Thermal analysis of magnetic shields for induction heating P. Sergeant 1,2 D. Hectors 3 L. Dupré 1 K. Van Reusel 1,3

Published in IET Electric Power Applications Received on 27th October 2008 Revised on 9th January 2009 ISSN 1751-8660 Thermal analysis of magnetic shields for induction heating P. Sergeant 1,2 D. Hectors 3 L. Dupré 1 K. Van Reusel 1,3 1 Department of Electrical Energy, Systems and Automation, Ghent University, St-Pietersnieuwstraat 41, Ghent 9000, Belgium 2 Department of Electrotechnology, Faculty of Applied Engineering Sciences, University College Ghent, Ghent, Belgium 3 Department of Electrical Engineering, KU Leuven, Kasteelpark Arenberg 10, Heverlee 3001, Belgium E-mail: peter.sergeant@ugent.be Abstract: For the design of magnetic shields for induction heating, it is useful to analyse not only the magnetic field reduction but also the temperature behaviour of the shield. The latter is heated by its electromagnetic losses and by thermal radiation from the workpiece. A coupled thermal electromagnetic axisymmetric finite element model is used to study the temperature of a shield for an axisymmetric induction heater, highlighting the effect of the radius, length, thickness and material of the shield on its temperature and magnetic shielding factor. Also the effect of frequency and workpiece dimensions is investigated. The model is validated by measuring magnetic induction, induced currents in the shield and temperature of the shield on the experimental setup. The temperature is unacceptably high for shields close to the excitation coil, especially if the shield length is lower than the workpiece length. Although the study is carried out for one specific induction heater geometry, the paper indicates the effect of parameters such as geometry, material and frequency on shield temperature so that the results are also useful for other induction heating configurations. 1 Introduction In magnetic shield design problems, optimisation goals are usually minimisation of the field in a predefined area the target region and minimisation of electromagnetic losses in a shield [1, 2]. However, it is also useful to include thermal aspects: a shield may obtain a high temperature because of electromagnetic losses and heat radiation from the workpiece. The temperature should be limited to avoid injuries when touching the shield. In this paper, the limitation of the shield temperature is an additional design goal. Thermal analysis is carried out for a copper passive shield and an induction heater with sinusoidal excitation current, but it is also applicable to pulsed induction heating [3] and to active shields [4]: in the latter case, the coils that compensate the magnetic field are also heated by electromagnetic losses and heat radiation from the workpiece. The structure of the paper is as follows. In Section 2, a coupled thermal electromagnetic finite element model (FEM) is presented to simulate the heating as a function of time and to determine the steady-state temperature of the workpiece. The temperature of the workpiece during the heating is experimentally validated for the unshielded case. In Section 3, a simplified thermal electromagnetic FEM calculates the temperature of the shield as a function of the radial position and length of the shield. The simplification reduces the CPU time so that many shield configurations can be evaluated within a reasonable computation time. In Section 4, the influence of the length and radius of the shield on losses, temperature and shielding efficiency is investigated. Also the radiated and electromagnetic contributions to the total power in the shield are indicated in this section. In Section 5, the temperature and shielding performance are studied for several shield thicknesses, material properties (conductivity, permeability), frequencies and workpiece geometries. In Section 6, the shield is optimised to obtain a given shielding efficiency without exceeding a limit temperature. IET Electr. Power Appl., 2009, Vol. 3, Iss. 6, pp. 543 550 543 & The Institution of Engineering and Technology 2009

