously measured by a spring weight balance [19].In this study, we attempt to increase the accuracy of force measurement and reduce the effects of friction by placing the measurement probe atop a high-precision digital balance.A conference paper describing the measurement method used in this work has been presented and published in the proceedings of conference 'Measurement 2023' [31].

Experimental setup
The setup of the experiment is shown in figure 1 and its electrical diagram is shown in figure 2. The position of the measurement probe was changed along the vertical direction (z-axis) inside the EML coil.A high-frequency induction furnace (MXBAOHENG LH-25 A) was used to power an EML coil made of copper tubing (8 mm in diameter) with water cooling.The power supply worked under resonance conditions with a frequency around 59.1-59.2kHz without a sample and up to 59.6 kHz with an Al sample inserted into the EML coil.The shape of the EML coil (figure 3) was chosen and optimised to ensure a symmetric shape and stable levitation of a molten Al sample.Said optimization, however, resulted in certain deformations in the EML coil geometry.This makes it difficult to accurately describe the geometry of the EML coil in simple parameters.However, approximate values for the position and the radius of the EML coil windings are given in table 1.The diameter of the windings was measured in the horizontal plane, perpendicular to the central axis of the EML coil.
The 3D model shown in figure 3 was made from reference photos in Blender software.Additional insulation was added to the primary circuit of the induction furnace to prevent electrical discharges during low-load operation.The measurement probe was made of heat-resistant plastic and was located beneath the coil.It was moved into the coil by a mechanical precision lifting platform.A stabilization device (a circular frame holding the measurement probe in a vertical position atop a precision balance by binding threads) was used to ensure frictionless weight measurement.The weight measurement was used to determine the lift force acting on the sample by comparing the weight of the probe before and after powering the EML coil.It was assumed, that the lift force acting on the probe coils (the only other metallic component within the measurement probe) was negligible and constant across all measurements, including measurements without a sample.A measurement scale beneath the lifting platform was used to determine the position of the sample and the turns of the coil.Figure 4 shows a schematic of the measurement probe.
The samples were metallic cylinders (Al, Cu, Sn, Ni) with a height of 6.00 ± 0.02 mm and diameter of 16.00 ± 0.02 mm with a hole 6.05 ± 0.05 mm wide and up to 4.0 ± 0.1 mm deep for the thermocouple.The mass of the samples was m Al = 2.92 g, m Cu = 9.71 g, m Sn = 7.83 g and m Ni = 9.69 g.The body of the measurement probe was made of polytetrafluoroethylene (Teflon).The probe coils were positioned to be at the same height as the sample.The vertical field probe coil was placed 0.5 mm from the surface (diameter of d z = 17 ± 0.25 mm) of the sample, and it had N z = 9 windings.The area of the vertical field probe coil was estimated to be A z = π d 2 z /4 = 2.27 ± 0.07 cm 2 .The radial field probe coil was positioned 1.5 mm from the surface (diameter of d r = 19 ± 0.25 mm) of the sample, the distance between the top and bottom sets of windings was h r = 5 ± 0.25 mm, and each set had N r = 2 windings.The area of the radial field probe coil was estimated to be A r = π d r h r = 2.98 ± 0.15 cm 2 .Both sets were positioned in opposite directions (see figure 4 inclusion).The probe coils were made of copper wire with Teflon insulation.The total diameter of the wire was 0.74 mm.For each measurement, the coil was powered for four seconds, and the weight (measured with a Kern 440-33 precision balance) and the magnetic field measurements (measured with a Siglent SDS1104X-E oscilloscope) were recorded in the third second.The oscilloscope also measured the frequency and RMS voltage in the EML coil, and the phase inversion was determined from the oscillograms.
The increase in temperature was determined as the difference between the starting temperature and the maximum temperature, which the sample reached after turning off the power.The sample was cooled to a temperature below 25 • C after each measurement.When the increase in temperature during the measurement exceeded 25 K, liquid nitrogen was poured into the cooling bath to accelerate the cooling process.Measurements were done with a z-distance step of 2 mm, starting from underneath the EML coil, and lifting the measurement probe through it.Every 5 steps, the measurements were repeated 5 times to estimate the measurement errors, which were calculated with their 95% confidence intervals.The position z = 0 was set to be in the center of the first (bottom) turn of the coil.
During this study, we ended up with the need to visually confirm the positions of the levitation and heating within the EML coil.The position of levitation was determined by recording the levitation of a molten Al sample (weighing approximately 16 g) in an EML coil with the digital camera 'Canon EOS 350D'.The position of the most intense heating was determined by vertically inserting a long Ni rod (3 mm in diameter) in the EML coil and recording its heating process after powering the coil, with the thermal camera 'Optris PI'.Regarding the radial field probe coil, its shape can be thought of as a cylinder (figure 5(a)), where the sum of the magnetic flow through the top (Φ Bz1 ), bottom (Φ Bz2 ) and side (Φ Br ) surfaces must be equal to zero.The radial field probe coil used in this work simultaneously measures two things: the magnetic flow through the side surface at position z Φ Br (z) and the magnetic flow difference between the top and bottom windings ∆Φ , since both quantities are the result of each other.We will divide the signal strength of the radial field probe coil by 2 to compensate for the influence of ∆Φ Bz (z) signal.

