COMPEL 27,1. The current issue and full text archive of this journal is available at

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1 The current issue and full text archive of this journal is available at COMPEL 7, 7 Magnetic shielding of buried high-voltage (HV) cables by conductive metal plates Peter Sergeant, Luc Dupré and Jan Melkebeek Department of Electrical Energy, Systems and Automation, Ghent University, Gent, Belgium Abstract Purpose To study the magnetic shielding of buried high-voltage (HV) cables by adding conductive metal plates on the ground surface above the cables. Design/methodology/approach The field is calculated with eight rectangular conductive plates above the cables, positioned with their long edge either parallel to the cables or transversal to the cables. Here, the circuit method is used. In this method, the shield is replaced by a grid of straight filaments in which the unknown currents are searched by solving an electrical circuit. Findings It is observed from the calculation results that it is important to have a perfect electrical connection between adjacent plates. In the area above the shield, an infinite contact resistance between neighbouring plates results roughly in double field amplitude compared to the situation with contact resistance zero. The positioning of the rectangular plates (parallel or transversal to the cables) has not much influence on the shielding. The shielding efficiency as a function of the shield size is studied as well. The circuit method is validated by measurements on an experimental setup at reduced scale. Research limitations/implications The circuit method is applied to conductive objects and not to ferromagnetic objects. Practical implications As the circuit method is rather fast also for 3D geometries with thin plates, the shielding of HV cables can be evaluated in a computationally more efficient way than by using, e.g. finite elements. Originality/value The circuit method is already described in the literature. The originality of this paper is the study by this circuit method of the effect of several parameters (size of the shield, contact resistance, orientation of the plates) on the shielding efficiency. Keywords High voltage, Magnetism, Circuit theory, Magnetic fields Paper type Research paper COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 7 No., 8 pp. 7-8 q Emerald Group Publishing Limited DOI.8/ Introduction High-voltage (HV) power cables generate a relatively high-magnetic field in a small area around the cables. The field should comply with the reference levels of the European Government on the limitation of exposure to electromagnetic fields (4/4/EC (European Parliament, 4), 999/59/EC (European Parliament, 999)). To obtain optimal field reduction with the least possible amount of material, it is necessary to know what the optimal size of the shield is, how the sheets should be positioned and how they should be connected to each other. This work was supported by the Fund of Scientific Research Flanders (FWO) projects G.3.4 and G.8.6, by the GOA project BOF 7/GOA/6 and by the IAP project P6/. The first author is a postdoctoral researcher with the FWO.

2 The simulations are carried out for the geometry shown in Figures and by using the circuit model explained in Section. The numerical values are given in Table I. The three HV cables are at distance D below the ground surface. The current in each of the three phases has amplitude I. In the simulation, the copper shield consists of eight plates of.5 4 m. The plates are drawn in thick line in Figure while the grid of the circuit method is shown in thin line.. Circuit model The circuit model is able to evaluate the shielding performance of any 3D magnetic field source by thin, conductive 3D shields. Thin means that the thickness of the Magnetic shielding 7 z P Measurement points Conductive sheets Ground d p z m y D b p d d Ds 3 x Figure. Geometry in the xz-plane of the three cables and the shield consisting of eight conductive sheets next to each other on the ground surface, and with longest edge parallel to the cables.5 m z (m).5.5 y (m).5 m Notes: Each plate (thick line) is divided in a 6 3 grid of filaments (thin line). The small crosses at the edges of the plates indicate where contact resistance is modelled and the circles at height z = z m = m are the field evaluation points. The HV cables are much longer than shown 4 Figure. Geometry in 3D of the three buried HV cables and the shield consisting of copper plates in transversal configuration (i.e. longest plate edge is orthogonal to the cables)

