Office of Naval ResearchContract N00014- 75-C-0648 NR-372-012
ELECTRICALLY SMALL LOOP ANTENNA LOADED BY AHOMOGENEOUS AND ISOTROPIC FERRITE CYLINDER-PART 11
By \,~ x
D.V. Girl aid R.W.P. Klig
July 1978
Technical Report No. 668
This document has been approved for pfblLc releaseAnd sale; Its distribution ii unlimited. Re prod uc tLon in
whole or In part in permitted by the U. S. Government,
Division of Applied Sciences
Harvard University Cambridge, Massachusetts
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1. REORT UMB. ECOPNTR ACTAOR OGRN NUMBER(S
Techns/ica Repr No. P.8 Vig
9. TITERO IG ORANnAIO NAMEtle AND ADDES REOR & PEIO COVERE-
HarvardIUniversity /a L9amT , Masahset
9I. CERONROLING ORGAICEIO NAME AND ADDRESS
Division f Ancliassiieiee
I16. COSTROIBUING STIC ATMEANT (oDDiREport
14. DOISTORIBUTION NSTATE MEN (o thDeaStract entferedt frmCnB t rollin 0 Idfferent from SEUIYCAS o hsreport)
6. DSTIUPTLEMNSTARYNTESET(fti eot
19. KEY WORDS (Continue on reverse side if necessary and Identify by block number)
magnetic current on finite ferrite-rod antennaapproximate 3-term solutionnumerical solution of coupled inteigral equationsexperimental verification
20, ABSTRACT (Continue on reverse side If necessary and Identify by block number)
.Two theoretical approaches are developed to determine the magnetic current dis-tribution on a ferrite cylinder of finite length that is center-driven by anelectrically small loop anteng'a carrying 'a constant current. Tile first methodmakes use of thle analogy betwe~en the ferrite rod antennn and the conducting cy-lindrical dipole antenn to erive an` integral equation for the magnetic currenton an infinitely permeable o'j7r- e') ferrite antenna that corresponds to the in-tegral equation for thle elect c'iccurrent on a perfectly conducting electric di-
I (Continued)
DID I A 1473 F~OITIOtJ OF I NOV A5 IS OFISCoLITr Ucasfe
SECUPIT'f CL ASSIFICATION OF THIS PACE (Vhq'n On~to Fnrifrrd)
SECURITY CLASSIFICATION OF THIS PA*9~(V cc fLe..E. td)
20. Abstract (Continued)
-pole antenna. In the limit h (ýV this integral equation is shown to agree with1that obtained previously in Part I for the infinite ferrite rod antenna. Con-tinuing to parallel the treatment of the electric dipole aInt~nna, the integralequation is modified by the introduction of an internal imp, , dance per unitlength of the magnetic conductor to account for values of pthat are large butnot infinite, and finally an approximate, three-term expre'asion is derived forthe current on an 'imperfectly conducting' magnetic conductor. The second, morerigorous theoretical approach obtains two coupled integral equations in terms ofthe tangential electric field and the tangential electric surface current fromindependent treatments of the interior (ferrite) and exterior (free space) prob-lems. The coupled equations are then solved numerically by means of the momentmethod, Finally the results of the two theories are compared with experimentalmeasurements made on eleven different antenna configurations. Teareetigood.
pj1
li
1
Unlasiie
SEUIYCASFIAINO HSPAEi.O." d
Office of Naval Research
N00014-75-c-0648 NR-372-012
ELECTRICALLY SMALL LOOP ANTENNA LOiDED BY A
HOMOGENEOUS AND ISOTROPIC FERRITE CYLINDER -- PART II
By
D. V. Girl and R. W. P. King
Technical Report No. 668
-his document has been approved for public release andsale; its distribution is unlimited. Reproduction invhole or in part is permitted by the U. S. Government.
July 1978
The research reported in this document was made possible throughsupport extended the Division of Fngineering and Applied Physics,Harvard University by the U.S. Army Research Office, the U.S.Air Force Office of Scientific Research, and the U.S. Office ofNaval Research under the Joint Servicew Electronics Program byContracts N00014-67-A-0298-OO05 and NOOOlG-75-C-O648.
Division oi Applied Sciences
"Harvard University . Cambridge, Massachusetts
TABLE OF CONTENTS
PAGE
1. INTRODUCTION ............................ 1
2. PROBLEM OF A FINITE FERRITE ROD ANTENNA .................. . . .. 1
3. FERRITE aS A PERFECT MAGNETIC CONDUCTOR ........ .............. 2
4. MAGNETIC CURRENT ON AN INFINITE ANTENNA ........ .............. 8
5. FERRITE AS AN IMPERFECT MAGNETIC CONDUCTOR ....... ............. 10
6. THE LIMITATIONS OF THE THEORETICAL FORMULATION ...... ........... 24
7. A MORE RIGOROUS TREATMENT OF THE FINITE ANTENNA ............ .... 32
8. NUMERICAL SOLUTION BY THE MOMENT METHOD OF THE
COUPLED INTEGRAL EQUATIONS ...... ................... ..... 45
9. EXPERIMENTAL MEASUREMENT OF THE MAGNETIC CURRENT . . ....... ..... 52
10. SUMMARY ............. .............................. ..... 72
APPENDIX A . . .. . .............. ......................... ..... 75
APPENDIX B ............ ....................... ........ ...... 77
APPENDIX C ............ ............................... ...... 81
APPENDIX D.. .. . .............. ... ............... . ..... ... 106
ACKNOWLEDGMENT ............ ............................. ..... 117
REFERENCES .................. ............................... 117
Ii
ELECTRICALLY SMALL LOOP ANTENNA LOADED BY A
HOMOGENEOUS AND ISOTROPIC FERRITE CYLINDER - PART II
By
D. V. Girt and R. W. P. Kinp
Division of Applied Sciences
Harvard University, Cambridge, Massachusetts 02138
ABSTRACT
The problem of a finite, ferrite-rod antenna has been treated theoreti-
cally by recognizing an analogy between the ferrite antenna. and the conduct-
ing cylindrical dipole antenna which has been studied extensively. Initially
the ferrite is idealized to be a perfect magnetic conductor and an Hall~n
type of integral equation (1] is obtained for the magnetic current. By al-
lowing the antenna height to approach infinity, the formulation is shown to
be consistent with previously obtained results for the infinitely long fer-
rite antenna [2]. Subsequently, the integral equation is modified appropri-
ately to treat the ferrite as an imperfect magnetic conductor, and the cur-
rent is obtained in the three-term form of King and Wu (3]. Because this
treatment relies rather heavily on a mathematical equivalence of the two
problems under idealized driving conditions, an alternative, more rigorous
formulation is presented. The result is a pair of coupled integral equations
in the tangential electric field (or magnetic current) and the circumferen-
tial electric current. The coupled integral equations are solved numerical-
ly. An experimental apparatus was fabricated to verify the solutions. Good
agreement is obtained for a range of parameters. 11te experiments were per-
formed for three values of 0 - 2 Ln(2h/a) - 8.5534, 7.4754 and 6.089. The
electrical radius ak0 ranged from .00132 to .01662.
il
1. INTRODUCTION
In an earlier report on this subject (21 the magnetic current on a fer-
rite-rod antenna was derived explicitly in the form of an inverpe Fourier in-
tegral. The driving loop loaded by an infinitely long, homogeneous and iso--
tropic ferrite rod was assumed to be electrically small so that it carried an
essentially constant current IO. When the ferrite rod is assumed to be of
infinite length, the magnetic current is equal to a definite integral which
is suitable for numerical evaluation. Two values of electrical radii, viz.,
ak0 • 0.05 and 0.1, were considered and for one of the cases the magnetic
current was plotted (2] for several values of the permeability of the ferrite
rod ranging from 10 to 200. The total magnetic current can be interpreted in
terms of a sum of transmission and radiation currents. If u and cr of the
ferrite rod are assumed to be real, the transmission current can be associ-
ated with eu unattenuated, rotationallv symmetric TE surface wave. It was
further found that the cutoff condition for this wave is that the electrical
radius akI be greater than 2.405.
In a practical situation, however, the antenna is of necessity finite
and electrically short as well, so that a new mathematical formulation along
with an experimental investigation is needed for the problem of a finite for-
rite-rod anteanna. Sections 2 through 8 present the two different theoretical
approaches used to determine the magnetic current distribution on tile finite
ferrite antenna; Section 9 describes the experimental apparatus and results.
2. PROBLEU OF A FINITE FERRITE-ROD ANTENNA
The present formulation is based on the analogy between the cylindrical
dipole antenna and the ferrite-rod antenna. The dipole antenna is made up of
a wire, rod or tube of high electrical conductivity and aay he driven by a
-1-•
-I- :I
-2-
two-wire line. Equivalently, a monopole antenna fed by a coaxial line corre-
"sponds to a dipole antenna through its image in a ground plane. In either
configuration, the driving source is represented by an idealized voltage or
electric field generator which mathematically takes the form of a delta func-
tion. Similarly, the ferrite rod antenna is fabricated from a material of
high magnetic permeability and is driven by an electrically small loop an-
tenna carrying a constant current. The loop is, correspondingly, represented
by an idealized current or magnetic field generator and takes the form of a
delta function. These similarities suggest approaching the problem of the
ferrite antenna by treating the ferrite rod as a good magnetic conductor.
J Initially, however, the ferrite is idealized to be a perfect magnetic conduc-
tor (G r ®) and, later, appropriate changes are made to account for the finite-
ness of the value of the permeability of the ferrite material.
3. FERRITE AS A PERFECT MAGNETIC CONDUCTOR
The analogy between the ferrite antenna and the dipole antenna is based
on the dual property of electric and raagnetic field vectors in Maxwell's
equations
(1)V
Figure I(a) shows an electrically small loop antenna of diameter 2a. The
loop carries a constant current and is assumed to be made up of a wire of in-
finitesimally small radius. The wire loop 1% loaded bt a ferrite cylinder of
height 2h. The ferrite is assumed to have an infinite permeability, in which
i
-zzh
-2o /u - =1
Ii.•'rk, ! kokk,
Szz-h _.
(a) (b)
FIG. I (o) ELECTRICALLY SMALL LOOP ANTENNA OFDIAMETER '2o LOADED BY A FERRITE CYLINDEROF HALF HEIGHT h AND SURROUNDED BYFREE SPACE.
(b) MATHEMATICALLY EQUIVALENT BUT PHYSICALLYUNAVAILABLE MODEL FOR THE ANTENNA SHOW-ING THE IDEALIZED CURRENT GENERATOR o8(p-() 8(z).
i _ .. ........._.
-4-
case the value of its dielectric constant C is immaterial in view of the na-r
ture of the driving source. Figure l(b) shows the mathematical model of the
antenna. Region I is the ferrite with parameters Or' Er k, and region II is
free space with constitutive parameters vO, co and wave number k Because0,
of the nature of the driving source and azimuthal symmetry, the non-zero com-
ponents of the fields are 11z, Up and E A time dependence of the form
exp(-iwt) is assumed. Because of the assumption r , Hz and Hl vanish in
region I. The ferrite is also assumed to be homogeneous and isotropic. Thus
the idealized driving source is taken into account by setting
-16(z) (for p a and <zi < h) (2)
Since both regions I and II have vanishing electrical conductivity and
there is no free charge, Maxwell's equations reduce to
V x (3a)
V - iI - U (3b)
0 • (3c)
v • U (3d)
It is required to solve (Ua-d) for the fields subject to the condition (2)
which states that the tanstential component of It is discontinuous by the true
electric surface cutrent at P - a and for [zi < h. In order to obtain an in-
tegral equation for the iagnetic current onl the antenna, an electric
vector potential A and a magnetic scalar potential 0 are defined and used.
I) -V (4)
The definition of A' is incomplete unless its divergence is also 3pecified.
UeUsing (4) in (3b) gives V (Ii + A )-0, from which the scalar magnetic
zi potential is defined by setting
in terms of the potentials, the fields are now given by
E (-1kW A (6a)
it leV~- (6b)
Substitution of (6a,b) into Maxwell's eqiuations (30) and (3a) gives
02* + V oe
(7a)
VA1 -er V IVv e + I (7b)
Equation (7) is a set of coupled equations for the potentials which may be
V .A + VC; -0()
upon using (8), (7) becomes
V2-VA -ut
(9a)
If (9) is solvedI for Lthe potentLialg, subject to suitable boundary conditions,
then Lite electromaFgnetic field is known everywhiere by making use of (6~).
However, for the problem ait hand, a z-component of electric vector potential
Is adequate for a complete solution so that A A On thesufc(v a
Izi < 1 of the antenna.* (610) then becomes
S C) z
Using (8). one can rewrite (10) as
-6-
(_i +k Ae = i(k 0 /vO)I06(z) (ii)
dz
where k. is the free space wave number and vO the velocity of light in free
space. This equation is identical to that for the z-component of the magne-
tic vector potential in the case of the dipole antenna [I, eq.(3.2.4)] and,
therefore, has a complete solution - like [1, eq.(3.2.12)] - which is given by
Ae(z) - (i/v 0 )[C cos |z + (Io/2)sin k IZI1 (12)
Equation (12) is an expression for the z-ct-aponent of electric vector poten-
tial in terms of the driving current le. However, the general formula for
2*
A (r) due to an arbitrary distribution of magnetic surface current K (r) can
be written as
4) r )(e /R) dS•
Si
In general, K (r) - K (r) + n I,(;), but because of the nature of the driv-
ing source P6() - 0, so that K Cr) K K(r - Cu I) - magnetic surface cur-
rent. Since rotational syusetry obtains, the total axial magnetic current
can be introduced with thin antenna approximation so that I (Z) - 2-aK (r)
_kh P
A:, (z) " Ao() - /4") 1*(z') dr.' (e 0"/) de/2,
where
s (z - Z') + (2a sin ('/2)"31
lettiar.
K (Z,X') I et /g S) d0s/2.
"givesAeC(z) - Ct0/4) J *(')K(.,7.') (z' (13)z
-7- i
Ae(z) was previously obtained in (12). Equations (12) and (13) together lead2;
to the required integral eqiation,
f I(z')Ks(z."') dz' = i4ir 0 [C coo koz + (,Ie2)sin-hsi k Izjj (14)
' • -h
with C 1/2 ( the free space characteristic impedance.
The integral equation (14) for the magnetic current on a finite, infi-
nitely permeable, ferrite rod antenna can be identified formally with the
similar integral equation fl, eq.(3.2.23)] for the electric current on a fi-
nite dipole antenna made up of a perfect metallic conductor. Comparing the
integral eouations for the two cases, one fin,4s that the driving voltage Ve
and the free space characteristic impedance 0 in the electric dipole case
are replaced by the deiving cutrent I e and the free space characteristic ad-
mittance (li/ 0 ) in the magnetic case. Commonly used metals like copper and
brqss are foutd to have sufficiently large electrical conductivities to justi-
fy the assumption of vanishing electric field inside the material of the di-
pole antenna so that an integral equation of the form (14) is adequate and
has been used to obtain the electric current distributioni. Furthermore, if
.tore accuracy is required, theories de exist for imperfectly conducting cy-
lindrical transmitting antennae. However, it Is questionable whether the in-
tegral equation (14) is directly applicable to the practical ferrite rod an-
tenna due to the relative permeability ranges of available ferrites. Whereas
the treatment of the imperfectly conducting dipole antenna is done for reasons
of improved accuracy, a similar treatment (vt larse but not infintte) for the
ferrite antznna appears to be a necessity. This forms the sublect of Section
i + !5.