Results
The measurement results for an Al sample, as well as a control measurement without a sample, are illustrated in figure 6.It should be noted that the independent variable (z) is plotted on the vertical axis since it represents the vertical direction in where the ∆P Al is the change of weight of the Al sample as logged by the precision balance and g = 9.81 m s −2 is the gravitational acceleration.It should be noted that the minus sign was introduced in formula (1) due to positive lift force resulting in a negative change of weight.The outputs we have from the oscilloscope are the RMS voltage U x y and frequency f x (measured by 'synchronization signal' output in figure 2), where indexes x and y respectively refer to the type of sample material (x = 0, Al, Cu, Ni, Sn; 0 refers to the control measurement without a sample) and the type of the measurement coil (y = z, r; vertical or radial field probe coil).For a tightly wound coil, the law of Faraday states that where Φ B is the magnetic flux [Wb], N is the number of turns in the coil and ε is the electromotive force induced in the coil [V].Using the outputs of the oscilloscope and assuming a sine wave, we can express the voltage induced in the probe coil as where the ( √ 2U x y ) is the amplitude of the wave [V].We can calculate the magnetic flux by combining equations ( 2) and (3): The average magnetic flux density B x y [T] can be calculated by dividing magnetic flux by the area of its corresponding probe coil: It should be noted, that B x y in this case is an averaged value of flux density across the area of the probe coil and that the actual magnetic field flux density is not entirely homogenous, especially in the upper and lower regions of the EML coil [24].In addition, the uncertainty of the measurements for the probe coil area adds systematic error to B x y values.In relative terms, for B x z it will result in an additional 3% error, and for B x r it will add an extra 5% systematic error.Further in the text, we will use B x y to refer to the amplitude of magnetic field Negative values of B x y will represent phase inversion.The horizontal grey lines in figure 6 represent the average positions where the center of the sample and the turns of the EML coil intersect.
While the measurements were done with a variety of materials (Al, Cu, Sn, and Ni), the shapes of the graphs, when scaled to the same height, were very similar.Therefore, we have left the graphs with Cu, Sn, and Ni measurements in the figure 1A of the appendix, while the maximum values of the measured values for all the materials are summarized in table 2. Since the temperature of the sample can affect the physical properties of the sample, Table 2 also includes the minimum and maximum temperatures of the samples during their respective measurements, as well as their estimated resistivity range, which was calculated from engineering tables [32].Cu sample generated the largest amount of lift force and the least amount of Joule heating.This can be attributed to its lower electrical resistance since a larger amount of eddy currents can be sustained with the same applied magnetic field.The amount of eddy currents can be estimated by the reduction of the observed magnetic field around the sample since eddy currents generate a magnetic field opposing the direction of the externally applied magnetic field.Cu sample shows the highest reduction of the magnetic field.Al sample performed quite similarly to the Cu sample-slightly more heating, slightly less lift force and eddy currents.Such differences were expected since Al has slightly higher resistivity.Sn sample had the highest resistivity, leading to lower lift force and eddy currents compared to Cu and Al, as well as the highest generated heat of all the samples.Ni sample generated much lower lift force (and eddy currents) compared to other samples, despite its resistivity being somewhere in between that of the Al and Sn samples.This, most likely, is due to the Ni sample being ferromagnetic.
We can calculate the amount of magnetic field deviation caused by induced eddy currents in the Al sample by subtracting the value of observed B Al y from the control measurement B 0 y .The deviation is The heat generated in the sample comes from Joule heating caused by the induced eddy currents.If we assume that the strength of eddy currents is proportional to their produced deviation of the magnetic field and that the power absorbed (or generated as heat) is proportional to the square of eddy current strength, we could say that the power absorbed is proportional to the square of the magnetic field deviation.If the amount Table 2.The maximum values for the measured lift force, temperature increase, and magnetic field flux density for all the measured samples, as well as the range of temperatures to which the samples were exposed during the measurements and the estimated resistivity range at those temperatures.of absorbed power is equal to the temperature increase in an Al sample ∆T Al and we consider the contribution of both the vertical and radial components of magnetic field, we get In figure 7 we present a comparison between the curves of the temperature increase and the squared values of the magnetic field deviation.It appears that the contribution of the radial component of the magnetic field to the heating of the sample is negligible, but ∆B Al z 2 in an Al sample matches quite well to the ∆T Al graph.Additionally, it seems that the resonance frequency f Al of the EML coil with an Al sample in it is linked to the power converted into heat by the sample.The resonance frequency increases from 59.2 to 59.6 kHz, as the heating of the sample reaches its peak.Since the heating of the sample depends on the position of the sample within the EML coil, it could also be said that the resonance frequency of the system depends on the position of the sample within the EML coil.This dependence has been previously reported for a system of conical EML coil and spherical Al sample [23].
