Passive Shielding of Stray Magnetic Fields

Passive Shielding of Stray Magnetic Fields

C. GohilCERN, Geneva, Switzerland

CLIC Beam Physics Meeting(11/10/18)

• nT tolerances have been simulated to remain within a 2% luminosity loss budget.

• CLIC 380 GeV:

• Will need mitigation → passive shielding.

CLIC Stray Field Tolerances

ML and BDS

101 102 103 104 105

Wavelength [m]

10�1

100

101

102

103

104

105

Tol

eran

ce[n

T]

Sine

Cosine

101 102 103 104 105

Wavelength [m]

100

101

102

103

104

105

Tol

eran

ce[n

T]

With TA Correction

No TA Correction

RTML transfer line

!2

Shielding Theory

Magnetic Shielding Mechanisms

that do not completely separate source and shielded regions. For closed topologies,the only mechanism by which magnetic fields appear in the shielded region ispenetration through the shield, while for open topologies, leakage may also occur.Magnetic fields may leak through seams, holes, or around the edges of the shield aswell as penetrate through it. The extent of the shield is an important factor whenconsidering open shields: the more the shield is extended, the better the shielding.However, if penetration exceeds leakage, an increase in the extent of the shield maybring little improvement in the SE. The extent of the shield plays an important rolealso for closed geometries, as it will be seen later. Besides, the shield thickness isanother key factor; if penetration is the dominant mechanism, a thicker shield resultsin improved shielding.

The material parameters of the shield cause two different physical mechanisms inthe shielding of low-frequency magnetic fields: the flux shunting and the eddy-current cancellation. The flux-shunting mechanism is determined by two conditionsthat govern the behavior of the magnetic field and the magnetic induction at thesurface of the shield: Ampere’s and Gauss’s laws require the tangential componentof the magnetic field and the normal component of the magnetic induction to becontinuous across material discontinuities. Hence, in order to simultaneously satisfyboth conditions, the magnetic field and the magnetic induction can abruptly changedirection when crossing the interface between two different media. At the interfacebetween air and a ferromagnetic shield material having a large relative permeability,the field and the induction on the air side of the interface are pulled toward theferromagnetic material nearly perpendicular to the surface, whereas on theferromagnetic side of the interface, they are led along the shield nearly tangentialto the surface. The resulting overall effect of the shielding structure is that themagnetic induction produced by a source is diverted into the shield, then shuntedwithin the material in a direction nearly parallel to its surface, and finally releasedback into the air. In Figure B.2 a, the typical behavior of a cylindrical shield placed inan external uniform magnetic field is reported.

The field map refers to a structure with internal radius a ¼ 0:1 m, thicknessD ¼ 1:5 cm, and mr ¼ 50 at dc (f ¼ 0 Hz). The SE is determined by the materialpermeability and the geometry of the shield. The shield in fact gathers the flux over a

(a) (b)

FIGURE B.2 Magnetic-field distribution for cylindrical shields subjected to a uniformimpressed field: (a) ferromagnetic shield; (b ) highly conductive shield.

284 MAGNETIC SHIELDING

that do not completely separate source and shielded regions. For closed topologies,the only mechanism by which magnetic fields appear in the shielded region ispenetration through the shield, while for open topologies, leakage may also occur.Magnetic fields may leak through seams, holes, or around the edges of the shield aswell as penetrate through it. The extent of the shield is an important factor whenconsidering open shields: the more the shield is extended, the better the shielding.However, if penetration exceeds leakage, an increase in the extent of the shield maybring little improvement in the SE. The extent of the shield plays an important rolealso for closed geometries, as it will be seen later. Besides, the shield thickness isanother key factor; if penetration is the dominant mechanism, a thicker shield resultsin improved shielding.

The material parameters of the shield cause two different physical mechanisms inthe shielding of low-frequency magnetic fields: the flux shunting and the eddy-current cancellation. The flux-shunting mechanism is determined by two conditionsthat govern the behavior of the magnetic field and the magnetic induction at thesurface of the shield: Ampere’s and Gauss’s laws require the tangential componentof the magnetic field and the normal component of the magnetic induction to becontinuous across material discontinuities. Hence, in order to simultaneously satisfyboth conditions, the magnetic field and the magnetic induction can abruptly changedirection when crossing the interface between two different media. At the interfacebetween air and a ferromagnetic shield material having a large relative permeability,the field and the induction on the air side of the interface are pulled toward theferromagnetic material nearly perpendicular to the surface, whereas on theferromagnetic side of the interface, they are led along the shield nearly tangentialto the surface. The resulting overall effect of the shielding structure is that themagnetic induction produced by a source is diverted into the shield, then shuntedwithin the material in a direction nearly parallel to its surface, and finally releasedback into the air. In Figure B.2 a, the typical behavior of a cylindrical shield placed inan external uniform magnetic field is reported.

The field map refers to a structure with internal radius a ¼ 0:1 m, thicknessD ¼ 1:5 cm, and mr ¼ 50 at dc (f ¼ 0 Hz). The SE is determined by the materialpermeability and the geometry of the shield. The shield in fact gathers the flux over a

(a) (b)

FIGURE B.2 Magnetic-field distribution for cylindrical shields subjected to a uniformimpressed field: (a) ferromagnetic shield; (b ) highly conductive shield.