2 Coupled thermal electromagnetic FEM to study the workpiece temperature A coupled axisymmetric thermal and electromagnetic FEM finds as a function of time the temperature distribution in the workpiece. Fig. 1 shows the axisymmetric geometry of the induction heater with properties in Table 1. The induction heating device is considered to be unshielded : the copper shield shown in Figs. 1a and 1c is not present for the determination of workpiece temperature. The thermal time-stepping model solves the equation for temperature T @T rc p @t r(lrt ) ¼ Q em (1) n (l 1 rt 1 l 2 rt 2 ) ¼ Q conv Q rad (2) where r is the mass density, c p the heat capacity and l the thermal conductivity. Equation (2) is the boundary condition on a boundary between media 1 and 2 with n being the normal vector. The workpiece is heated by resistive electromagnetic heating Q em, which is a volume power density (in W/m 3 ). It is cooled by convection Q conv and radiation Q rad, which are heat flux densities (in W/m 2 ) imposed on the workpiece surface. The convection is Q conv ¼ h(t T 0 ) with h ¼ 10 W/ (m 2 K) calculated from [5] and T 0 ¼ 208C the ambient temperature. Table 1 Geometrical and electromagnetic properties excitation radius r e 47.4 mm total length h e 104.8 mm number of turns t e 7 conductor radius w e 4.8 mm current I e 532 A rms frequency f 26.8 khz workpiece outer radius r w 25.0 mm length h w 90.0 mm magnetic permeability m w m 0 mass density a r w 7350 kg/m 3 electrical s w 0.799 MS/m conductivity a thermal conductivity a l w 30.3 W/(m. K) heat capacity a c p,w 667 J/(kg. K) emissivity a e w 0.5 0.97 shield inner radius r p (60 250 mm) length h p (0 400 mm) sheet thickness t p 1.00 mm electrical conductivity s p 50 MS/m target area magnetic permeability m p m 0 thermal l p 400 W/(m. K) conductivity b mass density b r p 8900 kg/m 3 heat capacity b c p,p 385 J/(kg. K) emissivity b e p 0.15 inner radius r ta 0.30 m width w ta 1.0 m cross section length h ta 1.0 m a At steady-state temperature of 11708C b At 808C Figure 1 Geometry of the shielded axisymmetric induction heating device with dimensions in Table 1 a Cross section of the geometry in the rz-plane. The target area the region where the field should be reduced is not shown b Unshielded induction heater with workpiece c Induction heater with shield in a wooden frame. The shield is a 1 mm thick copper plate with r p ¼ 0.135 m and h p ¼ 0.15 m The total radiated power P rad of the workpiece in stainless steel [6] is ð ð P rad ¼ Q rad da ¼ e w c s (T 4 T0 4 )da (3) A w A w where c s is the Stefan Boltzmann constant (5.670 10 28 W/ m 2 /K 4 )anda w the workpiece surface. The emissivity e w of the 544 IET Electr. Power Appl., 2009, Vol. 3, Iss. 6, pp. 543 550 & The Institution of Engineering and Technology 2009

workpiece was measured in an oven with a pyrometer and a thermocouple. The emissivity was observed to vary between 0.5 and 0.97 depending on the oxidation state. For the simulation of workpiece temperature, the value 0.97 was chosen because the workpiece was completely oxidised. The thermal conductivity l w, the mass density r w and the heat capacity c p,w of the workpiece are temperature-dependent expressions found from [7], causing the numerical model to be nonlinear. The values of these quantities as a function of temperature are shown in Fig. 2, and their values at the steady-state temperature of the workpiece are given in Table 1. To determine the total electromagnetic power P em, at every time step, a time-harmonic and quasi-static electromagnetic 2D FEM is solved with the vector potential as unknown: (1=m 0 )r 2 A þ jvs w A ¼ J e. Then, the electromagnetic power is ð ð P em ¼ Q em dv ¼ sv 2 jaj 2 dv (4) V w V w where V w is the volume of the workpiece. The electrical conductivity s is temperature dependent: (see Table 1, Fig. 2 or [7]. To validate the model, the time evolution of the temperature is simulated and measured by a sensor Thermovision A40, starting from a cold workpiece in stainless steel. The excitation current in the seven-turn induction heater was set to 532 A rms at 26.8 khz. The steady-state temperature of the workpiece of 50 mm diameter and 90 mm length is 11708C (simulated) and 12128C (measured) with time constants of 168 s (simulated) and 201 s (measured) as shown in Fig. 3. The small difference in steady-state temperature can be explained by tolerances on the measured emissivity and the electrical conductivity values. 3 Simplified thermal electromagnetic FEM to study shield temperature Secondly, the temperature of the 1 mm thick copper shield is studied as a function of its radial position and length. To Figure 2 Heat capacity c w, thermal conductivity l w and electrical conductivity s w of the workpiece in stainless steel, relative to their values at 11708C: c w ¼ 667 J/kg/K, l w ¼ 30.3 W/m/K and s w ¼ 0.799 MS/m Figure 3 Workpiece temperature has steady state of 11708C (simulated) or 12128C (measured); time constant is 168 s (simulated) or 201 s (measured) reduce the CPU time of the simulation, the coupling between the electromagnetic and the temperature model is broken. The solution method is the following. Firstly, an electromagnetic FEM is solved for a given shield geometry. Here, the workpiece is assumed to be at steady-state temperature, which means that the possible small influence of the shield on the workpiece temperature is neglected. It