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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
!4
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
!5
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) |
!6
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.
!7
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:
!8
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
!9
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
!10
• 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
!11
Analytical Calculations of Shielding Effectiveness
!12
• 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
!13
• 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
!14
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:
!15
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 |
!17
Comparison of Analytical and Opera2D Calculations
• Copper: S/m, .• Inner radius = 1 cm, thickness = 1 mm.
σ = 5.9 × 107 μr = 1
!18
Analytical Calculations• Copper: S/m, .• Varying inner radius:
σ = 5.9 × 107 μr = 1
Thickness = 1 mm
!19
Frequency = 50 Hz
• Copper: S/m, .• Varying thickness:
Analytical Calculations
Inner radius = 1 cm
!20
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
!21
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
!23
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.
!24
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.
!25
• Iron: S/m, .• Inner radius = 1 cm, thickness = 1 mm.
σ = 1 × 107 μr = 200
Comparison of Analytical and Opera2D Calculations
!26
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
!28
• 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
!29
• 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:
!30
LHC Tunnel Measurement
• With a recursive feedback (from ground motion simulations):
!31
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
!33
Homogeneity Inside the Shield
• Standard copper with inner radius = 5 cm and thickness 1mm.• Perfect shield:
!34
Homogeneity Inside the Shield
• Standard copper with inner radius = 5 cm and thickness 1mm.• 1 mm deformation:
!35
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.
!36
B0 = µiH0 + ⌫H20
B0 ' µiH0
Low Intensity Magnetic Fields
• Shape of a hysteresis curve in the Rayleigh region:
!37
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.
!38
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.
!40