SCIENCE

Electromagnetic shielding in a system of two slabs

S.M. PanasProf. E.E. Kriezis

Indexing terms: Electromagnetic theory, Eddy currents, Conductors and conductivity, Mathematical techniques

Abstract: In the paper, an attempt is made at theexamination of the shielding effectiveness of asystem of two conducting slabs, when the excita-tion is a current filament placed above the systemof the slabs. For the determination of the electro-magnetic field the Helmholtz equation is used.The term of displacement current is not omitted asis usually the case. The problem is solved understeady-state conditions. The solution is carried outin the Fourier transform domain kx and theinverse transformation is obtained by the use ofthe fast Fourier algorithm.

1 Introduction

The interaction of electromagnetic (EM) waves withsystems is a very important topic in electromagnetictheory. The use of shield topology arises in the protectionagainst high electromagnetic fields, nuclear EMP, radarbiological effects or other EM threats. The interaction isassociated with multiple levels of shields and leads toanalytical operations with considerable complexity.

The shielding technique is expressed by the electro-magnetic phenomenon, according to which the metalscatters the incident wave and attenuates the resultantfield in proportion to the expression (d/S)exp(—d/S),where d is the thickness of the shield and d = (o)fia/2)~112

is the classical depth of penetration.In this work, the incident field is produced by a

current filament that is placed above a system of twoconducting slabs, which form the shield. The time varia-tion of the excitation can be considered to be eithersinusoidal or in a time-domain pulse.

For the calculation of the induced current densitiesinside the slabs and the magnetic flux densities in the dif-ferent regions outside the slabs, the Helmholtz equationfor the magnetic vector potential is used.

In previous works [1,2], the problem of shielding wasexamined by using the transmission-line analogy, or anumerical solution in the case of a thin sheet. In thepapers [3, 4], the problems of shielding in enclosures andin cylindrical shells are treated, respectively.

The analytical method in this paper is based on thefundamentals of EM theory. This analysis includes thedisplacement current term. Of course, for low frequencies,this term does not influence the results.

The differential equation of the problem is spatiallyFourier transformed and the inverse discrete Fourier

Paper 6157A (S8), first received 16th July 1987 and in revised form 8thApril 1988The authors are with the School of Electrical Engineering, Faculty ofEngineering, Aristotle University of Thessaloniki, Thessaloniki, Greece

transform (IDFT) is obtained by applying the inverse fastFourier transform (IFFT) algorithm.

2 Development of equations

A current filament is placed above a conducting slab or asystem of two slabs (see Figs. 1A and IB) at a distance b.

U air

Fig. 1A One slab geometrical configuration

i

1 air

2 air

, y1 T

\ '

\f^y//////A

br

k<fcV////////A

^ X

/A6 air

Fig. 1B Two slabs geometrical configuration

The filament is parallel to the slab(s) and a current isflowing in it (them), where the current is the real part ofthe complex current

I(x, y, t) = /(x)/(>> - 6) exp (ja)0 t)z0

where

1 for y = b0 elsewhere

(1)

(2)

For the geometrical configuration of one slab, the mag-netic vector potential (MVP) A,{x, y)z0 for the fourregions satisfies the following differential equations:

for i = 1, 2, 4 (3)dx

dx2

dy2

57r = O^o ̂ ~ ^o /^Mi f°r J = 3 (4)

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The time variation is harmonic. The term col HEA{ is dueto the displacement current and can be neglected for thelow-frequency case.

By spatially Fourier transforming eqhs. 3 and 4 andsolving the associated ordinary differential equations, thegeneral solution of the MVP (Appendix 7.1) is written

i = Ciexp(Xy) for i = 1, 2, 4 (5)

and

At = Ct exp (yy) + D{ exp ( - yy) for i = 3 (6)

where At is the spatial Fourier transform of Afac, y), withrespect to x,

AjLK y) =c=

J, y)] = At{x, y)e~Jkx dx

Jy = (k2 - col fie + jco0fia)i/2

and

The constants C, and D, (i = 1, 2, 3, 4) of eqns. 5 and 6are determined for each region by applying the boundaryconditions, the continuity of the normal component ofthe magnetic flux density and the continuity of thetangential component of the magnetic field intensity.These boundary conditions can be expressed in terms ofthe MVP as follows:

dAi+1

dx

1 dA;

d^dx

i+lBy

1 C/T.J

~'tIi~dy~=zIi

and their transforms with respect to x:

1 dAi+1 1

dy dy

(7)

(8)

(9)

(10)

The constants Cx and £>4 are zero, so that the MVP van-ishes at y = ±oo. The field in regions 1 and 2 is the sumof the field due to the excitation and the field due to theeddy current distributed within the conducting material.In region 3, the field is only due to the eddy currentsinduced within the conducting slab.

The calculation of the constants C{ and D{ (Appendix7.2) leads to a system of equations of the form

At = T-ffi i = l , 2 , 3 , 4 (11)

where T = & x[I{x)~], and H, is the transfer function of alinear system in_which the input is / and the output is theM V P A{. The His are given by the following expressions:

~Xb

x e-Xy , /f0 -X(y-b)

, 2X(12)

(13)

(14)

(15)

452

where

Q = (16)

The components Bx and By of the magnetic flux densityexpressed as spatial Fourier transforms with respect to xare written (Appendix 7.3) as follows:

dyi = l , 2 , 4 (17)

Fig. 2 Linear system analogue

and

i = 1, 2, 4 (18)

The components of the magnetic flux density Bx , andBy ,, as functions of x, y and t, are given by the inverseFourier transform with respect to x (IFTX) as follows

Bx, i(x, y,t) = \ IFTX(BX<,) | cos (co01

By, i(x, y,t) = \ IFTx(By,,) | cos (co01

(19)

(20)

where </>, and 0, are the arguments of the IFTX of Bx , andBy i, respectively.

The current density in the slab is given by the expres-sion

J(x, y, t) = | J | cos (coo t + </>)z0 (21)

where J is derived from Maxwell's equation V x E =dB/dt and Ohm's law J = aE and is written | J | =| — jco0oA3\ and (f> is the argument of —jco0oA3 whereA3 is the IFTX [/43].

Considering now the geometrical configuration of thesystem of two slabs (Fig. IB), it is evident that six regionsare created. Forming the differential equations as eqns. 3and 4 and considering the expressions of their generalsolutions as eqns. 5 and 6, the following equations arewritten:

Ax =Dl

C2eXy

As = D5e

A6 = D6e

yt = (k2 -

y2 = (k2 -

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

Using eqn. 11 and eqns. 22-29 as well as the boundaryconditions, the quantities Ht are calculated:

~ }i0Qxp(-Xy)H

exp (Xb)(X - yj + exp {-Xb\\ -X

2X

i Qxp(-Xb)k5

1-yidiU(30)

(X - yt) exp (y^i) - (X + yx) exp (•

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Ho exp (-Ab)—7i \ —

x \tt ~ 7i)[exp (Ay) - exp (-Ay)] + exp (-Ay)

where

X I — - Vi) exp ( y ^ J - (A + yx) exp (-yxd^

(31)

H0Q\p(-Ab)

{ exp(exp (y

H4 = y!^0(l + &i) exp ( — Ab)[(A + y2 k2) exp [A(d2 — 3;)]

+ (A — y2 k2) exp [ — A(d2 + y)J]/K3 (33)

H5 = 2Ay1/z0(l + fci) exp [A(d2 — bj]

x [[y2 exp (2y2 d3) - A] exp ( - y 2 y )

+ (y2 -I- A) exp (y2 y)]/[K3 K4] (34)

H6 = 4Ayxy2 fio(l + kt) exp [Arf2 — Ad3 — Ab

+ li d$ + Ay]/[K3 • K 4 ] (35)

where

l) + (y2 + A)ed*y2+X)ey2d3(y2 ey2d3 -2 ey2d3(y2 ey2d3 - - (y2 + A)edl{y2 + k)

(37)

x exp \_-dM - yx) + Ad2-] + (kl7l - A)(A - y2 k2)

x exp [dY(A + yx) - Ad2~\ (38)

kA = ey2d3(y2 ey2d3 - Xe-y2d3)e-d2(y2~X)

+ (y2 + A)ed2{y2+X) (39)

k5 = e~y^ +(A- y ^ l + k^^A + y2 k2)eX(d2~dl)

+ (A-y2k2)e-«d*-d^/k, (40)

Considering eqns. 17 and 18 the IFTX of the componentsBx i and By i, of the magnetic flux density, we write

Bx< t = I IFT&,,) I By<,- = I IFTx(By>,) | (41)

and consequently

Bt = Bx i cos (co01 + 0,-)xo + By i cos (co01 + 0$o (42)

From eqn. 42 it can easily be proved that

x cos (20,- - 20J]1/2 • cos (2<o01 + a,) > (43)

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1/2

a,- =Bl : sin 26: + B2 .- sin 20,

Bx ,- cos 2</>j + J5y i cos 20,-(44)

The maximum instantaneous magnetic flux density willbe given by the expression

max,- | Bt | =

., + BJ. i + 2BX< t By< t cos (2^ - 20,-)] ̂ J (45)

Eqns. 41-45 are valid for both geometrical configurationsFig. 1A and Fig. IB.

3 Numerical treatment

For the numerical treatment of the magnitudes of thefield, which were defined in the preceding Section, thefinal Fourier transform inversion is applied to functionswhich are called general waveform functions. These func-tions are neither band-limited nor duration-limited. Theapplication of the inverse discrete Fourier transform(IDFT) to these functions introduces the truncation error(TE) and the aliasing error (AE) [6]. The parameters thatinfluence the two errors are the kind, the width of thewindow function and the number of samples.

There has not been a general criterion or method thatrelates the above three parameters to the decrease of AEand TE, except to choose a large truncation interval anda large number of samples. But, in an iteration process,we do not know a priori the step in which the processmust be stopped, unless we have already established acriterion of convergence of calculated values to the trueones. To do so, a method called the best-DC-valuemethod is developed [5]. According to this method, wefirst establish a truncation interval such that the TE isminimised. The criterion for the minimisation of the TEis the convergence of the zero frequency (DC) term of thediscrete Fourier transform (DFT) of the absolute value ofthe function, which will be truncated. The calculation ofthe DC term can be done by the use of a numericalmethod of integration. The reason we use the con-vergence of the DC term as measure of the minimisationof the TE is because the TE of the DC term of the DFTof the absolute value of the function is an upper boundfor the TE at other frequencies of the function, as can beeasily proved.

After the truncation interval has been established, theAE is decreased by increasing the number of samples. Aprogram based on the above considerations has beendeveloped for the calculation of DFT and IDFT ofgeneral waveforms. The DFT and IDFT are calculatedby means of the fast Fourier transform (FFT) algorithm.

4 Example

We consider the cases of one and two slabs made of alu-minum of relative magnetic permeability /zr = 1 and con-ductivity tx = 3 . 5 4 x l 0 " 7 S m " 1 . The excitation currenthas maximum value / 0 = 1 A and is placed at a distanceb = 0.1 m.

The distances d, dlt d2 and d3 of Fig. 1A and Fig. IBare d = dt = 8 mm, d2 = 9 mm and d3 = 10 mm.

453

We have computed the maximum values of B withrespect to t and x, for different distances y from y = — 20mm to y = — 30 mm, for the case of one slab and for twoslabs and for different frequencies. The Iog10|fi| isdepicted in Fig. 3 for/0 = 100-500 Hz and in Fig. 4 for/o = 10-50 kHz.

-15.9

-16.3

-16.6

-17.0

-17.4

-17.8

-18.2

100HZlOOHz

200Hz^ m 200Hz

400Hz500Hz500Hz

-30 -29 -28 -27 -26 -25 -24 -23 -22 -21 -20y.mm

Fig. 3 logl0 \B\for one and two slabs as function of y for frequenciesf0 = 100-500 Hz

1 slab2 slabs

-26.9-28.5-30.1-31.7

^ -33.4

E -35.0m"^-36.6

o -38.2

-39 .9

-41 .5

-43.1

10kHz

10kHz

20kHz

20kHz• 30 kHz

.40kHz30kHz

50kHz"40kHz

50kHz

-30 -29 -28 -27-26-25-24 -23 -22 -21 -20y.mm

Fig. 4 log10 \ B\for one and two slabs as function of y for frequenciesfo = 10-50 kHz

1 slab2 slabs

In Fig. 5, the maximum values of log1 0 |B| withrespect to t, for y = — 20 mm, are depicted as functionsof x for various frequencies from/0 = 10-40 kHz, for oneand two slabs.

In Figs. 6 and 7, the maximum values of log10|B|with respect to t, for y = — 20 mm and x = 0, aredepicted as functions of frequency for one and two slabs.

5 Conclusions

We must mention that in our computations we haveincluded the displacement current term, which, as it wasexpected, has no influence on the results for the range offrequencies we have used. For higher frequencies, the

454

-43.8

Fig. 5 loglQ \B\for one and two slabs as function ofx for frequenciesfo = 10-40 kHz

1 slab2 slabs

-26.9

-28.5

-30.1

-31.7

-33.4

-35.0 PCD

E \ \: \ \= N \

E N

I:

Fin Mil mi mi in

y = -20mmx=0

\

M i l I I I 1 11 1 11 11 1 I^»V

_-36.6

jf-38.2-39.8

-41.5

-43.110 14 18 22 26 30 34 38 42 46 ' 50

f0 .kHz

Fig. 6 logi0 \B\ for one and two slabs for y = —20 mm, x = 0 asfunction of frequency

1 slab2 slabs

10 14 18 22 26 30 34 38 42 46 50

Fig. 7 \B\for one and two slabs for y = —20 mm, x = 0 as functionof frequency

1 slab2 slabs

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influence of this term can be examined. We have not usedthe classical penetration depth value S which is anapproximation figure. So our calculations are exactwithout any simplification.

The shielding effectiveness for one or two slabs, as afunction of the excitation frequency, is obvious: i.e. lowermagnetic flux density for higher frequencies, Figs. 3-5, inboth x and y directions.

From Figs. 6 and 7, we can deduce that the shieldingeffectiveness of the two slabs increases with the frequencycompared to the single shield. From Figs. 3 and 4, we candeduce that shielding is a linear function of y. Finally, wemust note that the influence of the different parameters ofthe problem, such as b,dud2,d3,n and a, on the shield-ing effectiveness can be easily examined.

Concerning the experimental verification of theanalysis, we are preparing a device appropriate for themeasurement of the magnetic flux density.

Eqns. 49 and 50 are solved and eqns. 5 and 6 areobtained, where y = (k2 — col fie +jco0o{j)112 and X =

6 References

1 SCHULZ, R.B., HUANG, G.C., and WILLIAMS, W.L.: 'RF shield-ing design', IEEE Trans., 1968, EMC-10, (1), pp. 168-175

2 MEREWETHER, D.E.: 'Electromagnetic pulse transmission througha thin sheet of saturable ferromagnetic material of infinite surfacearea', ibid., 1969, EMC-11, (4), pp. 139-143

3 FRANCESCHETTI, G.: 'Fundamentals of steady-state and transientelectromagnetic fields in shielding enclosures', ibid., 1979, EMC-21,(4), pp. 335-348

4 KRIEZIS, E.E., and ANTONOPOULOS, C.S.: 'Low-frequency elec-tromagnetic shielding in a system of two coaxial cylindrical shells',ibid., 1984, EMC-26, (4), pp. 193-200

5 PANAS, S.M., and KRIEZIS, E.E.: 'Field calculation in conductingmedia by means of FFT algorithm'. Proc. ICEM-82, GT 1/4, 1982

6 BRIGHAM, E.O.: 'The fast Fourier transform' (Prentice-Hall, 1974)

For the configuration of two slabs in Fig. IB, an ana-logous set of ordinary equations of the form of eqns. 49and 50 is easily obtained, where the form of eqn. 49 isvalid for regions i = 1, 2, 4 and 6, and the form of eqn. 50is valid for regions i = 3 and 5.

7.2 Calculation of the constants C,and D,The constants that must be calculated for the fourregions of Fig. 1A are Cl9 C2, C3, C4, Dlt D2, D3 and£>4. As Cx and £>4 are zero there are six constants. Fromeqns. 5 and 6 we form the system:

At = D1exp(-Xy)

A2 = C2 exp (Xy) + D2 exp ( — Xy)

A3 = C3 exp (yy) + D3 exp {-yy)

At = C4 exp (Xy)

(51)

Applying the boundary conditions of eqns. 9 and 10,where

Jf = 0 i = 2,3,4 (52)

7=T=grrn j = l (53)

y = b (54)

y = b (55)

y = 0 (56)

we obtain

1 dA2

Ho dy ~

1 dAY

dy

7 Appendixes

7.1 General solution of the MVPAs the Fourier transform of region i, At{k, y), with respectto x, is given by

t = At(k, y) = - f00y)e~jkx dx (46)

the transformation of eqns. 3 and 4 are given by the ordi-nary differential equations:

-k2A{ +dy7

and

dy2

Eqns. 47 and 48 are written as

dy2

and

d2Ai

dy2

>- = (k2-

= (jco k2)At

i = l , 2 , 4 (47)

i=3 (48)

i = l , 2 , 4 (49)

i = 3 (50)

\i dy no dy

dA

dy \i dy= 0

y = 0 (57)

y=-d (58)

y=-d (59)

y (60)

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We substitute eqns. 51 to eqns. 54-59 and evaluate theexpressions for the corresponding values of y. After somealgebraic manipulation, we obtain the system

Dx exp ( — Xb) = C2 exp (Xb) + D2 exp ( — Xb)

XC2 exp (Xb) — XD 2 exp ( — Xb)

+ XDX exp(-Xb) =

C3 + D3 = C2 + D2

C4 exp (-Xd) = C3 exp (-Xd) + D3 exp (Xd)

/UC4 exp (-Xd) - noyC3 exp (-yd)

+ nQ yD3 exp (yd) = 0 /

Solving the system of eqns. 60, the six remaining con-stants C2,C3,C^, £>!, D2 and D3 can be found and sub-stituted into eqns. 51. The current / is a common factor

455

and can be factorised out to obtain eqn. 11, where Rt aregiven by eqns. 12-16.

The Fourier transform / of the excitation current / ineqn. 1 is given by

K y, t) =f(y - b) exp (jco0 t)SFx\l{x)-\ = b (61)

For our numerical application I(x) = Io d(x) where S(x) isthe impulse function and Io = 1 A. So ^"^[/(x)] = 1 and

= T(k,y,t) =exp (jco01) for y = b0 for y # b

(62)

The same procedure can be applied to the geometricalconfiguration of the two slabs in Fig. IB, and the corre-sponding expressions for H,- in eqns. 30-36 can be found.

7.3 Calculation of the components Bx and BY of themagnetic flux density

From the well known relation V x A{f = Bt, for eachregion i, it can be easily proved that

(63)

so

dyand

dA< (64)

Taking the spatial Fourier transform of eqns. 64 withrespect to x, we obtain

dy(65)

Substituting eqns. 11 into eqns. 65, eqns. 17 and 18 areobtained. Taking the real parts of the inverse spatialFourier transform of eqns. 17 and 18 with respect to k,eqns. 19 and 20 are obtained.

The total magnetic flux density in both the x and ydirections is given by eqn. 42, from which we calculatethe magnitude | B, |

I COS ((O0t + </>/)] 2

+ [By i cos (O)Q t + 0,)]2]1'2 (66)

From eqn. 66, we obtain eqns. 43-44. The magnitude | B{ \is a function of x, y, t and its maximum instantaneousvalue with respect to time in eqn. 45 is obtained fromeqn. 43 by replacing cos (2w01 + a,) = 1.

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