lEEE TRANSATIOKS ON ANTEWAS AND PROPAGATION, TOL. AP-14, NO. 6, NOVEhIBER 1966 737

Impedance of a Finite Insulated Antenna in a Cold Plasma with a Perpendicular Magnetic Field

Abstract-A variational formulation is developed for the im- pedance of a finite strip antenna embedded in a planar dielectric slab which is surrounded by a magnetoionic medium (cold electron plasma) with a static magnetic field impressed in a direction per- pendicular to the antenna surface. Closed form expressions are ob- tained in the limit of low frequencies, and for a short antenna in a uniaxial medium. The impedance becomes large at the plasma fre- quency, near the upper hybrid resonance frequency, and further resonances are observed near the gyro-frequency if the gyro-fre- quency exceeds the plasma frequency. The reactance of a short an- tenna is inductive at low frequencies, but becomes capacitive as the thickness of the insulation around the antenna is increased. For very thin insulating layers the wave number of the variationally approxi- mated current distribution exceeds v ' d z z ko (Q and o are the two diagonal elements of the permittivity matrix), where and e may have positive or negative real parts. However, this approxima- tion does not apply to current distributions along an insulated an- tenna. The present calculations are also compared with earlier work on antenna impedances.

I. INTRODUCTIOE:

A VARIATIONAL calculation of the impedance of an insulated cylindrical antenna immersed in a cold plasma with a longitudinal magnetic field

was recently reported [l] and the results were compared with earlier work in the subject area. In the past, in studies on antennas with a transverse magnetic field, the antenna impedance has been estimated using quasi- stationary approximations [2], there have been separate treatments of high frequencies [3] and low frequencies [4], and several calculations have considered the an- tenna resistance [SI , [6], but all involve restrictive as- sumptions about the antenna current distributions. The propagation coefficient of the antenna current distribu- tion has also been estimated using an approximate sta- tionary impedance expression [4], but the accuracy of this estimate has not been established.

In this paper the impedance of an insulated flat strip antenna is calculated for a cold plasma with a steady transverse magnetic field as shown in Fig. 1. Near the antenna an isotropic dielectric region is intended to ap- proximate the effects of an ion sheath formed around the antenna immersed in the ionosphere.

The variational impedance expressions developed previously [7] for a linear antenna in an arbitrarily stratified isotropic dielectric involve reflection coeffi- cients of the TE and TRI modes a t the dielectric bound-

The work reported here was supported in part by the Office of Aero- Manuscript received February 10, 1966; revised July 14, 1966.

space Research, U. S. Air Force, under Contract AF 19(628)-5718. The author is with the Applied Research Laboratory, Splvania

Electronic Systems, a division of Sylvania Electric Products, Inc., Waltham, Mass.

T

Fig. 1. Antenna geometry.

ary nearest to the antenna. In the present antenna geom- etry these reflection coefficients should consider the anisotropy of the surrounding magnetoionic medium. The fields in the anisotropic medium are expressed as double Fourier integrals [SI, [ 9 ] , and are described as a sum of the field components of the ordinary and ex- traordinary modes, which are characterized by different wave numbers for propagation in the z-direction. The field components of these two modes are separated into coupled TE and TM components, which are matched with the corresponding uncoupled TE and TM modes of the isotropic medium a t the plasma boundary. How- ever, the relative amplitudes of the ordinary and ex- traordinary modes depend on the boundary conditions near the antenna and the determination of the reflection coefficients is more involved than for an isotropic dielec- tric. The antenna current distribution is represented similarly as in [l], and also the antenna and plasma parameters of the subsequent numerical exampIes have been selected to be similar.

11. FIELD EXPRESSIONS IN THE PLASMA 3;IEDIUM

Assuming a suppressed exp (-id) harmonic time de- pendence of the fields for a source free medium, the elec- tric and magnetic vectors E and H satisfy Maxwell's equations with the dyadic permittivity defined as in (1) to (6) of [I]. The field components Fij in the anisotropic medium are related to their Fourier transforms Fij by the integrals

= (&)2 J-1 J - : ~ i j ( u , z ~ ) e i ~ + i ~ ~ e ~ i Z d a ~ (1)

where F = E or H, i = x , y, or z, and where subscripts j denote a mode characterized by a particular value of y j .

