2D and 3D magnetic shielding simulation methods and practical … · 2009-11-10 · 1 2D and 3D...

1

2D and 3D magnetic shielding simulation methods and

practical solutions

Oriano Bottauscio

Istituto Elettrotecnico Nazionale Galileo FerrarisTorino, Italy

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Outline

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

Summary

Part I – General principles of magnetic field mitigationPart II - Mathematical models for shielding problemsPart III – Magnetic material properties and influence of geometrical parametersPart IV – Examples of applications

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Part I: General principlesof magnetic field mitigation

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Concept of passive shieldingOne 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:

shield field Magnetic shield field Magnetics

withofabsence in=

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Two basic physical mechanismsTwo separate physical mechanisms can contribute to materials-based magnetic shielding.

1) Magnetostatic shielding, obtained by shunting themagnetic flux and divertingit away from a shielded region.

2) Eddy current shielding, obtained in presence of time-varying magnetic fieldsby inducing currents to flow whose effect is to "buckout" the main fields.

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Magnetostatic shieldingIt is realized by the introduction of ferromagnetic materials having high magnetic permeability, which create a preferential path for the magnetic fieldlines

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

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Magnetostatic shielding

Without shield

Shield

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Ideal cases: infinite cylinder and spherical shieldsFor cylindrical and spherical shields with relative permeability µr inner radius a and thickness ∆ theshielding factor in presence of a uniform magnetic field is:

( ) ( )

r

2

2r2

r

4144

11

µ++

−µ−+µ= rrs

ar

2∆=

Cylinder

Sphere( )( ) ( )

r

23

2r

rr

916128

12122

µ+++

−µ−+µ+µ= rrrs

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Ideal case: infinite cylinder andspherical shields

1E-4 1E-3 0.01 0.1 1 101

10

100

1000

Cylinder

s

∆/(2a)

µr = 10 µr = 100 µr = 1000 µr = 10000

Sphere

Large relative permeability and large ratio of thickness to diameter produce good shielding.

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Eddy current shieldingTime varying magnetic fields induce electromotive forces and, consequently, eddy currents are forced to circulate in theconductive 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 themagnetic flux density beyond the shields.

Obviously, this kind of shield is not effective ford.c. fields and its efficiency increases with thesupply frequency.

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Eddy current shielding

Without shield

Shield

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Ideal case: infinite cylinder shieldFor a long cylindrical shield with permeability µo, conductivity σ, inner radius a, and thickness ∆ in a sinusoidally varying field at angular frequency ω, the shielding factor is given by:

21 o ∆σµ

ω+=a

is

At the increasing of conductivity, radius, andthickness the shielding efficiency increases.

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From ideal to actual shieldsThe 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.

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Part II: Mathematical models for shielding

problems

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Peculiarities of shielding problems

Main peculiarities of the problem:o the shields usually have small thickness

with respect to other dimensions ⇒scale problem

o The field is usually not limited in a defined volume ⇒ open boundary problems

o Presence of significant electromagnetic effects

o Possible complex geometrical situations⇒ Analytical formula are not always available

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Possible approaches to simulation

o The solution of Maxwell equations is needed.o Standard Finite Element codes are usually not

adequate for two main reasons:o Open boundary domainso Scale problem introduced by thin shields

o Possible alternative approaches:o Analytical methods (only for simple geometries)o Hybrid Finite Element – Boundary Element

formulations (mainly for 2D open boundary nonlinear problems)

o Thin shield formulation (2D-3D open boundary linear problems)

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Analytical methodsTwo possible alternative analytical approaches:Separation of variables• Simple geometrical configurations:

• Closed cylindrical or spherical shields• Infinite planar shield

• More complex material properties (linearbehaviour)

Conformal mapping• More complex geometrical configurations• Idealized material properties:

• Ideal pure conductive (PES)• Ideal pure ferromagnetic (PMS)

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Separation of variables: Planar one-layer shield