-8-
4. MAGNETIC CURRENT ON AN INFINITE ANTENNA
The magnetic current I (z) obtained by solving the integral equation (14)
may be called a zeroth-order solution because of the assumption Or = The
integral equation (14) is for a finite antenna from which the zeroth-order
solution IQ(z) for an antenna of infinite length may be obtained. For the
sake of convenience, the integral equation is rewritten as
eSI•(z')K,(Iz - z'j) dz' - cn/co)A(z ( Cos ko + - sin k01)
-hi
As h * , the vector potential A e(z) is a traveling wave which may be obtainedz
by setting C = 1e/2i. In this case,
eik lIzI,,(z')K,(I - zIj) dz' - 2ur,010e (15)
Taking Fourier transforms of both sides of (15), one obtains
7 o de 7 - , ', . l I o o dz
The left side of the above equation is a convolution integral and on the
right side, the integration may be performed to obtain
, 2t4 0 10[2iko/(k 2 )2
2 2 - 2With Y 2 ko
i.O-41nt I'k Iy;K(T) (600 0 0
By recalling
* dz - z'j) ( i (e0 Its/Rs) dO'/2n-s
with R [(z - z')2 + (2a sin {|'/2)]21/2, it can be shown 13] that the
Fourier transform R(r) of the kernel Ks(z,z') is given by
• , . . ,' . . . : . . . .• j .- , • .• .• . . . . ., . . .
9 -9-
fe-i(z K (z) dz W 0 Jo(ayo)H 0)(ay0 ) (17)
Using (17) in (16), one obtains
I E 4ý 1e k 2J (ay )H~l) (ay)
The inverse Fourier transform may now be taken.
1()" - f !*(&)ei'Z d&
*~z) ~ 1 0 ko ei'Z d(,(I (Z) 1C) volts (18)
, --. :O~(aYOO ),"(.YO)
Equation (18) is thus an explicit expression in the form of an infinite inte-
gral for the current on an infinitely long, infinitely permeable, ferrite rod
antenna.
The problem of infinitely long ferrite rod antennas was formulated pre-
viously [2] in terms of differential equations and the Fourier transform of
this current was obtained, from (2, eq.(23)], to be
-iwa(u - 1)l,2val" (AY )i (aya (
"* -J 1(19)a[y Jo(ay )II (eay) - Y Oral (ayl )II, (ayo)]
where 2 2 _ E2 and y 2 k 2- _2. & is the Fourier transform variable, and
k, and k0 are the wave numbers in the ferrite and the surrounding free space
medium, respectively. The zeroth-order current on the infinite antenna may
be obtained from (19) by taking the limit o .
First, (19) may be rewritten as
Y .0 .(.y) YOIrI (,Yo) -
~:() -(iwl~i 0 2) F(ur -1)31 (my) "r 1 )"Ii(a1)(
-10-
As r -1 , the ratio [Jo(ayl)/Jl(ayl)] is finite co that the first term with-
in the brackets approaches zero. In this case4ý o0 0k0 v ( )() Y*() Ta 1, 1 (ayO)Jo(ayO)]
Y 'OJo(ayO)11 (ayO)
Furthermore, for a thin antenna, a small argument approximation may be used
for the Bessel functions in the numerator so that I.() reduces to
C4(oI'k2T(* =2 (1)
YOJo(aYO)}{O (ay0 )
from which
iczI*(z 2e e..di)tsI (a) 1-4 k0 f T dtT volt (20)... I00k J0(ay0)H(a
Thus, equations (20) and (18) are both independently derived explicit expres-
sions for the zeroth-order (ir - o) magnetic currents on an infinitely lung
ferrite rod antenna. In the limit of infinite permeability the two formula-
tions give the same result. This limit is, however, physically unrealizable
since a magnetic material with ur - does not exist and, hence, a modifica-
tion of the formulation which treats the ferrite as an imperfect magnetic
conductor is .equired. This modification has, once again, an analogue in the
electric case in the treatment of the imperfectly conducting cylindrical
transmitting antenna [3), (4].
5. FEiRRITE AS AN IMPERFECT MAGNETIC CONDUCTOR
In order to account for the fact that t0c relative permeability is large
but nut infinite ,the concept of 'internal impedance' is useful and suffi-
cient. With reference to Fig. 2, thn internal impedance per unit length of a
cylindrical magnetic conductor of circular cross section of radius a with its
axis along the z-axis of a system of cylindrical coordinates (o,4,z) may be
defined by
z (rm - ix.) - ý(p a)/1* (21)m z S
where Hz(p = a) is the tangential magnetic field at the surface, p a, of
the cond: ctor and I is the total axial magnetic current. Recalling the ex-
pressions of electric and magnetic fields in terms of the potentials
11 -v -A A (22a)
D- -V X A (22b)
V •e + VE;* 0 (22c)
che elecLric vector potential satisfies
(,2 +k)e k 0 (23)
where k is replaced by kI and I, for the two regions I and II shown in Fig. 2.
For the problem of a thin cylindrical conductor, the axial component of elec-
tric potential is sufficient to satisfy Maxwell's ecuations and the relevant
boundary conditions. Thus, the electromagnetic fields everywhere can be ob-
tained by setting A = zAe. With the vector potential being entirely axialz
and also because of azimuthal symmetry, (23) becomes
S+ 1 k2)Ac(p z) 0 (24)
3a 2 p P ~ ) ao 0
A product solution to (24) is sought in the followins form:
AeCq,z) - f (z)R (P) (25)
z z
i ~-12.-
4
JP
ii I
k kI ko
(V2+k)•)e=O (V2+ko)ýAeo
FIG. 2 A CYLINDRICAL MAGNETIC CONDUCTOR CARRYINGTOTAL AXIAL MAGNETIC CURRENT OF tz
AND IMMERSED IN FREE SPACE.
½!
-13-
Substitution of (25) into (24) leads to
d2
f 1d z+kf~R 2 f dR (26)
R __z +fz 02• To do
dz
Equation (26) may be rewritten as
d2f 1 12 d (27)
az dz2 Izt (o
The left side of (27) is a function of z alone, while the right side is a
function of p alone. Hence, they can be equal to each other for all possible
values of o and z only if they are both equal to a constant (say 2 which
may, however, be multivalued. Therefore,
d2f dR
-+(k2 _ )fz 0 and o do do + 2
dz2
After solving the foregoing differential equations for fz and R the axial
"vector potential in the two regions can be written down as,4
2) 2A 2k C"0" 1)
in region I
Ae (2 2) in region 1IA. eZZ(o,Z) C2 H0U O~x~ýk0-40Z
SBoundary conditions that are useful in determining the unknown constants In
tihe solution require the continuity of tangential • and 1 across the surface
p a; that is
I z (oa) 821( a)
1 " 1 ,1 (0 - a" 52'a2(o
In terms of the vector potential, the boundary conditions at the surface
a are
-14-
ae Me1 zi 1 z2
C1 ap C0 ap
2 2iW e i e 0 e
-A 40 - A1 k
2 zi I P . z2
1 0
Application of the boundary conditions yields
1 2 = 2 (1), 2 2 z1 CltIJu(Cla)exp(i A C - 1 ) . L O H 0a)exp - z) (28)
2r2 2iW 2 _2 '0 (1) 2 2 z) (29)-~~- C l%~a)exp(i fk 1 , ) ý CH 0 (y~)exp(if (291 k 2k 0 -
1i 0J Since (28) and (29) are valid for all values of z at all times, it follows
that
2! /k c 0 q (say) (30)
so that 4l 1 q2
and c~ 0 q2
Dividing (29) by (28) and rearrang-
ing , one obtains
11(1) t,012 2 10((la)to a 1 Co~ P~i 2 L0 J 0 -(r.a) (31)
Although, in general, an explicit solution is not possible, equations (30)
and (31) are theoretically sufficient to determine the unknowns t,, and 4(W
The two unknowns will be determined here by two methods.
Miethod 1:
An approximate solution is possible by allowing k1 to become very large.
Since q is fillite, I' k, and, therefore, t. is also large. Using this on
the right side of (31) gives
!4
• ] ~~.................... ....... •...................• •7i••• •' • . ... .... •• •
-15-
k2
Ei 0 tO aiCOIl k 21
Therefore,
(00 a) (32)
0 01)
Vquation (32) is satisfied by to 0 since it can be shown that the ratio
(1)(t 0 a)IH(01),(tOa) remains finite as t0 approaches the value of zero. It
then follows from (30) that
Thus the solutions are ti - i and to 0.
Mlethod 2:
liethod I may seem to be an oversimplification ant. hcnce, a slightly
-more rigorous method may be needed in some cases. is observed that (31)
may be ideutified with a similar equation obtained by sommerfeld 151 in the
problem on 'waves on wires.' Sommerfeld has developed an iterative form of
solution which may be used here,
In the limit of large 41 the right side of (31) becomes
2 :0 122 al0
ClO O In 7,o
0r ka
and is small, Since thle left side of (31) also has to be small, we have
2q/:1o U t) 2 U w i t h u ( Ybe
ca Iei )
-16-
2 aku in u v with v -- Ž
2 Or r r
Since in u varies slowly in comparison with u, it is possible to write
n+l1 u = v
thwhere un is the n approximation to u. The method is best illustrated by an
example. In the later part of this report, eleven different antenna config-
urations were used in the experimental determination of the magnetic current.
The example chosen here (antenua 1l) corresponds to the lowest value of the
SrEr product for the eleven cases.
Example: ak0 o 0.00166, Or = (18 + 1.036), ur 11.0.
Let u0 2v = ak i 4
r
This gives
•- Ul v -il.l Y - .0,in u0 (-9.115 + 14.712 -0 (.049 - i.095)
v = -ii.i l 0-4u2 v -i. 10 -10-4 (.0167 - 1.()931)u n =
2 u1 (-11.445 + 12.047)
Continuing the iteration
u3 . _v -il.l X 10-4 . -10"4(.0140 - i.0930)u3 n u2 " (1.5684 + 11.7478)
v -il.l X -1 -410U n u 3 (-11.5748 + 1.287 " 0- (.038 - .
and finally
vin u (-I.5748 x 1-4--- -0-4 (.0138 - 1.0930)
It is seon that this iterative procedure is rapidly converging and using the
above value of u,
-17-
(oa)2 -- u2 10-10(1.062 + 1.3225)
Furthermore, from (30)
(aq) 2 . (ak 0)2
-(ar 0) 20s0
2,7556 6 _O-10 (1.062 + 1.3225)
from which a1 may be calculated using
(at)2 (ak)2 (sq)2
(ak)2 - (ak)2
+ (at 0 )2
I 1 (a-O 12
ak 2
l - (.0056 - i.00001) + 10-7(2.159 + 1.651))
From the calculated values of (aý0)2 and (a 1)2, it is seen that the approxi-
mate solutions, i.e., C0 - 0 and CI ' k1, are quite satisfactory even when
IIrC I is as low as 10.
Therefore, the vector potential in the interior of the conductor is
given by
"A (o,) Al (P)A 1(Z) - CIJo(kI) exp(i 2 2Iz zi z
where t in the argument of the Bessel function is replaced by k1 in view of
the calculations of "Method 2." The constant C1 can be written in terms of
the potential on the surface so that
Ae (P) - A'=A • k o)/jo(ka)
Since ii, is proportional to A E
SJz
Hzl(p) H.1
(a)J 0 (k 1 P)/Jo(kIa)
The magnetization is then given by
z(p) (Wr - l)AzI(P) =(pr -)k l) (a)Jo0(kl1P)/J0 (kla)z r zi r z 01 1
from which the magnetic current can be obtained as
I =(p) - 2w p 0 Az(p')r' do'F' 0
A Iz (a) o
-2w0 (r - 1) f Jo(klP')p' do'
Performing the integration gives
S~~Iz() 2•0U~(•--• pll~la J J(k 0) (33)
The total magnetic current carried by the conductor is, however, given by
I() 2w1!15011(o) Ddor 0which becomes
* 251O(r - 1)'1 (a)I (r I J,(k__ a) (34)
From (33) and (34) the radial distribution of the msagnotic current in the in-
terior of the conductor is given by
* *,, J 1 (k 0)"!:- Iz) -I(a) jka)(35)
(a z a (a J(ka)
F Furthermore, having obtained the total axial magnetic current of (34), the
-19-
internal impedance per unit length defined in (21) can be written as
z -r -ix -H (p a)/II(a)
a~ zi
kI. _ 0 1(36)
2w m 2 T-( kpox m
where X (V r 1) =magnetic susceptibility;
w -radian frequency;
-4vm x1 him -permeability of free space;
Ia cross-sectional area of the conductor%
k, + in 1 k (B +N it,) wave number in the material of the
conductor.Eini ie
With Cr Wa + iv") le P~ and CrE (ti' + iC") 1 kr I
k k I r 11/2 expti(Q + 0 )121
so that
8 Oirr cosIOUICaEC/2 (37a)
and
a I lOur~rI1/ sint(O +~ Q)/2 (37b)
Wne may now use the internal impedance per unit length of a magnietic con-
ductor carrying an axial magnetic current to find an integral equation for
the magnetic current on a finite ferrite rod antenna. The axial component
AO(z) on the surface of a cylindrical antenna that has ait internal impedance
per a~i .carries anl axial current [*(z) . and is drivenm at z - 0
by adelta-function generator with anl mat of le, satisfies the following dif-
ferential. equation:
-20-
+ ~)Ae) zY ) - I6(z)1 (38)
If the antenna were made of a perfect magnetic conductor, zI = 0 because xmm
infinity so that (38) will reduce to (11). If the radius a of the antenna
and the free-space wave number k =/v0 2s/A satisfy the inequality
0 0 0
ak0 '< 1
then the vector potential is given approximately by
C hAz(z) (z')K(z,z') dz' (39)
"-hIf the equations (38) and (19) are formally identified with those for the im-
perfectly conducting, electric dipole antenna [3, eqs, (7) and (9)], it ise e
observed that P and 10 play the roles of &0 and V0 . King and Wu [3] have
developed a three-term solution for the electric current on the imperfectly
conducting dipole antenna which can he well applied to the present problem of
the ferrite as an imperfect magnetic conductor. The procedure used to obtain
the three-term solution will be described here briefly; for a detailed analy-
sis the reader is referred to (3].
The approximate kernel in (39) may be separated into real and imaginary
parts,
Y(z,z') (zz') - iK(z,z') - o /r
so thatcon k0 r sin kfr
K (Z'Z') ,r K I (t~z) r
2 1/2with r - [(z - z')2 + a2]1. The vector potential may also he divided into
4 two parts,
A'(,.) -A'(z) -IA'(,.)
-21-
wherekoc h cos kor
A(z)= 1-T z-h f k r dz' (40)-h z 0
e "kE h (z sin korA•(z) - - -h -----(z') dz' (41)
The properties of the two integrals are quite different. The kernel in (40)
has a sharp peak at k 0z - z' 0 and thus greatly magnifies the contribu-
tion to the integral due to current elements near z = z'. The current van-
ishes at the end but the vector potential A.e(h) has a small finite value so
that the difference in vector potential qhn,,ld vary closely like Tz(z).