738 lEEE TRANSACTIONS ON -4NTENNAS AhrD PROPAGATION NOVENBER

The relations between yj and the transform variables u and v u-ill be determined in the subsequent develop- ment. Substituting (1) above into (1) and (2) of [l], the transverse field components can be expressed in terms of 8, and ff, as

(3)

- 1

P2

- 1

P2

fl, = - [DU€O€3Ez + iuyf f ,] ( 4

H , = - [--W€0€3Ez + -ivyRs] ( 5 )

where p 2 = u 2 + v 2 , k Z 2 = k & - p 2 and k O = u d E o . Sub- stitution of (2) to ( 5 ) into the x and y components of (2) of [1] gives two homogeneous equations for the un- knowns E, and Hz. Solving for E, gives

[(p2 - 7’ - ko2E1)(€3Y2 + + = 0. (6)

Equation (6) of the transformed field components B,(u, v) contains factors p2, y2 and their squares. This equation corresponds to a fourth-order partial differen- tial equation for the field components E,(x, y, 8). Equa- tion (6) is satisfied for values of y which make the factor contained in the square bracket equal to zero. This fac- tor is rearranged into

( 3 4 + (” [ 2€1 -

+ - - = 0, ( 5 ) u 4 ::I

the solution of which is

* +qy(;)2]2+ €22 [l - y J 2 ] . (8)

Equation (7) can be recognized as the Booker quartic, and its two solutions represent the ordinary and the extraordinary modes. For (y/ko)-+O, (7) represents a cylindrical wave which propagates radially, i.e., trans- verse to the applied magnetic field with no variation in the z-direction. Its wave number has the values

where the subscripts + or - correspond to the + or - signs of (8). Follon-ing a convention used in the past

[ 9 ] , a solution of (8) which gives for transverse propa- gation (y = 0) a wave number p = d / & k o is designated as the ordinary mode. The plus sign in the solution (8) represents, therefore, the ordinary mode (subscript j = +), while the minus sign gives the extraordinary mode (j= -). For p+O, (8) represents a plane wave propagating in the z direction with

y+? = (- €1 k €Z)kO2. (10)

In the presence of a strong static magnetic field e2+O, and the anisotropic medium becomes uniaxial with ~ 3 f ~ 1 . The two solutions of (8) become

y+2 = - €lko2 + - p2 €1

€3

7-2 = - d o 2 + p2. (1 1)

When e2 = 0 and E~=EI, (11) also applies to an isotropic medium. The ordinary wave (j = +) corresponds to T M field components and the extraordinary wave (j= -) has TE field components, which are uncoupled by the medium, if e 2 = 0 .

Equation (8) characterizes only y2 and there is still an ambiguity of sign in the definition of y. For complex or positive real values of y2 the real part of y should be less than zero in order to obtain exponentially attenu- ated fields (1) for large values of z. For a lossless plasma y may be a pure imaginary. Under this circumstance the sign is determined from the requirement of outward power flow across the plasma boundary a t z=h. In the present paper the plasma is considered slightly lossy and the sign of y j is always determined from the require- ment Re yj<O for z>O.

The amount of coupling between the TE and TI\‘I modes in the anisotropic medium is determined by the admittance

With y j determined from the solution of (8), this ad- mittance is computed by substituting (2) to (5) into either the x or y component of (2) of [l]. A lengthy algebraic manipulation results in

Substituting (8) into (13) and letting €240, i t follows tha t Y+-ea2/e2+0 and p z i = O due to an excitation by E=+. Similarly, i t follows tha t I Y-1- co , and E=-= 0 due to an excitation by E,-. -

I I I . THE REFLECTION COEFFICIENTS The reflection coefficients of the TE and TM modes

in the isotropic region I zI 5 h are determined by match- ing the tangential field components across the plasma boundaries a t z = k. The transverse fields of the isotropic

1966 GALEJS: ANTEXNA LV MAGNETIC FIELDS 739

region of O s z s h are deduced from the scalar functions and (15) are reversed. A t the boundary z= - h , (20) t o (23) and the final expressions for R, and & in (24) and

\k = (:)’J x JZ4 ( u , t l ) e i u r + i w ( p z . + ~ , ~ - ~ ~ ~ ) d ~ ~ (14) (25) remain unaltered. However, E,j and B,j now refer

a = (i)’J-; JB(u, t l ) e i u ~ + i ~ ~ ~ ( p z . + Rbe-Yi~)d2& (15) applying the boundary condition H,(O-) =H,(O+)+ J,,

7r --m to their values at z = - h. After matching the E,, E,, and H z field components across the z = 0 boundary and

where J , is the x-directed current density on the an- tenna, it can be shown that the signs of E,j and Yj are reversed at z = - h. However, X = E,-/i?,+ remains the

