2D and 3D magnetic shielding simulation methods and practical solutions

1. 2D and 3D magnetic shielding simulation methods and practical solutions Oriano Bottauscio Istituto Elettrotecnico Nazionale Galileo Ferraris Torino, Italy

2. Summary • Part I – General principles of magnetic field mitigation • Part II - Mathematical models for shielding problems • Part III – Magnetic material properties and influence of geometrical parameters • Part IV – Examples of applications

3. Part I: General principles of magnetic field mitigation

4. Concept of passive shielding • One strategy for reducing magnetic fields in a specific region is to make use of material properties for altering the spatial distribution of the magnetic field from a given source. • A quantitative measure of the effectiveness of a passive shield in reducing the magnetic field magnitude is the shielding factor, s , defined as:

5. Two basic physical mechanisms • Two separate physical mechanisms can contribute to materials-based magnetic shielding. Magnetostatic shielding, obtained by shunting the magnetic flux and diverting it away from a shielded region. Eddy current shielding, obtained in presence of time-varying magnetic fields by inducing currents to flow whose effect is to "buck out" the main fields.

6. Magnetostatic shielding • It is realized by the introduction of ferromagnetic materials having high magnetic permeability, which create a preferential path for the magnetic field lines • A considerable reduction of the magnetic field is generally reached in the region beyond the shield • This is the only passive shield solution in presence of d.c. magnetic fields

7. Magnetostatic shielding Without shield Shield

8. Ideal cases: infinite cylinder and spherical shields • For cylindrical and spherical shields with relative permeability µr inner radius a and thickness  the shielding factor in presence of a uniform magnetic field is: Cylinder Sphere

9. Ideal case: infinite cylinder and spherical shields Large relative permeability and large ratio of thickness to diameter produce good shielding.

10. Eddy current shielding • Time varying magnetic fields induce electromotive forces and, consequently, eddy currents are forced to circulate in the conductive material. • Induced currents constitute an additional field source, which is superimposed to the main magnetic field. • The global effect is a compression of the flux lines on the source hand and a reduction of the magnetic flux density beyond the shields. • Obviously, this kind of shield is not effective for d.c. fields and its efficiency increases with the supply frequency.

11. Eddy current shielding Without shield Shield

12. Ideal case: infinite cylinder shield • For a long cylindrical shield with permeability μo, conductivity , inner radius a, and thickness ∆ in a sinusoidally varying field at angular frequency , the shielding factor is given by: At the increasing of conductivity, radius, and thickness the shielding efficiency increases.

13. From ideal to actual shields • The analysis of ideal shields having cylindrical or spherical shapes is useful as a first approach to understand the factors affecting the shielding mechanism. • Anyway, in most cases actual shielding configurations are far from these idealized geometries. • In order to reproduce actual conditions, more sophisticated models have to be implemented.

14. Part II: Mathematical models for shielding problems

15. Peculiarities of shielding problems • Main peculiarities of the problem: • the shields usually have small thickness with respect to other dimensions  scale problem • The field is usually not limited in a defined volume open boundary problems • Presence of significant electromagnetic effects • Possible complex geometrical situationsAnalytical formula are not always available

16. Possible approaches to simulation • The solution of Maxwell equations is needed. • Standard Finite Element codes are usually not adequate for two main reasons: • Open boundary domains • Scale problem introduced by thin shields • Possible alternative approaches: • Analytical methods (only for simple geometries) • Hybrid Finite Element – Boundary Element formulations (mainly for 2D open boundary nonlinear problems) • Thin shield formulation (2D-3D open boundary linear problems)

17. Analytical methods • Two possible alternative analytical approaches: • Separation of variables • Simple geometrical configurations: • Closed cylindrical or spherical shields • Infinite planar shield • More complex material properties (linear behaviour) • Conformal mapping • More complex geometrical configurations • Idealized material properties: • Ideal pure conductive (PES) • Ideal pure ferromagnetic (PMS)

18. r , , t y x Im L Separation of variables: Planar one-layer shield

19. y x h -I +I L Limit of the assumption of infinite shield Infinite shield (analytical) No shield Actual solution Infinite shield (analytical) No shield Actual solution

20. jv plane t +I jv0 jy plane z - I (0,j) z = -l z = -l +I jy0 u0 - z=+l  = 0  = 2 + x0 +l x - l Negligible tickness • Transformal mapping: planar shield w = complex magnetic potential PMS (perfect magnetic shield)   0, r  + PES (perfect electric shield)   +, r = 1

21. y x h -I +I L Limit of the assumption of PMS No shield Actual solution PMS h = 0.6 m No shield Actual solution h = 0.15 m PMS

22. 2D Hybrid FEM-BEM formulation • To handle open-boundary fields, the domain is fictitiously subdivided into: • An “internal” limited region i (including the shields) • An “external” unlimited region e (including field sources) ni i J0  e ne Magnetic field h is expressed as the sum of two terms: Shield effects Field of the sources

23. “Internal” region • Linearization of B-H curve by Fixed Point technique Introduction of magnetic vector potential a

24. “External” region • Green formulation applied to am:

25. p+1 ni ti BEM FEM te ne p Complete set of equations • By introducing continuity conditions at FEM-BEM boundaries we obtain:

26. Measurement point Shield Busbars Effect of magnetic material nonlinearity • Two busbars leading high current are shielded by a cylindrical Fe-Ni alloy • The material of the shield is modeled assuming: • linear behaviour (μr = 300000) • First magnetisation B-H curve

27. Effect of magnetic material nonlinearity

28. “Thin-shield” formulation:basic principle • The goal of this approch is to remove the shield thickness, by substituting the 3D shield with an equivalent 2D structure • This results is obtained by acting on two geometrical scales: a “microscopic” scale on the shield thickness and a “macroscopic” scale on the shield surface • Working on the “microscopic” scale the shield thichness is substituted by suitable interface conditions

29. w n (b) t 2 d v t u 1 (a) “Thin-shield” formulation: assumptions Working at the “microscopic” scale, the field behaviour inside the shield is assumed to depend only on the w coordinate • The two sides of the shield are indicated with (a) and (b), assuming n oriented from (a) to (b) Shield: Magnetic permeability  Electrical conductivity  An expression of H inside the shield is found: penetration depth C1, C2 = integration constants

30. “Thin-shield” formulation:I interface equation • Starting from the Maxwell equation: the I interface condition is obtained:

31. “Thin-shield” formulation:II interface equation • Considering an infinitesimalcylinder of volume V: the II interface condition is obtained:

32. w n (b) t 2 d v t u 1 (a) “Thin-shield” formulation:Resulting interface conditions • The resulting interface conditions: (I) (II) link the normal and tangential components of the magnetic field the two sides of the shield

33. “Thin-shield” formulation:Field equations on the shield surface • In the two homogeneous external regions (a) and (b), where field source are present, the magnetic field H can be written as: Curl-free reduced field Reduced scalar potential Source field

34. “Thin-shield” formulation:Multilayered screens • In presence of multilayered screens, the field equations obtained by the interface conditions can be generalized. For the generic i-th layer, the following interface conditions are deduced:

35. “Thin-shield” formulation:Multilayered screens • The layers are connected in cascade, by multiplying the matrices of each single layer:

36. “Thin-shield” formulation:Resulting FEM equations • The field equations on the shield surfaces are solved by FEM, discretizing the screens into 2D elements (for 3D problems) or 1D elements (for 2D problems). • The weak formulation (w=test function) leads to:

37. “Thin-shield” formulation:Integral equations • In the two external regions, integral equations can be written, applying the Green theorem: Side (b) FEM equations on shield surface n Side (a)

38. Codes available at IEN 2D Code PowerField (2D thin-shield formulation) Sally2D Code (Hybrid nonlinear FEM-BEM formulation) Sally3D Code (3D thin-shield formulation)

39. Part III: Magnetic material properties and influence of geometrical parameters

40. Ferromagnetic shieldsInfluence of material properties • The behaviour of ferromagnetic materials is defined by the first magnetisation curve • In principle, all ferromagnetic materials can be in principle used for passive shielding • In many applications (e.g. open shields) shielding devices are characterized by giving rise to low magnetic flux density values inside the materials • The shielding efficiency strongly depends on the value of the initial permeability (Rayleigh region)

41. Nickel-Iron alloys Nickel-Iron alloys (mumetal, permalloy) exhibits very high permeability (r~105). They are available as bulk or thin (up to 10 m) laminations. Their use is justified in the shielding of limited regions and when a high shielding efficiency is needed.

42. “Low cost” magnetic materials “Electrical steels”: iron, low carbon steel alloys, silicon-iron alloys (oriented and non oriented) with a thickness of some hundreds of micrometers. • GO Si-Fe alloys: 104 • Iron low carbon steel alloys, NO Si-Fe alloys: 102103

43. “Low cost” magnetic materials Rapidly solidified alloys: amorphous materials, nanocristalline materials, produced as ribbons with a thickness of some tens of micrometers. The initial relative permeability is: • about 105 for Co-based alloys (comparable to Ni-Fe alloys) • about 104 for Fe-based alloys

44. Ferromagnetic shieldsInfluence of material properties • The choice of the material is connected with the specific application: • For small volumes screening (e.g. shielding of electronic devices for compatibility reasons) high quality and high cost materials can be employed (e.g. Ni-Fe alloys) • For large scale screening other materials are more useful both for economical and technical reasons (e.g. Low carbon steel, Fe-Si alloys)

45. Ferromagnetic shields:Effect of field nonunformity =thickness, 2a=diameter Case1 Case2 Case3

46. Closed/U-Shaped shield =thickness, L=side Case1 Case2

47. Measurement point shield d source Measurement point shield d source Plane shield =thickness, L=side Distance of the measurement point = 0.5 L Distance of the measurement point = 0.1 L

48. Shielding factor: Plane ferromagnetic shields:Influence of material properties

49. shield d source y x Plane ferromagnetic shields:border effects

50. =thickness, L=side Measurement point shield d source Measurement point shield d source U-shaped shield Distance of the measurement point = 0.5 L Distance of the measurement point = 0.1 L