1 2D and 3D magnetic shielding simulation methods and practical solutions Oriano Bottauscio Istituto...

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2D and 3D magnetic shielding simulation

methods and practical solutions

Oriano Bottauscio

Istituto Elettrotecnico Nazionale Galileo Ferraris Torino, 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 mitigation

Part II - Mathematical models for shielding problems

Part III – Magnetic material properties and influence of geometrical parameters

Part IV – Examples of applications

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Part I: General principles of magnetic field

mitigation

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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:

shield field Magnetic

shield field Magnetics

with

ofabsence 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 the magnetic flux and diverting it away from a shielded region.

2) 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.

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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

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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 the shielding factor in presence of a uniform magnetic field is:

r

2

2r2

r

4144

11

rrs

ar

2

Cylinder

Sphere

r

23

2r

rr

916128

12122

rrrs

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Ideal case: infinite cylinder and spherical shields

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 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.

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Eddy current shielding

Without shield

Shield

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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:

21 o

a

is

At the increasing of conductivity, radius, and thickness the shielding efficiency increases.

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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.

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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 (linear behaviour)

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

0 1

)(02 )(cos2

4 M

mm

ytykmx dkxxke

IWB m

0 1

)(02 )(sin2

4 M

mm

ytykmy dkxxke

IWB m

k 20 21

21 jpk

2

11

1

WkW

r

12

kW r

tt eWeW 11 22

22 )1()1(

Imx

y r , , t

L

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Limit of the assumption of infinite shield

Infinite 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) 0, r + ))(log())(log(

2*

00 jtjtttttI

w

dz

dwH x Im

dz

dwH y Re

PES (perfect electric shield) +, r = 1

jt

jt

tt

ttIw loglog

2 *0

0

2

2

1

1

t

tlz

x

jy plane z

- l +l

+I

x0

jy0

Negligible tickness

= 0 = 2

jv plane t

-z = -l

z=+l

z = -l

+u0

jv0+I

- I (0,j)

w = complex magnetic

potential

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Limit of the assumption of PMS

h = 0.6 m

h x

y

+I-IL

PMS

Actual solution No shield

PMS

Actual solution No 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 i (including the

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

field sources)

i e

ni

ne

J0

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

mhhh 0

Field of the sources Shield effects

c

dvgrad00 jh

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“Internal” region

Linearization of B-H curve by Fixed Point technique

dsS

aa 1M

Introduction of magneticvector potential a

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“External” region

Green formulation applied to am:

abh curl00 0aaa m

c

dvja 000

0

ee

dsn

aadsn

aaa

eo

e

mo

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Complete set of equations

By introducing continuity conditions at FEM-BEM boundaries we obtain:

02

1

i e

oi ie

mo

i,ei,e

dsn

aadsn

aaa

niti

tene p

p+1

FEM

BEM

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Effect of magnetic material nonlinearity

Two 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 (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 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

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“Thin-shield” formulation: assumptions

The two sides of the shield are indicated with (a) and (b), assuming n oriented from (a) to (b)

v

w

ud

t1

t2

n

(a)

(b)

Shield:Magnetic permeability Electrical conductivity

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

f

1 penetration depth

H C C 1 2exp exp w w

C1, C2 = integration constants

1 j 1j

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 div

jHH HH

dd

dwjdw00

nBnE

)()(1 at

btSdiv HH

)()( b

na

no HH

j

2dtgh

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

V

(b)

(a)

n

t

0)()()(

)()()()( latba

dSdSdSdV bbaa

V

tBnBnBB

V

tdVdSlat

BtB)(

)(bnB)(a

nB

the II interface condition is obtained:

)()()()( bt

atS

o

an

bn div

jHH HH

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 div

jHH HH

)()()()( at

btS

o

an

bn div

jHH HH

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

Curl-freereduced field Source field

Reduced scalar potential

baSS

o

sn

anm

bnm graddiv

jHHH

2)(,,

baS

sS

o

anm

bnm graddiv

jHH

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.For the generic i-th layer, the following interface conditions are deduced:

)a(

i,tS

)a(i,n

i,i,

i,i,)b(

i,tS

)b(i,n

div

B

TT

TT

div

B

HH 2221

1211

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“Thin-shield” formulation:Multilayered screens

The 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

B

TT

TT

TT

TT

TT

TT

div

B

H

H

)a(

tS

)a(n

)b(tS

)b(n

div

BTT

TT

div

B

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:

SS

S

wdsHjwdsHHj

dswgradgrad

sn

bnm

anm

Sba

S

2o,,

o

wdsgradwdsHHj

wdsgradgrad

SS

S

Ss

ta

nmb

nm

Sab

S

H2,,o

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

dsdsP aaa nn )()()(

dsdsP bbb nn )()()(

Side (a)

Side (b)nFEM equations on shield surface

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Codes available at IEN

2D Code PowerField(2D thin-shield formulation)

Sally2D Code(Hybrid nonlinear FEM-BEM formulation)

Sally3D Code(3D thin-shield formulation)

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Part III: Magnetic material properties and influence 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 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)

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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.

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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.

-GO Si-Fe alloys: 104

-Iron low carbon steel alloys, NO Si-Fe alloys: 102103

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“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

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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)

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Ferromagnetic shields:Effect of field nonunformity

1

10

100

1000

/(2a)=0.1

r=10000

/(2a)=0.01

r=10000

s Case1 Case2 Case3

/(2a)=0.01

r=1000

Case1

Case3

Case2

=thickness, 2a=diameter

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Closed/U-Shaped shield

1

10

100

1000

/L=0.1

r=10000

/L=0.01

r=10000

s Case1 Case2

/L=0.01

r=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, r=1000

/L=10-2, r=1000

/L=10-2, r=10000

d

source

shield Measurement point

=thickness, L=side

0 2 4 6 8 10

2

4

6

8

10

s

L/d

/L=10-3, r=1000

/L=10-2, r=1000

/L=10-2, r=10000

d

source

shield Measurement point

Distance of the measurement point = 0.1 L


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Shielding factor: so BBk

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

6S

hiel

ding

fact

or -

k

X Coordinate (m)

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

Plane shield (d = 5 cm)

x

y

d

source

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, r=1000

/L=10-2, r=1000

/L=10-2, r=10000

d

source

shield

Measurement point

0 2 4 6 8 10

10

20

30

40

50

60

70

80

s

L/d

/L=10-3, r=1000

/L=10-2, r=1000

/L=10-2, r=10000



d

source

shield

Measurement point

=thickness, L=side

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U-Shaped ferromagnetic shields:Influence of material properties


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

When the shielding configurations make an angle (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 the magnetic flux flowing in the shield.

The lack of continuity gives rise to a significant reduction of the shielding efficiency, which mainly affects the high permeability materials.

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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

d

source

shield

-0.4 -0.2 0.0 0.2 0.41

2

4

6

810

20

Plane shield

s

x (m)

L = 0.6 m d = 0.1 m = 10 mm,

r=1000

= 1 mm, r=1000

= 10 mm, r=100

= 1 mm, r=100

Closed shield

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Design of the shielding for a MV/LV substation

A 3D computer model is implemented in order to identify the most important field sources

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LV board30T

[T/A]

125T

57.6T15T

[T/A]

MV board

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On the floor over the substation

8.4T

6.7 T

5 T

5.5 T

[T/A]

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LV board shielding

After shielding

Before shielding

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MV board shielding

After shieldingBefore shielding

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Double trefoil HV underground power line

1200

750

100

1500

3x1600 mmq cables

Dig boundary

Ground level

Junction area

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Possible shielding configurations in proximity of the junction area

2 mm Al shields

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

1 2D and 3D magnetic shielding simulation methods and practical solutions Oriano Bottauscio Istituto...

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Transcript of 1 2D and 3D magnetic shielding simulation methods and practical solutions Oriano Bottauscio Istituto...