low-frequency shielding effectiveness of a double cylinder enclosure

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-19, NO. 1, FEBRUARY 1977

Low-Frequency Shielding Effectiveness of a

Double Cylinder Enclosure

F. A. M. RIZK

Abstract-The low-frequency shielding effectiveness of a long dou-ble cylinder shield is determined through a solution of Maxwell's fieldequations. The shielding expression obtained is then compared withthe results obtained by both the circuit approach and the transmission-line analogy. The findings of the present paper are also compared withthe analysis by previous authors of the multishield problem. A digitalcomputer program for numerical evaluation of the effectiveness of adouble cylinder shield is developed and used to study the influence ofthe shield dimensions and material constants.

INTRODUCTION

S EVERAL methods are available for mathematical analysisof electromagnetic shielding effectiveness. The first

approach is by solving Maxwell's field equations subject toappropriate boundary conditions [1], [2]. Understandably,analytical solutions are available only for simple shields suchas a sphere or a long circular cylinder. A fundamental resultobtained by this method relates the external longitudinal mag-netic field Ho of frequency f and the corresponding internalfield Hi of a long circular cylindrical shield of internal radiusa, thickness d, and conductivity a, and permittivity p

Ho0yaap0-= cosh d +± sinhyd (1)Hi 2p

with y2 =jcog,w=2irf,j= T.A second method, called the planar shielding approach, is

based on the analogy between the equations of propagation ofplane electromagnetic waves through an infinite plane sheet,and the familiar transmission-line equations. This approachwas originally worked out by Schelkunoff [3], but significantlater developments were made by Schulz [41, [5] and hisassociates. Here the shielding effectiveness S is expressed as

S=A +R +B (2)

where A designates the absorption losses in the shieldingmaterial, R the reflection losses, and B the multiple reflectionlosses between the two interfaces of the plane sheet. Extensiveformulae and graphs of the above terms have been publishedby Schulz. Although in its original form this theory deals withan idealized situation, its practical use has been substantiatedconsiderably by the nonuniform shielding concept elaboratedby Schulz, where the different discontinuities and the mainshield are treated as parallel transmission lines.

Manuscript received April 21, 1975.The author is with the Hydro-Quebec Institute of Research,

Varennes, Quebec, Canada. (514) 652-8473.

The third method for shielding analysis is the circuitapproach. A basic investigation by Wheeler [6] showed that atlow frequency a spherical shield reduces the magnetic field bya factor Q, which is the quality factor of the equivalent coil.Further fundamental work by Miedzinski and Pearce [7],Miller and Bridges [8], as well as by Cooley [91 showed thatthe low-frequency magnetic shielding effectiveness of a cylin-drical shield could be expressed as

S= 201logl| Rs +

j sL |

Rs(3)

where L. is the equivalent inductance and R. is the equivalentlow-frequency resistance of the enclosure. Skin effect could betaken into account by multiplying R, by the factor x/d/6where 6 is the skin depth and d is the shield thickness. It hasbeen shown that [9] for a cylindrical shield at low frequencythe circuit approach yields identical results as those obtainedfrom field theory solutions developed by King [1]. In orderto account for the leakage flux (3) has been modified by Millerand Bridges [10] to take the form

S = 20log10l +jL

Rs ±jcoL1(4)

where LI is the leakage inductance of the shield.Multishield problems arise in several practical situations and

among these double shields are most frequent. Kaden [2] out-lined a general solution of the multishield problem, using atraveling wave approach. For cylindrical shields however,Kaden worked out in detail only an approximate solution fora double cyclinder shield, with the outer cylinder made of ahigh conductivity metal as copper or aluminium and the innershield made of a magnetic material as iron. With the symbolsused in this paper Kaden's expression reads

Ho 1172 sinh ldl sinh Y2d2Hi 2goyl

(5)

If, moreover, the shield thickness is smaller than the corre-

sponding skin depth, Kaden [2] obtained

- p1a2dld2i 2

(6)

where Ml, d4 are the permeability and thickness of the inner(iron) shield and 02, d2 are the conductivity and thickness ofthe outer shield, respectively.

14

RIZK: LF SHIELDING OF A DOUBLE CYLINDER

Harrison et aL [11 ] investigated the shiplding properties ofa cavity formed by two identical, infinite, nonmagnetic, im-perfectly conducting parallel planes. They also extended theanalysis to a double shield formed by these two infinite plates.After some practical simplifications, Harrison et al. obtainedfor the ratio of the outer (disturbing) to the inner magneticfields, with the symbols of this paper

-= , sinh 'yd[cosh yd + 'yA sinh yd]Hi jcoeo

z

d2 di

t~~~~~~~~~t~~~ ~~I a1(7)

where a is the conductivity of the metal, A is the spacingbetween the two shielding infinite planes.

In this paper, a more general formula for the low-frequencyshielding effectiveness of a long double cylinder shield ofarbitrary materials, thicknesses and diameters will be deducedby solving Maxwell's field equations. The solution will becompared with the transmission-line analogy, the- circuitapproach, and the above results by Kaden [2] and Harrisonet al. [11]. A digital computer analysis will illustrate the roleof the different parameters involved.

Fig. 1. Double cylinder shield.

some detail. For b < r < b + d2, the field equations are

CurlH = a2E

and

Curl E =-jWI12H

SOLUTION OF FIELD EQUATIONS FOR ADOUBLE SHIELD

Consider the two infinitely long coaxial cylinder shields ofFig. 1, with internal radii a, b and thicknesses dl, d2, and withmaterial conductivities a1, a2 and permeabilities k1, ,2,respectively. At low frequencies such that wavelengths are

much greater than the radius a, and for practical situationswhere the thickness of the shield d, is much smaller than theradius a, solutions for a single cylindrical shield obtained byprevious authors [ 1], [2], [9 ] yield

H(r) = const = Hi (8)

for 0 S r S a; while inside the shield material where a < r 6a + d,

H(r)=H1 Lcoshyl(r-a) + sinhyj(r-a)J (9)

where yj 2 = jcojil a1. With the skin depth 81 = 1/N/i ~j,

'yj and 81 are related by yj = (1 +1)/51'.At r = a + dl, H = H2 is given by

H2 = H cosh ldl + sinh 'yjd, (10)

which forH = Hz = H(r), reduce to

I d /dH\'r =7y22H

r dr\ dr /(14)

with y22 =jcoP2a2 -

The formal solution of (13) is given by [15]

H(r) =A1II(72r) +B1K.(72r) (15)

where IJ, and K,, are modified Bessel functions of the first andsecond kinds respectively, and Al, Bl, constants.

Noting that 'y2r = .V(r/52), with b < r < b + d2 it isobvious that our interest is in values of Y2r > 1, in whichcase the asymptotic expansions [151 of If and K,, substitutedin (15) yield the approximate solution

H(r) = [Al 'e'y2r+ Bl 'e-'2r1V'Y2(l5.a)

Al', Bl' being constants.Since we are normally dealing with thin shields, such that

d2 < b, then within the range b < r < b + d2, one can assume

\/r in (1 5.a) as being approximately constant. If, in addition,we express H(r) as a function of (r - b), (1 5.a) takes the form

H(r) = A 2eY2(r-b) + B2e-T2(r-b)Similarly, within the space between the two shields, a + d, <r < b; with the distance between the shields much smaller thanthe wavelength

H(r) = const =H2. (11)

The solution for the outer shield needs to be elaborated in

(16)

where A2 and B2 are constants to be determined from theboundary conditions.

Alternatively, (16) may be directly obtained from (14) ifthe variation of r in the term r(dH/dr) is neglected within theshield [2] for b < r < b + d2, a procedure which is somewhatless rigorous than the one adopted above.

(12)

(13)

15

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, FEBRUARY 1977

The boundary conditions are the following: at r = a + d,'dH\

a,E(a + dl) =dr / r=a+di

From (17), (9)

(17)

alr,E(a + d, ) = -[ yjsinh -yjd, +± ucosh'yldl Hi.L ~ ~ ~~2J

(18)Similarly, at r= b

IdH\a2E(b)=-() (19)

dr r=b

To determine E(b), we apply the induction theorem to thespace between the two cylinders

E(b) - 2rrb-E(a + dl) - 27r(a + dl)

=-jw[7rb2-r(a + dj)2] /i0H2. (20)

From which

2~ ~~~~(1E(b)=- I1-(_ H2

+ E(a + dl) * (a + dl) (21)

The external field H = Ho at r = b + d2 will then be related tothe inner field Hi by

Ho [ch d ±w10o0a s j

* jcosh Y2d2 +± o;2 [1( 1 )2]

* sinh 72d2 ±+ (a sinh '1d,2 bfY21O L

+ coshyldl sinhY2d2- (26)

This is the general expression of the low-frequency ratiobetween the external and internal longitudinal magnetic fieldsof the long double cylinder shield, as a function of the shielddimensions and material constants. The expression for thesingle shield obviously follows as a special case of (26) by sub-stituting d2 = 0.

SPECIAL CASES

i) At higher frequencies such that y1dj1 > I and 1y2d2 I>1, (26) simplifies to

Ho'=wl-+o a, a[I WloU2b[ a+dl)3Substitute for E(a + d1 ) from (18) in (21 )

E(b) = - b [1- ( )2] -a Hi

coshd',gl* fYl sinh yjd, + 2 coshldl | *

Substitute from (16) in (19)

+ (a + d)'Yl1a2b+Y2a1

(22)

With cvp0aa/2,yj1 I> 1, (27) reduces to

a2E(b) = -72A 2 + Y2B2 - (23)

From the continuity of the tangential magnetic field at r = b

H2=A2 B2- (24)

Solving (23), (24) for A2, B2 and substituting in (16) oneobtains

H(r)j=chdla .Hi=cosh yjd, + - ssinhy1d

Hi h 2,y,

*'|cohz2-b) ±I<X oU2b [(a+dli )22'y2 b/

'y1a92(a +dl) rsinh 72 (r-b) + [sinh yjd,

+ jwpo;1a cosh yd,] sinh Y2(r-b). (25)2,y,

Ho JjcOoaladl 51 ±I)oa2bd2 82= _ I +Hi 2(1 +j) d, 2(1 +j) d2

F[(a±d\d112 a±di 62a2b b 51a,

jcouoalad181 e 61 62

2(1 +±) dJ 4

If further 1 - ((a + dl)/b)2 is not negligibly small (28) couldbe approximated by

Ho Il,a,ad, 1 co11a2bd2 82

Hi 2 Nf2id1 2 N/?d2dl d2

[1 (a±l )2] 161 62 (29)

jwi-toalal e1d y22

L +T 2,yl24(27)

(28)

16


or in terms of the shielding effectiveness

S = 20 loglo (co 3oaladi 6 )2 NJ2id1/

C/copa2bd2 62+20(ogl 2 -\/d2 /

+201oglog1 [1- 1)] 2.04/

(30)

Since the term 20 log10 [1 - ((a + d1)/b)2] is always negative,it is realized that, under the above conditions (including a andb - (a + dl) much less than a wavelength), the resultantshielding is less than the sum of the individual shieldingeffectiveness of the two shields

If, on the other hand, there is no intermediate space, b =a + dl, and (28) reduces to

|Ho | cigoaladi 51HiIL 2 \J2dl

dl d2

Coa2ad2 62 e61 62+ 2 XJ~d2 J 4or

S = 20 loglo [ 2lor,ad, 6L 2 \/-2d1

wuoa2ad2 62 1±Fd±d2 0412 X,/~id2 L 61 62 J

The last term of (30) and (31) represents the absorption Iwhile the terms 61/Vdl, 62/V/Id2 account for theeffect. The symmetry in (31) indicates that, at highequencies, the two shields could be exchanged withoutencing the shielding effectiveness. As will be shown belovis not the case at lower frequencies.

ii) At lower frequencies such that yjd1 <1I and 172I1, then corresponding to (28) one obtains

-=~~ WIWA1alaal>| C)o2bd2

Hi \2 /l2

[lQ1a+d1i)22 a+dl

* ~~~IJXllo a2ad2. [It1U2did2 + 2

which differs from (28) essentially by the absence of abtion losses and skin effect.

With no intermediate space, corresponding to (31)obtains

-= 1±+j [ald + U2d2 ] +jcdld2i1 a2 -

Hi ~~2

Here it is obvious that a combination of high c2 and high Mlwill improve the shielding effectiveness. This effect will bemore pronounced at lower values ofP1,=0a/2,2ldl as will beshown by numerical computations.

TRANSMISSION LINE ANALOGY

Analogy has been established between the ratio Ho/Hi ofthe outer and inner longitudinal magnetic fields of a longcylindrical shield and the ratio Isl/r between the sending andreceiving end currents of an equivalent transmission line [12].This will now be extended to the case of a double cylindershield with intermediate space.

The system, represented as a four pole with constants A, B,C, D, consists of three cascaded transmission lines corre-sponding to the two shields and the intermediate space. Thetransmission lines are characterized by the propagation con-stants y1, Yo, 72, lengths dl, b - (a + dl), d2, and waveimpedances Zl, Zw, Z2 referring to the inner shield, inter-mediate space, and outer shield, respectively.

The terminating impedance Zr corresponds to the shieldedspace and has been previously [12] shown to consist of areactance jcoi0al/2. The equivalent four pole constants areobtained from those of the individual transmission lines bystraightforward matrix manipulations that are omitted here forspace limitation. The ratio I'I/r is obtained at:

-D+CZr (34)Ir th

Under the simplifying, but practically valid, assumptions that

(31) ZrYo [b-(a + dj)]

Zw

lossesskin and

r fre- Zjyj [b-(a + dj)]influ- <x this Zw

< and substituting for D, C, in (34), one obtains

=I[cosh y1d, +- sinh y 1d cosh 72d2r

+ z-7y [b-(a + dj)]sinhy2d2 |Z2 1Z2

jsinh-y1d, +±-rcosh ddl sinhY2d(32)

Withsorp-

one Z1=

(36)

(33) Z2 =UT2

(35)

17


AL0

co

7o= jiwViI7and

Zr = jcoa2

(36) takes the form

= [cosh ijd,+ 2 asinh yjd, cosh 72d2Lr 27yj 2Y

icoSoa2 b a + dl.,1a+ IClo-a2b sinh Y2d2 +712

7I2 b 72 Ol

* [sinh yjdj + cosh yldl] sinh 'y2d2-

(37)

Comparison between (37) and (26) shows that the transmis-sion-line analogy will be valid as a good approximation if thespacing between the two shields is much smaller than theshield radius, a condition which will often by satisfied inpractice. The two expressions are identical for b = a + dl, i.e.,with no intermediate space between the two shields.

CIRCUIT APPROACH

If the double cylindrical shield is treated as two cylindricalmagnetically coupled circuits, with radii a, b and qualityfactors Ql, Q2, respectively, straightforward manipulationsshow that

IHo a2-=(1 + jQl)(I +1Q2)--jQl .jQ2

which may be written as

Ho =(1 +jQl)

where

wL l

sl

a2 a2i2 1b2 b2 2

Q2 RRs2

one realizes that (39) is equivalent to (27) with the exceptionof

a) absence of absorption loss term in (39),b) neglecting unity with respect to ,ouora/212y in (39).

The absence of the absorption losses is natural in a circuitapproach and actually corresponds to practical situations withnoncontinuous shields where considerable field leakage is in-evitable. The second deviation is caused by considering fluxlinkages through the coil windings while neglecting flux link-ages within the shielding conducting material itself. Withvalues of (c,i0ajadj/2)(61/1V"2dj) > 1, such an approxima-tion is permissible.

If (1 -a2/b2) is not negligibly small, then with Q, > 1 andQ2 > 1, the term (a2lb2 )jQ2 in (39) will be negligible. Underthese circumstances and introducing skin effect, (39) takes theform

S=20loglo (Q \/-1 )+ 20 log10o (2

(2g) (40)

which is equivalent to (30) apart from the absence of theabsorption losses.

In using the expressions obtained from the circuitapproach, the effect of the finite dimensions of the enclosurecould be included in the inductance calculations [8], [10].Similarly the increase in the equivalent enclosure resistancesdue to discontinuities, intermittent bonding [15], etc., couldbe accounted for.

COMPARISON WITH PREVIOUS WORK

The results of the present analysis will, as far as possible, be(38) compared with the derivations of Kaden [2] and Harrison

etal. [11].Comparison of (6) and (33) shows that Kaden's formula (6)

corresponds to the last term of our expression (33). The factor2 is due to the fact that Kaden takes Ho as the amplitude ofthe incident wave, while in our analysis Ho corresponds to theresultant field magnitude at the outer surface of the shield. Itis clear that Kaden's analysis implicitly assumes that only thelast term of (33) is significant. This would be justified if

pOaal < 2p1d2u2 (41)

and

(42)

At higher frequencies, skin effect must be taken into con-sideration by increasing the equivalent shield resistances R81and R82 by the factors V/dl/61, \/d2/62, respectively.

Noting that for long continuous cylinders Q, = cog,,, a, ad, /2,Q2 = cop0 a2bd2/2, and taking skin effect into consideration,

While (41) may often be acceptable, it is easy to demonstratethat, for some interesting practical situations, (42) is far frombeing satisfied. Consider, for example, a high voltage test hallwith a = 40 m, d1 = 1 mm, p, = 1000 pu0; it follows thatPoa/2p1d, = 20. It is therefore concluded that Kaden's

18

M,,a < 2pldl.


formulae (5), (6) which yield a shielding effectiveness in-dependent of the shield radius are only valid if the shieldradius is so small that P1, oa/2tild, < 1, as, for example,in coaxial cables.

Expression (7), deduced by Harrison et al. [11], appliesonly to a double shield made of two identical infinite planeshields, each with thickness d, separated by a distance A andmade of a nonmagnetic good conducting material. As such, itcannot be directly applied to the problem of a double cylindershield treated in the present paper. However, (7) could be ob-tained from our expression (36), if Zr is taken equal to Z, =

Vp0/c0, which is appropriate for the plane shield terminacingimpedance, and by simultaneously assuming d, = d = d,b - (a + d) =A,g1p 2 = /lo,1 = a2 =o, yielding 1y=72 =7, Z1 = Z2-

Under these conditions (36) reduces to

-= [coshyzd +flg sinh yd [cosh yd + -yA sinh yd]Hi l'jeCo

(43)+ [sinh yd+ ^ coshyzd ]sinh yd.

If, moreover, we assume with Harrison et al. [11] that

tanhyd I>-==

Zw coE

(43) reduces to

-= 2a

sinh yd[cosh yd +yA sinh yd]H1 jcoe0

(44)

which is identical with Harrison's expression (7), except forthe factor 2 which relates to the definition ofHo as explainedabove.

NUMERICAL EVALUATION

The shielding effectiveness of a single-cylindrical shield canbe expressed [2], [12] in terms of two parameters: the ratiod/6 between the shield thickness and the skin depth, and theparameter P-,a/2pd. Similarly, it can be easily shown thatthe shielding effectiveness of a double cylinder shield; (26) canbe expressed in terms of four independent parameters d1/b1,d2/82, P1 -poa/2p1dl, P2 =p0ob/2p2d2. A digital computerprogram has been written for the numerical evaluation of (26).Although it is not possible to present the general expression interms of plane curves for arbitrary variation of the four param-eters, some special cases may be interesting.

Fig. 2 shows the results of shielding calculations for a

double shield, with the two cylinders of the same thicknessand material, within wide ranges of di/6l and P1 -and for afixed ratio of b/a. Similar curves for a single shield were previ-ously computed by Shenfeld [12]. Fig. 3 shows that for a

fixed value ofP1, the shielding effectiveness increases with theincrease of b/a, particularly at higher d1/11, i.e., at higherfrequencies.

100

80Omc>~60

40h

20

1.0d/ . p.u.

Fig. 2. Shielding effectiveness of long double cylinder shield in longi-tudinal magnetic field; both cylinders of same material and thick-ness with fixed ratio between inner radii.

1.0

Fig. 3. Shielding effectiveness of long double cylinder shield in longi-tudinal magnetic field; both cylinders of same material and thickness,for different ratios between radii. 1-No intermediate space. 2-b/a =1.02. 3-b/a = 1.04. 4-b/a 1.06. 5-b/a = 1.08.

Fig. 4 compares the shielding effectiveness of a doublecylinder shield consisting of a steel inner cylinder and a copperouter cylinder of equal thickness with a single steel cylinder ofthe same total thickness as the two shields. At lower fre-quencies the mixed double cylinder exhibits a clear superiority.

Finally, Fig. 5 shows the effect of the arrangement of thetwo cylinders. It is shown that for a steel-copper shield andparticularly for relatively small values of P8, it is advantageousover a wide frequency range to have the high conductivityshield adjacent to the disturbing field. Such practice has beenpreviously recommended by Kaden [2].

EXPERIMENTAL CHECK

We had the opportunity of checking (40) experimentally atIREQ's high voltage laboratory which is shielded againstelectromagnetic fields in order to allow sensitive partial dis-charge detection in high voltage insulation [14]. Magneticfield measurements were carried out inside and outside thelaboratory using a Stoddart NM25T radio noise meter. Thesource of interference was a remote radio station acting as aplane wave source with a frequency of 690 kHz. The labora-tory consists of a double cylinder rectangular enclosure withsteel shields of d, = d2 = 1 mm, equivalent radii [9] a =

31.1 m, b = 33.7 m and a length of 60 m. Shield inductance

I I II

P,=100 PO10 PI=1.0/ / //P~~1=O.1-

=~~~~~~~~~~~~~~~6 =1,,,,2,v-^A

19

a 11


aspects like end effects and intermittent bonding of the

shielding plates.Digital computation of the shielding effectiveness has been

carried out over a wide range of the parameters involved. Atlower frequencies, and only for shields of relatively small radii,a combination of a magnetic material shield and a non-magnetic high conductivity shield is shown to be superior to asingle magnetic material shield of the same total thickness.Conditions have been determined under which it would beadvantageous to install the high conductivity shield adjacent to

io the disturbing field in agreement with recommended practice.

Fig. 4. Magnetic shielding effectiveness of mixed shields, with nointermediate space. 1-Steel inner shield and copper outer shield. 2-Copper inner shield and steel outer shield. 3-Double steel shield. Ps,ds, bs refer to the steel shield.

100

80k

,, 60

40

20

0.1 101.0ds/8s

Fig. 5. Influence of shield arrangement on magnetic shieldingeffectiveness. 1-Steel inner shield and copper outer shield. 2-Cop-per inner shield and steel outer shield. PS, d5, bs refer to the steelshield.

calculations [13] took end effects into consideration whileresistance calculations accounted for the intermittent bonding[15] of the shielding plates. The individual shielding effective-ness values were obtained as 51 dB and 40 dB, and the result-

ant shielding effectiveness, at 690 kHz, by substituting in (30)

S=51 + 40 + 201oglo [1 )] 74 dB.

The measured value was 72 dB.

REFERENCES[1] L. V. King, "Electromagnetic shielding at radio frequencies,"

Phil. Mag. J. Sci., vol. 15, no. 97, pp. 201-223, Feb. 1933.[2] H. Kaden, Wirbelstri5me und Schirmung in derNachrichtentech-

nik. Berlin, Germany: Springer, 1959, pp. 71-91, 108-134.[3] S. A. Schelkunoff, Electromagnetic Waves. New York: Van

Nostrand, 1943, pp. 303-315.[4] R. B. Schulz, G. C. Huang, and W. L. Williams, "RF shielding de-

sign," IEEE Trans. Electromagn. Compat., vol. EMC-10, pp.168-175, Mar. 1968.

[5] R. B. Schulz, Shielding in Practical Design for ElectromagneticCompatibility, R. F. Ficchi, Ed. New York: Hayden, 1971,pp. 69-92.

[6] H. Wheeler, "The spherical coil as inductor, shield, or antenna,"Proc. IRE, 1958, pp. 1595-1602.

[7] J. Miedzinski and S. F. Pearce, "The performance of screeningrooms," Electronic Engineering, October 1950, pp. 414-419.

18] D. A. Miller and J. E. Bridges, "Geometrical effects on shieldingeffectiveness at low frequencies," IEEE Trans. Electromagn.Comput., vol. EMC-8, pp. 174-186, Dec. 1966.

[9] W. W. Cooley, "Low-frequency shielding effectiveness of non-uniform enclosures," IEE Trans. Electromagn. Compat., EMC-10, pp. 34-34, Mar. 1968.

[10] D. Miller and J. Bridges, "Review of circuit approach to calcu-late shielding effectiveness," IEEE Trans. Electromagn. Compat.,vol. EMC-10, Mar. 1968, pp. 5 2-62.

[11] C. W. Harrison, Jr, M. L. Houston, R. W. P. King, and T. T. Wu,"The propagation of transient electromagnetic fields into acavity formed by two imperfectly conducting sheets," IEEETrans. Antennas Propagat., Vol. AP-13, pp. 149-158, Jan.1965.

[12] S. Shenfeld, "Shielding of cylindrical tubes," ibid, pp. 29-34.[13] F. W. Grover, Inductance Calculations. New York: Dover,

1962.[14] G. Karady and N. Hylten-Cavallius, "Electromagnetic shield-

ing of high voltage laboratories," IEEE Transactions Power App.Syst., vol. PAS-90, no. 3, pp. 1400-1406, 1971.

[15] F. Rizk, Y. Gervais, and H. Luhrmann, "Performance of electro-magnetic shields in high voltage laboratories," presented at theWinter Power Meeting, New York, Paper T75 083-1, 1975.

[16] J. Irving and N. Mullineux, Mathematics in Physics and Engi-neering. New York: Academic Press, 1959.

SUMMARY

A general expression for the low-frequency shieldingeffectiveness of a long double cylinder shield in a longitudinalmagnetic field has been developed through a solution of Max-well's field equations. It has been shown that, if the distancebetween the two shields is much smaller than the shield radius,results of the transmission-line analogy will be equivalent to

the solution of the field equations.Apart from the absence of the absorption losses, the circuit

approach of the double cylinder shield yields results that are

found equivalent to those obtained from the field solutionand, furthermore, allows the inclusion of several practical

EDITORIAL SUMMARY

In the long history of electromagnetic shielding, various

technical papers have analyzed double shields, either for

restricted special cases or by means of analogy with trans-

mission equations. This paper extends the knowledge of low-

frequency shielding behavior for a long double cylinder bydirect solution for attenuation of the longitudinal magneticfield with Maxwell's equations using a digital computer pro-

gram.

,,

i0 - d,- d2 X

10 ,--/

!O , ,, 1 ,,

0. .

40

8

v) 6

1.0ds/8s, p.u.

//2

Ps= 0.1d1=d2

~~~~~~~~~~ I I I I 111i I

I , , , ,

U' ,.

20

O.A


Results (26)1 are obtained, subject to the following con-ditions:

1) the thickness of each shield is much smaller than itsinner radius (a, b of Fig. 1);

2) the smaller inner radius a is much less than the shortestwavelength;

3) the radial dimension, b - (a + d, ),between shields, isalso much less than the shortest wavelength.

Subject to these conditions, two important special casesshow that, for each wall thickness dl, d2 much greater thanthe skin depth 6,62,

1) shielding effectiveness of a double shield is alwaysless than the sum of the individual effectiveness ofeach separate cylinder (30) (the amount of differencecan readily be calculated from a term in (30));

2) for a laminated shield (no intermediate space), the twoseparate layers, even if composed of different mate-rials, can be interchanged without influencing theshielding effectiveness (31).

For each wall thickness much less than the skin depth,

1) absorption losses disappear (32);2) shielding effectiveness is better with a combination

of high outer-shield conductivity and inner-shieldpermeability than with the reverse combination (33);

3) this effect is more pronounced the smaller the ratioof inner-shield radius a to thickness d, (33).

A comparison of results using this approach (27) (for thinshields and radial dimensions much less than a wavelength)with the transmission-line analogy (37) shows the latter to bea good approximation if the spacing between shields is muchless than the shield radius, which is frequently true in practice.Both approaches yield identical results for zero spacing (lamin-ated shields).

'Note: In this paper Ho is not the incident field, but is the totalfield at the exterior surface of the shield, normally twice the incidentfield.

A comparison of results using the general solution for thisapproach (26) with the circuit approach (39) shows that thelatter a) neglects the absorption loss term and b) neglectsunity compared with (yo/p,l)(a/x/~6-). The former differencecorresponds to practical situations with noncontinuous shieldswhere considerable field leakage occurs. The second deviationis caused by considering flux linkages through the circuit-analogy coil windings (the interior space of the shield) whileneglecting flux linkages within the shielding material itself;this approximation is acceptable provided (po/pl,)(a/N/26 )> I(extremely low frequencies excluded). Several features of thecircuit approach are that a) the effect of finite dimensions ofthe shielding enclosure could be included in the inductivecalculations and b) the increase in equivalent enclosure resis-tances, due to discontinuities, intermittent bonding, etc.,could be taken into account.

Comparisons of results with prior effort of Kaden andHarrison et al. are made. Apparently, Kaden's formulas (5),(6) are valid only for (0,/u/,u)(aI2dL) < 1, as in coaxial cables.Harrison et al. consider a different case, i.e., a double shieldof two identical, infinite-plane shields; it cannot be directlycompared. However, with appropriate assumptions, identicalexpressions result.

The general expression (26) of this paper can be expressedin terms of two parameters for each shielding material, theratio d/6 between the shield thickness and the skin depth, andP = oa/2,2d, a size-permeability factor. These parametersare used to illustrate, in Fig. 2, results for cylinders of thesame thickness and material, given a fixed ratio of cylinderradii. Similar curves for a single shield were previously pub-lished by Shenfeld [12]. When the ratio of radii is permittedto increase, Fig. 3 shows the corresponding increase in shieldingeffectiveness.

Fig. 4 illustrates the superiority of a steel inner cylinderand copper outer cylinder of equal thickness to a single steelcylinder of the same total thickness.

Fig. 5 shows the advantage of having the high conductivitymember of a double shield adjacent to the disturbing field, apolicy previously recommended by Schelkunoff, Schulz,Kaden, and others.

RICHARD B. SCHULZ

21

low-frequency shielding effectiveness of a double cylinder enclosure

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