(16) can be determined by using the R, and Ra expressions (24) and (25) in conjunction with the boundary condi- tions at z=O. These calculations are summarized in the

using the equations

d 8’ same for both half spaces of the plasma. The ratio X dY

E, = iwpo - 9 + __ a dadx

d d ? E , = - iwpo- * + ~

d s dzdy ( I f ) appendix from which i t follows that

d 2 1 d p = wpod,’(ki?p’) (28)

(29)

(19) H , = - \k-- k;? - @ azay iwpo ax I = J-1 J - ~ J z e - ~ u ~ e - ~ ~ ~ d . r d p .

lvhere kt2=ciko2, y i=i - \ /k i2-z~2-~2, and ei=€,i(l+tan Si). After equating these field components to the field corn- This completes the formal specification of the reflection ponents computed from (2) to ( 3 , i t follows that coefficients. However, for the subsequent numerical

work i t is desirable to derive an explicit expression for r3 (20) combinations of reflection coefficients R, and Rb as they

~

yiB(e7ih - Rbe-yih) = - - ~ E - . 1 kz? - p 2 j Y j will be encountered in the impedance calculations. Sub- 1 stituting X=Ez-/E,+ into (24) and (25) and applying

a d ( e ~ i h + ~,e--t ih) = - - p . 23 (21) (12), i t follows that

P’ j

1 1 + R, LTa + P, 1 - Rb L y b f Pb y i A ( e y i h - &e-yih) = 7 yjH,j (22)

- -- - D 1 - R, ’

- - - - D (30)

P- I

where

(23) -Ya = c3yi2( Y+ - IT.-) cosh2 y j h

Solving (21) and (22) for Ra and (20) and (23) for Rb gives

where j = + or -. Noting that pzj is related to E,i by + E ~ Y ~ ( Y + E‘+ - 7- Y-) cosh2 ~ i h (12), (24) and (25) contain one as yet undetermined quantity B,-/Ez+ which will be designated as X.

The above calculation of R, and R b was carried out a for z > 0. For z < O the signs of yj in (1) and of y i in (14) P

(32) P a = @/a)Yi(Y+ - Y-1 k;Y--Ei (33)

(3 9 Pb = - E3yikz2 (+ - i)

740 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION NOVEMBER

and where Nb is obtained from N , by interchanging cosh y ih and sinh yih functions. It may be noted that N,, llTb, and D are even functions of the transform variables 21

and v, and that Pa and Pb are odd functions because they involve the ratio P / L Y of (27) and (28). The expressions for the reflection coefficients (24) and (25) with X of (26) and also the expressions (30) are valid for both positive and negative values of z.

For E ~ - + O , Y++O, I Y-[ -+m, but Y+Y- remains finite. Noting that Pa and Pb are negligible relative to N , and Nb, (30) can be simplified to

1 + R, yi - y- tanh yJz

1 - & y- - y i tanh -yih

1 + Rb &E2 + WYN+ tanh yih

- - (35)

(3 6) - - - 1 - Rb E~Y,-Y+ + E A 2 tanh yih

where y+ and y- are defined by (11). Equations (35) and (36) can be also derived directly from (24) and (25) by noting that the reflection coefficients Ra of the TE modes contain only the j = -terms and that the reflec- tion coefficients & of the TN modes have only the j = +terms in the limit of ~ ~ ' 0 .

I\?. AKTENNX DRIVING POINT IMPEDANCE

The driving point impedance of a flat strip antenna may be computed from (1) of [7] . The surface current J , is assumed to have only an x-component J,, which in addition is an even function about x = 0, y = 0. I t follows from the field expressions (14) to (16) and the boundary relations near the antenna (56) to (59) that

. [ JJ Jo(x, y) cos ux cos vydxdy dudv (37) T where

and p 2 = u2+v2. Only the even parts of F(u, o) function will contribute to the integral over u and v. The odd parts P, and Pb of (30) can be neglected and F(u, v) can be simplified to

expression for determining the complex amplitudes A and B , i t follon-s that ( A / B ) and Z are given by (48) and (50) to (54) of [l], where

and N , M = A or B. The accuracy of this impedance formulation is discussed in the last paragraph of Section 3 of [l].