Here, we will attempt to compare the methods found in the literature for calculating the lift force generated by the EML coil to our experimental results.To achieve that, we will compare the curve of the directly measured lift force F Al to curves calculated from the measured magnetic fields.The comparison will be made graphically and numerically (through the L 2 distance between curves).Since the measured curves in figure 6 have the same discrete z values, we will base our comparison on the sum of squared differences between the curves at each z position (both curves are scaled so that their maximum value in the positive direction is equal to 1).To normalize the result, the resulting number will be divided by the sum of squared values of F Al curve at each z position.The difference function L 2 between two curves X 1 (z) and X 2 (z) would look as where The most common method to estimate the lift force acting on a conductive sample within an EML coil is to multiply the vertical component of the magnetic field by its derivative along the z-axis, F ∼ B z × dB z /dz [24,25].It should be noted that the effect of magnetic fields induced in the conductive sample is usually ignored in such calculations.
To compare this approach with the experimentally measured lift force F Al , we will define a curve F 1 through B 0 z and its derivative (B 0 z was chosen instead of B Al z to exclude effects caused by the sample) where k 1 = 458.5 (mN m T −2 ) is a constant set to equalize the max values of this and F Al curve.A graphical comparison between curves F 1 and F Al is presented in figure 8.The difference function for both curves is L 2 F Al , F 1 = 8.061 × 10 −3 .An alternative approach we have encountered for calculating the lift force is to assume that the lift force F ∼ AP s , where P s -'the power absorbed by the metal, which goes to heat it' and A-an experimentally determined coefficient, which 'characterizes the degree of heterogeneity of the EM field (…) It was experimentally established that the greater its heterogeneity, the smaller the coefficient A, which theoretically can tend to zero, but is practically impossible to get it less than 0.2.' [33].We can relate P s to the square of deviation of the magnetic field ∆B Al 2 which has been shown in figure 7 to relate to the heating of the sample quite well.Interpretation of the coefficient A, however, appears to be non-trivial.We will assume that the heterogeneity of the magnetic field is proportional to the derivative of the vertical component of the magnetic field.The coefficient A has been said to be inversely proportional to the heterogeneity of the magnetic field and its range has been said to be dependent on the geometry of the EML coil.We will consider a coefficient A in a range (a, b) for each z value to be Here we will consider two different ranges (a, b) for the coefficient A: (0.4, 0.6) and (0, 1), resulting in two coefficients A 0.4, 0.6 and A 0,1 .A comparison of F Al to the absorbed power (∆B Al ) 2 multiplied by A 0.4, 0.6 and A 0,1 is graphically presented in figure 9.In the case, if the coefficient A was meant to be constant for a particular EML coil, (∆B Al ) 2 curve by itself is also included in the graph.
Figure 9 shows that the approach of relating the lifting force to the absorbed power is not applicable in our case.We considered the reasons why the idea of the lift force being proportional to the absorbed power was proposed.The conclusion we reached was that such an approach could only be valid in a sufficiently narrow region of the EML coil, where the curves of lift force and absorbed power are close to linear and the signs of their derivatives match.Let us say that if someone were only examining equilibrium points for the levitation of a sample, it is possible that they would end up looking at such a region.In figure 10 we show a zoom-in of figure 9 for the region (z-value range of 18-30 mm) of the EML coil, where stable levitation of the sample is possible because the lift force is positive and has a negative derivative (lift force increases if the sample drops down).As it was demonstrated by figure 10, it is possible to match the measured force F and curves based on absorbed power ∆B 2 by scaling in regions, where both curves are at least approximately linear.In this case, it would correspond to regions around z-values of 24 mm and −20 mm.Of course, this approach would not work in regions where one of the curves has a significant curvature (for example, a peak).
From the results shown in figure 8, it can be summarized that most of the bulk of the heating in the EML coil happens below the area, where stable levitation of a sample is possible.The levitation zone for the EML coil used in this study and the zone, where most of the heat is emitted, are illustrated in figure 11(a).The displacement of these two zones was visually confirmed by filming the levitation process of a molten Al sample and thermal imaging of a metallic wire placed inside the EML coil.The molten Al sample levitated approximately between the third turn and the counterturn (see figure 11(b)).The most intensive heating was observed between the first and second turns (see figure 11(c)).

