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)

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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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-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

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COMPUTER PROGRAMS USED TO SOLVE THE COUPLED INTEGRAL EQUATIONS (86a,b):

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00186 SCALES(III0.0

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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 ___