V. IMPEDANCE OF A THIN ANTENNA WITHOUT A DIELECTRIC LAYER

A . Quasi-Stationary Approximations In the low-frequency limit ko+0 and K O 2 will be

assumed to be negligible relative to the integration variables u2+v2=p2 in all of the previous expressions. The propagation constants yj of (8) simplify to y+ = - 4 / E l j E 3 p , y- = - p , which makes Y+ = 0. Substitut- ing this in the N,, Nb, and D expressions of (31) and (32), and letting h= 0 and ~ i = 1, it follows that F(u, v) of (39) becomes

The antenna impedance is computed from (54) of [ l ] with k.4 of the trial functions approaching zero, which implies a triangular current distribution along the an- tenna. This results in

where KA+O. The resistive part of 2, is computed by substituting the second term of (42) as F(u, v) in (43), which gives

where D, N,, and Nb are obtained from (31) and ( 3 2 ) . The antenna current density is assumed to be repre-

sentable by the trial function

Jz(x , y ) = { A sin [kA(Z - I x [ )]

+ B sin [ k d - I x I > I ]jol) (40)

with f ( y ) = 1 / ( 2 ~ ) =constant. Substituting (40) into (37) and utilizing the stationary character of the impedance

- .. -

The v-integrations become elementary by noting that (sin E V / E V ) = 1 over the range of v where du2+02#v, which applies strictly if E+O and u < u 0 is finite. The subsequent u integrals are evaluated using tabulated definite integ-rals. Thus,

1966 GALEJS: AhTENRT.4 IN MAGNETIC FIELDS 741

2iwpo (cos uz - 1)’ R, =

( 7 r k o l ) ’ V ’ Z 1 du Zt ‘2

.[- 3 - c - log ( E U ) 2 1

where C=O.57721 . . is Euler’s constant. The reactive part of 2, is computed by substituting the first term of (42) as F(zI., v) in (43), which gives

2wpo Joe (cos ul - 1)’ X , = - ~ du ~. ~~~

T212 21

The v-integral is evaluated using the same approxima- tions as in (44), and the reactance becomes

2wpo J 0 W d z c (cos ul - 1)’ x , z.5 - __

ii’l’ U.4

. [- 1 - 2

c - log (E.) . 1 (47)

The integral which is proportional to the constant part of the square brackets is evaluated directly. The remain- ing term of the integral is reduced to tabulated integrals with 2 2 and u2 in the denominator by repeating the inte- gration by parts twice. This gives

- 21

X, = - ?+/:[log- + log 2 - - . (48) 37r E ‘I 3

The antenna resistance (45) is inversely proportional to the length 1 and it compares closely to the resistance of the quasi-static approximation (64) of Balmain [2] and with (23) of Ament et al. [4]. A flat strip antenna of width w=2e corresponds to a cylindrical antenna of radius a , if a = e / 2 . For an antenna of Q = 2 log (41;~) = 8 the calculations of [a ] give a 25 percent higher resistance while the results of Ament et al. [4] show a 5 percent lower resistance than (45). These differences will become even smaller for larger values of Q. The antenna reac- tance X , is negative (i.e., inductive) in (48), which can be also seen from (46). X , is proportional to the antenna length I . The reactance (48) was computed from the first term of (42) which represents the TE modes. (Al- ternately, (48) can be obtained by integrating the first term of (49) for a uniaxial medium.) TE modes exhibit E, = 0 by definition; they do not depend on €3 and in the low-frequency limit they are the same as for free space around the antenna. The TE fields represent an equiva- lent magnetic dipole and they should exhibit an induc- tive reactance. The integration of the second term of (42) or (49), which represent the T M modes, gives a pure resistance in the limit of a lossless plasma at low frequencies.

B. A Short Antenna in a Uniaxial Medium In a uniaxial medium (with = 1 and eg = 0) y+2 and

y-2 are given by (11). Substituting (35) and (36) into (39) and applying (11) it is seen that

’[ ] (49) V 2 iU’V’/E~ko2 - p2

F ( u , u) = - ___ - ko2v’c 2p2 i d k o 2 - p’

where p2= u2+v2. In the low-frequency limit k0-+0, (49) reduces to F ( u , v) of the quasi-stationary approxi- mation (42). The quasi-stationary impedance approxi- mations (45) and (47) also apply to a uniaxial medium at low frequencies. Separate approximations applicable to a uniaxial medium will be derived only for frequencies w>w,.

The antenna impedance is computed by substituting (49) into (43) and by assuming k-4 = dzko for purposes of simplifying the integrations. For w>w,, € 3 > 0 , and the current distribution is triangular for sufficiently short antennas. Both terms of F(u, v) contribute now to the resistance and to the reactance of the antenna.

The first term of F(u, v) contributes to the antenna resistance only for p< ko and the second term for p 5 . \ / < K O . Over this range of p, cos ul -cos kJ = ( k A 2 - , z j 2 ) (1?/2) and sin EV = ev. After introducing polar coordinates p = du2+vz and q5 = tan-l(v/u), the double integrals are changed into products of two elementary integrals and

R, = (15 + 5€3)(koL)’. (50)

This expression for the antenna resistance has been given by Seshadri [6] also.

The first term of F ( u , Y) contributes to the antenna reactance for p > k o and the second term for p>v/&o. The contribution of the second term over the range of p given by k o > _ p 2 v / & can be shown to be proportional to (ko1)2 and will be neglected. Both terms of F ( u , v) will be considered only for p l k o . F(u, v) of (49) is now simplified by assuming that 1 +e3 = 2 d G r which will be most accurate for e3 = 1. This gives

(COS U l - COS kA1)2 sin EG (51)

(u2 - k A 2 ) d / u z + 2’ - ko’

The v-integrals are evaluated using the same approxima- tions as in (44), which gives - (COS U Z - COS kAb)’ x, = -

V‘ €3 (nkol)’ 2wpo

d‘ ( 2 2 - ha2)

The integral which is proportional to the constant term of the square brackets is evaluated directly. The

71.2 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION NOVEMl3ER

remaining term of the integral which involves a loga- rithmic function is reduced to tabulated integrals after expanding the denominator in partial fractions and changing the variables of integration to y = (zc/ko) f 1. This results in

In the high-frequency limit E3+l and (53) gives nearly the same capacitive reactance as a standard equation for antennas in free space, where the two terms (0.5 -2 log 2) are replaced by (- 1).

For a slightly lossy plasma c3 is complex and (53) exhibits a small real part

which should be added to Rs of (50).

VI. DISCUSSION OF NUMERIC-AL RESULTS

A . Uniaxial Mediu.m The magnetoionic plasma degenerates into a uniaxial

medium with €1 = 1 and €2 = 0 in the presence of strong magnetic fields ( w , / w , + ~ ) . For such a medium the antenna impedance is shown in Fig. 2 , where the left- hand scale of the resistance applies for w<w, and the right-hand scale for w>w,. The calculations are made for antennas of frequency dependent length X/20 and X / l O (indexes 1 and 2) and for very small (h= lo-%) and for larger (h=X/100) thicknesses of the insulating layer around the antenna (indexes p and i), with X = free space n-avelength. The antenna width 26 and length 21 are such that Q = 2 log ( ~ Z / E ) = 8. The medium has slight losses with an average collision frequency v =O.OOlw,. The calculated resistance is nearly constant for w > up, and increases suddenly and reaches its maxi- mum value for w=wp. The resistance is decreased for lower values of w or higher values of up. The resistance is proportional to 1 for w>w, and it is inversely propor- tional to 1 for w<w,. For very thin insulating layers (h= 10-6 X) the antenna reactance is inductive for w<wp, but it is capacitive for w>o,. The reactance reaches its maximum value at w = up, and is numerically equal to the resistance. The same general impedance behavior is also observed from the approximate im- pedance expressions (45), (48), (50), and (53), which are indicated by heavy lines in Fig. 2. The closed form approximations tend to agree more closely with reac- tance data computed for a sinusoidal current distribu- tion (54) of [l], which are indicated by thin lines in Fig. 2. This indicates that the current distribution along the antenna may differ from the usually assumed sinus- oidal or triangular distributions, and these differences appear to be more pronounced for frequencies w<w,. An insulating layer around the antenna of h = X / l O O (index i in Fig. 2) has little effect for frequencies w>o,.

I t decreases the magnitude of the impedance a t w=wp, but the reactance of the antenna becomes capacitive and the magnitude of the resistance is decreased for frequencies w<w,.

Variational approximations to the antenna current distributions are computed from (51) of [l] and are shown in Fig. 3. For frequencies w <up the current dis- tribution is attenuated along the antenna when the insulating layer is of negligible thickness, but an insulat- ing layer of a thickness Iz = X / l O O tends to make the current distribution more triangular. The insulating layer has the opposite effect a t w=wp, and the insulating layer has no effect on the current distributions for w>w,. According to Ament et al. [4], the currentdis- tribution along the antenna should exhibit the wave number (E3)0.25ko. A simple sine wave with this wave number is indicated by thin lines in Fig. 3. The observed current distributions are generally more attenuated than is indicated by the sine wave with a number k,.