Discussion
The results of our measurements confirm that Joule heating in a conductive sample inside an EML coil is directly proportional to the power absorbed by the sample (figure 7).It is proportional to the square of B, which is absorbed in the sample (or compensated by the magnetic field induced by eddy currents in the sample).It was also shown that the vertical component of B multiplied by its derivative closely matches the force acting on the sample, even if B values were from the measurements in the EML coil without a sample (figure 8).However, we believe that ignoring the effects of the sample on the magnetic field in EML coils should not be done moving forward with EML models.
Since the force acting on a sample within an EML coil fundamentally comes from the Lorentz force interaction between eddy currents and the magnetic field, it would be more logical to use the deviation of the vertical B component, which was used to generate the eddy currents, instead of the full vertical B component generated by an EML coil.Based on that assumption, here we will propose a few alternative ways of calculating the lift force acting on a sample in an EML coil.The most direct way to determine the lift force through Lorentz interactions between the eddy currents and the magnetic field would be to express the strength of eddy currents perpendicular to the z-axis through the deviation of the vertical component of B and multiply it by the radial component of B. We will define a curve F 2,Al as where k 2 = 5.488 × 10 5 (mN T −2 ) is a constant set to match the maximum values of this and F Al curve.
In the absence of magnetic field distortions introduced by a conductive sample, the radial component of B within an EML coil should be equivalent to the derivative of the vertical component of B, since they both are related to magnetic field lines changing direction relative to the z axis.With that in mind, we will define curve F 3,Al by replacing the measured radial component around an Al sample with the derivative of the vertical component of B (measured without a sample) where k 3 = 1.012 × 10 3 (mN m T −2 ) is a constant set to equalize the max values of this and F Al curve.The graphical comparison of curves F 2,Al and F 3,Al to F Al is shown in figure 12.For a numerical comparison, L 2 F Al , F 2,Al = 1.163 × 10 −2 and  The curves calculated from equations ( 10)-( 13) all are similar to the force measured, and we could make the fit in the levitation region even closer by adjusting the scaling of the curves.It should be noted that the calculated curves F 1,Al , F 2,Al and F 3,Al are similar enough to each other to be considered equivalent within the margins of error.However, the difference between the aforementioned curves and the directly measured force curve F 1 appears to be statistically significant, albeit very slight, at our chosen confidence level (95%).The most obvious difference between them is the relative amplitude of the maximum and minimum values of the curves.If we define a function If the minimum values of the curves are scaled to the same value, the directly measured lift force shows a noticeably higher amplitude in the positive region.It should also be noted that the z-distance between maximum and minimum values for F Al curve appears to be slightly larger (by 1.186 ± 0.937 mm), compared to the calculated curves (in this case, F 1 ).We believe that the differences between the calculated and measured force curves are caused by the geometric effects of the sample.One of the possible explanations is the displacement of the density of the electron cloud in a vertical direction due to Lorentz force.The electrons in the sample are forming the eddy currents which interact with the magnetic field in the horizontal direction, so their shift up or down would also displace the position of the 'effective center' of the sample.This could also explain the differences in the extreme height for the measured force, since the top part of the sample has more area to generate eddy currents, compared to the bottom part, where the hole for a thermocouple was located.However, we do not have an accurate estimate of the magnitude of this effect.Another geometric explanation is related to the inhomogeneity of the magnetic field along the vertical direction of the sample.Due to field changes across the size of the sample, the induced eddy currents would exhibit similar inhomogeneity in the sample.Consequently, the interaction with the EML coil would produce spatially varying force density in the sample.In the case of non-linear field inhomogeneities, the integral force on the sample would not be equal to the point value estimated at the center of the sample.This in turn would lead to a shift between predicted force values from field distribution and experimentally measured values.Further investigations are required to determine the exact magnitudes of these effects and their applicability in this scenario.
Based on the measurement results of this study, we can also provide some suggestions for EML coil optimization for different purposes.If we seek to maximize the lift force for a given amount of power supplied to the coil, it would be preferable if the amount of power going towards eddy current generation in the horizontal plane (∼ ∆B z ) and the amount of power going towards magnetic flux generation in the horizontal plane (∼ B r ) would be equal.In the case of the EML coil used in this study, ∆B z value is noticeably higher compared to B r (see figure 6), so increasing the radial component of the magnetic field we could expect to increase the lift force of EML coil.The radial component of magnetic flux could be increased by widening the conical angle of the first 3 windings.Additionally, the 4th winding (opposite direction) being with a smaller radius compared to the 3rd winding serves to 'compress' the radial magnetic field in the smaller area along the vertical direction.This raises the maximum value of lift force produced by the EML coil used in this work in comparison to truly conical coils, as described in [20].If we seek to maximize the heating of the sample placed in the EML coil (as, for example, it was done in papers [20,21,23]), it would be advisable to place the sample as close as possible to the area where most of the heat is emitted (figure 11(a)).Reducing the distance between coil windings would result in reduced distance between the levitation zone and 'heating area', therefore increasing the heating efficiency.This could also be achieved by choosing the amount of sample which levitates as close as possible to the bottom part of the levitation zone.It should be noted that attempting to increase one of the EML parameters, other parameters, including the stability of the sample, may suffer as a result.