Therefore,
(Ol ,e Ae * A(4
1/cO)[A(z)- A(h) I z)'(z ) R 'l•() (42)
where T is the approximately coustant value of T(z) defined at a suitable
reference value of z. However, in the second integral in (41) the rather
flat behavior of (sin k0 r)/k 0 r with k0 r allows the following approximation:
khr khrsin kor 2 sin 0rcos-2 kor
0 0.- 0-- kor co -1---
which is useful over a ranpe kor n". This approxireition leads to
kaA'(z) Ae(0) cos O0 (43)
1 2
where A0(0) is a constant given hy
A'(0) I- --(-- dz' (44)
If equation (42), rearranped in the form Ca(z) ý4,/YiO )IA (z) - A(h), is
substituted in the differential equation (38), one o!,t'tns:
-22-
82 2)\ei i 2 ed -n + k) Ae)- A'(h)] 1 4.ýOkO.zF-[4R(.) -AR(h)] k kA (h)
21,6 (z)(45)
A complex constant k may now be dnfined by
i
k2 (a + ia)2 ~( 4r (46)0
Using (43) and (46), (45) becomes
Sk)[Ae(.)Ae(h)]--i(k -k2
A ) (k2A'(h) - k A'(h) + k
2e(
Ze Z c lehi+kl
(47)The integral equation of (39) may now be written in the form
(A -) A5(h)] - - I(z')K (z,z') dz' (48)
where the difference kernel K8 is given by
ik r ik rh
K (z,z') -K(z,z') -K(h,z') e
with r ((z - ') 2 3 2 / an~d r h - Kb - Z') 2+ a 2 1/. If the differen-
tial equation (47) is solved for the vector potential difference and tile
solution is substituted for the left-hand side of (48), an integral equation
for the magnetic current onl the ferrite a~ntenna is Obtained, viz.,
tJ I*(Z,)K (z,z') -z T---- 0 (1/smC,(/21k.+11` -Dcos kh F6~k
(49)where for ease of reference, the same notation as in King and Wu (31 is
employed and the various factors on the right side are given by
-k sin k(h - Ii)
-23-
kohU' - uk + D cos --
k 2 k20 0i~~2 (kI 2)~h ~~)k - - 0 i 0
ko k - ko]4
Fkz cos kz -cos kh
0o 0h0z-Cos -2- Cos-y-
Following the procedure as in (3], an approximate formal solution to the in-
tegral equation may be written in the form
I Vjz Ukz FD' (50)
where the coefficients IV, IU and ID are obtained by a numerical procedure.
Letting TU . I /IV, T* - 1*/IV and evaluating IV. one may write (50) as
~ ~~1 v ... Iz Z e o
I(S) -kdR 0 kh (sin k(h - Izj) + Tu(cos t - co kh)Tz(cos ko c
+ T*(cos _" CosT)] (51)
where k is redefined by
(k2 - (B+ is)2 k + (52)k0 dR
and YdR is given by the integral expressiondR
f sin k(h - 1z'I)KdR(z.z') dz' 4 sin k(h - Izl)tdR (53)
Thus, equation (51) is the required expression for the total magnetic current
on the antenna from which the admittance can be obtained to be
-24-
Y C - iB 1 0 ohms
- 12k 0 0 [sin kh + T*( - cos kh) + TD(I - COs 1h)] (54)
Note that, because of an earlier approximation of the imaginary part of the
kernel, equations (51) and (54) are valid representations for the magnetic
current and admittance only when koh < 5n/4.
The existing computer programs for the imperfectly conducting dipole an-
tenna due to King, Harrison and Aronson [4] have been modified for use on the
IBM 370/155 computer system of the Joint Harvard/M.I.T. Batch Processing Cen-
ter. Appendix B includes a listing of the Fortran IV programs that compute
the magnetic current distribution and the admittance of the finite ferrite-
rod antenna when the ferrite is treated as an imperfect magnetic conductor.
6. THE LIMITATIONS OF THE THEORETICAL FORMULATION
The present formulation is based on an analogy between the ferrite-rod
antenna and the conducting cylindrical dipole antenna. Because of the symme-
try in Maxwell's equations, a set of scalar magnetic (0*) and electric vector
(P) potentials was defined and used in formulating the finite ferrite-rod
antenna problem. It is considered useful to determine the existence of these
potentials for the infinite antenna and thus provide some justification for
their use in the finite antenna problem.
The electromagnetic fields in both regions for the case of the infinite-
ly long antenna were determined previously [2] to be:
Region 1,0 < p<a:
i • l~ ,• = iwpU l ellOn ) (y~a)Jl(ylp)/D(E)0 1
"-25-
1. . + aey (1
jy)(YOa)Jo0 (y )ID(p)
jz~p{ - .. L- ra@IP{@ +. E CIP,•5/P] = ao].1
R! 0P(O• W-11 (/PEl(p,ý5 ial e H~l)(yoa)Jl(ylp)/D(Q)
P1 01 aI0 1 ''1
1 (,) zl(p,{) - 0 (55)
Region II, p > a:
- le a) (Y01) 0
Rz2 (p, ) - a 0l01rJ((Y1 a)11 01 (yOp)/D(ý)
-- 2 (p,[) , i &I;{J,(y a)H1l)(YoO)/D(I )
o2pt) , Ep2 (ot) - ER2 0p,0) , 0 (56)
where
D(M5 I a(YiJ(Yl 1a) N•1)(Y0 a) - YOrj1 (Yla)11I0 () 0 a)]
The actual field quantities may be obtained by applying the Fourier inverse
formula to the above transformed fields. It can be verified easily that the
above field quantities satisfy the following transformed boundary conditions:
i) Tangential t 0(a2 ,) ( 41(a- O (07a)
ii) Tangential '1 Rz2 (a,{5 - ii1 (a",) o .Io (57b)
iii) Normal B. B(+ .) 2 P1 (a,&) (57c)
iv) Normal D: Zero in both regions
$ is a scalar magnetic potential and has a non-gero value in both re-
gions. For the infinitely long antenna, the only non-zero component of is
-26-
the z-component so that ;A= A, The potentials may be derived either fromz
the already known electromagnetic fields or from an independent solution of
the following wave equations with suitable boundary conditions:
(V2 + k 2)Xe(PZ) 0 (V2 + k2
) *(pZ) = 0
The equations reduce to
Region •I, 0 < < aei
22
Region II, p > a:
Using a Fourier transform pair, the above equations become
22 I
L-•, -3 , (k - 2) ^ 2 (n,')-0
With a change of variable the above equations can be recognized as Bessel
equations with the following solutions,
X (p,&) "Po(Ylo) for 0 < o < a
Qp. 11 (1) (, for P > aZ2 0 0
-27-
where Yo (k 2 2 1/2 and y2 (k2 C2)12whr 0 (k-i n 1-( 1 )
The boundary conditions (57a,b), axpressed in terms of the electric vector
potential, become
WE~)Xl~-O~ (a ClW1a2+•1p (58a)
2 2 -e + 2 2 -e _,e(8)(iWY0/k0)Az 2 (a ,E) - (iwyl/kl)Azl(a ,Z) (58b)
By applying the boundary conditions and determining P and Q, the electric
vector potential can be written as:
e -iWU Cal 0nH (Y 0 a)J 0 (ylp)/yiD(t) for 0 < p < a
(59)
I, -iwYj) (Yt0aJ()( Y D(0) for p > a
Similarly, by solving the wave equation for the scalar magnetic potential
the solution can be obtained as:
-*e (1)fo
(P ia1 0 H1 (y 0a)J 0 (ylp)/yiD(C) for 0 < p < a
(60)
2 ia&I PJ1(yja)U17 (yp)/Y0D(&) for 0 > a
The boundary conditions satisfied by ; (p% .) at the surface p - a are:
0 1 1 (61b)
It can also be verified that the potentials satisfy the gauge condition,
aAe(p.z)l3z iw••c*(pz) - 0 in both regions
{z
-28-
The potentials of (59) and (60) can also be obtained from the electro-
magnetic fields of (55) and (56) by making use of the following relationships
in both regions:
E (pz) = (l/r)aAe(p,z)/3z ; Hz(p,z) -a4 (p,z)/3z + iwAe(pz)
and
3Aze(p ,z)/~z - *-D iWCO (p,z) = 0
The above analysis verifies that when the antenna is infinitely long,
both the scalar magnetic and electric vector potentials exist. They are both
discontinuous across the antenna surface and satisfy respective wave equations,
appropriate boundary conditions, and the gauge condition.
In the case of the finite antenna, however, a precise knowledge of the
vector potential in the two regions is not necessary to derive an approximate
integral equation for the magnetic current. What is required is the electric
vector potential on the surface of the antenna. To determine this, an inter-
nal impedance per unit length is defined and used to obtain the three-term
solution for the magnetic current. Usinp the computer programs described and
listed in Appendix B, the magnetic current was evaluated for a range of param-
eters. The current distribution was studied as a function of the four inde-
pendent parameters, viz., ir'; p" or Q pr/pr; h/X or koh; and ak0 or Qr r r r 0 khadkof
2 Zn(2h/a). In this study the value of the dielectric constant of the fer-
rite was fixed at 10.
The ranges of the four parameters were as follows: Pr' 1 0, 100, 1000;r
Q.I lto Q 1 I00; h/X0 - .1 to h/A0 . .5; and ak0 . .001 to ak0 . .1. Typi-
cal results of the computations are shown plotted in Fig. 3. The quantities
!Pr ako, h/A 0 and 0 are varied, respectively, in Fig. 3a-d, while in each
case the remaining three paraneters are kept constant.
} --•- - - -.-..-
1.00
'51.
0.8X 0/.20.
0.6 ako .01 0.8 hX~
(zh t 0.60.4h (z/h)
0, 0.
0.2 "000.2
00
0 0 100 150 200 0.0 0>
I*(Z)/Ie I_* (volts/amp) 0 5 10 25 30
(a) VARYING Mrl (b VARYING a ko
1.0- a ko 0.01 1.0- ak0 =.01M~r 1000+10 /4r 100.O
0.8- to0.8 - h/x 0o -O.2
(ih) (zh)
0.6- 0.6
0,00.4- 0 0.4 -10, 10
-100, 10.2 0.2
0.0c --- _ 0'.00 40 80 120 160 0 10 20 30 40 50 60
II*WZ/1oe I -'(volts/amp) *I(Z)/ Ie I_' (volts/amp)
(c) VARYING h/>X0 (d) VARYING 1-
FIG. 3 PLOT OF THE MAGNITUDE OF NORMALIZED MAGNETIC CURRENT(*I(Z)/IleI) AS A FUNCTION OF NORMALIZED DISTANCE (z/h) FOR
VAR IOUS PARAMETER RANGES. (Er 10+ 0 FOR ALL THE CASES)
-30-
In Fig. 3a for fixed height, radius, and ratio 0, the magnetic current
on the antenna is seen to increase with the real part of the relative perme-
ability. A similar behavior is observed in Fig. 3b for increasing antenna
radius and fixed height, permeability and 0. A comparison between Fig. 3a
and Fig. 3d shows that a large value of viI produces a greater inc"-ase in ther
magnetic current than a high Q ratio; in fact, an increase in Q for Q < 50 is
seen to reduce the magnitude of the magnetic current. To interpret Fig. 3c,
it is useful to examine the behavior of the propagation constant k on the an-
tenna, given by
k - 6 + in - ko[l + i(4ziCO/kOIFdR) 1/2
If the dimensionless parameter 4 (
4Czmo/ko) is introduced, this expression
becomes
k = a + in - k0 (l + i$i/y )R1/2i dR
Despite the fact that dR is itself a function of k, an efficient iterative
method can be used to determine the value of the propagation constant. By
substituting for zia from (36) the following expression for i is obtained:
2iakI 0 (ak1)(ak0)x
1)
$I becomes positive imaginary for the cases plotted in Fig. 3c where ak1 is
real. This makes the propagation constant k on the antenna pure imaginary
which leads to an exponentially decreasing magnetic current. For most prac-
tical ferriteas the positive imaginary part of dominates, which makes the
attenuation constant a significantly larger than the phase constant B. This
can also be seen in the experimental results reported in Section 9.
-31-
At this stage it is considered useful to summarize all the approxima-
tions and assumptions involved in the derivation of the integral equation in
(38) with (39). The ferrite was first treated as a perfect magnetic conduc-
tor (•r " infinity) and the integral equation in (14) was obtained. This ex-
pression was later modified by adding an intrinsic impedance per unit length
for a practical ferrite that is an imperfect magnetic conductor and finite.
The basic assumption that the radius be small, i.e., ak <0 1, was made. An
implied approximation was introduced when the impedance per unit length z
derived originally for the infinitely long magnetic conductor, was used for
the finite antenna. Its use can be justified as follows. For an infinitely
long magnetic conductor, the transverse distribution of electric vector po-
tential is independent of the axial distribution. It is reasonable to assume
that this remains the case when the conductor length is large compared to the
radius, so that the intrinsic impedpmce per unit length derived for the infi-
nitely long conductor can be used directly for antennas of finite length. A
further question arises concerning the discontinuity of the electric vector
potential across the antenna surface. It has been established that the elec-
tric vector potential is discontinuous across the antenna surface when the
antenna is infinitely long. It is reasonable to conclude that the discontin-
uity exists even when the length of the antenna is finite. The derivation of
the integral equation for the magnetic current or the tangential electric
field requires a knowledge of the electric vector potential on the surface
p a, which has apparently two values. This problem is not peculiar to the
ferrite-rod antenna but also exists in the analogous resistive electric di-
pole antenna, In either case, the value of the vector potential used is that
obtained by approaching the antenna surface from the surrounding medium. It
is believed, however, that the discontinuity in the vector potential is ail I
-32-
consequence of the way in which the vector potential was defined and can be
overcome with the introduction of a suitable scale factor in the definition.
The approach of treating the ferrite as an imperfect magnetic conductor
relies on the mathematical equivalence of the two analogous problems. One
cannot escape the fact, however, that while there are two pieces of conductor
separated by a slice voltage or electric field generator in the case of an
electric dipole, the magnetic conductor in the ferrite problem is a single
continuous rod driven on the outside surface. A delta function, although un-
physical, is a mathematical convenience in either case.
In view of the above discussion, a more rigorous analysis which does not
invoke the analogy with the electric dipole is developed and presented in the
following section.
7. A MORE RIGOROUS TREATMENT OF THE FINITE ANTENNA
Since the total magnetic current Iz (z) is linearly related to the tan-
gential electric field E (a,z) by the relation
I* (z) --2 naF. (az)
the following procedure seeks to derive an integral equation for E (aez) by
solving the ferrite-interior and free space-exterior problems.
Interior Problem. The interior problem consists of a ferrite cylinder
of height 2h driven at the center by a constant-current loop. The driving
condition will be accounted for after the interior and exterior problemb are
solved. The diameter of the rod and of the loop is 2a and the restriction
ak0 << I is satisfied in order to maintain a constant current I0 in the driv-
ing loop. Given a cylindrical coordinate system (o,o,z) and after eliminating
Sfrom Maxwell's curl equation and imposing azimuthal symmetry, one obtains
-33-
for the electric field
2 1 + (k - )+ (P,) 0 (62)
with k1 ko(Prcr)1/ 2 , I1 <_ h, 0 < p < a, and E,(pz) E (p,-z).
Solving (62) by a separation-of-variables technique gives
*2 2,2 2 1/2E (p,z) .An cos((n + l/2)z/h]J l(P k1 -I (n + 1/2) /2h 111 (63)
with the coefficients An given by
A2 2 _ E (az') cos[(n+ I/2)wz'/b) dz'
hJ1 {(a( n+A-1/2) it /h I-h c5(+l2'1
This procedure aims to determine the tangential magnetic field IIz(a,z)
from independent treatments of both the interior and exterior problems and
then to require that their difference equal -16(z), the true electric sur-
face current. Thus, Hz(a ,z) can be obtained from the above by using
Ilz(P'z) (l/iWP1 )[PE (P,z)I3P + E (P.z)/p)
-(i/iwpUh)I n11 (. dz' E$(az') cos((n + 1/2)sz'/b,])
J pk2 _n+12) 2,f 2 11/2Jo ( 21- (n + 1/2)2. /b2I 15 cos((n + 1/2)sz/h] 1e(ak~ - (n + 1/2)252 1/h2]I2
Sk 2 (n + 1/2)2s2/h2 1/21 (64)
It should be pointed out that, as a first approximation, R (e,z) is made!.
-34-
to vanish on the top and bottom surfaces defined by Izi h and 0 < p < a,
thus neglecting all the fringing fields at the ends of the antenna. In prac-
tice, this condition nearly prevails for antennas with heights large compared
to the radius (h >> a).
Exterior Problem. The exterior problem is concerned with the free space
surrounding the ferrite rod which extends from (0 < P < a) and (-h < z < h)
for all 4. It is equivalent to solving the problem with the ferrite removed
but with the tangential electric field on the surface E (az) for Izi < h re-
quired to be the same as that used in the interior problem. With an assumed-ist
e time dependence, the governing equations are:
V " -. ý: 0(65a)
V s i.V i0 0 '1 (65b)
V: v - 0 (65c)
V D - o (65d)
SFrom (65d) in free space, one may define an electric vector potential
-- V
so that
0
This leads to
•- -V.* + iw
The exterior problem may be "odeled by a cylindrical surface of radius a
S---
-35-
that extends from z - -h to z - h. This surface, when immersed in free
space, has the following boundary conditions valid for jz< h:
E (a+,z) = f+(z) E E 4(a,z) (66a)
E (a,z) - f'(z) - 0 (66b)
Substituting for I 3nd in (65b), one obtains
-V x V x e + iW0 c 0 V* + koIle . 0 (67a)
(V2
+ k2 A, - V(v . + POiO;,*) - VX (67b)
If the Lorentz gauge is satisfied, the Lorentz factor X (and the right-hand
side of (67b)] is zero. The equations may now be specialized to the problem
at hand. There is rotational symmetry in the problem and the non-zero quan-
tities are E and 0*. Equations (65a,b) for the different com-
ponents become
(allp / az ll /30) -iWC0 rO (68a)
iWI•o0 11 3 -aE/3z t68b)
iWUOlZ -4• (OE,) (68c)
These three equations are true everywhere excep, on the surface p
and jz[ < h. To make the equations valid on the surface, one has to intro-
duce the surface conditions into the above equations. In addition to the
7 4 conditions in (66a,b), there is an electric current (W) on the surface as
well as a large axial magnetic field. Thus, (6da-c) hecome
. .4
-36-
(OR/az - 31,1 /Dp) + 6(p - a)K (z) -iv 0 EE (69a)
r iwI•o0 H . --3E az (69b)
iwVOH + 6(p - a)E, = (3E /Do + E /p) (69c)
In terms of the potentials, the fields are given by
H -3C*/-:' P
i .• H -@ * / 3 z + i w A eR z 3z
From the preceding equation,
:(PZ) - CO r i(pA ) ,p' (70)
zP
It is now required to set up an equation for Ae(P,z).
2 2e(V2 + k2)Ae(p,z) I a + D + k 2 ] C ( p,z) dp'
C' /I +C DE(P',z)/3z dP '"Op t;[l(Pz)/Dp + E ý(p~)P. ] 0 aO
2o+ Cok• L W¢p,z) dot
With (69b,c) this becomes
(V 2 + k 2)A"e(o.z) c tO~iuttolz(P.z) + S(P - )E (p,Z)l
-iw Vo C.O 7 311 ( ,p ' ,z ) / 3 -do ' + c o % k 2 , o , o
SP P
| Using (69a) gives
-37-
(V2 + k2)Ae(p,z)- 311 (p' ,z)
0 zz,. ioeOjz(p,z) + c0S(p - a)EO(P.z) - iw1iOC 0 j 1-iwCoE(PO'z) + DO0 P
6(p' - a)K (z)] do' + COko f E(p',z) do'
iwipOtOH z(Pz) + t0 o(p - a)E$(Pz) - kf ,z) dp'
2 W6(,,Z) do'
i~i C0K (z) •6(p' - a) do' - ipPOC0Hz(P Z) + coko P
C 0 6(p - a)E CP ,z) + iwU 0 C K0K(z) S (p' a) do'p
The p' integral may be performed:
1 if O<o<a
f 6(p' - a) do' M e(a - o) p
po +!0 if 0> a
Therefore, finally
2 2 ~ a)E (P,z) +iwU~ct 0 K 4(z)11(a (1
If (71) is formally identified with (67h), it is seen that the second term on
the right in (71) corresponds to the Lorentz factor term. The Lorentz gauge
is satisfied (x - 0) in the exterior repion (p > a) but is not satisfied in
the interior region. Furthermore, by differentiating with respect to p
x v + -U00 (p,r)/3z - iwuoCOf*(pz)
it can be shown that x is independent of o and a function of x only, i.e..
X(z), which leads to.-
-38-
i(iwPi0 E K K(z') dz' if 0<p< azz
x(Z) = •Ae (p,z)/3z - *•a¢(p~)=
I0 if > +
It is thus seen that the Lorentz condition is satisfied on the exterior
but not in the interior. This is because of the presence of the transverse
electric current in the ferrite medium. This situation can be contrasted to
an electric dipole antenna (thin or thick), where there are no magnetic cur-
rents to make the Lorentz condition invalid.
It is now required to solve (71) for the electric vector potential. The
equation becomes
+ + -L2 + k 0 A (p,z) - c06(o - a)E (P,z) + iWij K (z)H(a - p)
This equation can be solved with the use of Green's theorem and the principle
of superposition. Thus,
Ae (O•z) AQE(0,z) + AK (o,z)
where
ik 0As (pz)--(C/4#) -d d'/2w dn' 2'p' dz' E ' - a)(etR W /R)
-190 0 -
- -(aco/2) I dz' E (a.z')K(z - z',P)
with
ik R
K(z -Z,*P) 2w R~-j -(2-71
and It ((z ) + p2 + a2 2pa cos 0,] A/2. (O,) will be used later
• -- Az~~)w1 eue later I
•:• " :• .,,.," ,'" ". . . . . . . . . . . . . - . . .:" " ,L-. . .. •
L - 39"-
to obtain A: 1 (a,z). Similarly,
ik0R
[I (Pt .(i(.)JOe/4 U) -~do'/2ir dp
t 2%P' dz' K,(z')fl(a W )e I~h
-(i~tI0 0 /2 ) I dz' K(z')M1 (Z - ,)
-h
where
!" I("z'O=•d OW p (73)"H ( d' a Re
sard R1 2(z - z)2 + 02 + 02 2op' cos $'I
Thus, the total electric vector potenxtial is
h
A(0,z) = -(aeO/2) f dz' E (a.Z')K(z -Z )
,(ijO&t/2) dzI Y@ (z')b 1 (z - z' ,) (74)
[ a+
By specializing (74) for p a and p - a and by making use of (70) and
k• (66a,b), one obtains
E (a,z) -(ano/ 2 ) - dz' E (a,z') 3K(z Z',P)/(-
_(imPOC02) dz' K,(z') 0 W (z - S' ,o) Io a-
Returning to (74), one may now obtain for the tangential magnetic field
U (p,z) - (iw/kI I k0
**1 1z
...
...
,..
-40-
Thus,•2 2)h
Hz(p,z) (a/2)(i/ij 0 )( I•+ k0 I dz' E,(az')K(z - z',p)
+ (1/2) (--.+ ko dz' K (z')M(Z - z',p) (76)
2+
Dz h
Once again, on the exterior surface p = a
•rl 2 + 2• h•
Hz(a+,z) (a/2)(1/iwp0) (2 ko I da' E(az') (a z'2,a+)a~a
0\~
0-h
+( -2) - _2 dz' E(az') K(z- z',a ) (77)+ (1/2) + k dz' K(z')?i(z
It now remains to use (77) and (64) to obtain the integral equation.
Integral Equation for E (a,z) and KW(z). The required integral equations
for the unknown quantities may be obtained from the results of (64) and (77)
for the interior and exterior problems by requiring that
[ •H (a 'z) - Hz(a" z) = -1,6(z)
This givews
+(/2 kli~ 2) dz' E,(a~z')K(z z' ,a) + (1/2) (K- +k2
x dz' K 'z')M (1 - Z',a) + i/WIh a j dz' E (a,z')
2Jot(k• . p2 )
1/2]
X o~z) os(pa) Jofk 2 2 1/2] (a(k, P)']j 1 6I~(z)
(78a)
with p (n + 1/2)11h.
H,
El -41-
The other equation to be satisfied simultaneously is (75), which is re-
produced here for convenience:
C 0 E(a'z) = /(a~o!2) f dz' E,(az') aK(z - z',p)/)p - 7a+q -h
I " h- (iwjOcO/2) f dz' KW(z') 3Ml (z ',p) p (78b)
•I :-h p-a"
The two kernels K(z - z',p) and Ml(Z - z',p) appearing in the coupled inte-
gral equations above are defined by (72) and (73) respectively.
It cn he verified easily that in the limit h ,* the integrals in
(78a,b) become convolution integrals, that the two equations decouple and
that the expression for E (p,z) on the surface p = a is in complete agreement
with the results presented in Part I [2, Eqs. (17) or (18)].
Returning to the coupled integral equations in (78a,h), it is seen that
there are three kernels. First of all, the kernel on the right-hand side of
(78a) will be examined carefully. The kernel is made up of an infinite series
which is clearly divergent since, for large values of n, it behaves like n.
Although strictly not valid, the operations of summation and integration will
be interchanged for the purpose of examining the series. The interchange is
reversed at a later stage so that, in effect, all the steps ari valid.
It is convenient to define the kernel M(z,z') on the left-hand side of
(78a) as:
2 2)112 2 1/2
M(z,z') G • cos(pz') cos(pz) o 212 [a(k1 - p1) 11
where p - (n + l/2)s/h.
As was pointed out earlier, this series is divergent and, hence, it is
useful to write it as the sum of two series, using the first two terms in the
* ~ ..........
-42-
asymptotic form. For large values of n, the series behaves like
J (iap) I 0 (ap)Z cos(pz') cos(pz) (ia) (iap) -, cos(pz') cos(pz) y (ap)
I
-• cos(pz') cos(pz) (ap) 2 os(p') Cospz [ +• 1 a
8ap
We now write
M(zz') P(z,z') + Q(z,z') (79)
with
J0[a(k2 - p 2)i/2]0 1 2 2 1/2
(80)
and
Q(z,z') (ap + 1/2) cos(pz') cos(pz) (81)
Equation (79) along with (80) and (81) is exact because it only adds and sub-
tracts the first two terms in the asymptotic form. Now P(zz') can be writ-
tan as:
P(z,z') - • An cos(pz') cos(pz)
with An given by the term in the square brackets in (80).
If all the coefficients A wore equal, P(z,z') would be a delta function;
but this is not the case. In view of the differential operator on the left-
hand side of (78a), it is helpful to remove a similar factor from P(z,z'),
This is easily accomplished by solving an equation of the form:
( (2/z 2 + k2)f(z) cos(pz)0
it3Through the use of Green's function (or by any other method), one can obtain:
f(z) - a 1 coa(koz) + a 2 sin(koz) + (1/k0) f sin~k 0(z - z')Icos(pz') dz'I 0Note that, without any loss of generality, the constants a 1 and a 2 can be set
equal to zero and the integral on the right performed to obtain:
f(z) (cos(pz) -Cos(k~)/k 2 )2
so that
a2 2co ~Z) cos(k Z)
P(z,z') z- .+ k 0) A- i .- coa(pz') (82)
In (82) it appears that one of the terms in the series will be equal to infin-
ity if p - k0 . This condition is equivalent to h/A - 1/4, 3/4, 5/4,.....
This is not the case, however, because of the numerator end the fact that, as
p -* k0, the term in the square hrackets in (82) approaches [z sin(k z)1/2k.
Returning to (81), it is found that, since Q(z,zl) is an odd series, its
most divergent part is identically equal to zero, so that
Q(z~z') -(1/2) cos(pz') cos(pz) -(h/2)6(z -z') (83)
Using (82) and (83) with (79) in the integral equation (78a), one obtains
+ k) [f dz' B (a,z')K(z - zl,a) + -j- - )M t~(' (z- Z',a) dz']
a30z ~ 2 0 _4'
a pr 2 r 11 az 2 -h(Continued)
-44-
00 roso<,z)>- cos~0:<,) 1 osp<>,>)
ns-- i
Rearranging terms gives
2 h +iw 0 hf-.22+k2)I- dz' EL(a'z')Kl(z z a') +- • KO(z')Ml(z z ',a) dz'\az2 /L. a -
- E• i.UoI i(F) - (a,z)] (84)
where the combined kernel Kl(z - z') is defined as
iikRik Ra cos(pz) -cos(k 0 Z)
K (z z,_) W e 2 A co1,(pa')S-T a 2 hrh n-- L k p-r0(85)
with the coefficients A given byn
2 2)1/2A.j 0 [s(k1 -p 2- 2 1/? a 1
A- 2 2 1/2 a(k- p 1 - ap -1-
and p (n + ,/2)n/h.
The second integral equation from (78b) is:
h iuu0 h11f -d-<' E" "' -2- K Z'<< '"°! -+ 2 - d<' K (az (z Z' P)2... a a I-a+ _ h 0 a 1 'jp..J-
E E4 (a,z)
The coupled integral equations can now be written in a short-hand notation
suitable for numerical evaluation:rA
a32 + • )[ 11 d a' a ,< ,> <,( Z ,') + C, f <,, , f<' ) I ( a,<(Z- Z' ,, C 6(z)+ C3E, (S)
(86a) .
V:
L-(:•.. L .,'-<.T ,•,,.+,, ,,-, .:
-45-
h h1
K 1 (z
C _f dz' E (z')K2(z -z') + r5_ z ,z'M(-' E,(z) (86b)
where the electric surface current I (z) = 2ral,(z). Also, the kernel
K (z - z') has been defined previously in (85), and
I2I iw 0 /2yra C2 = -2i /a1/a/a
C4 -a/2 ; C5 =i0/4a
The factor (1/2) on the right-hand side of (86b) comeas from the discontinuity
in the derivative of the K2 (z - z') kernel, viz.,
K2(z-z') " K(z -z',p)[ ~+# K(z- Z',p)pa
M2 (z - z') - if(Z- z' ,p)=+ I- Ml(Z - z' P)
8. NUMERICAL SOLUTION BY THE MOMENT METHOD OF THE COUPLED INTEGRAL EQUATIONS
The differential equation (86a) can be solved to obtain:
hi hf dz' E,(z')K (z - z') + c, f dz' I (z')MI(z - Z')-h _h
(87a)z
"C6 cos(k 0 Z) + C7 sin(kolzj) + C8 I dz' E,(z')sin[ko(z - z')]0
Similarly, from (86b)
Sh bif dz' EH(z')K2 (z - Z') + C9 dz' I (zt)m2 (z - z') C C101E4 (z) (87b)
-h -
where C6 is unknown and determined numerically by employing the end condition
tI,(h) - 0, and
V
-46-
C1 iwpO/21ra 2 ; C7 C2 /2k 0 -iw voI/ak0
2 2C8 = C3 /ko l/apr k0 ; C9 = I5/C4 = i 0UO/27ra ; C 10 1/2C4 . -1/a
It is now considered useful to examine the four kernels in (87a,b) and to ob-tain their Fourier transforms. Thus,
K - z') - - - 2 An [cospz- s(Pz)a
2 )rh n- • k 2 o
with p - (n + i/2)7/h. The Fourier transform of K(z - z') is given by R(Q)2 2 2
0 00J0(ayo)Hl(ay 0 where Y 0 k -
2~ 2 2 ~ 1/2a ik0[(z-z')2+p24. 2-2pp' cos 0')I/z ,P) f /dp' e _____,e
(ir 0 z(-))2+ 2
+p 2o2 c. 7os ,
RI(E,P) i f ll1)(p yo)Jo(P'yo) p' dp'
where
-P> larger of p and p'
P<. smaller of p and 0'
which leads to
(ira/Y0 )Jo(PYo)Il ()(ay 0 ) if p > p'
(Iffe/Yo)11(1)(pYo)J1(aY) if p < p,
It is easily seen that
2pi ', + 1 ap) (a- 1 0 0Sa
-47-
Finally,
where
KRao) :iJo(YO)H (1)(p Y)
with
pc - smaller of 0 and a
> larger of p and a
so that
iJoO(PY O)iI() (aYo) for 0 < a
inJo(aYO)H& (py 0 ) for P > a
Therefore,
*.1YJ(ay )Il (y 0
To begin the numerical procedure, it ia recognized that, because of the
evenness of E (z) and I (z), the integrals ranging from -h Zo h may be con-
verted as follows:
h h
f E0(z')K , 2 (z - Z') dz' - F (z')IK1 (z" z') + v1 , 2 (z + z')] dz'
-h 0
Similarly,
hi h
lI(z')M ,2 (z - z') dz' f f I (z')EMi, 2(z - z') + i 1,Ziz + z'A dZI-h2 1,2 1"2
-48-
Now the interval from 0 to h can be subdivided into n+l panels. Within
each panel the unknown quantities E (z) and I (z) are approximated by con-
stants and the constant value is assigned to a location z which corresponds
to the center of the panel. With the length h = 2nt, each panel is of width
2t except the first and last panels which are of width t.
z { 1 z 2 z3 z4 z5 zn-l Zn Zn+l
0 t55 t 7t 9t-3 nh
1 2 3 4 5 n-l n n+l
Panel ii
-The locations at which the unknown quantities are determined are Riven by
1z - (21 - 2)t with I 1, 2, 3, ... , (n+l). Typically,
hf E (z')(K (Z - z') + K (Z + z')] dz'
0
il " f+ f + f + +' + f z:('[I• ') + Kl(z+ z')] dZI
0 t 3t (2n-3) t (2n-1) (
In each of these intervals E (z) is approximated by a constant value so that
(2J-l)t izKI(1,I)" (Y [(zl + zI) + K (az z') }dZ' , d., K(r)e .:I
-e' d ~ (2J-1)t ' { cos(pa1 ) - OI)+ .•d. Ar o cou (pz,') :
a2 rh (2J-3)t kn--0
.}
- - .-----~-..-.."-------
-49-
By substituting for K(C) and carrying out the z' integration, one obtains
K (1,J) = K(I,J) + B(I,J)
where
K(I,J) = (4/n) f (inJ 0 (ay0 )l1o0) (ayO)][(sin Ct)/I]{cos[2C(I+J-2)t]
+ coa[2C(I-J)t]) dC
cos[2pt(I-l)] - cos[2kot(I-i)]B(IJ) A [2
+ [ainrpt(2J-l)) - sin[pt(2J-3)]+1~
L(hlan
The integral in K(l,J) is evaluated by s~itablv deforming the contour
from the real axis to a contour that wraps around the branch cut. When this
is done, K(I,J) for IOJ can be written in the form
K(IJ) - f f(x) eaX dx0
where f(x) is a complex function of a real variable x. The integrals are
evaluated using a 10-point Gauss-Laguerre quadrature method. The special
case of diagonal elements (I-J) can be written in the form
S21zl - (&)I(sin .t)IC](e2 + 1) d& - (1/2)K(l,l) + T(J)
where
IIT(I) - f iR(C)[(oin OCt)/t d&
KbI *|
-50-
with z = 2(1 - l)t, I = 1, 2, ... , (n+l). K(l,l) is evaluated as an inte-
gral on the real axis because of the absence of the exponential decay factor,
using 10-point Gauss quadrature routines. T(I) can once again be put in a
form suitable for Gauss-Laguerre quadrature by a deformation of the contour
that wraps around the branch cut at • = k0. Care is taken in evaluating the
first and last panels' inte.ratiui;!, ecause of their ha]f normal width. What
is discussed for kernel K(z - z') or R(&) is essentially true with the calcu-
lation of the elements corresponding to the three other kernels.
Referring back now to the three terms on the right-hand side of (87a),
viz.,
C6 cos(k 0z1 ) + C7 sin(k0 jzlI) + C8 fl dz' E (z') sink (Z- z')]0 7 ik 0(z10
the first and last terms, containing respectively the unknowns C6 and E (z),
are moved to the left-hand side. For example,
C6 cos(kozI) - C6 cosokG(21 - 2)t]
C8 f dz C 8 - C8 f z ] C81 + + + C dz't0 0 t (21-3)t
We now define
PtA(I,P) I sin(k0 (z - z') dz' ( (1/k 0 ){cosakot(21 - P - 2))
(P'-l) t
- cos(k 0 t(21 - P - 1)])
It is seen that when the term associated with C8 is moved to the left-
hand side, it affects only th6 lower triangle elements of K1 (1,J) and not the
upper triangle elements, thus renderi•g the K(IJ) matrix elements not equal.
1~
-51- .
to KI(J,I), Extending these calculating principles to (87b), and using the
fact that I (h) I (I-n+l) 0, one can finally set up the following matrix
equation:
1 12 . a,n+l eln+2 . 1,2n+l al,2n+2 V1 G1
1
an+l,l . n+l,n+l an+l,n+2 ... n+1,2n+l anl1,2n+2 11n+1 n+l
a a .. a a aISn+2,1 n+2,n+l n+2,n+2 n+2,2n+l n+2,2n+2 1
1 0I
in IIV VI
I 0; i 0
a2n+2,1 *" a2n+2,n+l a2 n+ 2 ,n+ 2 ... e2n+2,2n+l I a2n+2,2n+2 C6 0
where the elements on the right-hand side are given by G(1) C • ain(k 0 (21-2)tj
with I 1 1, 2, *.., (n+l).
The magnetic current I*(z) Is easily obtained from the solution of the*
system of linear equations by using Iz(z) - -2saE (z) volts. The computer
programs are included in Appendix C and the results are plotted and discussed
in the next section.IK- -
' 1 -- -"'":" ' " • • • .,.. .i.• •-, •, %•,••• ,•.•,:• ?,V•':" ', t•' '• ;• :;, < "i I
-52-
9. EXPERIMENTAL MEASUPRMENT OF THE MAMNETIC CURRENT
The magnetization current is essentially the time rate of change of the
magnetization vector (A) integrated over the antenna cross section. The ex-
perimental procedure, however, determines the total axial magnetic flux with
the use of a shielded loop placed coaxially over a driven loop which is loaded
by a ferrite cylinder. Suitable modifications to the theory have to be made,
therefore, before the computations can be compared with the experimental re--
sults. These modifications and the assumption of azimuthal symmetry on which
they are based are discussed in detail in Appendix D.
Ferrite materials that are available coimmercially have been used in this
experimental Investigation. Table 1 lists the initial permeability a' (i.e.,
the slope of the B-H curve for small H) and the applicable frequency range
for a variety of ferrite materials, grouped under their respective suppliers.
Ferrites #C-2050 of Ceramic Magnetics, Inc. and #Q-3 of Indiana General were
selected for use in the 5-100 M1Hz frequency range. Toroidal samples of thle
#C-2050 material were obtained and its properties (p' and Q) measured as a
function of frequency by means of a Q-meter. The quality factor Q of the
ferrite material is defined by
1 "2V X stored energy 88)Q "lOSS factor r r energy dissipated per period, 271w
The measured vsluru of Q and a' for the ferrite material #C-2050 are shownr
plotted in Fig. 4(a) as a function of frequency together with the values
supplied by the manufacturer. Fair agreement is observed between the two.
The imaginary part ii" of the relative permeability can be calculated easilyr
using (88) and measured values of u, snd 0. 11Te manufacturer-supplied values
of p' and O for the ferrite material 00-3 are shown in Fig. 4(h). The valuesr
of the relative permittivity c used In the theoretical calculations werer
•I1 TABLE 1. List of Commercially Available Ferrite Materials
Source #I: Ceramic Magnetics, Inc., Fairfield, N.J.
Type of Initial
Manufacturer Frequency
Material Code # 1, Range
Mn-Zn MN-31 DC-10 2800 Up to 10 MHz
Mn-Zn M-31 DC-20 3300 Up to 10 MHzNi CN-20 800 300 KUz - 2 MHzNi CH-2002 1500 1 KHz - 1 MHzMn MN-30 4000- Up to 500 K10z
Mn MN-60 6000 Up to 600 K~z
Mn MN-0O 9500 Below 1 MHzC-2010 200-300 Below 15 MHz
C-2025 150-200 Below 15 MHzC-2050 100-150 Below 20 MHz
C-2075 25-50 Below 50 MHzCMD-5005 1400 Up to 10 MHz
N-40 15-20 Up to 100 MHz
Source #2: Indiana General, Keasbey, N.J.
Ni-Zn Q-1 125 Up to 10 MIz
Ni-Zn Q-2 40 Up to 50 MHz
Ni-Zn Q-3 18 Up to 200 MHz
Source #3: Fair-Rite Products Corp., Wallkill, N.Y.
Ni-Zn 30-61 125 200 K/R: - 10 M}Lz
Source 54: Ferroxcube Corp., Saugerties, N.Y.
Ni-Zn 4C4 125 Up to 50 Mh1z
Mn-Zn 3D3 750 Up to 5 M4z3B9 1800 Up to 5 MNz
Mn-Zn 3B7 2300 Up to 1 MIz
Source #5: National Moldite Co., Inc., Newark, N.J.
H-Grade 125 @ I MK1z tip to 20 MHz
Source #6: Stackpole-Carbon Co., St. Marys, Pa,
Grade 24 2500 Uip to 100 KUzGrade 27A 1000 Up to 800 KUz
Grade 9 190 Up to 2 Mi0zGrade 11 125 Up to 6 MK1zGrade 12 35 up to 80 Ritz
Grade 2285A 7.5 Up to 300 'Ulz
'A
-54-
- 0- 0
z z2F- U)
C0 u
z <_ LL J
-f) z
I Z
0i w
U- 0
00
o~~ 0 0Qh
M- 0 c
In
p Z0
o ~0 -
0.0
00 (Y 'hiA
I -55-
also supplied by the manufacturers. It can be seen from Fig. 4(a) that the
value of Q for the ferrite material #C-2050 is nearly constant (2!00) up to
abouL 2 Mflz and then falls off rapidly to less than 10% of this value at 20
MIz. Similarly for the material #Q-3, Fig. 4(b) indicates a nearly constant
value of Q (_500) up to about 15 Mbiz, beyond which it decays to %_50 at 200
Three antenna cores were fabricated, as photographed in Fig. 5(a)-(c).
Cores (a) and (b) are made of material #C-2050, while core (c) is made of
material #Q-3. Cylindrical rods of 5/16" diameter and 5.25" height were used
along with adhesive tape to fabricate core (a) of 2" overall diameter and 21"
height. Core (b) has the same height as core (a) but is comprised of five
cylindrical rods of I" diameter and varying lengths. Core (c) is formed from
three cylindrical rods of .625" diameter and 7.5" length for a total height of
22.5". As was pointed out earlier, cores (a) and (b) are useful for frequen-
cies up to about 20 14hz, core (c) up to 200 MIz.
The three cores were used in various antenna configurations in which an
"electrically samll loop antenna is loaded by a finite cylindrical ferrite rod.
"'he antentla parameters for the eleven different cases are tabulated in Table 2.
For antennas numbered I through 3, measurements were made at frequencies of
i0, 50 and 100 MHz, respectively. The electrical radius ak0 of the driven
loop raeges from .00166 to .01662. Antennas numbered 4 through 7 were oper-
ated at frequencies of 5, 10, 15 and 20 Millz, respectively; the electrical
radius ranged from .00132 to .00531. The operating frequencies for antennas
numbered 8 through 11 were the same as for the previous set but the radius
"was doubled.
It can be seen in Table 2 that the value of ak 0 does not exceed 0.017
for any of the eleven antennas. This ensures the validity of the assumption
-56-
<I; c~04
H N(N H(%4
II i
~I-) .J El
-57-
TABLE 2. Antenna Parameters
#Q-3 Material (Supplier: Indiana General)
2a - 0.625", 2h - 22.5", S - 2 tn(2h/a) " 8.5534
Antenna vr 'r - " h ak z r + jxr r rW h/A0 0 m m m
1 18 - j.036 .00952 .00166 .00797 - j3.7637
2 19 - J.0544 .04762 .00831 .00214 - J.70979
3 20 - J.0890 .09525 .01662 .001577 - J.33443
#C-2050 Material (Supplier: Ceramic Magnetics, Inc.)
i) 2a - 1", 2h - 21", Q - 2 In(2h/a) - 7.4754
4 100 - Jl.0 .00444 .00132 .0051 - j.50479
5 115 - J2.55 .00889 .00265 .0049 - J.21892
6 125 - J12.50 .01333 .00398 .01341 - J.13270
7 135 - J67.5 .01778 .00531 .03747 - J.07394
ii) 2a - 2", 2h - 21", P " 2 tn(2h/a) - 6.089
8 105 - JO.63 .00444 .00265 .001275 - J.12611
9 120 - J2.4 .00889 .005319 .001225 - J.05456
10 150 - J15. .01333 .007979 .003352 - j.03292
11 140 - J42. .01778 .01064 .009368 - J.01815
-58-
that the driven loop be electrically thin in the theoretical calculation of
antenna currents. The height of the monopole antenna (h/X 0 ) ranges from
.00444 to .09525 so that the longest dipole is nearly (1/5)-wavelength long.
