NASA Technical Memorandum 105837

Damping and Scattering of ElectromagneticWaves by Small Ferrite SpheresSuspended in an Insulator

Gerald W. EnglertLewis Research CenterCleveland, Ohio

August 1992

NASA

https://ntrs.nasa.gov/search.jsp?R=19930003178 2020-04-20T05:04:41+00:00Z

DAMPING AND SCATTERING OF ELECTROMAGNETIC WAVES BY SMALL FERRITE

SPHERES SUSPENDED IN AN INSULATOR

Gerald W. EnglertNational Aeronautics and Space Administration

Lewis Research CenterCleveland, Ohio 44135

SUMMARY

The intentional degradation of electromagnetic waves by their penetration into a media comprisedof somewhat sparsely distributed energy absorbing ferrite spheres suspended in an electrical insulator isinvestigated. Results are presented in terms of generalized parameters involving wave length and spheresize, sphere resistivity, permeability, and spacing; and their influence on dissipation of wave power byeddy currents, magnetic hysteresis, and scattering is shown.

SYMBOLS

S^ a2w/L.I r

'moo a2witolr

• radius of sphere

a Qcoefficients in equation (8)

B magnetic flux density

b pcoefficients in equation (7)

Cacosh (2a Re ^/—jp)

cospeed of wave travel

Cacos (2a Im

D electric displacement current

constant defined in equation (6)

E electric field vector

e 2.71828

H magnetic inductance

H(1) Hankel function of first kind

Im imaginary part of a complex quantity

J Bessel function of first kind

K power loss constants in equation (2)

k complex wave number

N number density of spheres per unit volume

P power

p parameter wplr

Re real part of a complex quantity

VMcoM

W

Sa sinh (2aRe V]p)

Shi hyperbolic sine integral

Si sine integral

s average spacing between center of spheres

sa sin (2alm djp)

t time

Un An — 1

Ugn (µn1 2 + ( 1 — 2 Re µn)

x distance in direction of wave travel

a2 Re jp

V I PI

p 2 Im V,P

V I PI

A wavelength

it permeability inside spheres, µ' — jµ"

µn normalized permeability, µ/µ o

µ o permeability of material in which spheres are immersed

T resistivity

W angular velocity

Subscripts

A applied wave

ec eddy current

by hysteresis

Superscripts

real part of complex quantity

" imaginary part of complex quantity

INTRODUCTION

Ferromagnetic oxides with their low electrical conductivity and large skin depth can be effective inabsorbing electromagnetic wave energy (refs. 1 and 2). Their low weight densities, approximately 60 per-cent that of iron (ref. 3), make them attractive for flight applications. Spheres will be representative ofthe absorbent particles in the model compositions herein. Eddy currents are induced in the spheres bythe applied electromagnetic waves resulting in dissipation of field energy by electrical resistance. Mag-netic hysteresis phenomena cause additional losses. This evolves from the continual distortion of thecrystalline structure of the Weiss domains and spin reorientation of the bound electrons induced by thewaves (ref. 2, p. 16).

2

The intensity of interaction and resulting scattering of a plane electromagnetic wave by a conduct-ing sphere falls off rapidly as the cube of the outside distance from the center of the sphere (ref. 4). Thisallows approximation of the multiple sphere problem by a succession of binary interactions except in thecase of closely packed spheres.

Absorption of electromagnetic wave energy by small individual ferrite spheres was treated in refer-ence 5. Two algebraic equations were derived expressing dimensionless power dissipation parameters foreddy currents and for hysteresis loss phenomena. These equations include complex magnetic permeability,µ, normalized by the permeability of the surrounding media, µ o , and a parameter of variables, a2wµo/rinvolving angular wave frequency, w, sphere radius, a, sphere resistivity, r, as well as µ o . The presenteffort also includes power loss by inelastic scattering of the incident waves by the ferrite spheres and istreated by use of a complex wave number, k, (ref. 6).

Particles used in composites to isolate electronic equipment, as well as in wall coatings to dampunwanted electromagnetic radiation, are usually much smaller than the radiation wave length. At suchlow ratios of particle size to wave length diffraction plays a minor role and has been omitted.