284 MAGNETIC SHIELDING

Flux-Shunting Eddy-Current Cancellation

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Magnetic Shielding Mechanisms

• Which of the mechanisms is dominant depends on:• Material properties:• Electrical conductivity, .• Magnetic permeability, .

• Properties of the external magnetic field:• Frequency, .• Amplitude, - implicitly through the permeability.

• The reduction in magnetic field also depends on the shield geometry: radius and thickness.

σμ = μrμ0

ω = 2πfHi

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Shielding Effectiveness and Shielding Factor

• There are two common measures:

SE(ω) = 20 log10|Hi(P0, ω) ||H(P0, ω) |

SF(ω) = 20 log10|H(P1, ω) ||H(P2, ω) |

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Calculation of Shielding Effectiveness

• Two methods:• Analytical solutions of Maxwell’s equations.• Only possible for simple geometries - infinite sheets,

cylinders and spheres.

• Electromagnetic FEM codes (E.g. Opera2D, CST Microwave Studios).• Allows calculation for more complicated geometries.

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Analytical Calculations of Shielding Effectiveness

Hinc

HrefHtrans

Hslab

x = 0 x = t x

Z0 ZS Z0Hslab(0) =

2ZS

ZS + Z0Hinc

1) First interface:

2) Slab:

3) Second interface:

Hslab(t) = Hslab(0)e− tδ

Htrans =2Z0

ZS + Z0

2ZS

ZS + Z0e− t

δ Hinc

• Infinite sheet:

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Analytical Calculations of Shielding Effectiveness

Hinc

HrefHtrans

Hslab

x = 0 x = t x

Z0 ZS Z0

• Infinite sheet:

Htrans =pe− t

δ

1 − qe− 2tδ

Hinc

4) Accounting for multiple reflections:

q =ZS − Z0

ZS + Z0

Z0 − ZS

ZS + Z0

Htrans = pe− tδ Hinc

+pe− tδ (1 − qe− 2t

δ )Hinc

+pe− tδ (1 − qe− 2t

δ )2Hinc + . . .

p =2Z0

ZS + Z0

2ZS

ZS + Z0

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Analytical Calculations of Shielding Effectiveness

SE = 20 log10(Z0 + ZS)2

4Z0ZS+ 20 log10 |e

tδ | + 20 log10

(Z0 + ZS)2 − (Z0 − ZS)2e− 2tδ

(Z0 + ZS)2

Reflection-loss term Absorption-loss term

Htrans =pe− t

δ

1 − qe− 2tδ

Hinc

Multiple reflection-loss term

SE = − 20 log10Hinc

Htrans

⟹ SE = − 20 log101 − qe− 2t

δ

pe− tδ

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• Infinite sheet:

Analytical Calculations of Shielding Effectiveness

• Transmission Line approach:• The calculation is analogous to a transmission line.• Shield is equivalently modelled as a lossy segment of a

transmission line.

dVdx

= − ZI ↔dEdx

= − jωμH

dIdx

= − YV ↔dHdx

= − (σ + jωϵ)E

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Analytical Calculations of Shielding Effectiveness

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• Transmission Line approach:• In a medium the electric and magnetic fields are related by

the intrinsic impedance:

• Each layer of a shield has an associated impedance.

Z =EH

Analytical Calculations of Shielding Effectiveness

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• Transmission Line approach:• The impedance is transported through a layer with a

transfer matrix:

• The magnetic field is calculated with:

Hinc

Htrans=

21 + Zβ

T = (T11 T12T21 T22), Zα =

T21 + T22Zβ

T11 + T12Zβ, Zβ =

T11Zα − T21

T22 − T12Zα

Analytical Calculations of Shielding Effectiveness

• Infinite sheet:

• Infinitely long cylinder:T11 = (γb){I′�1(γa)K1(γb) − I1(γb)K′�1(γa)} T12 =

μ0

μ(γb)2{I′�1(γb)K′ �1(γa) − I′�1(γa)K′ �1(γb)}

T21 =μμ0

γbγa

{I1(γa)K1(γb) − I1(γb)K1(γa)} T22 =(γb)2

γa{I′�1(γb)K1(γa) − I1(γa)K′ �1(γb)}

γ =1 + j

δ are 1st order modified Bessel functions of the first and second kind.

I1, K1

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T11 = cosh ((1 − j)tδ ) T12 = − ZS sinh ((1 − j)

tδ )

T21 = −1ZS

sinh ((1 − j)tδ ) T22 = cosh ((1 − j)

tδ )

are inner, outer radii.a, b

Analytical Calculations of Shielding Effectiveness

• For more details:

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J. F. Hoburg, “A Computational Methodolgy and Results for Quasistatic Multilayered Magnetic Shielding”, IEEE TRANSACTIONS

ON ELECTROMAGNETIC COMPATIBILITY, VOL 38, (1996)

Shielding Calculations

Penetration Factor• To describe how good a magnetic shield is I will use a

quantity called the ‘Penetration Factor’.• Penetration Factor = Transfer Function.