3 COMPEL 7, 7 shield is small compared to the penetration depth. The thickness of the copper plates in the simulations is 3 mm. This means that the circuit method is applicable at 5 Hz as for this frequency, the penetration depth in copper is 9.3 mm. An application example of the method is given in Clairmont and Lordan (999). The magnetic shield as well as the source conductors in this case the three power cables are replaced by a grid of wire filaments (the mesh ). Each wire has its own resistance, self-inductance and mutual inductance with all other filaments. In each node, the filaments are electrically connected. This results in an electrical network in which the unknown currents can be found by solving Kirchhoff s laws. Once the currents in all filaments are known, the final step is to calculate in the considered point the magnetic field due to each filament by Biot-Savart s law. The total field is the superposition of all source and shield current contributions. For each filament, the resistance, the self-inductance and the mutual inductances must be calculated. The resistances are those of bars made from the same material as the shield and with a diameter chosen such that the sheet and the equivalent mesh of filaments contain the same amount of material. Think of the solid sheet being remelted into a grid structure. The self induction of the filament with length l and radius r is calculated by the classical formula (Clairmont and Lordan, 999): L ¼ m l 4p l ln r g with r g the geometric mean radius of the filament: r g ¼ re /4. For the mutual induction between two straight filaments in 3D space, an exact formula was published in Campbell (95). The next step in the circuit method is to solve an electrical circuit with n n nodes and n t branches. The calculation of all resistances R and all inductances L results in the matrix [R þ jvl ], which is a n t n t matrix. The equations: ½R þ jvlš_i t þ _V i ¼ _V t ðþ ðþ give the relation between the n t voltages over all branches _V t and the currents _I t in these branches. Underlined symbols represent phasors in the frequency domain. The vector _V i symbolizes the contribution of the source (induced voltages). As no external currents are supplied in the nodes, the sum of all currents in a node is zero: Quantity Simulation Experiment Description Table I. Dimensions and phase current used for the simulations and for the experimental setup at reduced scale D.5 m. m Distance between cables D.5 m. m Depth below surface D s.5 m.4 m Distance origin to cable 3 z m. m.3 m Height of evaluation points I 9 A 5. A Phase current amplitude l p 4. m.6 m Plate length b p.5 m.3 m Plate width d p 3. mm 3. mm Plate thickness N 8 Number of plates

4 ½TŠ_I t ¼ where [T] isa(n n ) n t matrix, with one row for each node except the reference node. The element T ij is if branch j starts in node i, if branch j arrives in node i and in all other cases. If equations () and (3) are combined with _V t ¼½TŠ T _V n, then the following equation is obtained with the branch currents and node voltages as unknowns: " ½R þ jvlš ½T T # 3 " Š _I t 4 5 ¼ _V # i ð4þ ½TŠ ½Š _V n ð3þ Magnetic shielding 73 Finally, the Biot-Savart law: _HðrÞ ¼ Z 4p V _JðsÞ ðr sþ jr sj 3 dv ð5þ yields for every branch the magnetic field caused by the calculated current. The volume integral is for a straight filament reduced to a line integral. The current density _J is replaced by the current _I t ð jþ for filament j. 3. Simulation results 3. Unshielded cables We consider the three cables as shown in Figure and Table I, however without the shield. equation (5) yields the flux density in the point (,, ) in absence of conductive and/or magnetically permeable materials in the neighbourhood: _Bð; ; Þ ¼ _B ð; ; Þþ _B ð; ; Þþ _B ð; ; Þ 3 ¼ m _I D x d z þ m _I pd s D s D s pd x þ m _I 3 D x þ d z ð6þ pd s D s D s The currents with amplitude I in the cables are assumed to be sinusoidal: _I ¼ Ie ðp=3þj, _I ¼ Ie j and _I 3 ¼ Ie ðp=3þj. 3. Refinement of the grid The studied shield consists of eight plates of 4.5 m with a total surface of 4 4m or 6 m. The word shield represents the combination of eight plates. For every plate, the number of filaments along the length is denoted n l and the number of filaments along the plate width is n b. Figure 3 shows the influence of n b and n l on the field amplitude in a point P at one meter height above the shield center. For the parallel configuration, the number of filaments along the width n b is important. The choice n b ¼ is unacceptable: the plate is modelled along its width (x-direction) by only one filament that has the same width as the plate itself. Owing to the coarse discretization in x-direction, the y-axis currents cannot choose the position they have in reality. Consequently, the shielding efficiency is underestimated. The too low n b cannot be compensated by a very high n l, as the calculated field for n b ¼ and increasing n l converges to a wrong value. For n b ¼ 3 or higher, the curve converges to

5 COMPEL 7, 74 Figure 3. Influence of the grid refinement on the calculated magnetic flux density in the point P (,, ) n l Parallel, n b = Parallel, n b =3 Parallel, n b =5 Transversal, n b = Transversal, n b =3 Transversal, n b =5 Notes: Each copper plate is divided along the length in n l filaments and along the width in n b filaments an acceptable value for B. In the parallel configuration, firstly the important parameter n b should be chosen sufficiently high. Secondly, the less important parameter n l tunes the accuracy. For n l ¼ 6, the field amplitude in the considered point is overestimated by about percent compared to corresponding amplitude in case of a very fine grid. For the value n l ¼ 6 chosen in the simulations, the error is smaller than percent. For the eight plates, the solution of the 544 unknown node potentials and the,39 unknown currents in the filaments requires on a GHz PC a computation time of about 4 min per shield and almost 3 s per point where B should be evaluated. In the transversal configuration, the refinement along the length n l has more influence than n b. For the transversal configuration, n b ¼ 3 and n l ¼ 6 is a good choice. The overestimation of the field amplitude in the (worst-case) point P (,,) is also less than percent, which can be considered as sufficiently accurate for shielding applications. The conclusion can be quoted from Clairmont and Lordan (999) in order to determine the maximal allowed cell size: as a general rule, it appears that at distances greater than the largest dimension of the cells, the results match measurements very well. For an evaluation distance z m of m, n l should be at least 4 for a 4 m long copper plate. Is extra refinement possible, then refinement in the direction transversal to the cables is more important than refinement parallel to the cables. 3.3 Contact resistance For the parallel configuration (like in Figure ), the current distribution of Figure 4 is obtained. The line thickness is proportional to the current in the filaments. In the case of a high-contact resistance between the plates (Figure 4(a)), the currents cannot flow from one plate to another. Consequently, the currents flow mainly along the outer edge of every plate because this trajectory encloses the largest surface.

