Electromagnetic stirring, magneto-electrophoretic separation, and induction heating are based on the application of alternating magnetic fields. The frequencies of the electric currents to be supplied to the coil systems may span a range from as low as a few Hz for stirring up to a few kHz for heating. To conceive of using three different coils supplied by three respective generators for the three tasks of stirring, separation, and heating seems impossible because of the following reasons:
Ideally, one coil system will be used, which is fed by a superposition of the three frequencies. This is difficult because of the wide range of frequencies from stirring to heating. Dual-frequency inverters, e.g. known form surface hardening of gears, usually cover one order of magnitude in difference between both frequencies. In the SIKELOR project, the difference will be about two orders of magnitude.
Theoretically, such a superimposed solution can be done using an appropriately dimensioned filter circuit between the power supply and the coil system. The difficulty lies however in the fact that any coil system has a fixed inductance. Given the wide frequency range, the material requirements for capacitors and inductances (the latter include potentially transformers) to accomplish resonance conditions may turn out to be unaffordable. Thus, building a low power model aims at investigating whether a single-coil solution is feasible in terms of effort.
To build such a model, the range of frequencies and the inductance have to be known. The next two Sections 2 and 3 address this problem. Commercial frequency synthesisers can be used to model the final programmable wave form power supply. One of these synthesisers will be operated at the lower (stirring) frequeny, and the other one at the upper (heating) frequency. The major task formulation is to feed these two frequencies, stemming from different sources, into the same coil. How to do this, with the above mentioned filter circuits, is described in Section 4.
The low power model is to be build with a coil having the same inductance as one of the 6 side-coils in Demonstrator II. In the demonstrator, there will be influence among the side-coils due to the proximity effect. Moreover, the other coils will carry current. This sophisticated interaction will be studied in later work packages and is neglected here.
In this stage of the project, the final design of the side-coil system is not yet known, also this will be subject of further work packages. Nonetheless, the fixed geometries of crucible, graphite susceptor, insulation, and vacuum chamber, in conjunction with the demands for achievable field strength to create sufficiently strong Leenov-Kolin force and potential parameters of the power supply, restrict the variety of potential designs. Figure 1 below shows a quite likely variant of how the side-coils in Demonstrator II may look like.
The two spiral windings of the rectangular flat coil will be made from copper pipe 18 mm in diameter. The ratio between the inner and the outer diameter is determined by a spacing of 8 mm between the windings. For this geometry of conductors, the inductance needs to be calculated. In general, inductance calculations are not a trivial task. Nowadays, in a time when electromagnetic finite element numerical simulation has become state-of-the-art, the inductance is readily available from the energy contents of the magnetic field. This will, of course, be done in the course of the Project as soon as the precise geometry will be known. For the time being, an analytical expression is sought which delivers reasonable results and with the help of which the influences of changing this and that parameter can be quickly estimated.
An approximation for the inductance L based on the method of current sheets can be found in :
There, µ is the permeability, n the number of turns, davg the average diameter (dout + din)/2, ρ = (dout - din)/(2 davg) the fill factor, and C1, C2, C3, and C4 are constants depending on the shape of the conductor. Given the conductor's geometry from Fig. 1, davg = 844 mm, ρ = 0.0521, and n = 2. The constants for a rectangular coil are C1 = 1.27, C2 = 2.07, C3 = 0.18, and C4 = 0.13. Insertion in the Equation (1) yields an inductance of L = 9.94 µH.
The authors of  also provide a simple formula derived from the celebrated work of Harold A. Wheeler . This modified Wheeler formula reads:
The two constants for a square coil are K1 = 2.34 and K2 = 2.75. The induction calculated with Equation (2) yields LMW = 8.68 µH, which is about 13 % less than that from the current sheet approximation. Wheeler himself stated that his formula, which was only for circular spiral coils, is correct within 5 % for coils with dout - din > 0.2 davg. The square coil in Figure 1 has a ratio of dout - din ≈ 0.027 davg, which indicates that no precise results should be expected. However, agreement to within 15 % between Equation (1) and Equation (2) justifies using such simple formulas as a rough estimation. In this respect, it is noted that the usual mechanical tolerances in manufacturing such a coil will also result in a uncertainty of some per cent.