was observed from experiments that this is acceptable. Also the electrical conductivity of the shield is independent of temperature, which means that the temperature of the shield in the electromagnetic FEM is assumed to be constant: as the temperature of the shield is much closer to the ambient temperature than the workpiece temperature, the change of material parameters as a function of temperature is rather small. Secondly, the heat problem (1) and (2) is solved by a transient FEM in the shield only, instead of the whole domain of Fig. 1a. By modelling only the shield, the number of unknowns is very small compared to the number of unknowns in the electromagnetic problem that describes the whole domain, and the evaluation time is short. The transient thermal model of the shield has the following source terms. The negative source terms that cool down the shield are applied at all its boundaries: free natural convection in air Q conv,p ¼ h(t T 0 ) and heat radiation to infinity Q rad,p ¼ e p c s (T 4 T 4 0 ) wherein e p is the emissivity of the shield. The positive source terms that heat the shield are the electromagnetic losses Q em and the heat radiation Q rad from the workpiece to the shield. Q em in the shield is found from a linear time-harmonic electromagnetic model with (full) geometry of Fig. 1a. The broken coupling between the thermal and the electromagnetic FEM means that the latter assumes temperature-independent material properties, chosen based on a workpiece temperature equal to the steady-state value found in Section 2 (11708C). P rad is the total power radiated from the workpiece to the shield through the air gaps between the excitation coil windings. In the thermal FEM of the shield only, P rad is a heat flux boundary condition on the shield edge that is illuminated IET Electr. Power Appl., 2009, Vol. 3, Iss. 6, pp. 543 550 545 & The Institution of Engineering and Technology 2009

Figure 4 Radiation from workpiece section A w,i to shield section A p,j by the workpiece as shown in Fig. 4. P rad ¼ e w e p c s X 6 i¼1 X N j¼1 F wp,ij ¼ cos2 a ij pr 2 ij F wp,ij A w,i (T 4 i T 4 j ) (5) A p,j (6) The surface of the shield is divided into a sufficiently large number N of segments with total surface A p. The surface A p, j is illuminated by surfaces A w,i, i ¼ 1,..., 6 (the number of gaps between the seven excitation coils). The height of the surfaces A w,i is determined by the space between the adjacent excitation coils and the angle a ij, see also Fig. 1c. The experimental validation of this model is carried out in three ways, for the same excitation current as in Section 2 (532 A at 26.8 khz) and for a shield with radial position r p ¼ 0.135 m and length h p ¼ 0.150 m. Firstly, the steady-state temperature of the shield is measured and simulated. Both agreed well at 83.1 and 87.78C, respectively. Regarding the shield, it was observed that the emissivity of the used copper samples strongly depends on the temperature; during the experiment, the emissivity of copper was identified in real time by measuring the temperature both by a thermocouple and by the Thermovision sensor that uses the emissivity to determine the temperature. The emissivity at 83.18C shield temperature turned out to be 0.15. This value was used in the simulations. Secondly, the induced currents in the shield were simulated and measured by a Rogowski current probe, for five values of the excitation current. For all five measurements, the measured induced current was about 11% higher than the simulated one, for example for 219.2 A excitation current, the measured and simulated shield currents were 186.7 and 167.4 A, respectively. The difference may be caused by the fact that the numerical model is axisymmetric, whereas the shield in the experimental setup has soldered zones with different resistivity and holes for the power supply of the inductor (see Fig. 1c). These soldered zones in the axial direction and holes cannot be taken into account into the 2D model. Figure 5 Measured induction (Exp.) and simulated induction (FEM) in the target region for the unshielded and shielded induction heater for 180.7 A excitation current The third method of validation is the comparison of the magnetic flux density that is simulated and measured (by a field meter Maschek ESM-100) in the target region. In Fig. 5, it can be seen that in the absence of shields, the measured and simulated curves are almost coinciding. With the shield present, the deviation increases with increasing radial distance, probably because the lower field levels are influenced by the magnetic field of the generator. Fourier analysis was carried out to avoid disturbing fields that have other frequencies. As a reference, field curves are shown for a small shield at short distance to the excitation coils (r p ¼ 0.10 m, h p ¼ 0.05 m) and for a large shield at large distance (r p ¼ 0.20 and 0.30 m). 