ANALYSIS

The essential power balance equation for absorption of electromagnetic energy by means of eddycurrents, P eC , and magnetic hysteresis P hy , induced in ferrite spheres can be expressed as

1 a 5A ABA aBA 1 aB2 1 (3)EA + HA = HA = _ —N (P eC + Phy)2 at at at 2µA at

The applied electric and magnetic fields contribute equally in the initial power source. Here H A and BA

are the magnetic induction and flux density of the wave whereas E A and DA are the electric field inten-sity and electric displacement vectors respectively. N is the number density of spheres per unit volumeof composite. Power densities P eC and Phy are equal to constants Kec and Kh respectively timesthe square of the magnetic field of the applied wave, B A , as in equations (33) and (35) of reference 5.

Including an additional power loss term, P . = BA K s to represent elastic scattering of the incident

wave, and therefore further loss of return signal, gives

2dBA - 2^AN

(Kec + Khy + KS)dx

B 2 coA

where c o = dx/dt is approximated by the speed of wave travel in the carrier material in which the spheresare immersed.

Integrating both sides of (2) gives the stopping power formula

BAN = BA (6 ) e -AA N (Ke,: +Khy +Ka)

x/c(3)

(2)

3

By (ref. 5) the constant, K eC , for eddy currents, is

KeC _ PQC - 37rW 2a5 µ n 1 2 s^I Sa IM + sa Re - (Ca - C

B 2 r-9Re sal ^IA

and for hysteresis

2 5 " C + ca a5 – 08a – Ca Sa sK Phy _ 9^rW a µ ^ 1L.1 I.^I a 2 + ,(j2 – 1 a+ 1 a ^ a _ a + _ aby BA r^ 6 ,^— 3 IsV I I.4,I3/2

a 0

2 2 2 2

+ 4 + p - a a Shi(2Re V s^ ) - 4 + a - p Si 2 Im3 6 2 3 6 2

where parameter sad = a2Wµ[r, normalized complex permeability µn = (µ' - jµ")/µ o , j = v'-I, and µois permeability of the media in which the spheres are suspended.

The constant -Q in the denominator of equations (4) and (5) is

-9 = U , I Si I(^a +ca) +LUG +2Im (sa7 µn) + I I2-2Im s^ J1Ca-Cal

– I gn[ (Sa + sa) Re S-It – (Sa – sa) IM VV

(6)

- (Sa - sa ) Re (U. .W 47) - (Sa + sa) IM (Un sw V sad! )

where Un = An - 1, U2 = Iµ n 1 2 + (1 - 2 Re µ n), p = WµI r, Ca = cosh (2a Re 3,gyp),

Ca = cos (2a Im Vp), 4a = sinh (2a Re (j), and s a =sin (2a Im ).

Utilizing the scattering cross section, a, of page 236 of reference 7 or page 449 of reference 8 gives

Kg = P s = co = arc

E (2Q + 1)(Ia0I2 + IbQ12)B 2 2µA µAk2 Q-1A

where a Q and b Q are expressed in Bessel functions of complex argument as

a = _ J,+1/ 2(ka) b p

- (Q - 1)J, +1/2 (ka) - kaJp - 1/2(ka)

QH(1) / (ka) (Q - 1)H (1) / (ka) - kaH (1) (ka)

P+1 / 2 9+1 / 2 9-1 / 2

(4)

(5)

(7)

(8)

4

and

2 2

k2 =

[ W J

+ jWµ = 27r

+ w( + jlL,)c T A T1

Three coefficients were used in each of the rapidly converging series of equation (7). They can beexpressed as

_ _ sin(ka) - ka cos(ka)alka[sin(ka) - j cos(ka)]

3 cos(ka) + 1 - 3

sin(ka)

a2ka (ka) 2 = ,

ka cos(ka) + 1 - 3 2 sin(ka) + ka sin(ka) - 1 - 3 2 cos(ka)(ka) (ka)

1 - 51 cos(ka) + 5! _ 31 sin(ka)a3 = 8(ka)2 8(ka)3 ka

(11)t t i

1 - 5 ' + j 3' - 5 ' [cos(ka) + j sin(ka) ]8(ka)2 ka 8(ka)3

bl - [l - (ka) 2 ] sin(ka) - ka cos(ka)(12)

dkaIka - j [( ka) 2 - 1] }

b2 [3(ka) 2 - 6] sin(ka) - [(ka) 3 - 6ka] cos(ka)

Idka (ka) 3 + 6ka + j[6 - (ka)2 J

(13)

and

b3 -= (ka) 4 sin(ka) + 6(ka) 3 cos(ka) - 21(ka) 2 sin(ka) - 45[ka cos(ka) - sin(ka)]

r l(14)

ejka 2 (ka) 3 + 45ka - j r (ka) 4 - 18(ka) 3 - 3(ka) 2 + 45kaJ

(9)

(10)

5

by use of pages 966 and 967 of reference 9.