∫ ∑+∞

=

−−− −⋅⋅π

µ⋅

Φ−=

0 1

)(02 )(cos2

4 M

mm

ytykmx dkxxke

IWB m

∫ ∑+∞

=

−−− −⋅⋅π

µ⋅

Φ=

0 1

)(02 )(sin2

4 M

mm

ytykmy dkxxke

IWB m

k== 20 γγ21

21 jpk +=γ

2

11

1Wk

Wr

=µγ=

12 γ

µ= kW r

tt eWeW 11 22

22 )1()1( γγ −⋅−−⋅+=Φ

Imx

y µr , σ , t

L→∞

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Limit of the assumption of infinite shieldInfinite shield (analytical)

Actual solution

No shield

h x

y

+I-IL

12

=h

L

62

=h

L

Infinite shield (analytical)

Actual solution

No shield

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Transformal mapping: planar shield

PMS (perfect magnetic shield) ⇒ σ → 0, µr → +∞ [ ]))(log())(log(

2*

00 jtjtttttIw +−−−−=π

⎟⎠⎞

⎜⎝⎛−=

dzdwH x Im ⎟

⎠⎞

⎜⎝⎛−=

dzdwH y Re

PES (perfect electric shield) ⇒ σ → +∞, µr = 1 ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−π

=jtjt

tt

ttIw loglog2 *

0

0

2

2

11

ttlz

+−=

x

jy plane z

- l +l

+I

x0

jy0

Negligible tickness

α = 0 β = 2π

jv plane t

-∞z = -l

z=+l

z = -l

+∞u0

jv0 +I

- I (0,j)

w = complex magneticpotential

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Limit of the assumption of PMS

h = 0.6 m

h x

y

+I-IL

PMS

Actual solutionNo shield

PMS

Actual solutionNo shield

h = 0.15 m

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2D Hybrid FEM-BEM formulationTo handle open-boundary fields, the domain is

fictitiously subdivided into:o An “internal” limited region Ωi (including the

shields)o An “external” unlimited region Ωe (including

field sources)Ωi Ωe

ni

ne

J0∂Ω

Magnetic field h is expressed as the sum of two terms:

mhhh += 0

Field of the sources Shield effects∫

Ωψ×=

c

dvgrad00 jh

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“Internal” regionLinearization of B-H curve by Fixed Point technique ( ) rbbh +ν=ζ= FP

ra +ν= curlFP

( ) ∫Ω

= dsS

aa &&1M

( )

( )( ) ∫∫∫

∫∫

ΩΩΩ

Ω∂Ω

⋅−⋅−σ+⋅=

=⋅×+⋅ν

iii

ii

dvcurldvdvcurl

dsdvcurlcurl miFP

wrwaawh

whnwa

&&M0

Introduction of magneticvector potential a

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“External” region

Green formulation applied to am:

abh curl00 ν=ν= 0aaa += m

∫Ω

ψµ=c

dvja 000

( ) ( ) 0=∂

ψ∂−+ψ∂∂−−ζ ∫∫

Ω∂Ω∂ ee

dsn

aadsnaaa

eo

e

mo

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Complete set of equationsBy introducing continuity conditions at FEM-BEM boundaries we obtain:

( )( ) ∫∫∫

∫

ΩΩΩ

Ω

⋅−⋅−σ+⋅=

=∂∂

ν+⋅ν

iii

i

dvcurldvdvcurl

nal

dvcurlcurlie

miFP

wrwaawh

wa

&&M0

0 2

( ) ( ) 021 =

∂ψ∂−+ψ

∂∂−− ∑ ∫∫∑

Ω∂Ω∂ i eo

i ie

mo

i,ei,e

dsn

aadsnaaa

niti

tene p

p+1

FEM

BEM

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Effect of magnetic material nonlinearityTwo busbars leading high current are shielded by a cylindrical Fe-Ni alloy

Shield

Busbars

Measurement point

The material of the shield is modeled assuming:

-linear behaviour (µr = 300000)-First magnetisation B-H curve

0.1 1 10 100 10000.0

0.2

0.4

0.6

0.8

B (T

)H (A/m)

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Effect of magnetic material nonlinearity