4 Influence on shielding performance and shield temperature of shield radius and length The electromagnetic and radiated power P em and P rad in the shield depend on the position r p and length h p of the shield as shown in Figs. 6a and 6b. Also the field reduction caused by the shield and the shield temperature depend on r p and h p (Figs. 6c and 6d). In the following paragraphs, we discuss these four quantities. P em decreases slightly with increasing h p, and decreases strongly with increasing radius r p (Fig. 6a). Concerning the effect of h p, for example in a shield with rather small r p ¼ 0.10 m, the electromagnetic heat P em is 100 W for h p ¼ 0.05 m and 70 W for h p ¼ 0.40 m. This is a change of only 30% between a very low and a very high shield. Concerning the effect of r p, for example in a shield with a medium length of h p ¼ 0.20 m, P em is about 200 W for a shield close to the excitation (r p ¼ 0.08 m) and only 6.5 W for a shield at large distance (r p ¼ 0.20 m). This is 30 times 546 IET Electr. Power Appl., 2009, Vol. 3, Iss. 6, pp. 543 550 & The Institution of Engineering and Technology 2009

Figure 6 As a function of the radial position r p and the length h p of the 1 mm thick passive shield a Electromagnetic power P em b Radiated power P rad c Average magnetic induction in the target region d Average temperature in 8C The excitation current is 532 A. The experimental shield is indicated by a cross less. To have low electromagnetic losses, the shield should be at a large distance from the source. The radiated heat P rad is usually much higher than P em at the considered workpiece temperature (Fig. 6b): it increases more or less linearly with h p if h p is lower than the inductor length (0.1 m), and saturates for higher h p to a constant value of almost 300 W. At r p ¼ 0.10 m, the radiated heat is 205 W for h p ¼ 0.10 m; it is 270 W for h p ¼ 0.20 m, and 285 W for h p ¼ 0.30 m. The radiated heat decreases slightly with increasing radial distance. To have low radiated power to the shield, the shield should have a low length. The average flux density B avg in the target region with dimensions in Table 1 is low if the shield is high and close to the induction heater (Fig. 6c). At r p ¼ 0.10 m, doubling the shield length from h p ¼ 0.10 to 0.20 m reduces the average field in the target area about seven times. Increasing the radius from r p ¼ 0.10 to 0.20 m at length h p ¼ 0.10 m causes the average field to increase by a factor of 1.75. The field reduction of three shield configurations can be seen as a function of radius in Fig. 5. Also in this figure, it can be noticed that the field reduction the main goal of the shield improves considerably if the length h p increases: the small shield at short distance to the excitation coils (r p ¼ 0.10 m, h p ¼ 0.05 m) reduces the field less efficiently than the larger shield at large distance (r p ¼ 0.20 m, h p ¼ 0.30 m): in the region r. 0.4 m, the B-values are at least ten times lower if the large shield at large distance is used. The reason for this is that if the shield is not closed, that is if the shield does not enclose the source completely, the magnetic field can reach the target area without passing through the shield; it can bypass the shield via regions above and below the shield. For good field reduction, the shield should have a small radius and a large length. When increasing r p, the length h p should increase more or less proportionally to achieve the same shielding performance. The temperature distribution in the shield is almost uniform because it is thin and a good thermal conductor (Fig. 6d). Therefore shield temperature can be studied by using the average shield temperature instead of the entire temperature distribution in the shield. Fig. 6d shows the average steady-state temperature in the shield. It can be IET Electr. Power Appl., 2009, Vol. 3, Iss. 6, pp. 543 550 547 & The Institution of Engineering and Technology 2009

observed that the temperature decreases with increasing r p, because both P em and P rad decrease. With increasing h p, the temperature of the shield decreases although P em þ P rad increases. This is because that the larger surface of the shield causes better cooling by convection and radiation. To obtain a shield with low temperature, it should have a high radius and length. 5 Influence on shielding performance and shield temperature of shield material, workpiece geometry, shield thickness and frequency Other material for the shield with different electrical conductivity and magnetic permeability than copper affects the electromagnetic power, but also the radiated power because of a change in emissivity. Only taking into account the change in electromagnetic power, Fig. 7 shows the steady-state temperature of the shield for several conductivities and permeabilities. The shield dimensions are r p ¼ 0.135 m and h p ¼ 0.150 m, which are the dimensions of the experimentally tested shield. Copper with s ¼ 50 MS/m and m r ¼ 1 almost results in the lowest possible temperature, 87.78C. If the copper material is replaced by a ferromagnetic electrical steel with s ¼ 5.9 MS/m and m r ¼ 372, the temperature increases from 87.7 to 238.88C. The maximum in the curve can be explained as follows. Electromagnetic losses are proportional to I 2 /s. For low s, the induced voltage in the shield can be assumed to have a constant amplitude, independent of s. This means that the induced current I is more or less proportional to s. An increase of s causes a more or less linear increase of P em I 2 /s. For high s, the current amplitude in the shield can be assumed to have constant amplitude, independent of s: a further increase of s reduces P em because the induced currents see less resistance. Concerning the workpiece geometry, Fig. 8a shows that the average temperature in the shield increases with increasing outer radius r w and length h w of the workpiece. The reason is that the radiated power shown in Fig. 8c increases because it is proportional to the workpiece surface: P rad A w r w h w. The electromagnetic power however see Fig. 8b decreases as more eddy currents in the workpiece reduce the stray field of the induction heater, that is the field that is not coupled to the workpiece. To calculate P em,we assumed that the shield does not influence excitation current amplitude and workpiece temperature. It is noticed that when assuming constant excitation voltage amplitude instead of constant excitation current amplitude, the excitation current will increase so that the magnetic stray field remains strong even in case of a larger workpiece. To summarise, a larger workpiece will increase shield temperature if the radiated heat is dominant, and will decrease shield temperature if electromagnetic heat is dominant. For the induction heater considered in this paper, radiated heat is dominant. If the shield thickness t p is changed, slight influence is observed on the temperature. P rad is independent of t p.for shields thicker than the penetration depth d (0.44 mm in Figure 7 Average temperature in 8C in the 1 mm thick copper shield (r p ¼ 0.135 m and h p ¼ 0.15 m) as a function of the electrical conductivity s p and the magnetic permeability m p The position of the experimental copper shield is indicated by a cross. The radiation P rad is assumed to be constant (constant workpiece temperature and emissivity of the shield) Figure 8 Influence of the outer radius r w and the length h w of the workpiece on the shield temperature a Average shield temperature b Electromagnetic power P em in the shield c Radiated power P rad from the workpiece to the shield 548 IET Electr. Power Appl., 2009, Vol. 3, Iss. 6, pp. 543 550 & The Institution of Engineering and Technology 2009

copper at 26.8 khz), field reduction and P em are almost independent of thickness. For shields with t p d, field reduction is less efficient. If the frequency is higher than 5 khz the frequency where d equals the shield thickness of 1 mm P em increases with the square root of the frequency: the induced currents flow in a thinner region resulting in an increased AC resistance. For lower frequency, P em is quadratic with the frequency, but in amplitude negligible compared to P rad. P rad shows almost no change in the hypothetic case that the workpiece temperature remains 11708C. However, increasing the frequency also increases the power in the workpiece, its temperature and the power radiated to the shield. The average field in the target area decreases with increasing frequency up to 5 khz; for higher frequency, B avg remains approximately constant. 6 Optimisation of the shield for good shielding efficiency without exceeding the temperature limit An optimisation is carried out to find a shield geometry that reduces the average magnetic field norm in the target region to maximally 1 mt without exceeding a shield temperature of 1008C. The parameters to optimise are the shield position r p (in the range 0.06 0.25 m) and the shield length h p (in the range 0 0.40 m). This is a multi-objective optimisation problem. The first objective aims at achieving sufficient field reduction. This means that the rectangular search domain in Fig. 6c is restricted to the area above the line B ¼ 1 mt. The second goal about the temperature means that in Fig. 6d, the search domain is