shielding of magnetic fields by eddy...

Payne : Eddy Current Shielding

1

SHIELDING OF MAGNETIC FIELDS BY EDDY

CURRENTS

© Alan Payne 2016

Alan Payne asserts the right to be recognized as the author of this work.

Enquiries to [email protected]


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TABLE OF CONTENTS

1. INTRODUCTION ........................................................................................................................ 3 2. MAGNETIC FIELDS ................................................................................................................... 3 3. THEORY FOR MAGNETIC SHIELDING BY A SINGLE PLATE ............................................. 4 4. THEORY FOR SHIELDING BY A BOX..................................................................................... 7 5. COMPARISON WITH PUBLISHED BOX MEASUREMENTS ................................................. 9 6. COMPARISON WITH PLATE MEASUREMNTS .................................................................... 10 7. HIGH PERMEABILITY MATERIALS ..................................................................................... 11 8. MESH SHIELDS ........................................................................................................................ 17 9. PERPENDICULAR FLUX ......................................................................................................... 19 10. CONCLUSION .......................................................................................................................... 20

APPENDIX 1 : MEASUREMENT OF ATTENUATION THROUGH METAL SHEETS ....................... 21 APPENDIX 2 : DETERMINATION OF DILUTED PERMEABILITY ................................................... 21 APPENDIX 3 : BOX RESISTANCE ........................................................................................................ 23


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SHIELDING MAGNETIC FIELDS BY EDDY CURRENTS

It is known that thin metal films can provide very good shielding of alternating magnetic

fields, even at low frequencies. The mechanism is often thought to be due to eddy currents

since these are known to produce a magnetic field which opposes the incident field, but

no accurate theory is available. A theoretical analysis is given here and is shown to give

excellent agreement with independent published measurements, including those of wire

mesh and magnetic materials such as steel.

1. INTRODUCTION

The shielding of alternating magnetic fields is often assumed to require a metal with a high permeability

such as iron, or require a conductor which is very thick such that it exceeds the skin depth in the metal.

However it has been shown that a thin aluminum foil can provide a high level of attenuation at frequencies

where it is much thinner than a skin depth (ref 1 Weston). The explanation for this is that eddy currents are

induced in the conductor by the incident field and these produce a magnetic field which opposes the applied

field. No accurate theory has been presented to date for the cancellation which eddy currents can produce,

and such a theory is given here. This gives the shielding provided by a single plate and this is then extended

to enclosures.

2. MAGNETIC FIELDS

2.1. Magnetic Fields and Electromagnetic Fields

There can be some confusion between a magnetic field and an electromagnetic field, and the difference is

outlined as follows.

When a direct current (dc) is passed through a wire there is a magnetic field around the wire and an electric

field between its ends. If the current is alternating these fields are still present and they also alternate, but a

new field is generated because the electrons in the wire are now accelerating. This new field is called an

Electro-magnetic Field (EM). Close to the wire this field is very weak compared with the magnetic and

electric fields, but as the distance from the wire increases the magnetic and electric fields decrease at a

higher rate than the EM field. They are equal in amplitude at a distance of about 1/6th wavelength, and

beyond this distance the radiated field dominates. For instance at 1 MHz the fields are equal at a distance of

about 50 meters.

In this article it is only the alternating magnetic field which is being considered.

2.2. The Electric Field around a Magnetic Field

When a loop of wire encloses a changing magnetic field an emf can be measured at the open terminals of

the loop. This effect is well known and is the basis of transformer action, but what is less well known is that

the emf exists whether the wire is present or not. Around every magnetic field there is an electric field, and

the wire is a device for measuring this. So when a magnetic field is incident upon a metal plate, an electric

field is induced in the plate and this drives a current whose magnitude is limited by the resistance of the

conducting path.

The magnitude of the electric field is given by Lenz’s law, which states that the induced emf is equal to the

rate of change of magnetic flux e = - dφr / dt volts.


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3. THEORY FOR MAGNETIC SHIELDING BY A SINGLE PLATE

3.1. Introduction

When an alternating magnetic field is perpendicular to a conducting plate, eddy currents are generated in

the plate as shown below :

Figure 3.1.1 Circulating currents and Resultant current

Here the magnetic flux is represented by 25 discrete flux concentrations and around each one there is an

electric field which causes a circular current. The direction of these currents is such that they produce a

magnetic field which opposes the applied field. Over most of the plate these currents cancel to produce a

resultant current around the edge, which opposes the applied field. A very good video demonstration of this

is given in ref 2, where an iron plate is placed inside an induction coil and the edges of the plate become red

hot while the rest of the plate remains black. (NB the audio explanation with this video says that the effect

is due to skin effect, rather than eddy currents).

In contrast to the above, this paper considers flux which is tangential to the surface, as shown below :

Figure 3.1.2 Tangential magnetic flux

The above shows a single plate, and the analysis is extended to a box structure in Section 4. Surprisingly

this tangential analysis gives very good agreement with published measurements, even when these appear

to be due to a perpendicular field. This is discussed more in Section 8.


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3.2. Theoretical Analysis

The configuration to be analysed is shown below. A plate of length ℓp, width wp, and thickness t is

illuminated by a tangential magnetic flux.

Figure 3.2.1 Analysed configuraton

The flux is shown entering the edge of the plate on the side designated as the width. Note that the width of

the plate is not necessarily the shortest dimension but is defined here as the edge into which the flux is

directed. The length of the plate is in the direction of the flux path through the plate.

This flux induces an emf in the plate and this leads to a circulating eddy current, the magnitude of which is

determined by the resistance of the path which the current follows. In turn this eddy current generates a

magnetic field which opposes the incident field, and partially cancels it. The resultant field is therefore

lower than the incident field and it is assumed that it is this resultant field which appears on the other side

of the plate. The resultant flux density Br is given by :

Br = Bo-Be 3.2.1

where Bo is the incident flux density

Be is the flux produced by the eddy current

The first thing to be determined here is the flux produced by the eddy current Be. As shown in the Figure

3.2.1 the cross-section of this current flow is very long and narrow, and it extends down the whole length of

the plate ℓp. There are therefore two parallel current sheets carrying current in opposite directions and it is

assumed here that the flux density is the same as that from two parallel wires. (This cannot be justified on

purely theoretical grounds but it does lead to an equation which agrees extremely well with published

measurements).

It is seen from Figure 3.2.1 that the two current sheets are spaced by a distance somewhat less than the

thickness t, and let this be t’. The flux density is then assumed to be :

Be= µo µrm i / (2π t’) 3.2.2

The relative permeability µrm is unity for most metals, such as copper and aluminium. Permeable metals

such as iron and steel are considered in Section 7.


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The emf e induced in the loop is due to the resultant flux φr :

e = d φr / dt 3.2.3

which for sinusoidal excitation, and area of loop wp t’ will be :

e = j ω φr = jω wp t’ Br 3.2.4

For a loop resistance R, the current i induced in the loop will be i= e/R :

i = j ω wp t’ Br / R 3.2.5

Substituting this into Equation 3.2.2 :

Be= j µo µrm 2 π f wp t’ Br / R/(2 πt’)

= j µo µrm f wp Br / R 3.2.6

Notice that flux due to the eddy current Be is independent of t’.

Normalising to incident flux density Bo, gives Be’ (= Be / Bo), and similarly Br

’, and then Equation 3.2.6

becomes :

Be’ = j µo µrm f wp Br

’ / R 3.2.7

From Equation 3.2.1 and normalising to Bo :

Br’ = 1 - Be

’ 3.2.8

Br’ = 1 - j [µo µrm f wp Br

’ / R]

Br’ + j [µo µrm f wp Br

’ / R] =1

Br’ = 1/ [1+ j µo µrm f wp / R] 3.2.9

The resistance R is equal to ρ ℓ /A. Notice that the width and length of the plate are defined with respect to

the flux, and for the current these are reversed and so R is equal to ρ 2 wp /( ℓp t/k), where t/k is the width of

the conducting area.

Equation 3.2.9 then becomes :

Br / Bo= 1/ [1+ j (µo µrm f ℓp (t/k) / (2ρ ) ] 3.2.10

The phase angle is given by :

Θ = tan -1

(- µo µrm f ℓp (t/k) / (2ρ ) radians 3.2.11

(k shown later to be equal to 4)

[this comes from y=1/(1+jk) = (1-jk)/[(1-jk)(1+jk)]= (1-jk)/ (1+k2). The angle of this is tan

-1 (-k/1)]

It is conventional to express the shielding performance as the ‘shielding effectiveness’ (SE) and this is the

inverse of Equation 3.2.10. Taking its modulus gives :

|SE|plate = [1+ {(µo µrm f ℓp (t/k) / (2ρ )} 2

]0.5

3.2.12

Using this equation good agreement with experiment and with published measurements was obtained with

k=4, so the average conducting width is ¼ of the conductor thickness. So we can imagine that the current is


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constant at its maximum value to a depth of t/4 from each surface, and zero at other depths. More likely it

could change linearly from one surface to the other as shown below:

Figure 3.2.2 Current in Conductor Cross-section

With k=4 Equation 3.2.12 becomes :

|SE|plate = [1+ {(µo µrm f ℓp t / (k1 ρ )} 2

]0.5

3.2.13

where µo = 4π 10-7

H/m

µrm is the relative permeability (normally unity)

f is the frequency in Hz

wp is the width of the plate in m

t is the thickness of the plate in m

ρ is the resistivity of the conductor

ℓp is the length of the plate in m

k1 =8 for a plate (see later for a box)

When the frequency is high enough that the factor {(µo µrm f ℓp t / (k1 ρ )} is much greater than unity

Equation 3.2.13 approximates to :

|SE| ≈ (µo µrm f ℓp t / (k1ρ ) 3.2.14

This shows that at high frequencies the shielding effectiveness is proportional to frequency, so the

attenuation through the conductor increases at 20 dB per decade of frequency.

4. THEORY FOR SHIELDING BY A BOX (ENCLOSURE)

The above analysis gives the shielding of a single plate, and that analysis is extended here to a box

structure.

Consider two plates placed one behind the other and having short plates joining them at the top and bottom.

Initially it is assumed that the sides are open. Current will still flow within the front face as shown in Figure

3.2.1, but in addition there will be current flowing around the whole box as shown below.


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Figure 4.1 Prototype shielding box

The incident flux is shown entering between the two plates, and this induces the current shown by the

dashed line. Conduction in the plates now takes place over their full thickness t, rather than in t/4 for a

single plate. If the depth dp is small there will be only a small resistance in the connecting plates and the

shielding provided by this current will be 4 times that of the single front plate (ie +12 dB).

A more practical box will have a greater spacing between the plates (depth dp) and so there will be extra

resistance from the connecting plates top and bottom, and this will reduce the improvement over a single

plate. However this practical box will also have conducting sides and these will carry some of the current,

and improve the shielding. This is considered in Appendix 2 where it is shown that the shielding

effectiveness of a box is given by Equation 3.2.13, but with a modified value of k1, as below :

|SE|BOX = [1+ {(µo µrm f ℓp t / (k1 ρ )} 2

]0.5

4.1

where µo = 4π 10-7

H/m

µrm is the relative permeability (normally unity)

f is the frequency in Hz

wp is the width of the plate in m

t is the thickness of the plate in m

ρ is the resistivity of the conductor Ω.m

ℓp is the length of the plate in m

k1 = = 1/ [1/8 + 1/ {1.4 (1 + dp / wp)}]

dp is the box depth

Notice that if the box depth dp is very large compared with the width wp then k1 ≈ 8, and the shielding

reduces to that of the front plate alone.

The SE is often expressed in decibels, defined as:

SE = 20 Log10 (|SE|) dB 4.2


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25

30

35

40

45

50

0.01 0.1 1 10 100

Att

enu

atio

n d

B

Frequency KHz

Weston Measurements of Aluminium Foil Enclosure

Published

Calculated

5. COMPARISON WITH PUBLISHED BOX MEASUREMENTS

5.1. Weston (shielding with Aluminium foil)

Weston (ref 1) measured the attenuation through an enclosure made from aluminium foil, and having

dimensions of 1m x1 m x 1m and thickness 0.0176 mm. He illuminated the enclosure with a tangential field

from a multi-turn loop locate 0.5m from front face, and detected the flux inside the enclosure with another

loop mounted at distances of 0.3 m, 0.5 m and 0.8 m from the front wall. The received signal was then

compared with the signal received in the absence of the enclosure for these 3 distances. For the receive loop

at 0.3 m he measured the following (blue curve), taken from his figure 6.1 :

Figure 5.1.1 Comparison with Weston’s measurements

Also shown is the attenuation as calculated from Equation 4.1, and the correlation is seen to be very good

over the whole frequency range from 30 Hz to 80 KHz.

For reference the skin depth at 80KHz is 0.3 mm, so the conductor thickness at 0.0176 was much less than

the skin depth.

5.2. Heinonen et al (Low Frequency 50 Hz)

Heinonen et al (ref 3) also measured the attenuation through an enclosure, and this had dimensions of 2 x 2

x 2 m, and made of 99% pure aluminium. This had a door-way and to reduce its leakage a corridor was

placed inside the shield. Shielding at 50 Hz was required and so the aluminium was very thick at 45 mm (4

skin depths at 50 Hz). They measured the field inside the box in 3 orthogonal directions and found ‘at least

40 dB in half of the enclosure area’. Equation 4.1 gives 41 dB.

5.3. Hoeft & Hofstra (Equipment case)

Hoeft & Hofstra (ref 4) measured the attenuation through a ‘generic equipment case’ having dimensions of

432 x 495 x 216 mm, and made of drawn aluminium 2.3 mm thick. They measured the attenuation over a

frequency from 10 KHz to 10 MHz, and the comparison with their measurements (taken from their figure

7) is shown below:


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0

20

40

60

80

100

120

0.01 0.1 1 10A

tten

uat

ion

dB

Frequency MHz

Hoeft & Hofstra Measurements on Aluminium Equipment case

Published

Calculated

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5

10

15

20

25

30

35

40

0 0.02 0.04 0.06 0.08 0.1

Att

enu

atio

n d

B

Frequency MHz

Copper plate 76.5 x 152 x 0.26 mm

Theory

Measuremnts

Figure 5.3.1 Comparison with the Hoeft & Hofstra measurements

In applying Equation 4.1, it was not clear from the published paper which dimension was the width and

which the length (as defined at the beginning of Section 3.2). However these dimensions are not very

different in their experiment, and so the average of the two was used (432+495)/2 = 464 mm for ℓp.

It is seen that the correlation is quite close except at the lower frequencies. However their measurements

may be unreliable at these frequencies, as evidenced by the unlikely dip at 0.012 MHz.

It is significant that the skin depth is only 0.026 mm at their maximum measurement frequency of 10 MHz,

so the conductor thickness was 88 skin depths, and yet the eddy current theory still holds.

6. COMPARISON WITH PLATE MEASUREMNTS

6.1. Single Metal Sheets

No published measurements were found for single plates, and so were carried-out by the author. For these

the coupling between two coils having their axes parallel was measured both with and without a metal plate

between the two coils. The experimental procedure is described in Appendix 1 and measurements were

made on sheets made of aluminium, copper and lead, and all gave good agreement with the Equation

3.2.13. As an example the copper results are shown below :

Figure 6.1.1 Attenuation through Copper Plate


11

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10

15

20

25

30

35

40

0.5 5 50 500

Att

enu

atio

n d

B

Frequency KHz

Steel plate 152 x 75 x 0.4 mm

Theory assuming µ =1

Measurements

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0.00 0.05 0.10 0.15 0.20 0.25

Phas

e sh

ift (d

egre

es)

Frequency MHz

Phase shift across Aluminium foil

Measured phase across aluminium foil

Theoretical phase

The copper was assumed to have a resistivity of 1.68 10-8

Ωm and a permeability of unity. Notice that the

frequency axis is linear not logarithmic.

6.2. Phase shift through the conductor

To test the equation for the phase shift (Equation 3.2.11), this was measured with a vector network analyser

through a sheet of aluminium foil of thickness 0.015 mm with the following results

Figure 6.2.1 Phase shift through Aluminium foil

It can be seen that the agreement is very good, and within the expected experimental error (especially the

error in the measurement of the thickness of this very thin foil).

7. HIGH PERMEABILITY MATERIALS

7.1. Introduction

Iron and steel are often used as screening materials and are different from other metals in having a very

high permeability, ranging from around 50 to many thousands. This improves the shielding considerably, as

shown by the following measurements on a steel plate :

Figure 7.1.1 Measured attenuation through a steel plate


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0

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10

15

20

25

30

35

40

0.5 5 50 500

Att

en

uat

ion

dB

Frequency KHz

Mild Steel plate 152 x 75 x 0.4 mm

Theory

Measurements

The measurements are shown in red (see Appendix 1 for more details), and Equation 3.2.13 is shown in

blue. For the purpose of this comparison the relative permeability of the steel is assumed to be unity.

If the permeability is assumed to be equal to 11 and constant with frequency, Equation 3.2.13 gives :

Figure 7.1.2 Permeability assumed to be 11

The correlation has now improved considerably. However a permeability of 11 is unrealistic for steel,

which is likely to have a permeability of several hundred. This apparent reduction in permeability is shown

later to be due to dilution caused by the flux passing not just through the metal but also through a long air

path.

At low frequencies the measured attenuation is higher than that predicted. This is because the equation

considers only the attenuation due to eddy currents, and does not include the additional effect of

ferromagnetic shielding, whereby the metal concentrates the flux within itself at the expense of the

surrounding air. This is considered below along with dilution, but firstly the change in material

permeability with frequency is considered.

7.2. Material Permeability v Frequency

Materials with a high permeability owe their magnetic properties to the ability of the material to organize

itself into magnetic domains. In an un-magnetised material these domains are oriented randomly and so the

net magnetisation is zero. When a small external field is introduced some domains move at the expense of

others, increasing the overall magnetization. In this way materials can have permeabilities of many

thousands. The domains have mass and so at high frequencies the movement is reduced in amplitude, and

has associated with it frictional energy loss.

So the material permeability has a real component µ’ and an imaginary component µ’’ and the overall

permeability is given by :

µm = µm’ - j µm’’ 7.2.1

Notice that the convention here is for the reactive component µm’ to be real and the loss component µm’’

to be imaginary, the opposite to the normal circuit representation.

The graph below shows these characteristics :


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100

150

200

250

300

350

0.01 0.1 1 10 100

Per

mea

bili

ty

Frequency KHz

Permeability

µ' µ''

Figure 7.2.1 Permeability µm’ and µm’’

In the above example the real component (blue curve) has an initial permeability µm’ of 300, and a

relaxation frequency fm of 1 KHz. The equation which describes this characteristic is:

µ’ = (µmi’ - 1 ) / [1+ (f/ fm)2] + 1 7.2.2

where µmi’ is the initial (low frequency) permeability

fm is the magnetic relaxation frequency of the material

At high frequencies this equation is asymptotic to unity, and this is typical of many magnetic materials

especially ferrites. However some workers have measured a higher asymptote for steel, and so the above

equation needs to be modified to:

µ’ = (µmi’ - µ’∞) / [1+ (f/ fm)2] + µ’∞ 7.2.3

where µ’∞ is the high frequency permeability

For instance Bowler (ref 5) measured µ’∞ = 78 on his steel sample.

The loss component shows a resonant characteristic with an amplitude equal to that of the real component

at the frequency fm. Its equation is :

µm’’ = µmi’ (f/ fm) / [1+ (f/ fm)2] 7.2.4

Later in this article the diluted permeability is calculated, and it is important to be aware of the distinction

between the material permeability µm’ and the diluted permeability µd’.

7.3. Ferromagnetic Shielding

A strict mathematical analysis of ferromagnetic shielding is very difficult and so the literature gives useful

approximate equations for the shielding effectiveness of common box shapes such cylinders, spheres and

cubes (ref 6). For instance the shielding effect of a cube is often given as :

SE ≈ 1+4/5 (µ’ - 1) t/a 7.3.1

where µ’ is given by Equation 7.2.3

t is plate thickness

a is the edge length of the cube, or the length of the spatial

diagonal for other shapes


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15

20

0 2 4 6 8 10 12 14 16 18 20

Dil

ute

d P

erm

ea

bilit

y

Factor x

Diluted Permeability

μr =10

μr =30

μr = 100

μr =300

μr = ∞

NB The above equation is normally written as SE ≈ 1+4/5 µ’ t/a, but this clearly fails when µ’ =1.

For the other shapes the equations are similar and have the form :

SE ≈ 1+ K (µ’ -1) t/ l 7.3.2

where l is a length or diameter

K is a constant characteristic of the shape.

It has not been possible to find the derivation of these equations or anything in the way of experimental

proof, and the definitions of K and l are rather vague.

Yashchuk et al (ref 7) have used the idea that a permeable shield shunts the flux away from the area being

shielded. They say ‘ .. the shielding efficiency depends on the relation of the effective reluctance of the

ferromagnetic ‘yoke’ …. and that of the inner space…’.

From these reluctances they derive an equation similar to Equation 7.3.2, but again they cannot supply a

strict definition of the dimensions K or l , and it seems that these can only be found by experiment.

7.4. Permeability Dilution

7.4.1. Dilution

Figure 3.2.1 shows the incident magnetic field passing through the air and into the conductor, and the

induced flux also follows a similar path. So both flow partly through the metal and partly through air so that

the effective permeability is lower than that of the metal alone. This dilution of the permeability has already

been analysed by the author for ferrite rod antennas (ref 8) and that work is adapted to cover the situation

here. It is shown in Appendix 2 that the diluted permeability is given by :

µdiluted = (1 + x) /(1 + x/ μ’ ) 7.4.1.1

where μ’ is given by Equation 7.2.3

x = 5.1 [ℓp / deff +0.45 ]/[1+ 2.8 / (ℓp / deff + 0.45)] ℓp is the length of the plate in the direction of the flux

deff = [ 4 wp t /π] 0.5

wp is the width of the plate into which the flux is directed

t is the thickness of the plate

In the above equation notice that if μ’ is much larger than x the diluted permeability is approximately equal

to 1+x. Since x is generally less than 16 the diluted permeability is hardly ever greater than 17, no matter

how large the material permeability. This is shown below :

Figure 7.4.1.1 Diluted Permeability


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1

10

100

1000

0.1 1 10 100

Pe

rme

ab

ilit

y

Frequency KHz

Increased Frequency Range

Material Permeability

Diluted Permeability

7.4.2. Increased Frequency range

A consequence of the dilution is that the diluted permeability rolls off at a greater frequency than the

relaxation frequency of the material. So the shielding effectiveness is reduced by dilution but is maintained

to a higher frequency. An example is shown below for a material with an initial permeability (µmi’) of 300

and a relaxation frequency of 1KHz (red curve) :

Figure 7.4.2.1 Increased Frequency Range

Also shown is the frequency response if the diluted permeability (µd) is 16, and this has a higher relaxation

frequency of 4.6 KHz. The diluted relaxation frequency, fd , is related to the material relaxation frequency,

fm by (from Equation 7.2.2) :

fd ≈ fm (µr / µd )0.5

7.4.2.1

7.4.3. Overall Equation for Permeable Materials

The overall equation for permeable materials is the sum of Equations 4.1 and 7.3.1:

|SE |= 20 Log10 [1+ {(µo µd f ℓp t / (k1 ρ )}2 ]0.5

+ 20 Log10 [1+4/5 (µ’ - 1) t/a] dB 7.4.3.1

where µd is the diluted permeability given by Equation 7.4.1.1

µ’ is the relative permeability of the material (Equation 7.2.3)

a (see Section 7.3) (for the other parameters see Equation 4.1)

7.5. Comparison with Measurements of Steel Shields

The author has found only one readily available reference which gives measurements of steel shields and

this is by Hoeft and Hofstra (ref 4 ) They used an off-the-shelf RFI equipment case with dimensions of 356

x 406 x 165 x 1.9 mm, having continuous welded seams, and a hinged lid gasketed with wire mesh and

held down by seven screw-down clamps. The attenuation through this case was measured for frequencies

between 10 KHz and 100 MHz and for various tightening torques on the screws. For the highest torque

they measured the following (in red) :


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0.5 5 50 500

Att

enu

atio

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Frequency KHz

Mild Steel plate 152 x 75 x 0.4 mm

Theory

Measurements

0

20

40

60

80

100

120

0.001 0.01 0.1 1 10 100A

tten

uat

ion

dB

Frequency MHz

Mild Steel Box 356 x 406 x 165 x 1.63 mm

Theory

Published Hoeft et al

Figure 7.5.1 Steel Equipment Case

The theoretical curve is shown in blue (Equation 7.4.3.1). The material resistivity and permeability were

not known and so the values used were those measured by Bowler (ref 5 ) on alloy 1018, of resistivity 19.3

10-8

Ωm, µmi’ = 250, µ’∞ =78, and fm= 5 KHz (Equation 7.2.2). However the permeability values are not at

all critical here because the diluted permeability is only 2.5. The value of ‘a’ in Equation 7.4.3.1 was

assumed to be equal to 565 mm, the length of the spatial diagonal of the case. However the accuracy of this

could not be assessed because its major effect is at frequencies below those measured.

The agreement is very good over most of the range and is surprising in that it indicates that µ∞ is

maintained in steel at frequencies up to at least 100 MHz. At 0.01 MHz the agreement is poor but the

reason for this is not known.

The agreement here is a powerful validation of the dilution given by Equation 7.4.1.1.

It is interesting to compare the above shielding with that which would be obtained if the box had been made

of aluminium of the same thickness. The resistivity would drop by a factor of 7.3, and the permeability by

2.5(diluted) giving aluminium an overall shielding advantage of 7.3/2.5 = 3 (9 dB). However at low

frequencies (lower than those in Figure 7.5.1) the ferromagnetic shielding provided by steel would give it

an advantage over aluminium of around 6 dB.

The author’s measurements of a steel plate are shown below, along with Equation 7.4.3.1:

Figure 7.5.2 Author’s measurements of steel plate

For the theoretical curve the permeability was not known. The optimum match with the measurements was

provided with µmi’ = 250, µ’∞ = 20 and fm = 1KHz (Equation 7.2.3), and this is shown above. The factor ‘a’

was chosen as 55 mm since this gave best match with the measurements. This highlights a major problem


17

with validating shielding equations with permeable material, that the magnetic parameters of the material

are often not known.

8. MESH SHIELDS

8.1. Woven Mesh

A woven mesh with overlapping wires is shown below :

Figure 8.1.1 Wire mesh

It is assumed here that Equation 4.1 still applies but that the resistivity is increased because there is not full

conductivity over the whole area.

There are two possibilities for the current flow:

a) The flux is exactly in the direction of one set of wires so that no current flows in these and all the

current flows in the perpendicular set.

b) The flux is at an angle to the grid and current thus flows in both sets of wires.

In calculating the effective resistance in each case it is convenient here to calculate the equivalent thickness

of solid plate which gives the same resistance. Taking the first case this thickness is then equal to volume of

conducting metal over a square metre divided by one square metre. If in the above figure the wire has a

radius of r and a centre to centre spacing of s, then :

The number of conducting wires per sq m = 1/s

So volume of metal per sq m = π r2/s

So effective thickness, t = metal volume/ area = π r2 /s 8.1.1

This ignores the increase in the wire length due to weaving, and this is approximately equal to (1+(2r/s)2)0.5

.

This is generally quite small and for instance if the ratio r/s is 0.2, the increased resistance is only 0.6dB.

For the second case all wires are now conducting, so the number of conducting wires is 2/s. If it is assumed

that the flux is at an angle of 450 the effective length of the wires is increased by √2 on that calculated

above giving :

Effective thickness, t = √2 (1+(2r/s)2)

0.5 π r

2 / s 8.1.2

The above ignores any current between wires across their contact points because the voltage across these

contacts is zero when the angle is 450. At other angles there will be a potential difference, and the resultant

current will tend to equalise the overall currents in the wires towards that of the 450 condition.

In the design of magnetic shields weight is often a problem, and some authors have considered mesh as a

method to reduce weight. It is therefore useful to consider the screening which would be provided by a

solid plate having the same mass of conductor as the mesh. If all the metal were used to make a plate, the


18

0

10

20

30

40

50

60

0 0.05 0.1 0.15

Att

enu

atio

n d

B

Frequency MHz

Kuhn et al Measurements of Copper Mesh (Side 300 x 300 mm)

Published

Calculated

volume of metal per sq m would be π r2 2/s, so the thickness would be t plate = 2π r

2/s. This is greater than

Equation 8.1.2 by √2, so the shielding provided by the plate would be greater by 3dB.

8.2. Perforated Mesh

A perforated mesh is shown below, and this is expected to have a similar shielding effect as the woven

mesh.

Figure 8.2.1 Perforated Mesh

8.3. Comparison with published Measurements

Kuhn et al (ref 9 ) measured the attenuation through a woven copper mesh having a mesh size of ≈ 0.564

mm and a wire radius of ≈ 0.1 mm (NB they give the mesh size as 0.364 mm, but their image of the mesh

shows this as the dimension of the open space between wires).

They made this mesh into a shielding box with dimensions of 300 x 300 x 100 mm, by fixing it to a

wooden frame. The whole box was placed inside a Helmholtz coil, and a loop within the box detected the

internal magnetic field. They measured the attenuation with the 300 x 300 side facing the field and the

following results are extracted from their figure 14 :

Figure 8.3.1 Comparison with measurements by Kuhn et al

Also shown is the prediction from Equation 4.1 using the effective plate thickness given by Equation 8.1.2.

The agreement is very good, especially when compared with the prediction made in the published paper

which is in error by 22 dB (based on Kaden’s method).

They also measured the shielding with the small side (300 x 100 mm) facing the field and found the

following (taken from their figure 12):


19

0

10

20

30

40

50

60

0 0.05 0.1 0.15

Att

en

uat

ion

dB

Frequency MHz

Kuhn et al Measurements of Copper Mesh (Side 100 x 300 mm)

Published

Calculated

Figure 8.3.2 Comparison with measurements by Kuhn et al

Here the agreement is poor, and this could be due to measurement problems. The Helmholtz coil they used

was not much deeper than their box and the photographs in their paper indicate that in this orientation the

front and back faces of the box were within the coils themselves. This could mean that the field intensity

changed when the box was inserted. In contrast, for the measurements shown in Figure 8.3.1, the box was

turned at 90 degrees and then no part of the box was close to the coils, and this might explain the better

correlation.

[For guidance on the use of the Helmholtz coil, the ANSI/ASTM t 4 standard states, "The Helmholtz coil

diameter shall be at least three times the length of the test specimen or four times its diameter, or both."]

Another potential source of error is the resistance between one face of the box and another face. No special

measures seem to have been taken by the authors to minimise this resistance and indeed they suggest that

this resistance could be significant. However the good correlation shown in Figure 8.3.1 indicates that this

was not an important factor, at least in that experiment. However it must be realised that the path of the

current flow was different in the two experiments.

[The effect of poor contact can be gauged from the experiments by Hoeft and Hostra (ref 4) who measured

a 10 dB improvement in SE when the screws holding the lid of their enclosure were tightened from ‘finger

tight’ to 5 in lb, and a further improvement of 10 dB when tightened to 15 in lb].

9. PERPENDICULAR FLUX

The theoretical analysis in this paper is for tangential flux, and it has been shown to give good predictions

of published measurements. As far as can be ascertained in most of these measurements the illuminating

flux was indeed tangential, but there was one exception : the measurements by Kuhn et al on mesh shields.

These were notable in being conducted in a Helmholtz coil and therefore the flux seemed to be directed

perpendicular to the face. However their Helmholtz coil was not large enough to give a constant field and

so there has to be some doubt about the validity of their measurements. Nevertheless the equations here

gave a good prediction of the shielding effectiveness (Section 8.2). The reason for this is not understood but

is possibly related to the fact that magnetic fields cannot be attenuated only re-directed, so that the effect of

the eddy currents could be to re-direct the field to be always tangential with the plate.


20

10. CONCLUSION

The theoretical shielding equations presented here assume that the direction of the magnetic field is

tangential to the front face of the box, and these equations give very good predictions of published

measurements even when these are not obviously of a tangential field. The shielding is proportional to the

frequency, the plate thickness, the material conductivity and the box dimension in the direction of the

magnetic field.

For magnetic materials such as iron and steel their relative permeability improves the shielding but by a

factor much less than the material permeability, and an equation is given for this reduced permeability.

It is shown that the magnetic shielding due to eddy currents applies even at frequencies as high as 100

MHz, and this is in contrast to the widely held view that at high frequencies the shielding of magnetic fields

is provided by skin effect (although this may be true for the shielding of electromagnetic fields).

The theory is extended to mesh, and in essence this is done by averaging the resistance of the mesh. An

important implication from this is that there is no weight advantage in using mesh, indeed for woven mesh

its shielding can be 3 dB less than a plate of the same weight.


21

APPENDIX 1 : MEASUREMENT OF ATTENUATION THROUGH METAL SHEETS

Measurements were made of the attenuation through small metal sheets, approximately 75 x 150 mm, using

the jig shown below :

Figure A1.1 Measurement jig

The two coils were identical and taken from ferrite loop antennas. They were 11 mm in diameter, 22 mm

long and had around 260 turns. They were spaced by approximately 20 mm centre to centre. One coil was

excited by a signal generator and an amplifier, and the voltage output of the other was measured on an

oscilloscope.

At each frequency the output on the oscilloscope was measured both with and without the metal plate and

the ratio taken as the attenuation through the plate. A major problem with this set-up was that with the plate

in place the maximum attenuation which could be measured was limited to about 20 dB because of

spurious coupling at a level of about -26 dB. Despite considerable investigations the source of this coupling

could not be positively identified but is possibly capacitive coupling around the edges of the plate.

APPENDIX 2 : DETERMINATION OF DILUTED PERMEABILITY

The effective permeability of a steel shield is reduced because the flux has to flow partly in air. This effect

had already been investigated by the author for cylindrical magnetic materials (ref 8), where the diluted

permeability was found to be given by Equation 7.4.1.1.

The geometry here is very different, with the permeable material being a sheet : very long and wide

compared to its thickness.

To evaluate this some flat slabs of ferrite were procured having dimensions measured as 11.92 x 4.04 x

61.3 mm. One of these was wound with 8 turns of copper strip having a width of 5.06 mm and thickness

0.07 mm and its inductance Lf measured as 1.93 µH. To evaluate the effect of inserting this into a coil it

was necessary to know the inductance of the coil without the ferrite, and this was determined by winding an

identical coil onto a wooden former cut to the same size as the ferrite. This was measured as Lo = 0.069 µH,

so the effective permeability was 28.1, since the diluted permeability is equal to the ratio of the

inductances.

This experiment was repeated with 4 of the above ferrites taped side by side to give an overall size of 48 x

4.04 x 61.3 mm, and wound with 8.5 turns of the same tape. These gave Lo = 0.37 µH, and Lf = 4.96 µH so

the diluted permeability in this case was 13.4.


22

The two coils measured, with and without ferrite, are shown below :

Figure A2.1 Measured Flat Coil

It was anticipated that the equation for the diluted permeability of the core with a cylindrical cross-section

(Equation 7.4.1.1) would also hold for the rectangular cross section if the cross-sectional areas were

equated (ie the same amount of ferrite), so π d2/4 = wp t, where wp is the width and t the thickness of the

metal plate. This gives the effective diameter of the rectangular core deff as

deff = [ 4 wp t /π] 0.5

A2.1

The experiments give :

Calculated Measured

Sample Lo Lf deff Length/deff Rod µdiluted Rectangle µdiluted

1 slab 0.069 1.93 7.8 7.9 28 28.1

4 slabs 0.37 4.96 15.7 3.9 13.6 13.4

The very good agreement between the last two columns confirms Equation A2.1. (The values calculated

from Equation 7.4.1.1 assumed that µf =200, a likely value for the ferrite used).

NB the measured values of inductance are very low and so it was necessary to subtract the inductance of

the leads. However this alone is not sufficient because there is flux in the air coil which is not much

affected by the ferrite, and the inductance due to this must also be subtracted. This flux is due to the

component of the conductor helix in the direction of the coil length, and is equal to that length. So in

measuring the lead inductance to these was added a straight conductor of the same overall length as the

coil.


23

APPENDIX 3 : BOX RESISTANCE

A basic box structure is shown in Figure A3.1. Initially this has no sides and these are considered later.

Figure A3.1 Box Dimensions

A current i1 is induced within the thickness of the front face (Figure 3.2.1) and another, i2, is induced

around the whole box as shown in Figure A3.1. If the resistance of the front face from top to bottom is R,

the two currents are determined by the equivalent circuits shown below.

Figure A3.2 Equivalent Circuit of box

The current i2 flows through the front face, the back face, and the top and bottom faces. The later have a

resistance of R d/w, where d is the depth and w the width as defined in the figure above. The total current

enclosing the flux is( i1 + i1), and the effective resistance RT is V/( i1 + i1 ). So


24

0

1

2

3

4

5

6

0 1 2 3 4

No

rmal

ise

d R

eist

ance

Box depth d/w

BOX RESISTANCE V BOX DEPTH

Rt/R

Derived from published measurements

RT = 1/ [1/{8R} + 1/ { R + 2R (d/w) +R}] A3.1

Normalising to R gives :

RT /R = 1/ [1/8 + 1/ { 2 + 2 d/w}] A3.2

When sides are added to the box some current will flow in these and if it is assumed that this reduces the

resistance of all the plates equally, and by a factor k2 we have :

RT /R = 1/ [1/8 + 1/ {k2 (2 + 2 d/w)}] A3.3

It is very difficult to determine the value of k2 theoretically but the best match with published

measurements is with k2 =0.7, giving :

RT /R = 1/ [1/8 + 1/ {0.7 (2 + 2 d/w)}] A3.4

This equation is plotted below :

Figure A3.3 Normalised Box Resistance

Also shown are the values of RT /R (= k1) which optimize Equation 4.2 for the published measurements

quoted in this report :

Weston (d/w =1),

Heinonen et al (d/w =1),

Hoeft et al (d/w= 0.47 and 0.46 (steel)),

Kuhn (mesh) (d/w = 0.33)

Equation 3.2.13 now becomes for the box :

|SE|plate = [1+ {(µo µrm f ℓp t / (k1 ρ )} 2

]0.5

A3.5

where k1 = 1/ [1/8 + 1/ {1.4 (1 + d/w)}]


25

REFERENCES

1. WESTON D A : ‘ Shielding a room using Aluminium foil’, EMC Consulting Inc, 2006.

http://www.emcconsultinginc.com/docs/foilroom_full.rep.pdf

2. YOU TUBE : https://www.youtube.com/watch?v=3uatBD2_P9s),

3. HEINONEN et al : ‘Properties of a thick walled conducting enclosure in low-frequency magnetic

shielding’ J. Phys. E, Sci Instrum, Vol 13 1980 http://www.bem.fi/edu/doctor/lekkala/p1.pdf

4. HOEFT L O & HOFSTRA J S : ‘ Experimental and Theoretical Analysis of the Magnetic Field

Attenuation of Enclosures’, IEEE Trans. on Electromagnetic Compatibility, Vol 30 No3, Aug

1988 . http://www.spiraemi.com/references/pdf/hoeft_enclosures_1988.pdf

5. BOWLER N : ‘Frequency-dependence of Relative Permeability of Steel’. CP820 Review of

Quantitive Nondestructive Evaluation Vol 25, 2006.

http://home.eng.iastate.edu/~nbowler/pdf%20final%20versions/conferences/QNDE2005Bowler.pdf

6. SEKELS BROCHURE:

http://www.sekels.de/fileadmin/PDF/Englisch/06_Brochure_Magnetic_Shielding.pdf.

7. YASHCHUK et al : ‘ Magnetic Shielding’, Optical Magnetometry Chapter 12, Cambridge

University Press, 2013. ( http://www.ee.bgu.ac.il/~paperno/2012%20Shielding-chapter-proof-

1.pdf )

8. PAYNE A N : ‘The Inductance of Ferrite Rod Antennas’, http://g3rbj.co.uk/

9. KUHN et al : ‘ Validation of a measurement method for magnetic shielding effectiveness of a wire

mesh enclosure with comparison to an analytical model’ Adv. Radio Sci, 11, 189-195, 2013.

http://www.adv-radio-sci.net/11/189/2013/ars-11-189-2013.pdf

Issue 1 : May 2016

© Alan Payne 2016

Alan Payne asserts the right to be recognized as the author of this work.

Enquiries to [email protected]

http://www.emcconsultinginc.com/docs/foilroom_full.rep.pdf

https://www.youtube.com/watch?v=3uatBD2_P9s

http://www.bem.fi/edu/doctor/lekkala/p1.pdf

http://www.spiraemi.com/references/pdf/hoeft_enclosures_1988.pdf

http://home.eng.iastate.edu/~nbowler/pdf%20final%20versions/conferences/QNDE2005Bowler.pdf

http://www.sekels.de/fileadmin/PDF/Englisch/06_Brochure_Magnetic_Shielding.pdf

http://www.ee.bgu.ac.il/~paperno/2012%20Shielding-chapter-proof-1.pdf

http://www.ee.bgu.ac.il/~paperno/2012%20Shielding-chapter-proof-1.pdf

http://g3rbj.co.uk/

http://www.adv-radio-sci.net/11/189/2013/ars-11-189-2013.pdf

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