6 Magnetic shielding 75 y (m) (a) y (m) (b) Notes: The line thickness of a filament is proportional to the amplitude of the induced current in the filament. For clarity, the refinement parameter n l was reduced from 6 to 6 while n b = 3 remains unchanged. The contact resistances between adjacent plates are in (a) high i.e.. Ω and in (b) low, i.e. Ω Figure 4. Current distribution in the eight copper plates parallel to the HV cables (the cables and the longest plate edges are vertical in the figure)

7 COMPEL 7, 76 The thick lines in the middle of the shield correspond with high currents in adjacent sheets having the same amplitude but opposite direction. Their amplitude is about 6.4 A in the vertical filaments around the point (,, ) for 9 A phase current in the HV cables. For contact resistance zero, Figure 4(b) shows that these two opposite currents cancel each other, resulting in two thin lines in the middle of the shield. The maximal current is now found along the outer edge of all plates together. The currents choose the outer filaments y ¼ m and y ¼ m along the x-axis. To return along the y-axis, many parallel filaments are chosen, which explains why no thick lines occur in vertical direction. The maximal current of. A is found in the points (,, ) and (,, ). The currents in filaments with the same thickness in Figure 4(a) and (b) have comparable amplitudes. The conclusions are similar when observing the transversal configuration (like in Figure ). Figure 5 shows the magnetic flux density norm in several points at z ¼ m above the shield. If the sheets are perfectly connected (resistance R c ¼ ), the shielding improves especially in the region above the shield. In the region next to the shield however at large distance from the cables the influence of the contact resistance is rather limited. Here, all curves are close to each other except a reference curve obtained by finite elements. The shield modelled by D finite elements is infinitely long, and this has significant effect on the shielding even at large distance. Consequently, it is necessary to use a 3D method as the D finite element model is not suitable for modelling this shielding application. Both for the parallel and the transversal configuration of the plates, an excellent electric contact between adjacent sheets is necessary for good shielding. 3.4 Sheet configuration In the parallel configuration, the rectangular sheets are positioned with their long edge parallel to the cables. In the transversal configuration, the longest edge of the sheets is orthogonal to the cables. For contact resistance R c ¼, Figure 5 shows that the B Parallel, R c = mω Parallel, R c =. mω Parallel, R c = mω Transversal, R c = mω Transversal, R c =. mω Transversal, R c = mω D Finite Element Method Figure 5. Magnetic flux density along the x-axis at m above the eight sheets for parallel and transversal 6 4 configurations and contact.5.5 resistances R c

8 difference is negligible as the two configurations represent the same shield. For R c ¼ mv a high resistance there is no difference either. Only for intermediate R c ¼. mv the parallel configuration is slightly better. We can summarize that the influence of the sheet configuration is negligible. 3.5 Shield dimensions To study the influence of the shield dimensions, the original configuration of eight sheets with a total dimension of 4 4 m is modified without changing the number of sheets. For high-contact resistance, Figure 6 shows that the shielding improves for larger shields. Nevertheless, the field reduction does not exceed a factor two even for the largest shield of 8 8 m. The lack of electrical contact cannot be compensated by increasing the size of the shield. For contact resistance zero, the field is reduced up to six times for the largest shield. The width of the shielded zone is roughly determined by the width of the shield unless the shield is very short (along the y-axis). In the latter case, the length also influences the curve as shown in Figure 7(a), where the shielding is given in the point (,, ) for shields with different lengths as a function of the width. The maximal field amplitude occurs above the edge of the shield and not above the center, although the unshielded field is maximal above the shield center. Magnetic shielding Distance between the cables and the shield If the shield is considered to be fixed on the ground surface (z ¼ ), the distance D shows the depth at which the cables are buried. The quantities that are not mentioned, have the default values of Table I. In Figure 7(b), the influence of this distance is studied. In the curves (), the position of P is changed together with the changing D,in order to obtain a constant distance between the cables and the field evaluation point P. Consequently, the unshielded field B is constant. For increasing D, the shielding seems to improve. For the curves (), the evaluation point P is fixed. The shielded field is mitigated in the most efficient way for small D. It is favourable to put the shield very B, R c =. Ω, R c = Ω 4, R c =. Ω 4, R c = Ω 4 4, R c =. Ω 4 4, R c = Ω 8 4, R c =. Ω 8 4, R c = Ω 8 8, R c =. Ω 8 8, R c = Ω Figure 6. Magnetic flux density along the x-axis at m above the eight sheets in parallel configuration for several dimensions length width of the shield and for several contact resistances R c