Next consider the generalized distance to reduce B A(x)/BA (0) to 1/e and let number densityN = 1/s3 , where s is the average spacing between centers of spheres. The exponent in equation (3)written in terms of dimensionless parameters, then reduces to —1 when

3a wx _ 1 / r Pec + I'hy)

(a2wµA/ r) + A/a

(2Q + 1)(^ag1 2 + IL-01 (15)

s c w2a5B2 2(ka)2 1

The individual contributions of eddy currents and hysteresis are given in reference 5 over wide ranges of

d.-

RESULTS

The generalized distance of wave travel, (a/s) 3 wx/c, in the composite to reduce B A(x)/BA (0) to 1/eis plotted versus µ'/µ o and µ" /µ o for fixed values of a2wµ o/r on three-dimensional plots of figures 1(a)to 2(h). The range of generalized distance decreases with increase of parameter a2wµ o/r for absorption(figs. 1(a) to (d)) as well as for scattering when holding A/a constant: figures 2(a), (e), and (g) withA/a = 103 and figures 2(b), (f), and (h) with A/a = 105.

The peaks of the surfaces move to higher µ' /µ o as a2wµ o/r and/or A/a decrease. Pronouncedconcave trough-like contours indicate the minimum distance of absorption whereas evident convex patternsgive the maximum distances for reduction of scattering. The dashed lines shown are merely diagonalswhere µ' = µ".

As A/a is decreased from 105 the ridges gradually rotate their directions from diagonal to ones moreclosely parallel to the µ" /µ o axis. With further decrease of A/a the contours flatten out to nearly con-stant values of generalized distance. This is shown over an especially wide range of A/a in figures 2(c)to (f). This trend also appears as a 2wµ o/r is reduced with A/a held constant. For example, note thechange in curve pattern between figures 2(g), (e), and (a).

The generalized distance to reduce BA (x)/BA M to 1/e by absorption is nonsymmetric about thedashed line due to the lack of symmetry in the contribution from magnetic hysteresis. The results forscattering are symmetric when like values are selected for µ' and µ".

CONCLUDING REMARKS

By use of generalized parameters the seven independent variables: sphere radius a, complex per-meability µ' - jµ", resistivity r, mean distance between spheres s, electromagnetic field strength BA,and wave length A times frequency combine to (1) a generalized distance of wave travel, (a/s) 3 wx/c, toreduce the applied magnetic field ratio, B A (x)/BA(0) to 1/e of its initial strength, (2) a power absorp-tion parameter a2 wµ o/r, and (3) magnetic permeability ratios µ' /µ o and µ" /µo.

The range of generalized distance for reducing B A(x)/BA(0) to 1/e by absorption decreases withincrease of a2wµ o/r over its range of 10"7 to 10 while the peaks of these plots reduce from the order of107 to 103 . The dashed lines shown are merely diagonals where µ' = µ".

6

The three-dimensional plots of generalized distance by scattering, however, are symmetrical aboutthe dashed lines. Also an additional parameter, wave length divided by sphere radius, A/a, enters as anindependent variable for these plots.

REFERENCES

1. VonAulock, W.H.: Handbook of Microwave Ferrite Materials. New York: Academic Press, 1965.

2. Smit, J.; and Wijn, H.P.J.: Ferrites. New York: John Wiley & Sons, 1959.

3. Schlicke, H.M.: Essentials of Dielectromagnetic Engineering. New York: John Wiley & Sons, 1961,p. 75.

4. Smythe, W.R.: Static and Dynamic Electricity. 3rd ed., New York: McGraw-Hill, 1968, p. 377.

5. Englert, G.W.: Parametric Study of Power Absorption from Electromagnetic Waves by Small FerriteSpheres. NASA TP-2949, 1989.

6. Stratton, J.D.: Electromagnetic Theory. New York: McGraw-Hill, 1941, p. 392.

7. Panofsky, W.K.H.; and Phillips, M.: Classical Electricity and Magnetism. 2nd ed., Cambridge, MA:Addison Wesley, 1955, p. 236.

8. Kerr, D.E.: The Radar Cross Section of Isolated Targets. Propagation of Short Radio Waves,D.E. Kerr, ed., M.I.T. Radiation Laboratory Series 13, 1951, p. 449.