No shield L NL No shield L NL0.1

1

10

100

1000

Current = 100 kA

B (µ

T)

Current = 10 kA

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“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 twogeometrical 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 bysuitable interface conditions

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“Thin-shield” formulation: assumptions

The two sides of theshield are indicated with(a) and (b), assuming noriented from (a) to (b) v

w

ud

t1t2

n

(a)

(b)

Shield:Magnetic permeability µElectrical conductivity σ

Working at the “microscopic” scale, the field behaviour inside the shield is assumed to depend only on the w coordinate

µσπ=δ

f1 penetration

depth( ) ( )H C C= + −1 2exp expγ γ w w

C1, C2 = integration constantsγδ

=+1 j 1−=j

An expression of H inside the shield is found:

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“Thin-shield” formulation:I interface equation

Starting from the Maxwell equation:

the I interface condition is obtained:

( ))()()()( at

btS

o

an

bn divjHH HH −

ωµζ=+

∫∫ ⋅ω−=⋅×∇dd

dwjdw00

nBnE

( ))()(1 at

btSdiv HH −

σ= ( ))()( b

na

no HHj +

σζωµ−=

( )2dtgh γσγ=ζ

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“Thin-shield” formulation:II interface equationConsidering an infinitesimalcylinder of volume V:

V

Ω(b)

Ω(a)

n

t

0)()()(

)()()()( =⋅+⋅+⋅−=⋅∇ ∫∫∫∫ΩΩΩ latba

dSdSdSdV bbaa

VtBnBnBB

∫∫ ⋅∇=⋅Ω V

tdVdSlat

BtB)(

)(bnB)(a

nB−

the II interface condition is obtained:

( ))()()()( bt

atS

o

an

bn divjHH HH +

ωµη=−

( ) γγωµ=η 2dtghj

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“Thin-shield” formulation:Resulting interface conditionsThe resulting interface conditions:

link the normal and tangential components of the magnetic field the two sides of the shield

( ))()()()( bt

atS

o

an

bn divjHH HH +

ωµη=−

( ))()()()( at

btS

o

an

bn divjHH HH −

ωµζ=+

v

w

ud

t1

t2

n

(a)

(b)

(I)

(II)

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“Thin-shield” formulation:Field equations on the shield surfaceIn the two homogeneous external regions (a) and (b), where field source are present, the magnetic field H can be written as:

)()( ssm grad HHHH +ϕ−=+=

Curl-freereduced field Source field

Reduced scalar potential

( ) ( ) ( ) ( )( )baSS

o

sn

anm

bnm graddivjHHH ϕ−ϕ

ωµζ=++ 2)(

,,

( ) ( ) ( ) ( ) ( )( )( )baS

sS

o

anm

bnm graddivjHH ϕ+ϕ−

ωµη=− H2,,

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“Thin-shield” formulation:Multilayered screensIn presence of multilayered screens, the field equations obtained by the interface conditions can be generalized.

v

w

u di

t1 t2

n

a

b

For the generic i-th layer, the followinginterface conditions are deduced:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡)a(

i,tS

)a(i,n

i,i,

i,i,)b(

i,tS

)b(i,n

div

BTTTT

div

B

HH 2221

1211

ii

iii, jT

η−ζηζ

ω= 21

12 iii, jT

η−ζω−= 2

21

ii

iii,i, TT

η−ζη+ζ== 2211

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“Thin-shield” formulation:Multilayered screensThe layers are connected in cascade, by multiplying the matrices of each single layer:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⋅⎥⎦

⎤⎢⎣

⎡⋅

⋅⎥⎦

⎤⎢⎣

⎡⋅⋅⎥

⎦

⎤⎢⎣

⎡=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

)(1,

)(1,

1,221,21

1,121,11

,22,21

,12,11

,22,21

,12,11)(

,

)(,

....

..........

atS

an

ii

ii

NN

NNbNtS

bNn

div

BTTTT

TTTT

TTTT

div

B

H

H

⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅⎥

⎦

⎤⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡)a(

tS

)a(n

)b(tS

)b(n

divB

TTTT

divB

HH 2221

1211

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“Thin-shield” formulation:Resulting FEM equationsThe 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:

( ) ( )( )( ) ( )( ) ( ) ∫∫

∫

ΩΩ

Ω

ζωµ−=+

ζωµ

+⋅ϕ−ϕ−

SS

S

wdsHjwdsHHj

dswgradgrad

sn

bnm

anm

Sba

S

2o,,

o

( ) ( )( )( ) ( )( ) ( ) wdsgradwdsHHj

wdsgradgrad

SS

S

Ss

ta

nmb

nm

Sab

S

∫∫

∫

ΩΩ

Ω

⋅=−η

ωµ

+⋅ϕ+ϕ

H2,,o

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“Thin-shield” formulation:Integral equationsIn the two external regions, integral equationscan be written, applying the Green theorem:

( ) ( ) ( )∫∫ΩΩ

⋅Ψ∇ϕ−Ψ⋅ϕ∇=ϕ dsdsP aaa nn )()()(

( ) ( ) ( )∫∫ΩΩ

⋅Ψ∇ϕ+Ψ⋅ϕ∇−=ϕ dsdsP bbb nn )()()(

Side (a)

Side (b)nFEM equations on shield surface

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Codes available at IEN2D Code PowerField(2D thin-shieldformulation)

Sally2D Code(Hybrid nonlinearFEM-BEM formulation)

Sally3D Code(3D thin-shieldformulation)

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Part III: Magneticmaterial properties andinfluence of geometrical

parameters

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Ferromagnetic shieldsInfluence of material properties

The behaviour of ferromagnetic materials is defined by the first magnetisation curve

In principle, all ferromagnetic materials can bein 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 (Rayleighregion)

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

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“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.

0.0 0.1 0.2 0.3 0.4 0.5 0.6100

1000

10000

Oriented silicon-iron alloy(Transversal direction)

Nonorientedsilicon-iron alloy

Oriented silicon-iron alloy(Rolling direction)

low carbon steel

Rel

ativ

e P

erm

eabi

lity

Magnetic flux density [T]

-GO Si-Fe alloys: ∼104

-Iron low carbonsteel alloys, NO Si-Fe alloys: 102÷103

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“Low cost” magnetic materialsRapidly 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

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Ferromagnetic shieldsInfluence of material propertiesThe choice of the material is connected

with the specific application:For small volumes screening (e.g.

shielding of electronic devices forcompatibility 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)

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Ferromagnetic shields:Effect of field nonunformity

1

10

100

1000

∆/(2a)=0.1µr=10000

∆/(2a)=0.01µr=10000

s Case1 Case2 Case3

∆/(2a)=0.01µr=1000

Case1

Case3

Case2

∆=thickness, 2a=diameter

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Closed/U-Shaped shield

1

10

100

1000

∆/L=0.1µr=10000

∆/L=0.01µr=10000

s Case1 Case2

∆/L=0.01µr=1000

Case1

Case2

∆=thickness, L=side

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Plane shield

0 2 4 6 8 101.00

1.25

1.50

1.75

2.00

2.25

2.50

s

L/d

∆/L=10-3, µr=1000

∆/L=10-2, µr=1000

∆/L=10-2, µr=10000

dsource

shield Measurement point


0 2 4 6 8 10

2

4

6

8

10

s

L/d

∆/L=10-3, µr=1000

∆/L=10-2, µr=1000

∆/L=10-2, µr=10000d

source

shield Measurement point

Distance of the measurement point = 0.1 L


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Shielding factor: so BBk =

A B C0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

BC

Sch

ield

ing

fact

or -

k

Test point

low carbon steel NO Fe-Si GO Fe-Si (RD) GO Fe-Si (TD)

A

Plane ferromagnetic shields:Influence of material properties

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Plane ferromagnetic shields:border effects