9 COMPEL 7, l p = m l p = m l p = 4 m l p = 8 m Shield width (m) (a) B, () B, () B, () B, () Figure 7. Magnetic flux density B with shield or B without shield in (,, ) above a shield as a function of (a) the shield width and length l p ; (b) the vertical distance between the shield and the HV cables; (c) the distance between adjacent cables D (m) 6 4 (b) Distance cables d hs (m) (c) Notes: For the curves (), the distance between the cables and P was kept constant so that the vertical position of P varies; for the curves () the point P is kept fixed so that the distance between the cables and P varies B B close above the cables, because few flux lines tend to go around the shield. The latter flux lines reach the region above the shield by using the space besides the shield. For smaller D-values, the shield behaves approximately like an infinitely large plate. For an increasing distance between adjacent cables, Figure 7(c) shows that both the unshielded field and the shielded field in the point P increase almost linearly. 4. Experimental validation and conclusions The shielding of HV cables by conductive plates has been experimentally verified on a scale model with two plates and dimensions in Table I. Figure 8 shows that the correspondence between measurements and simulations is good.

10 B, Num B, Exp Parallel, Num, R c =Ω Parallel, Exp, R c =Ω Parallel, Num, R c =Ω Parallel, Exp, R c =Ω Transversal, Num, R c =Ω Transversal, Exp, R c =Ω Magnetic shielding Notes: The curves show the numerical results (Num); the markers show the experimental data (Exp) Figure 8. Magnetic flux density along the x-axis at.3 m above the two sheets for several configurations and for several R c. We can conclude that, in order to obtain good shielding with a shield above buried HV cables, the shielding plates should be electrically connected. Moreover, the shield should be sufficiently large and the vertical distance between the cables and the shield should be minimal. References Campbell, G. (95), Mutual inductances of circuits composed of straight wires, Physics Review, Vol. 5 No. 6, pp Clairmont, B.A. and Lordan, R.J. (999), 3-D modeling of thin conductive sheets for magnetic field shielding: calculations and measurements, IEEE Transactions on Power Delivery, Vol. 4 No. 4, pp European Parliament (999), Council recommendation 999/59/EC of July 999 on the limitation of exposure of the general public to electromagnetic fields ( Hz to 3 GHz), Official Journal of the European Union, L 99, pp European Parliament (4), Directive 4/4/EC of the European parliament and of the council of 9 April 4 on the minimum health and safety requirements regarding the exposure of workers to the risks arising from physical agents (electromagnetic fields), Official Journal of the European Union, L 84, pp. -9. About the authors Peter Sergeant was born in 978. In, he graduated in electrical and mechanical engineering at the Ghent University, Belgium. In 6, he received the degree of Doctor in Engineering Sciences from the same university. He joined the Department of Electrical Energy, Systems and Automation, Ghent University in as Research Assistant. From 6, he is postdoctoral researcher for the FWO. His main research interests are numerical methods in combination with optimization techniques to design nonlinear electromagnetic systems, in particular actuators and

11 COMPEL 7, 8 magnetic shields. Peter Sergeant is the corresponding author and can be contacted at: Peter.Sergeant@UGent.be Luc Dupré was born in 966, graduated in electrical and mechanical engineering in 989 and received the degree of Doctor in Applied Sciences in 995, both from the Ghent University, Belgium. He joined the Department of Electrical Energy, Systems and Automation, Ghent University in 989 as a Research Assistant. From 996 until 3, he has been a postdoctoral Researcher for the FWO. Since, 3, he is a Research Professor at the Ghent University. His research interests mainly concern numerical methods for electromagnetics, especially in electrical machines, modeling, and characterisation of magnetic materials. Jan Melkebeek was born in 95 and graduated in electrical and mechanical engineering in 975. He received the Doctor in applied sciences degree in 98 and the Doctor Habilitus in electrical and electronical power technology in 986, all from the Ghent University, Belgium. He was a visiting Professor at the Universite Nazionale de Rwanda in Bultare, Rwanda, Africa in 98 and a visiting Assistant Professor at the University of Wisconsin, Madison in 98. Since, 987, he has been a Professor in electrical engineering at the Engineering Faculty of the Ghent University. Since, 993, he has also been the Head of the Department of Electrical Power Engineering. His teaching activities and research interests include electrical machines, power electronics, variable frequency drives, and control systems theory applied to electrical drives. To purchase reprints of this article please reprints@emeraldinsight.com Or visit our web site for further details: