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

Englert, Gerald W.

English

NASA Technical Reports Server

NASA Technical Memorandum 105837
Damping and Scattering of Electromagnetic
Waves by Small Ferrite Spheres
Suspended in an Insulator
Gerald W. Englert
Lewis Research Center
Cleveland, Ohio
August 1992
NASA
https://ntrs.nasa.gov/search.jsp?R=19930003178 2020-03-17T10:26:35+00:00Z
DAMPING AND SCATTERING OF ELECTROMAGNETIC WAVES BY SMALL FERRITE
SPHERES SUSPENDED IN AN INSULATOR
Gerald W. Englert
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
SUMMARY
The intentional degradation of electromagnetic waves by their penetration into a media comprised
of somewhat sparsely distributed energy absorbing ferrite spheres suspended in an electrical insulator is
investigated. Results are presented 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 scattering is shown.
SYMBOLS
S^	 a2w/L.I r
'moo	 a2witolr
•	 radius of sphere
a Q	coefficients in equation (8)
B	 magnetic flux density
b p	coefficients in equation (7)
Ca	cosh (2a Re ^/—jp)
co	speed of wave travel
Ca	cos (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
VM
coM
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
a	
2 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 in
absorbing 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 of
the absorbent particles in the model compositions herein. Eddy currents are induced in the spheres by
the 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 the
crystalline structure of the Weiss domains and spin reorientation of the bound electrons induced by the
waves (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). This
allows approximation of the multiple sphere problem by a succession of binary interactions except in the
case 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 for
eddy 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/r
involving angular wave frequency, w, sphere radius, a, sphere resistivity, r, as well as µ o . The present
effort also includes power loss by inelastic scattering of the incident waves by the ferrite spheres and is
treated 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 damp
unwanted electromagnetic radiation, are usually much smaller than the radiation wave length. At such
low 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 eddy
currents, 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 volume
of composite. Power densities P eC and Phy are equal to constants Kec and Kh respectively times
the 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
2
dBA - 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 spheres
are 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 S
a IM	 + sa Re	 - (Ca - C
	
B 2 	 r-9Re sal
	 ^I
A
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 +	 _ a
by 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 Im
3	 6	 2	 3	 6	 2
where parameter sad = a2Wµ[r, normalized complex permeability µn = (µ' - jµ")/µ o , j = v'-I, and µo
is 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) + L
UG +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)
	
Q	
H(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 be
expressed as
_ _ sin(ka) - ka cos(ka)
al	
ka[sin(ka) - j cos(ka)]
3 cos(ka) + 1 - 3	 sin(ka)
a2	
ka	 (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)
I
dka (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 density
N = 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
3
a	 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/e
is 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) with
A/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. Pronounced
concave trough-like contours indicate the minimum distance of absorption whereas evident convex patterns
give the maximum distances for reduction of scattering. The dashed lines shown are merely diagonals
where µ' = µ".
As A/a is decreased from 105 the ridges gradually rotate their directions from diagonal to ones more
closely 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 the
change 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 the
dashed line due to the lack of symmetry in the contribution from magnetic hysteresis. The results for
scattering 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, to
reduce 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 with
increase of a2wµ o/r over its range of 10"7 to 10 while the peaks of these plots reduce from the order of
107 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 about
the dashed lines. Also an additional parameter, wave length divided by sphere radius, A/a, enters as an
independent 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 Ferrite
Spheres. 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: Academic
Press, 1965, pp. 966-967.
7
107 3Ic
^N
105
U
103
	
101
	D
10" 1
 ro
d
10-3 y
	
102	 0
1t	 101	 11'0
10010-1	 Peke
`naty
^ o \ma9
(a) a2wµ0
 T =10-7
1
90.S
910
@did
xU3
104 _
ai
103
10 2
 mN 103
3°
105
 M
q10
mU
103
 m
_N
10	 N
d
to - '	 ac0 2	 0
"V-0
ryk, 10010"1	
pettaG_
o\^ag\aaty
(b) a2wµ0/T =10-5
1(
9°a
xU3
0-
103 
IV 10
ai
U
	