The value of the relative permeability Ur - Jir from Fig. 4 and the in-
iternal impedance zm per unit length calculated using equation (36) are also
listed in Table 2. In this section an ejwt time dependence is implicit and
is more convenient. Due account of this change in notation has been taken
iin using (36) to calculate zm. It is observed that for all antennas consid-
ered, the internal impedance is largely reactive.
For each of the three ferrite cylindrical cores described above, a set
of driven and measuring loops was fabricated. A photograph and represents-
tive line drawing showing the construction of the loops are shown in Fig. 6.
The six loops were all constructed from commercially available microcoaxial
cables ending in a modified BNC connector. The driven and measuring loops
are placed coaxially in the experimental setup, as can be seen in the photo-
graph in Fig. 7 and the block diagram in Fig. 8. The short lengths of micro-
coaxial transmission lines leading away from the two loops are at right
angles to one another in the horizontal plane so that any inductive coupling
between the two is minimized. The signal source used in this experiment was
either a 61R-1001A (5-50 Mhz) or an )IP-32000 (10-500 Mhz) oscillator. When
the CR-lOOlA oscillator was used for measurements with antennas #4 through
11, the power amplifier was not needed. The HP-230B power amplifier was
used only in conjunction with the |IP-3200B oscillator for measurements on an-
tennas 01 through #3. The source frequency was accurately measured using an
IIP-5240 electronic counter. A signal proportional to the total axial magnet-
ic field was induced in the receivinp loop. An IIP-8405A vector voltmeter was
used to detect and record this signal (B). The reference signal (A) to the
1111
-.59-
P I 1 1 1 11 1 1Il IIF11 t !
Jill 2 461
FIURITHE-AR FDIE AND ASURING10P LN
-60-
LILI
A RR -2414 ELECTRONIC
h
POWýER AMPLIFIER 2
OSCI LLATORG.R7-IOO1A (5-60OMHz)/H.P.-32OOB (10-500MHZ)
FIG. 8 BLOCK DIAGRAM OF THE EXPERIMENTAL SETUP FOR
MEASUREMENT OF MAGNETIC CURRENT ON THE
I FERRITE ROD ANTENNA.
:177.
-62-
vector voltmeter was provided from a coaxial T. The vector-voltmeter read-
ings were recorded as a function of the axial distance z from the driving
loop. In this manner the amplitude and phase of the magnetic current distri-
bution were obtained for the eleven antenna configurations described in
Table 2. The unnormalized data are given in Table C-2 of Appendix C.
Computer programs, described and listed in Appendices A and B, were
utilized in calculating the magnetic current distributions for the eleven
cases. As discussed earlier, the theoretical calculations are based on a
treatment of the ferrite rod as an imperfect magnetic conductor. The theo-
retical and experimental current distributions are shown graphically in Figs.
9 through 11. Also appearing in Figs. 9 - 11 are Tables 3, 4 and 5, respec-
tively; these show the calculated values of input admittance Y = Iz(0)/lIz 0
(G + JB)* ohms and input impedance 7. = l/Y* mhos. The antenna numberinp
scheme used in the figures and tables corresponds to that given in Table 2.
The values of Q - 2 Wn(2h/a) for the antennas in Firs. 9 - 11 are, respec-.
tively, 8.5534, 7.4754 and 6.089.
The agreement between the theory and experiment is seen to be good. The
antennas used here are relatively short and, consequently, the current dis-
tribution is seen to be nearly triangular. As may be expected, the largest
deviation of the experimental values from the theoretical computations occurs
at either end of the antenna (z/h - 0, 1) s•d especially at the driving point.
For this reason the raw experimental data were :tormalized in most cases to
the theoretical calculations at a point nearly a third the distance from the
driving point to the end of the antenna. In the case of antennas 01 and P2,
there appears to be a kink in the experimental values for the phase of the
current distribution. This is believed to be due to the stacking of individ-
ual ferrite rods by means of adhesive tape, This is not seen in the magnitude
!
-63-
(ý079ý 3.63 . I-j.0083
I,2ON
-- A0ss 0 7 i 0 55 1 01 0 1-0- 50 10
110)I 1.1tvO /00F) 4I01R005P
00757009D S ABLE-3 cALCULATED (3-TERM THEORY)
MTEN0 A 0¾ N0iO Y AND Z' FOR THE THREE
I10 15 O-1. 5S-441
7010
J ANTENNAS.
i oil O04
~-..THE.ORETICAL 3S-TERMI THEORY1
.. EXPERIMENTAL
(FERRITE SOURCE0 MATERIAL 0#3 SUPPLIED BY INDIANA GENERAL)
FIG, 0 PLOT OF MAAGNITUIDE AND PHASE Osý THE MAQ4NETIC CURRENT ALONG4
THE ANTENNA IE fj 2j,,2h/0lBaS534)
-64-
Ow,0 :2 N ON
N OAN* 00
N0 0 Z Zd -
-o _ . - z- - N
ow-.,.N x
N � �0 � 8
NN 0-
z* tflttt.. N
) 8 8 NU
0 U
z0
8 8 N
K 0'
N .- U.,
8 8 8N :.� 6w,
8
_______ C V02. 8 � 0
I .---..------
-65-
wll
434
~ 9.3
curves because of the relatively low magnitude values. The overall agreement
of the theory and experiment was used in deciding the point of normalization.
The coupled integral equations in (86a,b) in the two variables, the tan-
gential electric field E (z) and circumferential electric current 1 (z), were
solved numerically by the moment method on a Sigma-7 computer system. The
method itself has been discussed in Section 8; the computer programs are
listed in Appendix C. Table 6 contains a description of all the subroutines
used in this computation. The basic philosophy of this method is to reduce
the set of coupled integral equations to a system of linear algebraic equa-
tions. The standard routines [61 for solving a system of linear equations
were modified to handle complex variables. The results of these computations
are plotted in Figs. 12 through 14. As before, the experimental data have
been normalized at a point approximately or.e third the distance from the
driving point (z - 0) to the end.
The magnetic current I*(z) is easily obtained from the solution for the
tangential electric field using the relation I (z)- -2naE (z) volts per unitA *
current In the driving loop. The input parameters Y and Z are also tabu-
lated end the tables are included in the figures showing the magnitude and
phase of the magnetic current. In all eleven cases the phase is nearly con-
stant, since the antennas are electrically short in free space, and most of
the magnetic current is in phase quadrature. The agreement between the ex-
periment and the theoretical calculations is very good Including near the
source. This was to be expected because the coupled integral equations
(86ab) in two variables comprise a far more accurate and independent theo-
retical formulation of the problem than the approxiiate integral equation
(39) which relies rather heavily on an analogy between the ferrite rod an-
tenna and the resistive cylindrical dipole antenna.
-67-
TABLE 6
LIST OF SUBROUTINES USED IN SOLVING THE COUPLED INTEGRAL EQUATIONS (86a,b)
PROGRAM NAME PURPOSE
MAIN Computes E (z) and I(z) by solving the coupled
integral equations (86ab).
BSLSML Computes Bessel functions J0 (z) and J1 (z) for
Izi < 10.0 with 5 figure accuracy.
BESH Computes Hankel functions Hll)(z) H(2) (z)
111)(z) and H1 (z)W.
QGLIO 10-point Gauss-Laguerre quadrature routine.
QGIO 10-point Gauss quadrature routine.
i FCTK Computes the integrand for K(z - z') for I • J.
FKIIR, FKIII The same, for I - J - 1.
M1II The same, for I - J 0 1.
FCTDI, FMIIlR, Computes the integrand for Ml(z - z') for I1 J,
P1IIII, F1III I - J 1 1, and I - J 0 1, respectively.
FCTK2, FK211R, Computes the integrand for K (z - z') for I1 J,K2(
PK(211I, FK21I I = J - 1, and I - J V 1, respectively.
FCTM2, FN211R, Computes the integrand for 12(z - z') for Iq J.
FM2111, FM2II I - J -1, and I - J 0 1, respectively.
AUX Auxiliary function used in computing the above
integrands.
SERIES Computes the infinite series part of the kernel
K (z - a').
DECOMP, SOLVE, Programs used in solving the linear system of
IMPRUV, SING algebraic equations.
ANGLE Computes the phase angle of complex variables
I(z) and E (z).
S !
V>
Ii
F 0
� �t� I a
"4 0 01
01-0140 0
I 010
1�10�00I 0 0
1�*
t *0 � 00 o�
L � I .x�**.4 a a -II
'0 Oi4oo 0.4
*0*� " 0
Is.
�Li
~, -69-
11 t-/~,
I : I
A ___"A01 1
, i0
LI
- -, -.
0 .5
-71-
TABLE 7
INPUT ADMITTANCES AND IMPEDANCES OF THE ELEVEN ANTENNAS OBTAINED FROM SOLVING
THE COUPLED INTEGRAL EQUATIONS (86a,b)
Antenna # Input Admittance Input Impedance(ohms) (mhos)
1 .003 + j 4.99 .00012 - J.20040
2 .03 + J30.22 .00003 - J.03309
3 .10 + j62.32 .00003 - j.01605
4 .03 + j 9.23 .00035 - j.10834
5 .14 + J20.13 .00035 - J.04967
6 .97 + J29.81 .00109 - j.03351
7 6.21 + J45.62 .00293 - J.02152
8 .02 + J23.55 .00004 - J.04246
9 .16 + J44.27 .00008 - J.02259
10 1.06 + J60.02 .00029 - J.01666
11 4.26 + J75.32 .00075 - J.01323
).I
-72-
10. SUGMARY
An electrically small loop that carries a constant current and is loaded
by a homogeneous and isotropic ferrite rod has been called the ferrite-rod
antenna. In Part I [2] of this report the ferrite rod was assumed to be of
infinite length and the problem was treated using a boundary-value approach.
In a practical situation, however, the antenna is necessarily finite and
often electrically short so that a new mathematical formulation was needed,
* [ along with an experimental investigation, for the problem of a finite ferrite
* Erod antenna. With this current distribution known precisely, other quanti-
ties of interest can be derived from it.
Although, in terms of physical mechanisms, the ferrite-rod antenna can be
compared with the dielectric rod antenna, there exists a complete analogy be--
tween the ferrite antenna and the conducting cvlindrical dipole antenna.
This analogy is based on the dual property of electric and magnetic vectors
in Maxwell's equations. The electric dipole antenna has received consider-
able attention from researchers in the past and, therefore, a treatment of
the 'magnetic analog' of the dipole antenna is considered useful. Based on
this analogy, an integral equation has been derived for the magnetic current
on the finite ferrite-rod antenna. As expected, the integral equation is
identical in form to the corresponding equation for the electric current on
the dipole antenna. This derivation was based on the assumption that the
value of the relative permeability Ur of the ferrite material equals infinity.
In effect, the ferrite is treated as a perfect magnetic conductor aq when in
the 'electric case' the antenna material is assumed to have an infinite elec-
trical conductivity a. However, in practice, a material with "r equal to in--
finity does not exist and, furthermore, over a useful frequency range the nr
value is not high enouRh to justify using the perfect conductor approximation.
S~.*----- -..,**....,
For this reason, the integral equation had to be modified. The modification
was achieved by defining the internal impedance per unit length of the magne-
tic conductor to be the ratio of the tangential magnetic field to the total
magnetic current flowing in the magnetic conductor. An approximate, 3-term
expression for the magnetic current was then obtained in a manner paralleling
the procedure used by King and Wu to solve for the electric current on the
imperfectly conducting dipole antenna. It was found that for commercially
available ferrites the internal impedance per unit length was largely reac-
tive so that the propagation constant k (- 6 + in) on the antenna had a large
imaginary part. The predominance of the attenuation constant a makes the
magnetic current very small and, thus, one is led to conclude that the prac-
tical ferrite-rod antenna is not a very efficient radiator.
The treatment of the ferrite-rod antenna as an analog of the resistive
electric dipole antenna relies rather heavily on the mathematical equivalence
of the two problems under idealized driving conditions. For this reason, an
alternative derivation of the integral equation for the tangential electric
field o• the ferrite surface was developed. This derivation led to a pair of
coupled integral equations in terms of the tangential electric field and tan-
gential electric surface current. The coupled integral equations were solved
numerically by the moment method and the magnetic current obtained from the
tangential electric field. It was also verified that in the limit h . - the
equations decouple and are in complete ap.reement with the results of the
theory for the infinite antenna.
Since the magnetic current is proportional to the total axial magnetic:; field, a simple experimental apparatus was built to measure the magnetic cur-
rent distributicn on several antenna configurations. A praphical comparison
of the theoretical and experimental results has been presented. Although the
i *1!
-74-
three-term solution has been shown to give good results for antenna lengths
k h < 5m/4, the antennas used in the experiment were much shorter &nd the0-
near-triangular distribution of currents was verified. The frequency re-
sponse of the properties of available ferrite materials and the practical
limitations on the size of the ferrite rods made it difficult to construct
antennas of longer length.
In conclusion, while ferrites have been used extensively at microwave
frequencies and up to several MegaHertz, the fact that ferrites are now be-
coming commercially available in the range 30 - 300 Mdz should lead to useful
applications. In situations in which the physical size of the antenna must
be kept small, a loop antenna has limited usefulness because of its low effi-
ciency and radiation resistance. The insertion of a suitable ferrite core
offers the advantage of both improved efficiency and increased radiation re-
sistance. Although, in theory, an increased radiation resistance should sim-
plify the problem of matching the antenna to its associated circuit, in prac-
tice there remains a severe pronolem, This has been discussed by Dropkin,
Motzer and Cacheris (71 who made measuremenits of the receiving characteris-
tics of a cylindrical ferritc-rod antenna at a freeuency of 75 MIz. They
conclude that both ferrite-core and air-core loops can be described by simi-
lav equivalent circuits. These circuits have resonant properties and each of
the lumped circuit elements can be identified with a physical quantity char-
acterizing the antenna. The improved efficiency and the increaaed radiation
resistance which were determined experimentally can be attributed directly to
an increased magnetic flux passing through the loop. Thev also make the in-
teresting observation that with a dielectric cylinder (rr - i0, 1 1 r '), the
size of the ferrite used had no effect on the air-loop properties. This was
because the loop used was small enough to act as a magnetic dipole.
APPENDIX A.