restricted to the region right of the contour line 1008C. It can be seen that the experimentally built shield (indicated by a cross) is approximately the best solution. Of course, an optimisation routine should find the solution without scanning the whole domain. We used the Matlab routine fgoalattain to solve this problem. The optimisation routine used 32 function evaluations and returned the optimal solution that is close to the experimental shield: r p ¼ 0.1208 m and h p ¼ 0.1578 m, resulting in B avg ¼ 1.0 mt and T avg ¼ 1008C. 7 Conclusion When shielding the magnetic stray field of an induction heating device, the shield temperature increases as a result of electromagnetic resistive heating and radiation of heat from the workpiece. Although a good layout of a shield depends on the geometry, excitation current, frequency and workpiece temperature of the induction heater, some general considerations can be given that are valid for a variety of induction heating devices. The two thermal sources electromagnetic power and radiated power should be studied separately. The electromagnetic power caused by induced currents in the copper shield almost does not depend on workpiece temperature. For electromagnetic power, the shield material is very important; in the typical frequency range of induction heating (from about 1 to 400 khz), the electromagnetic power is rather small in a good conducting, non-ferromagnetic shield such as copper. It is very large in a ferromagnetic steel shield with lower conductivity than copper and a small penetration depth. The electromagnetic power decreases weakly with increasing length of the shield and strongly with increasing radial distance between the shield and the workpiece. The shield thickness should be at least equal to the penetration depth to avoid many field lines passing through the shield. However, making the shield much thicker does not improve the shielding efficiency much if the shield is not completely closed, that is if many field lines can reach the target area via the region above or below the shield. The radiated heat depends strongly on the workpiece temperature, geometry of the workpiece and the excitation coils that partially surround the workpiece. In the considered induction heater with a workpiece temperature of about 12008, the radiated heat is dominant for most shield configurations. If workpiece temperature is quite low or if workpiece radiation cannot reach the shield, radiated heat may also be completely negligible. The radiated heat increases strongly with the length of the shield until the shield length is the same as the length of the workpiece and becomes more or less constant for higher lengths. The radiated power decreases weakly with increasing radius of the shield. The cooling of the shield is most effective if it has a large surface, that is a large radius and length. For the same total heat power, the temperature will be lower for a shield with a large surface than for a shield with a small surface. Taking into account all the above considerations, a shield with good field reduction and rather low shield temperature can be achieved if the shield is rather high but at a large radial distance from the workpiece. 8 Acknowledgments This work was supported by the FWO project G.0082.06, by the GOA project BOF 07/GOA/006 and the IAP project P6/21. The first author is a postdoctoral fellow with the FWO. 9 References [1] ÖKTEM M.H., SAKA B.: Design of multilayered cylindrical shields using a genetic algorithm, IEEE Trans. Electromagn. Compat., 2001, 43, (2), pp. 170 176 [2] SERGEANT P., DUPRÉ L., DE WULF M., MELKEBEEK J.: Optimizing active and passive magnetic shields in induction heating IET Electr. Power Appl., 2009, Vol. 3, Iss. 6, pp. 543 550 549 & The Institution of Engineering and Technology 2009

by a genetic algorithm, IEEE trans. Magn., 2003, 39, (6), pp. 3486 3496 [3] SHENKMAN A., BERKOVICH Y., AXELROD B.: Pulse converter for induction-heating applications, IEE Proc. Electr. Power Appl., 2006, 153, (6), pp. 864 870 [4] SERGEANT P., DUPRÉ L., MELKEBEEK J.: Active and passive magnetic shielding for stray field reduction of an induction heater with axial flux, IEE Proc. Electr. Power Appl., 2005, 152, (5), pp. 1359 1364 [5] KAKAC S., SHAH R.S., AUNG W.: Handbook of single-phase convective heat transfer (Wiley, New York, 1987) [6] SIEGEL R., HOWELL J.R.: Thermal radiation heat transfer (Taylor & Francis Inc., New York, 2002, 4th edn.) [7] MILLS K.C., SU Y., LI Z., BROOKS R.F.: Equations for the calculation of the thermo-physical properties of stainless steel, ISIJ Int., 2004, 44, (10), pp. 1661 1668 550 IET Electr. Power Appl., 2009, Vol. 3, Iss. 6, pp. 543 550 & The Institution of Engineering and Technology 2009