PF =|H ||Hi |

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Comparison of Analytical and Opera2D Calculations

• Copper: S/m, .• Inner radius = 1 cm, thickness = 1 mm.

σ = 5.9 × 107 μr = 1

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Analytical Calculations• Copper: S/m, .• Varying inner radius:

σ = 5.9 × 107 μr = 1

Thickness = 1 mm

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Frequency = 50 Hz

• Copper: S/m, .• Varying thickness:

Analytical Calculations

Inner radius = 1 cm

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σ = 5.9 × 107 μr = 1

Frequency = 50 Hz

Analytical Calculations

• Analytical calculations are only valid for non-magnetic or linear materials.

Material Conductivity (S/m)

Relative Permeability

Copper - LHC 3.57E+09 1

Copper - Standard 5.96E+07 1

Aluminium 3.77E+07 1

Steel - LHC 1.67E+07 1

Steel - Standard 1.46E+06 1

Inner radius = 1 cm, Thickness = 1 mm

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Magnetic Materials

Magnetic Materials• Non-magnetic materials

have .

• Linear magnetic materials obey:

• Ferromagnetic materials, e.g. iron, steel, nickel, exhibit hysteresis:

μr = 1

B = μH = μrμ0H

Hysteresis Curve of Iron, Bozorth

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Permeability• Non-magnetic materials:• Permeability is independent of the external field strength.• Shielding effectiveness is independent of external field

strength.

• Ferromagnetic materials:• Permeability depends on the strength of the external

magnetic field.• Shielding effectiveness will depend on external field

strength.• Behaviour should be captured in the hysteresis curve.

• Opera2D can simulate ferromagnetic materials.

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Permeability of Ferromagnetic Materials

• Magnetic permeability measured as a function of the external DC field strength:

K. Tsuchiya et. al., Proc. EPAC’ 2006 (Edinburgh, Scotland, 2006) pp 505–507.

• Permeability of room temperature iron for B=40 uT is ~200.

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• Iron: S/m, .• Inner radius = 1 cm, thickness = 1 mm.

σ = 1 × 107 μr = 200

Comparison of Analytical and Opera2D Calculations

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Stray Field Power Spectrum

LHC Tunnel Measurement

Material Conductivity (S/m)

Relative Permeability

Steel (LHC) 1.67E+06 1

Copper (LHC) 3.57E+09 1

Iron 1E+07 200

Mu-Metal 1.7E+07 10,000

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• Pinside(⍵) = Poutside(⍵)T2(⍵)

• Measurement taken on 31/01/18 at Point 2.• Different materials with inner radius = 1 cm, thickness = 1 mm:

LHC Tunnel Measurement

Material Conductivity (S/m)

Relative Permeability

Steel (LHC) 1.67E+06 1

Copper (LHC) 3.57E+09 1

Iron 1E+07 200

Mu-Metal 1.7E+07 10,000

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• Pinside(⍵) = Poutside(⍵)T2(⍵)

• Measurement taken on 31/01/18 at Point 2.• Different materials with inner radius = 1 cm, thickness = 1 mm:

LHC Tunnel Measurement

• With a dead-beat feedback of unity gain:

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LHC Tunnel Measurement

• With a recursive feedback (from ground motion simulations):

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Additional Comments

Air Gaps• Air alone doesn’t attenuate

magnetic fields very well.

• Air gaps introduce:• Additional interfaces - more

reflections.• Larger inner radii - more

effective shields.

• Improves overall effectiveness, but the effect is small.

Air

Shields

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Homogeneity Inside the Shield

• Standard copper with inner radius = 5 cm and thickness 1mm.• Perfect shield:

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Homogeneity Inside the Shield

• Standard copper with inner radius = 5 cm and thickness 1mm.• 1 mm deformation:

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Low Intensity Magnetic Fields

• The behaviour of a ferromagnetic material in a low strength magnetic fields is govern by Rayleigh’s law:

• Initial permeability is usually extrapolated from measurements.

• Shields will be in the Earth’s DC magnetic field.• Outside of the Rayleigh region.

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B0 = µiH0 + ⌫H20

B0 ' µiH0

Low Intensity Magnetic Fields

• Shape of a hysteresis curve in the Rayleigh region:

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K.H.J. Buschow, Concise Encyclopedia of Magnetic & Superconducting Materials,

2nd Edition

Near and Far-Field

• The previous calculations all assume the magnetic field is ‘far-field’ - plane waves.

• Measurements with near-field sources tend to have lower SE.• To be investigated.

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Summary and Future Work• Models for attenuation of magnetic fields exist.• Valid for ‘linear’ materials.• Needs to be experimental verified.

• Behaviour of ferromagnetic materials must be investigated.• In simulations and experimentally.• Permeability must be understood.

• The power spectrum of stray fields must be measured.• Enables determination of the shielding required.

Aside: LHC Beam Screen• Modelled as an infinitely

long cylinder:• Inner radius of 2.2 cm.• 50 µm copper.• 1 mm steel.

• At different energies the conductivity of the copper changes (due to magnetoresistance).• Leads to different curves.

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