102 	 10
_N
10 1
 dN
19 	 110°
	10 Ot	 dqP 102	102	 (7
10 1 	 Iwo	 ^^4	 1	 101	 ^ I^001	 10°	 eap`Uty P	 °	
nSdy6^%^	 °	 10 etmea
.4.10° 10 " 1
	acy Pe	 4.10 10 - 1 	 ^y po \caa9^a	
o \^a9`a
(c) a2WµVT =10-3	(d) a2wµ0/T =10-1
Figure. 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
11
1 Do X02
'4100 10-1 \^yty Pe
o \^a9
(a) a2wµ0/, =10-7 , Va=103.
1
9
M_
CUM
N
c
ro
N_
N
12
v
0
3I°
9I
RL^'
w
103 Cl
CU
N_
U
101 N
v
10-1 y
102 (7
101	 101	 p t^°
100
 ev
k. 100101
	
e^
4
,ell
aaN P
0
(b) a 2wµ 0/T =10-7,)L/a=l 05.
9a
S
xIU3
o^(^N
UC
ro
ro
m
d
CO
xI U
*1,D2 102
102
100 w101103
10- 9100
102	
°'A
101	
101D1 `^	 ^ ;IV'°
ao	 e
1001 0-1.4.10010'10
o \^ a9^a 	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 U3
M.-.t^N
U
101
N_
10-1
N
10- v
2	 102	 c7
101	 101l 	 P1re
i.	 100 Oox)\
4. 10010_1 aH ve
^eo ^^a9
a2wµ0/, =10-5 , a/a=105.
xIU3
IeIN
1 30 y
c
w
N
101 a
N
10 -1
 d
)2 	102	 c7
1	 t01	 No610
F
-"loo  a 1 "\", V
' 100 10'1	 peke
..	 \each
o \ma9
(e) a2wµ 0/T =10-5 , \/a=103.
1C
d
103
10
x3 IU
fI^IN
d
VC
10-2 roND
10-
10- d
102 (7
	10 1V ^^ I I	
100
1101
	IP e
	
VI I	 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
U
UC
100 m
=o
a10-2 
N
10-4 (D
02	 m
C7
101
1
1	 ty1^a
100 ^e
10-1 
aN pe
11
mQ
1
Figure. 2—Concluded
10
Form ApprovedREPORT DOCUMENTATION PAGE OMB No. 0704-0188
Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,
gathering and maintaining the data needed, and completing and reviewing the collection of information. 	 Send comments regarding this burden estimate or any other aspect of this
collection of information, including suggestions for reducing this burden, to Washington Headquarters Services. Directorate for information Operations and Reports, 1215 Jefferson
Davis Highway, Suite 1204, Arlington, VA 	 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 	 20503.
1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
nu,1 1 ust 1992 Technical Memorandum
4. TITLE AND SUBTITLE 5. FUNDING NUMBERS
Damping and Scattering of Electromagnetic Waves by Small Ferrite
Spheres Suspended in an Insulator
W U — 505-62-526. AUTHOR(S)
Gerald W. Englert
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESSES) 8. PERFORMING ORGANIZATION
REPORT NUMBER
National Aeronautics and Space Administration
Lewis Research Center E-6363-1
Cleveland, Ohio 44135-3191
9. SPONSORING/MONITORING AGENCY NAMES(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING
AGENCY REPORT NUMBER
National Aeronautics and Space Administration
Washington, D.C. 20546-0001 NASA TM— 105837
11. SUPPLEMENTARY NOTES
Gerald W. Englert, NASA Lewis Research Center, Cleveland, Ohio. Responsible person, Gerald W. Englert,
(216) 433-5887.
12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE
Unclassified - Unlimited
Subject Category 32
13. ABSTRACT (Maximum 200 words)
The intentional degradation of electromagnetic waves by their penetration into a media comprised of somewhat
sparsely distributed energy absorbing ferrite spheres suspended in an electrical insulator is investigated. 	 Results are
presented 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 scattering
is shown.
14. SUBJECT TERMS 15. NUMBER OF PAGES
Magnetics 12
16. PRICE CODE
A03
17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT
OF REPORT OF THIS PAGE OF ABSTRACT
Unclassified Unclassified Unclassified
NSN 7540-01-280-5500
	
Standard Form 298 (Rev. 2-89)
Prescribed by ANSI Std. Z39-18
298-102


Damping and scattering of electromagnetic waves by small ferrite spheres suspended in an insulator

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server