Conclusion
EML-based crystal growth methods are a promising avenue for fabricating high-purity crystals for a variety of applications.Until now, known efforts in the application of EML have resulted in the crystallization of only small amounts of material.Better simulation models are necessary to improve EML-based crystal growth methods and optimise EML coils for different tasks and larger amounts of material.Our results provide valuable information, contributing to a more effective interplay between high-precision measurements and theoretical models and simulations of EML coils.In future research, we plan to continue our efforts with different configurations of coils as well as the geometry and materials of samples.By using inputs from carefully performed experimental observations, we show that the maximum heating in an EML coil is emitted in the area below the levitation zone, an effect not reported earlier.

Figure 2 .
Figure 2. Electrical diagram of the experimental setup.

Figure 3 .
Figure 3.An image of the 3D model of the EML coil-top view (top) and front view (bottom).The 3D model file can be found in the supplementary files.

Table 1 .
The approximate positions along the z-axis and coil winding diameter (figure2) of the turns of the EML coil.For more precise information regarding the EML coil geometry, see figure3.Winding No.Position along the z-axis (mm)

Figure 5 .
Figure 5. (a) Schematic of magnetic flow through the surfaces of the radial field probe coil as a cylindrical shape; (b) schematic of the contribution of magnetic flow to the EMF of the radial field probe coil from the top, bottom, and side surfaces (only one turn per winding displayed).

6 .
The values of the vertical and radial components of the magnetic flux density in an empty EML coil (B 0 z , B 0 r ) and with an Al sample (B Al z , B Al r ), as well as temperature increase ∆T Al and lift force F Al acting on the Al sample.

Figure 7 .
Figure 7.The squared magnetic flux density deviation around the Al sample in comparison to the temperature increase in the Al sample, as well as the resonance frequency at different z positions in the EML coil.

Figure 8 .
Figure 8.A graphical comparison of the measured lift force F Al 0 to an estimated force curve F 1 .

Figure 9 .
Figure 9.A graphical comparison of the measured lift force F Al on Al sample to curves based on the absorbed power (deviation of magnetic field squared) and coefficient A.

Figure 10 .
Figure 10.Zoom in on the levitation region of figure 9 in the levitation region with curves scaled to fit the modulus of F Al .

Figure 11 .
Figure 11.(a) A graph of the lift force and Joule heating for an Al sample with marked zones for stable levitation and highest emitted heat.(b) A levitating Al sample inside the EML coil with 3 turns and 1 counterturn.(c) Thermal image of a metallic wire inside an EML coil when heating first becomes visible to the thermal camera.(Video files for images (b) and (c) can be found in the supplementary materials).

Figure 12 .
Figure 12.Graphical comparison of the measured lift force on an Al sample in an EML coil with some calculated curves.

L 2 F
Al , F 3,Al = 5.894 × 10 −3 .Compared to the curve F 1 , curve F 2,Al is 1.44 times worse fit to F Al , while the curve F 3,Al gives 1.37 times better result.

Figure 1A .
Figure 1A.The graphs of the lift force F, temperature increase ∆T and vertical/radial components of the magnetic flux density Bz/Br for metal (Al, Cu, Sn, Ni) samples in an electromagnetic levitation coil.The values are given in normalized values, the absolute values are given in table 2.