9. Gradshteyn, I.S.; and Ryzhik, I.M

Table of Integrals, Series, and Products. New York: AcademicPress, 1965, pp. 966-967.

7

107 3Ic

^N

105U

103

101 D

10" 1 rod

10-3 y

102 0

1t 101 11'0

10010-1Peke

`naty^ o \ma9

(a) a2wµ0 T =10-7

190.

S

910@did

xU3

104 _

ai103

10 2 mN 103

3°105 M

q10

mU

103 m_N

10 N

dto - ' ac0 2 0

"V-0

ryk, 10010"1pettaG_

o\^ag\aaty

(b) a2wµ0/T =10-5

1(

9°a

xU30-

103 IV 10aiU

102 10_N

10 1 dN19 110°

10 Ot dqP 102102 (710 1 Iwo ^^4 1 101 ^ I^0

01 10° eap`Uty P ° nSdy6̂%^ ° 10 etmea

.4.10° 10 " 1acy Pe 4.10 10 - 1 ^y p

o \caa9^ao \^a9`a

(c) a2WµVT =10-3(d) a2wµ0/T =10-1Figure. 1—Generalized distance to reduce B A (x)/BA (0) to 1/e by absorption.

8

xu3

103

102

100

102

1V'o.101 10 1.0, w

111 Do X02

'4100 10-1 \^yty Peo \^a9

(a) a2wµ0/, =10-7 , Va=103.

1

9

M_CUM

Ncro

N_

N

12

v0

3I°

9I

RL^'w

103 ClCU

N_

U

101 N

v10

-1 y

102 (7

101 101 p t^°10

0 ev

k. 100101e^

4,ell

aaN P0

(b) a 2wµ 0/T =10-7,)L/a=l 05.

9a

S

xIU3o^(^N

UCro

rom

dCO

xI U

*1,D2 102

102

100 w101103

10-9100

102 °'A

101101D1 `^ ^ ;IV'°

ao e

1001 0-1.4.10010'10o \̂ a9̂a o ^^a9`P

(c) a2,µ0/-, =10 -5 , Va=10. (d) a2,µ0/, =10-5 , Va=102.Figure. 2—Generalized distance to reduce B A (x)/BA (0) to 1/e by scattering.

9

x U3M.-.t^N

U101

N_

10-1N

10- v

2 102 c7101 101l

P1re

i. 100 Oox)\

4. 10010_1 aH ve^eo ^^a9

a2wµ0/, =10-5 , a/a=105.

xIU3

IeIN

1 30 ycwN

101 a

N

10 -1 d

)2 102 c71 t01 No610

F

-"loo a 1 "\", V

' 100 10'1 peke.. \eacho \ma9

(e) a2wµ 0/T =10-5 , \/a=103.

1C

d

103

10

x3 IU

fI^IN

dVC

10-2 roND

10-

10- d102

(7

10 1V ^^ I I100

1101 IP e

VIII y abyK+w

100 10-1 Qe^e\^a

o ^^a0

(h) a2wµ0/T =10-3, ),/a=l 0'.

103

9

102

3°

f^N

UUC

100 m

=oa10-2 N

10-4 (D02 m

C7

101

1

1 ty1^a

100 ^e10-1

aN pe

11

mQ

1

Figure. 2—Concluded

10

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Damping and Scattering of Electromagnetic Waves by Small FerriteSpheres Suspended in an Insulator

W U — 505-62-526. AUTHOR(S)

Gerald W. Englert

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National Aeronautics and Space AdministrationLewis Research Center E-6363-1Cleveland, Ohio 44135-3191

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National Aeronautics and Space AdministrationWashington, D.C. 20546-0001 NASA TM— 105837

11. SUPPLEMENTARY NOTESGerald W. Englert, NASA Lewis Research Center, Cleveland, Ohio. Responsible person, Gerald W. Englert,(216) 433-5887.

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13. ABSTRACT (Maximum 200 words)

The intentional degradation of electromagnetic waves by their penetration into a media comprised of somewhatsparsely distributed energy absorbing ferrite spheres suspended in an electrical insulator is investigated. Results arepresented in terms of generalized parameters involving wave length and sphere size, sphere resistivity, permeability,and spacing; and their influence on dissipation of wave power by eddy currents, magnetic hysteresis and scatteringis shown.

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Magnetics 1216. PRICE CODE

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