COMPUTATION OF MAGNETIC CURRENT AND ADMITTANCE OF FERRITE ROD ANTENNA
The approximate magnetic current distribution as given by (51) is
I() -k----- - sin k(h - lzi) + T*(cos kz - cos kh)z ~ dR cs kh,(inW
, kz koh+ T(cos -0-- cos -0-)]
D 2 2
where TU and TD are given by
TU (CVE" CDEV)/(CUED CDEU)
T (CUEV CVEU)/(CUED CDEU)
with
CU - (i - (k2Ik2)](fdUR - TdR)(1
- cos kh) - (k 2/k2)dUR Cos khU~ -(I d(k (/- C0 MY') dRR u Rh
k h~dUI ( -c 0
k h koh
CD d 4d( - -)- [I - (k /k2)]TdR(l - cos0-1)- ) J2 2D 4 2
EU -(k /k )RdUR Cos kh + (i/4
)vdUI 2 + V (h)
k h• ~ED "-(1/4)Td 0o -•+V(h) i
ED Cos"~% 2 + Y
k hEv - "(l4)d o - (h)
The I functions appearing in the above expressions are defined as follows"
S'dR(h) 11 </)/2/2
If dR (h 1/4) -t2 k 0 h < 3v/2
71
-76-
h cos k 0 r cos kOrh
Td(z) csc k(h - Iz) h in k(h - _z'_) dz'-h [ r h
h Cos k O0 kOrhjdzTR [1 coo kh]- if (cos kz' - cos kh):f r krro e dhdU 0 5wy -- (Co , dzg-h L r0 rh
ir ikrhkoh h kozi koh eik 0rO A
[I (Os cog ' 0)[ e dz'dD Co 2 j 22 r r
koh h ssin kor sin k r(d I - 1 - C o s _ ._- 1 f s i n k (h - z ' j) d z 'dI - 2 j -h r 0 r h
koh h sin k~r 0 sin kOrhdl (I - - I (COB kz' cos kh)[ dz
where the propagation constant k is given by
k - ko[l + (i4vvozi/ko dR)31]2
with C 376.7 2 the characteristic impedance of free space. However, since
"•dR is dependent on k. an iteration procedure is used. To begin with,k 1 as
given by
kI k0{l + (14vy zi/k0 T )] 1/2
k1 + 0OdRO
is determined. With k1 substituted for k, TdRl is computed and then k is
evaluated using
k - ko[l + (i4aoniz/koVdRl)]UZ
This new value of k is used in evaluating all the 1 functions and the current
• • distributions.
i7
-77-
APPENDIX B
This appendix contains a listing of the main program and all associated
-ubroutines. The main prosram accepIts as inputs h/1X0 , Q and zm and computes
the input impedance Z , amittance Y , and magnetic current distribution as a
function of distance (z/h) along the antenna. Subroutine NINTG employs
Simpson's rule for integration to evaluate the functions. The various inte-
grands are calculated using the subroutines FCTP(Y), FCTO(Y) and FCTI(Y).
AM- COpy flnRwISli 70ODDO .....
F--ThT IV ALtflL ?i XALIT A.NEN5 I0T
g TESTU FAlA CELULT ET T EAGEATk KCLUGEt TEEPO AETA
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ITj lo I o - V
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-81--
APPENDIX C
This appendix opens with two tables containing experimental information.
The first, Table C-1, lists the values of the various antenna constants (di-
mensions, ferrite characteristics, frequency, etc.) for the eleven antenna
configurations studied experimentally. Table C-2 gives the raw measures data
(unnormalized) for the magnetic current distributionson the eleven antennas
as a function of z.
The appendix concludes with a listing of the computer programs used to
solve the coupled integral equations in (86a,b). The procedure used, i.e.,
the moment method, was discussed briefly in Section 8. The coupled integral
equations are reduced to a system of linear algebraic equations which are
then solved for the unknown variables. An unknown constant in the integral
equation is also determined in the numerical procedure by imposing the end
condition at z - 11. The magnetic current I1(z) is easily computed from the
solution of the tangential electric field E (z) by using Iz (z) -2maE(Z)
"volts per unit current in the driving loop.
-82-
(N (N LA (N LA (No 0 0 0 0 0 32 0 0 -4 LAo 0 0 0 0 0 0 0 0 0 0
X n n n n en en .rC 'en re3 t I I I I I
(N 32 CA (N 0 H H CO 32 3' (N
.7 (N H (N 32 3' CC 0 32 34 0 0
(N 0 0 H (N 0 H (N 32CO 3' 3) 32 3' 0 N (N 0 (N
LA CO 32 H N N (N 3' (N (N 32 3' 0 (N CO (N�4 3' 0 0 H 3'(N *(N (N (N (N (N (N �. (N (N (NI I I I I I I
0 (N 0 CO 3' �II (N 0 (N CAN (N
ri CO H 32 3' CO 3' (N 3' 3' (N (N32 CO H (N 32 .-C (N 32
3' H (N (N CO 3' (N 3' (N 3210 3' (N 3' (N 3' 3' (N '0 (N 3' I
C 3 0 0 H 0 0 0 0 0 0 0 H
Ad H CO CO H (N (N LA (N LA 0 0 0 0 0 C� 0 0 0
J ICC
(N (N (N LA LA LA LA 3' 3' 3' C:00 .4
LA LA LA N N N N CO CO 0) C)
CO CO CO N N N N CO CO CO 3' C_______________________________________________________________ 3
UCA LA LA LA 3' 3'.. 3' 40 3' H H H .4 . C0 0 0 C(N (N en (N (N � H (N .- e� C)
IC LA LA
CC . H H H H H H (N(NI) (N (N (N (N (N (N (N (N (N (N (N
C (N (N (N
'.4 ______________________________________ .4
CC LA LA LA IA34 (N (N (N H H H H (N (N (N (N(NI) 3' 3' 3' IC
Ce .4 0)
(.4
3' 32 3' - LA LA LA (N 32 LA (N4 I 4CC LA CO . LA . 3' .4 32
0 0 0 re . (N N . (N (N 1-CSt . . . I (N .4 3' Ce (N e
0 ee 1-' en 0 (N (N '-C I 0 0I I I 0 I I I LA 0 /1 CtCO 3' 0 .4 LA LA LA 0 (N HH H (N H (N en (N H
.4 LA LA HH (N . C')
54 .4 .4 .-4 0 �4 H H H
C.� H re (N .4 H .-C 0 - H .-C ..eI I H
'P0 .4 .- e (N .44 (N
tO 0 0 0 LA 0 0 0 LA 0 LA 0
________________________________
(N CCC * LA '0 N 3' 3' .4 (N
If
-83-
TABLE C-2 Unnormalized experimental data (Refer to
Table C-I for the numbering scheme and antenna constants).
Antenna Al Antenna #2 Antenna #3
zcms Mag Phase Mag Phase Mag Phase0.1 - 38 -50 -
0.2 130.0 30.4 - - -
0.3 100.0 30.2 31.5 -48.8 - -0.5 - - - - -
0.6 - - 25.5 -47.9 - -
0.8 - - 22.0 -47.5 - -
1.0 77.0 30 20.5 -47.2 82.5
1.5 - - 16 -47.0 57,0
2.0 49.0 30 13 -46.6 44 -169.23.0 33.0 30 9.1 -46.4 28.5 -1784.0 23.0 30 6.4 -46.5 19ý0 -1825.0 17.0 30 4.7 -46.5 13.5 -1846.0 12.5 30 3.5 -46.5 10.0 -1857.0 9.25 30 2.65 -46.5 7.7 -1858.0 7.0 30 2.0 -46.5 6.2 -1859.0 5.3 30 1.45 -46.8 5.6 -183.5
10.0 4.0 30 1.0 -47 ? ?
11.0 3.30 30.5 .8 -47.5 3.1 -18412.0 2.65 30.6 .63 -48.0 2.35 -185.513.0 1.95 30.6 .51 -48.5 1.85 -18614.0 1.75 30.6 .41 -49 1.45 -18615.0 1.45 31.2 .34 -50.2 1.20 -18616.0 1.10 31.6 .285 -51.0 1.02 -18617.0 .9 32.8 .240 -52 .86 -18618.0 .75 32.6 .205 -52 .70 -18619.0 .62 33.0 .175 -52 .59 -18620.0 .51 33.6 .140 -52 .50 -186
21.0 .42 35 .41 -18722.0 .36 38.2 .35 -18623.0 .31 38 .30 -186.524.0 .27 38 .26 -188.525.0 .22 38 .23 -18826.0 .19 38 .205 -190
-84-
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-86- THIS PAGE is BES QUAJI
nw copy FURKSIM1TOCDO
COMPUTER PROGRAMS USED TO SOLVE THE COUPLED INTEGRAL EQUATIONS (86a,b):
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0024 lp=IPM)l0025 SIZE= CABS(ULIIPKI))SCAL§ES(IPI0026 IF(StIF-IIIG) 11,11,100027 10 RIG=STIZE _______________________
00i28 InXpIV=I0029 ___11 CONT INUE ___
0030 IF01GIO 13#12,130031 _12 CALL SIN-G (2)-0032 GO TO 17003313 IIXI-I 4,540034 14 J=.IPSb(K).0035 1_ _ IPS (K.I=IPS I IXPIV)____0036 1 PS( IOxfIV ) jý003? - 15 KP =IPSIKI003P PIVOT-IItIKP.K)0039 KP1=K41 _ __ ____________________
0040 600 Y6 I=KPX,N0041 IP=IPSIII0042 EI4=-ULI IP,K)/PIVoT0043........UL(I,)-I0044 00 16 ,J=KP1,N0045 U-3L(iPJI=1UIIIP,J)4EM'ULIKPJ)
C INNER LOOP.0046 16 CONTINUE
*0047 17 CONTINUE0048 KP=IPS(N)
-0049___ --- I__ F (CABS (UL (KP, N) L-19,1 J1 A90050 18 CALL SINGCZ)005.1 19 RETURN005? END
-102- YA CPY AIME TOIS~ DDC .*
FORTRAN IV G LEVEL. 21 SOLVE DATE-= 75195
-00_______ -SUBROUT INE ~-______0002 DIMFNSION TPS(35)0003 COMPLEX UL135,35),B(35),X(35),SU'l0004 COMMON TAKOTPY,TKOTIPS0005 N=NN0006 C NP1= N+1
0007 IP=TPS(1;)0008 X(I) =B(IP)0000q 003 2 1=2,N0010 IP=ips(I)0011 IM1=I-10014 l2 SM( 0. , .) ___________
00 13 - - D 0 I '1 J~, IM0014 1 SUM=SUM+ULIIP,.J)*X(J)0015 2 X(Il=P(IP)-SUIJ
C0016 [P=IPS,(N)0017 ____ X(N)= X(Nl/ULIIP,N)____0018 --- 'nf '4 IBACK=2,N
C I GCUS (N-1) ....0020 IP=IPS(1I
j0021 IP1I41~0022 SJ~0. 'o.I __
11023 Dn 3 J-IP1,.J-0024 3 SUM-StlM4IIL(IP,Jli`X(J)0025 4 X11)=IXII)-SUM)/UL1IP#I)0%026 Rt:Tk)RN0027 END
FO0RTRAN TV G LEVFL 21 ANGLE DATE u75195
__ 0001FUNCTO10N ANCLECX,YL.
0003 IrMX5'5O,500,4500004 500 IF(Y) )50,3000250Ono,, 300 ANWl -o0.
01111 I Y0 AWA U l? 7U.C
0001i T10
0011 450 IH~Y) 455,454,4530012 454 A NI r.1Uo0. 00013 p r T1:10N0014 453 At, O.F -C OfF T A ANy T AN-...-X)"'001"¶ Rf: TiIP'l0016 4i55A(,r -IFTAA(-/)#0.0017R 11'J,)18 550 XNE-X001 . )I Y)551.,'i5l,55700?0 553 ANWiP1 F- g. 0
002? 552 a N G1 f 0 0. C 0 fF T A AN, Y/ XN I
Jv ' 4 554 6 NGLl:,1l80.0,COFFTATANl-YXN1)
0026 Vl
-THlIS PACE~ IS Mi~T QUALITY ICZICAO"-103- FRUI copy Fum.Ishi1 'D __-
FORTRAN IV G LEVEL 21 1 MPRUV_. -DATE- 75195 _
0001 -_____StU BRLi)T I NF. IMPPUV CNN t.A,0U1. tat-iJPILSJ. __________
0002 COMMON TAKOTPYtTK0TTPS0003 COMPLEFX A( 35,35)hOL (35,35) tB(35) X( 35)9.R(35) ,DXf35.),_T_0004 COMPLFX*16 SUM,AIJ,XJ
a ~~C _USES ABS( ),AMA Xl I )tALOG 10(I0005 - MNN
0006 EPS=1.OE-80007 - - ITMAN(=16
C **EPS AND ITMAX ARE MACHINE DEPENDENT.
0008 XNORM=0.0
0009 XNORMnAMAXl(XNOR'iCABS(X(IH))
0011o~ IF(XNOR.M) 3 .2,3 --
0012 2DIGITS = -ALOGIO(EPS)0013 GO TO 10
0014 3 DO 9 ITFR=1,ITMAX0015 - D00 5 -I -=1 ,N0016 SUM=D.O0017 00 4 J=1.N__0018 AIJ=A(I,J) -- -0019 XJ--X( JI00_k20 4 ,SUH=SUM+AIJ*XJ________
001sum=RmI-sum-__ _____
0022 __ 5 R( I)=SUMC **IT IS ESSENT IAL THAT A(ItJ)*X(J) YIELD A DOUBLE PRECISIONC RESULT AND THAT THE ABOVE + AND - BE.0OUBLIE_,PRECISLON.**
0023 CALL SOLVE(N,UL,R,DX)
0025 0O 6 1-1,W'0026 T=X(II0027 X(II'.X(II +PX(IIOM ~ DXNORM-AMAXI(DXNORM, CABS(X(tikT1I)3029 6 CONTINUE
0031l 7 DIGITS=-ALOGI0(A'4AXI(OXNORM/X'JORI4EPS)I0032 IF(OXNORM-F.PSk*XNORM) 1001009
0033 9 CONTINUEC ITERATION DIn -NOY CONVERGF
0034 CALL SING(31,~35 10 RTURN
0036 END
-104- T
SU~p~,I1 ,c ~ Q1L CMI MJ, THI~S PkGE IS BFST QUAI4I
IMLIfLCIT COW1L EX 49(C), REAL W(T)3 DI1iNSlbm. CSUM(30) OlN
4TJP-FLOAT(MJ)4.2-1.~17 -TJ3-FLOAT(Mj)x2. -3.