To built a coil with such an inductance for the low-power model, Figure 2 from Wheeler's work is considered.
derived by Wheeler gives the inductance of such a coil, in terms of the dimensions in inches and inductance in µH.This author stated that the formula is correct to within 1 % for coils with b > 0.8·davg. For b = davg = 1 inch, a numer of n = 13 turns yields LW = 8.9 µH. As this values is in between those calculated with the current sheet approximation and the modified Wheeler formula, such a simple small coil can be used to test the low-power model.
Initially, 3 different frequencies for stirring, separation, and heating were considered. It turned meanwhile out from analytical estimation of the Leenov-Kolin force (c.f. ), first numerical calculations by the University of Greenwich, and comparison with a successful small-scale model experiment , in which SiC particles were separated in a silicon melt, that the frequency for heating and separation can be likely the same. Presently, heating in the iDSS furnace at the University of Padua is done at 6 kHz, which would be also a good choice for separation. It is to date however unclear whether this frequency will be used in Demonstrator II. The reasons are (i) that the optimum heating frequency is lower. With the existing coil system, e.g., 3 kHz can not be used owing to strong vibration of the copper pipes. Since new coils have to be installed anyway, this problem may be solved. (ii) Shielding by the graphite susceptor becomes the stronger the higher the frequency. It deservers further examination to find the optimum compromise between that shielding, the thickness of the layer within the melt from which particles are separated, and the magnitude of the Leenov-Kolin force that also depends on frequency. (iii) The general structure of the current source is not yet know. Depending on which variant described in Deliverable 2.2.: Feasibility study for side-coil system of Demonstrator II will be implemented, the power supply may significantly influence the maximum possible frequency.
What can be said to date is that the upper frequency will be in the range from 3 to 6 kHz, and that it will be fixed. As there will be no need to vary this frequency once it is determined, the low-power model will simplify in that the upper frequency can be fed to the coil through a high-pass filter adjusted to that frequency. Since (ii) and (iii) strongly favour the lower frequencies and the issue in (i) will most probably be solved, the upper frequency to be tested in the low-power model is chosen as 3 kHz.
Also the low frequency for stirring is not precisely known yet. For stirring in the large crucible in Demonstrator II, it will be certainly less than mains frequency. The optimum, i.e. most efficient frequency for stirring will be determined in model experiments at the Helmholtz-Zentrum Dresden-Rossendorf. It is to be expected that there will be also a fixed frequency, depending on the shielding parameter. This means that the high degree of variability in frequency, up to free programmable wave form, may turn out not to be needed. In principle, it should be sufficient that the magnetic field in Demonstrator II is composed of two fixed transients. The low-power model can hence be designed to supply two frequencies to the coil described above in Section 2. Without loss of generality, the principle can be demonstrated with the upper 3 kHz argued above and mains frequency of 50 Hz for stirring, which will be a wide range of two orders of magnitude.
The simplest variant of supplying two frequencies to the same coil is to generate them by two generators, respectively, and combine them with a network made from capacitors and inductances. Depending on the required quality (distortion of wave forms) and allowed/acceptable cross-talk, more or less effort has to be spend. Again the most simple variant is a network of 4 reactive components, i.e. two capacitors and two inductances for the two frequencies, respectively. A possible realisation is shown in Figure 3 below.
Four solutions/combinations exist in general since either of the high and low frequency branches may be either a series or a parallel resonance circuit. The diagram in Figure 3 above shows the realisation with a series resonator for the high frequency (top), and a parallel resonator for the low frequency (bottom).
In the low-power model, the transformer between the filter and the coil (to the right) can be omitted. This transformer may be present in the final power supply to transform down the voltage and to increase the current. For the demonstration in the low-power model, it is not needed. The same holds for the transformer in the high-frequency branch, the secondary winding of which determines the high-pass filter characteristics. In the low-power model, it could be replaced by an inductance, i.e. by a small coil.
The actual model differs distinctly from any of the four possible variants with four reactive components, one of which is shown in Figure 3. As to be seen in Figure 4 below, it gets by with the three reactive components L1, C1, and C2, with C2 being common to both high and low frequency branches. C2 establishes a parallel resonator together with the load coil L2, which is then serially connected via a capacitor to the high frequency signal V2 and via an inductance to the low frequency V1. The resistors R1 and R2 limit the otherwise too high currents in the resonance case.
In  (c.f. Figure 3), the task formulation was, on the one hand, very similar to the present one: the superposition of two fixed frequencies in a single coil. But Esteve et al. were concerned with the induction hardening of gears, which demands frequencies in the range of, say, tens of kHz and hundreds of kHz. It is for this reason that the system investigated in  is qualitatively different from the present one:
As we shall see below, this has severe consequences. To that, first the resonance case is considered. Resonance was implemented in , the simple equations for the present circuit in Figure 4 are derived as follows.
Given the prescribed values of L2 = 8.6 μH and ω2 = 2π·3 kHz, C1 + C2 = 327 μF. With the values C1 = 10 μF and C2 = 300 μF taken from Figure 4, the resonance condition is fairly matched. This can be seen in Figure 5, which plots the impedance Z simulated with LTSpice.
Resonance for the low frequency depends on C2 and L1. The demand to block the high frequency from generator V2 requires L1 >> L2. With that, L1 drops from the resonance condition leaving C2 the only means for adjusting resonance. As there is the limit on C2 imposed by the resonance condition for the high frequency, resonance cannot be achieved. Even this limit would not exist, unrealistically high values would be required to reach resonance. The question is whether the fact that resonance for the low frequency cannot be reached restricts the applicability of the circuit? The answer is no since the frequency is that low that the impedance is almost solely determined by R and L1. This can be seen in the plot of the impedance for V1 contained in Figure 6.
The photo in Figure 6 below shows the filter circuit and the coil. As described above, the little coil has about the same inductance as the comparably hugh coil in Demonstrator II will have.
Two commercial frequency synthesizers were used to feed the transients of different frequency to the filter circuit. Several measurements with different ratio of the input voltages were done. The upper frequency was always 3 kHz, for the lower one mains frequency and 100 Hz were used. The voltage drop across L2 was monitored on an oscilloscope. Unfortunately, the very limited current of the frequency synthesizers did not allow to measure it with an clamp on ammeter. Since introducing a shunt in series to the coil would have modified the results, the system was additionally simulated with the electronic circuit simulator LTSpice. The figures below show an excellent agreement of the measured vs. the simulated voltages. Thus validated the electric current, which is responsible for the magnetic field, is believed to be correct in the situation. That current nicely shows that two quite different frequencies can be fed to a single coil.
Before showing the results of the superposition of the frequencies, it is instructive having a look at the cross-talk between the transients, i.e. the performance of the filters. In Figure 6 the voltage drop across R2 (c.f. Figure 4) can be seen. All minima and maxima of the 3 kHz sinusoidal signal have the same amplitude, which means that no modulation is visible. The low frequency is nicely blocked by the low capacity of C1.
The analogue in the low frequency branch is depicted in Figure 7. Some spikes in the voltage signal across R1 are visible indicating that the high frequency is not perfectly blocked by L1. It is however not expected that this will cause problems for the current control in the power supply to be developed. In this concern it is reminded that the filters are of second order. The lower the order the broader the resonance. A broad resonance would allow varying the frequency without the need to change filter components.
Figure 8 shows that the superposition of two transients of quite different frequencies to feed one coil is possible. The frequencies were 50 Hz and 3 kHz and the output voltage of the two frequency synthesizers was the same. The next figure shows that the ratio of the amplitudes can be easily changed by adjusting the output voltage of the synthesizers. In Figure 9, the amplitude of the voltage output of the high frequeny synthesizer was three times reduced.
The relatively broad resonance of the second order filter allows a certain variation of a frequency without drastically changing the amplitude. It is to be seen in Figure 10 that the amplitude ratio in the case of a lower frequency of 100 Hz does not differ significantly from that in Figure 8 for 50 Hz.
Figures 11 and 12 finally demonstrate the excellent agreement between the experiments and the electrical circuit simulations done with LTSpice. The parameters, i.e. frequencies and voltages, correspond to those in Figure 10. Figure 11 plots the voltage across L2 and, because the experiments do not allow to measure the quite small currents, the plot in Figure 12 provides also such a current transient.
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