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.30

2

4

6

Shi

eldi

ng fa

ctor

- k

X Coordinate (m)

s = 1 mm s = 5 mm s = 10 mm

Plane shield (d = 5 cm)

x

y

dsource

shield

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U-shaped shield

0 2 4 6 8 10

2

4

6

8

10

12

14

16

s

L/d

∆/L=10-3, µr=1000

∆/L=10-2, µr=1000

∆/L=10-2, µr=10000d

source

shield

Measurement point

0 2 4 6 8 10

10

20

30

40

50

60

70

80

s

L/d

∆/L=10-3, µr=1000

∆/L=10-2, µr=1000

∆/L=10-2, µr=10000



dsource

shield

Measurement point


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U-Shaped ferromagnetic shields:Influence of material properties

A B C0

2

4

6

8

10

12

Shi

eldi

ng fa

ctor

- k

Test point

low carbon steel NO Fe-Si GO Fe-Si (RD) B

C

A


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U-shaped/Closed ferromagnetic shields:Influence of air-gaps

When the shielding configurations make anangle (U-shaped, closed), the assumption of a perfect material continuity is a condition unattainable in practice.

The angle, realized by approaching two different laminations, introduces unavoidable airgaps in the path of themagnetic flux flowing in the shield.

The lack of continuity gives rise to asignificant reduction of the shielding efficiency, which mainly affects the highpermeability materials.

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A B C0

2

4

6

8

10

12Material: GO FE-Si (RD)

Shi

eldi

ng fa

ctor

- k

Test point

conf. (i) conf. (ii) conf. (iii) conf. (iv)

BC

A

Ferromagnetic shieldsInfluence of air-gaps


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Ferromagnetic shields:“Source-side” shielding

x

y

d

source

shield

The influence of thickness and permeability is evident for “source” side shielding

“source” side

x

y

dsource

shield

-0.4 -0.2 0.0 0.2 0.41

2

4

68

10

20

Plane shield

s

x (m)

L = 0.6 m d = 0.1 m ∆ = 10 mm, µr=1000 ∆ = 1 mm, µr=1000

∆ = 10 mm, µr=100 ∆ = 1 mm, µr=100

Closed shield

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Design of the shielding for a MV/LV substationA 3D computer model is implemented in order to identify the most important field sources

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LV board30µT

[µT/A]

125µT

57.6µT15µT

[µT/A]

MV board

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On the floor over the substation

8.4µT

6.7 µT

5 µT

5.5 µT

[µT/A]

0

2

4

6

8

10

P icco de ll'induzione in

funzione de lla d istanza

dalla parete (va lori in 10 -6T)

B [1

0-6T

]

50 cm25 cm0 cm

3.9

5.4

8.4

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LV board shielding

2.22.01.81.61.41.0 1.2

1.4

1.6

1.8

2.0

1.2

asse x

asse

z

56.94 -- 65.00 48.88 -- 56.94 40.81 -- 48.88 32.75 -- 40.81 24.69 -- 32.75 16.63 -- 24.69 8.563 -- 16.63 0.5000 -- 8.563

1.2

1.4

1.6

1.8

2.0

2.22.01.81.61.41.21.0asse x

asse

z

After shielding

Before shielding

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MV board shielding

0 1 2 3 4 5

1.8

1.5

1.0

0.0

0.5

-0.5

asse x

asse

y

0 1 2 3 4 5

1.8

1.5

0.0

0.5

1.0

-0.5

asse x

asse

y

2.475 -- 2.800 2.150 -- 2.475 1.825 -- 2.150 1.500 -- 1.825 1.175 -- 1.500 0.8500 -- 1.175 0.5250 -- 0.8500 0.2000 -- 0.5250

After shieldingBefore shielding

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Outline





Double trefoil HV underground power line

1200

750

100

1500

3x1600 mmq cables

Dig boundary

Junction area

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Possible shielding configurations in proximity of the junction area

a) b)

c)

e)

d)

f ~ h)

g)

2 mm Al shields

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Three dimensional plot around the junction area for shield configuration c) c)

2D and 3D magnetic shielding simulation methods and practical … · 2009-11-10 · 1 2D and 3D...

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Transcript of 2D and 3D magnetic shielding simulation methods and practical … · 2009-11-10 · 1 2D and 3D...