LoCCON~-4./(tIPY-CmuR
12 5 P4(Tth*8)~1-PY/TH)13
Z. CZC KIXCS&R1(CS)
16 CZo=CZ17CALL 6SLSML(t4C2O,CJO)
18 CALL 6SLSML(iCZi,C.)j)15TAP=TA.TP20CCEFT-(CJOVCZ/CJt)7TAF-O.5
22 TC?ýC6S(2. -7kOT.T13 )1SF S TN I T-T"3
'. TS2ýSIW(TpT-TJ3 )
26 7l'S=ITNt.5.()~TAKO4-2-TAN .2)
'ri-C A5( SICumLK))T 2 CA13 I CSUM(IL.4)IJ ~~30 lAs
.71 ~ ~ ~ ~ ~ ~ G TO(Iku..Eo)4 3 2032 K.K~t
34 IF (s.LE-26) Go To L635 w~ile (6,16)'36 '16 FORlMAT (5", '48888 (AUT~I8MI UNSA?3SF46D183 IN2 TeRms.,i)
38 lkvl35 15 Go 10 54.0 20 CS'CCON8CSVM(1)
`1.00VTIN~I AIJk(N1,Mj.x14r1?'rt2 IV'.ICIT C0MP
tLiK,fVCI,RCAL.h(T)
'3 COHMSN TAXO.(M-A?,TPY,T*A,.V,CA41,T83,TTIO,flXO1(O,34ji,244ij-l
6 IJI.FLOAT (Jl)
7TJ311t2,%Al (J,
CVit( 1..H TkOT
12 TIJ'tLOAT(1.t)
It. mfJ ?.Mi1 *.M-IJ-6
Af T^ ,0W 1 E2't9 T1.t:XPt '1 f'3~)
21c2.(t't-cCI*.TJt)
2, C4'CEt.FC-CKoT-TJt)
2 Tq:Rt.At(CFttt
29
-105- THIS PAGE 1-5 RET Q U A L tTy rs :cW ~yiQu Copy FUMIRaisj TYon DDC
$A' aT~s GLI0(L J,FCT,TVR,TYI)
3 C IMPLICII CdlIPLEA.BCC),QEALMACT)
A C 10PITGOS-AVRr QUAORATURE FORNULA OF S.S.P. (10M)
4CALL rc7(r,J,-rA,T69,TGI)7 TP *991182Ee-12 TGR
.9TXA3I99LS59to CALL. FCT( 1,J,TTGR106J)1
1YI.Cyli-.i6495~t -S ~TGI13 *-L1%L14 CAL'- CT(1,J, TA 1G1ZTC,)
T5YYR-YYT.42%93IAC-6o G'6 tYjýTy1't.qak9314E-6 AfTGl17 ')T -8 3 -
CALL RCTCIJ,TA.TGRT6I)19 TYR-TY~t.ee2sqa3Z3E4 %T64
20 YI.TYIt.282592
3E-4 . TG
2<CAL L 6C3 (I, JJT)(,T$, TGI>5TYR=TY~t.lS3Oo64E-3 A G
'yif"tY.t7A30094C-3 A TG?CX ý. ý55'2496
34CALL FCI(1,J,TX,TGR,TrGI)27 TY2&TY~t-.oŽ901AS157 * 766
28 YIcTYIj .6Q9S5L75 *T62
SoCALL PC T(1,j, TX, TGP, TG1trYAATIYR t. 06
24687 46 A 7GR
CAL L FCI1( 1, J,1XIGRI7Gi)%TYR'YYl~t.2160G83 A TGR
36 7yyi-TI fli.-Z I80513 * T'30
33CAL L fC7(1),J137,TGM,TCI.Tv.7t'vpj.1to I 199 TG
CAL L FCI (I, J,TX, 169,T61)1yt'f( 30&4411 A G 16
44 T YhIYI t.302ý413
2 1 6068*7T (1&i ,lA (t Lf 0 Pow ;N DECOMIPOSE::t3 12 8 1Ar&1 (3'IL~tAWL' IN61 DE COMPOSE. ZERO 0rvIVE IN S',(
1. 3 507MMT 11AOAO CONYL.76ENCL1- W'*lCUV. HARIGX is NEALty SIARULMj.
6 1 WRII(6Ii'7 60 TO;08 2 WRITE (6-12)9 60,10 ic,10 3 Ws It (6.8%1 10 PiTUVW
laEN
-106-
APPENDIX D
BASIS FOR MORE ACCURATE COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS
With reference to Fig. 8, the voltage V(z) induced in the receiving loop
is proportional to the tangential electric field, viz.,
V(z) = f Bz dS - -2uaE 4 (a,z)
This relation, however, assumes that there is azimuthal syassetry and that the
current in the receiving loop is negligible. These assumptions are investi-
gated in detail.
Azimuthal Symmetry. The value of ak0 for the eleven cases studied in
the experiment ranges from .00133 to .01662, whereas lak1I has values between
.02 to about .41. It is well known that a loop in free space ha& nearly con-
stant current if ak 0, 1 .1. If this criterion is applied, all of the eleven
cases are rotationally symmetric. However, if Iak1 j < .1 is the criterion,
rotational symmetry is clearly absent in some of the cases. Because of the
unique locution of the driving loop on the surface p - a, a simple analytical
criterion on the radius is not possible to ensure rotational symmetry. For
this reason, rotational symmetry was ensured experimentally.
As shown in Fig. V,-1, a shielded loop of 3/16 in. diameter was fabri-
cated and used to measure the voltage induced by the radial magnetic field as
a function of azimuthal coordinate for eight of the eleven coses. The larg-
aet deviation from a constant clue was found to be less than 51. The three
antennas not tested had lower values of jakl than those tested. The experi-
mental measurements established rotational symmetry conclusively for all of
the cases considered in the experimental study.
Measurements were then made of the total axial magnetic current on a
-107-
(ii
0~
z
o- w
I-I
tO
>11i0
0 _
z0.
-108-
dielectric rod made of Teflon, as shown in Fig. D-l(c). The experimental
parameters are summarized below:
Diameter of the driven loop 1.0. inch
Diameter of the Teflon rod 1.0 inch
Diameter of loop for axI.al measurements - 1.0 inch
Diameter of loop for radial measurements = 3/16 inch
Frequency = 20 MHz
The axial magnetic current distribution was measured with and without the
Teflon rod present. The results are tabulated in Table D-1 and plotted in
Fig. D-2. As one might expect, the dielectric rod (c r - 2.2) has very little
effect, and the measurements taken with it present do not differ significant-
ly from those taken with it absent. In-both cases, rotational symmetry was
confirmed experimentally. The characteristics of the Teflon rod antenna are
similar to those of the air-core antenna because the loop used in the experi-
ment was small enough to act as a magnetic dipole.
Correction to Experimental Data. An answer was then sought to the im-
portent question, what is the voltage V(z) induced at the terminals of the
receiving loop? The receiving loop is loaded by the vector voltmeter which
ihas a nominal impedance of 100 K-ohms shunted by a 2.5 pf capacitor. The
frequency in this experiment varies from 5 to 150 Miz so that the vector
voltmeter impedance has a range of values. An analysis based on circuit
theory can be carried out to determine accurately the voltage V R(z) measured
by the vector voltmeter. A diagram of the two coupled circuits is in Fig.
D-3. The various quantities shown in the figure are:
-iwtVe- Oscillator vcltage
-109-
TABLE D-1: Unnormalized experimental data of total
axial magnetic current with an air core and also a teflon
rod. (This data is plotted in Figure D-2).
Air Core ak =.00532 Teflon, er=2.2, ak0=.00792z [
cms Mag Phase Mag Phase
0.3 11.0 35 11.0 400.5 8.4 35 8.5 401.0 4.5 35 4.35 401.5 2.4 35 2.58 40.12.0 1.5 35 1.60 40.2
2.5 .98 35.4 1.02 40.33.0 .65 35.5 .695 40.43.5 .45 35.6 .500 40.64.0 .33 35.6 .365 40.74.5 .245 36' .265 40.7
5.0 .18 36 .205 40.75.5 .15 36 .155 40.76.0 .135 36.2 .130 40.86.5 .105 36.2 .10 417.0 .085 36 .083 41
7.5 .07 36 .067 418.0 .06 36 .056 418.5 .05 36 .048 419.0 .04 36 .041 41
10.0 .03 36 .03 41
cli L) 10-
LLo LL 0a
LLJ x .-
N- N L -j
U X C Z
LIl
-I----~II-- LLZ
0w~c
iI*-
U)
ow/a N
(0~~ rA t I
?U) 00-
C0
IC-j
X XX X )X )( )(C~X X XX X )W
NL
IT
1
IR•-3 •ZL VR(Z)
V| 2
Zg, ,-; .)
V21
FIG. D-3 THE TWO COUPLED CIRCUITS
-112- 1"A Generator impedance
IT . Current in the transmitting loop
V2 1 = V12 - Fictitious generator due to coupling
IR = Current in the receiving loop
ZL = Vector voltmeter impedance
The two mesh equations can be written as
V - 1T(Za + zg) -1R
7 M (Mz-)
o 1 T (-z M ) + I R(Zs + Z L) (D-2)
"where Z is the mutual impedance( Z. the self-impedance of the two loops.
"•- ' From (D-1),
IT- (V + IRZ )/(zs + Z)
Substituting this into (0-2) gives
(V + I R~ =z M)(7S z )-- + IR(Zs + Z
or
R (a+Rg W (Z + ZL) .M-/(Za + z
The measured voltage VR(z) - IRAL in Fig. D-3 Is, therefore,
( (M + ) + L) - + 7.)]
)[ - - I (D-3)I ~ + z/Z I\ + z /Z~ 1-
1 7ý/(% + Z )(78 + zgs a L
-113-
The theoretical computations from the integral equation account only for
the mutual coupling and ignore the generator impedance 7g the loading of the
receiver by the vector voltmeter impedance ZL' and the change in the current
in the transmitting loop due to the nearness of the receiving loop. The cor-
rection terms can be identified in (D-3) as follows:
v R VR(Z)m d C1 X C2 x C3 (z) (D-4)(z measured Icalculated
where
C . Correction due to generator impedance.
C2 - Correction due to the loading of the vector voltmeter.
C3 (z) - Correction due to secondary coupling (Lenz's Law).
It is observed that the correction terms CI and C2 are independent of the
separation between the tdo loops. Since the measured VR(z) is only relative
in the present study, C3 (z) is the only correction factor which is oignifi-
cant. It is (from the right-hand term in (D-3)]:
C3 1 -t Z- M(z)/(Zs+ Z )(7M + ZL)
As a first approximation the generator impedance Z is set equal to zero. Z
is the impedance of the vector voltmeter transferred to the pap in the receiv-
ing loop. The impedance of the vector voltmeter is composed of a 100 k-ohm
resistor shunted by a 2.5 pf capacitanceviz.,
Z/VVM = R(I + JwCR)
with R * 105 ohms and C - 2.5 x 1012 farads. can be calculated usingL (L
•~ ~ (Z+ JRc tan Od)
Z L R= Rc +c- JZ VVM tan Od (D-6)
-114-
where
R - 50 ohms - Characteristic impedance of the linec
1/28 8
O•rt
C Dielectric constant of Teflon FEP
d = Distance from the vector voltmeter output terminals to the gap inthe receiving loop
An approximate value for C3 (z) can now be computed using (D-5) with Zg
set equal to zero. However, ZM - the mutual impedance between two loops that
are in the near zone of one another - is still unknown. An excellent analy-
sis of the mutual impedance of two loops in air is available [8). In the
present case, however, the mutual impedance is required when the ferrite is
present. If the permeability U. in King's analysis [8] is replaced by the
permeability vI of the ferrite, an approximate value for the mutual impedance
is obtained, viz.,
9M(z) O(jl;1a2/2){([4/(z
2 + 42)l2 j[(2/A2
- l)K(v/2,a)
- (2/A2)F(a/2,a)1 - jda
2k1/3) (D-7)
where
w - Angular frequency; kI - Propagation Constant in the ferrite - ____
I Permeability of the ferrite; c1 - Permittivity of the ferrite
a - Radius of the two loops
z - Distance separating the two loops
2 a 2 a . 2z2 2
A sin a4a1AZ + 4a
n/ 2 d ýK(v/2da) f 2 2 Elliptic integral of the
0 (1 sin in first kind
• : -115-
and
ir2 2 ~2 1/2F(n/2,a) f (1 - sin ain ) dtp Elliptic integral of the
0 second kind
Fortran IV computer programs for computing the elliptic functions were
available in the Scientific •uhrouttne PacN:, of IB" 3((). luS, Lite correc-
tion factor C3 (z) was computed as a function of the separation distance z us-i3ing equations (D-5), (D-6) and (D-7). This factor was then used to correct
the experimental data for a comparison with the theoretical results. A typi-
cal comparison of the theory with corrected and uncorrected experimental data
is shown in Fig. D-4 for the specific case of antenna #6. The uncorrected
experimental data depart from the theoretical curve rear the drivine point,
0 < t/h < .25. The vector voltmeter impedance ZL at the gap for the antenna
configuration under consideration (antenna 1f16) is 7L . 2.88 - J534.7 ohms.
The corrected experimental data obtained when this value of 7 is used to
compute the correction factor are plotted in Fip. D-4 and are neen to deviate
less from the theory than the uncorrected values, The correction factor does
not account for the entire discrepancy, however. This is in part because of
the approximations made in computing the correction factor and in part be-
caust. of the lack of an accurate value for 7. For this reasotn, the correc-
tion factor wal also computed for a ranpe of real and imaginary parts of Z
Two representative cases are shown in Fig. P-4. htey illustrate that a pre-
cise knowledge of could improve the accuracy of the correction factor ap-
plied to the experimental data enld, thus, minimize the discrepancy with
theory near the driving point. Away from the driving point (z/h .25) the
agreement is seen to be very pood.
i! !
-116-
1.0 ANTENNA #6
125 -j 12.5a K0 =.003989
.75- Koh =.0837
THEORETICAL(Z/h) .50 -UNCORRECTED EXPERIMENTAL
ZL=(0_j10)hZL (0- j 1000) ohms
.25- Z VVM (at gap) ~(2.88 -j 534.7)
010 10 20 30 40 50 55
ICE (a,z)/I~lI -(iVolt! amp)
FIG. D-4 MAGNITUDE OF THE VOLTAGE RECEIV'EDBY THE MEASURING LOOP
-117-.
ACKNOWLEDGMENT
The authors wish tc thank Dr. G. S. Smith and Mr. Neal Whitman for their
help in the design and construction of the experimental apparatus. Thanks
are also due Dr. D. H. Preis for painstakingly reading the manuscript and Ms.
Margaret Owens for her a. aistance in the preparation of this report.
REFERENCES
(1) R. W. P. King and C. W. harrison, Jr., Antennas and Waves: A ModernApproach. Cambridge, Mass.: M.I.T. Press, 1969, pp. 143-149.
[2] D. V. Girn, "Flectrically Small Loop Antenna Loaded by a Homogeneous andIsotropic Ferrite Cylinder - Part I," Technical Report No. 646, Divisionof Engineering and Applied Physics, Harvard University, Cambridge, Mass.,July 1973.
[3] R. W. P. King and T, T. Wu, "Thd Imperfectly Conducting CylirndricalTransmitting Antenna," IEEE Trans. Antennas Propagat., vol. AP-14, No. 5,pp. 524-534, May 1966.
[4] R. W. P. King, C. W. Harrison, Jr., and E. A. Aronson, "The ImperiectlyConducting Cylindrical Transmitting Antenna; Numerical Results," IEEETrans. Antennas Propegaet., vol. AP-14, No. 5, pp. 535-542, May 1966.
[5] A. Soumerfeld, ElectrodyiArmics. New York, N.Y.: Academic Press, 1966,pp. 177-185, 3rdf Eiti'n.
[6] G. E. Forsyth and C. B. Moler, Conmuter Solution of Linear AlgebraicSystems. New York: Prentice-H&ll, 1967.
(7] H. A. Dropkin, E. Metzger, and J. C. Cacheria, '"VF Ferrite AntennaRadiation Measurements," Diamond Ordnance Fuze Laboratories, Washington,D.C., Tech. Report No. TR-484, July 1957.
(8] R. W. P. King, Fundamental Electromagnetic Theor!. New York: DoverPublications, 1963, Ch. VI, Sacs. 13 and 14.
:L2
5= - ET ___
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