B. Antenna Impedance for a Finite Xagnetic Field The impedances are computed for an antenna length

1 = c / o p ( I = X / 2 a a t w=w,). The antenna has a finite radius (Q= 8) and the surrounding medium has slight losses (v /w, = The calculations are made for a normalized gyrofrequency of the plasma of Q c = w c / w p =0.5 and 1.5, and are shown in Figs. 4 and 5, respec- tively.

For w = 0.50, and for antennas with very thin insulat- ing layers (h= 10%) the resistance decreases with in- creasing values of w/w, as indicated in Fig. 4. I t has low values for w>w, and exhibits finite peaks at w =up and w = w, where the upper hybrid resonance frequency w, = dw,"+w,". The larger peak occurs a t w =up. The reactance is inductive (negative) for w <up. I t becomes capacitive in the vicinity of w, and inductive again a t higher frequencies where the antenna is electrically long. This impedance behavior is quite similar to the imped- ance shown in Fig. 3 of [l] for an antenna aligned with a static magnetic field, except that the low-frequency reactance is of opposite signs and there are more pro- nounced impedance peaks a t w = ~ , in the present computations. The resonance peak which was observed for w<wc with a parallel magnetic field is completely absent in the present calculations for a transverse mag- netic field. This suggests that the antenna remains electrically short with a transverse magnetic field, although becomes very large in the vicinity of w=wc. For larger thicknesses of the insulating layer (h=X/100) the impedance exhibits a resonance peak near w =0.7wp, which was also observed for a parallel magnetic field (see Fig. 3 of [l I). The resonance peaks which occurred a t w =up and w, for thin insulating layers are lowered in frequency. The reactance is now capaci- tive for w <w, and i t becomes somewhat more inductive for w>wu. The numerical calculation showed that the reactance changes sign in the vicinity of the resonance peak a t w =0.7wp, but this could not be shown in the

1966 GALEJS: -4NTENN-4 I N T MAGNETIC FIELDS 743

IO I I I I I 1 1 1 1 I I I I I I I I I ] 0. I 0.3 1.0 3.0 IO

NORUALIZED FREQUENCY - ufu

10K

1K

NOPMALIZED FREQUENCY - u/up

Fig. 2. Impedance of a short antenna in a uniaxial medium.

-I Y ~ o 0.2 0.4 0.6 0.8 1.0

DISTANCE -

0 0.2 0.4 0.6 0.8 1.0

DISTANCE - ~2

: 0.2 0.4 0.6 o.a 1.0

DISTANCE - z.2

e o 0.2 0.4 0 6 0.8 1.0

DISTANCE - r Y

h = - - -h = 10- ’~ r i n [ ~ J B - z g sin(kl&)

k s = bj]o .25ko

Fig. 3. Variational approximations to the current distributions of antennas in a uniaxial medium. 1 =X/10.

744 IEEE TIWNSACTIONS ON ANTENNAS AND PROPAGA!ITON NOVEMBER

t/ 'I I l l I III I ID

.c UJ 1 t (-X) i I I 1 I I I l l I

0. I 0.3 Q, I "" 3

N O W M I Z E D FREQUENCY - u/r

I I I I I I I 1 I

0.1 0.3 "c 1 "u 3

NOWAALIZED FREQUENCY - u/u

Fig. 4. Antenna impedance for Qc=w,/w,=0.5.

IOK

- CLOSED FORM APPROXIMATION

SINUSOIDAL CUBRENT OISTi IWi ION l K

m

5 =

IC0 z ; - "7 Y E

I K \ \ \ \

L \ \

\ \

IO

Y --- x

I -X)

1 1 0. I 0.3 I "c "" 3

NORMALIZED FREaUENCY - Y u I , NOPWLIZED FREQUENCY - x u IP Fig. 5. Antenna impedance for ~&=o, /w ,= l .5 .

1966 GALEJS: ANTEhTNA IN M A G h W I C FIELDS 745

frequency scale used in Fig. 4. The quasi-stationary approximations (45) and (48) are also indicated in Fig. 4. The numerically computed resistance agrees very closely with the closed form expression, and the closed form reactance agrees better with the impedance calcu- lations based on sinusoidal field distribution (54) of [I] , which are also indicated in the reactance plots of Fig. 4.

For we= 1 . 5 ~ ~ and thin insulating layers, the imped- ance curves shown in Fig. 5 exhibit similar peaks at w=wp and near w=w,, but there is a further resonance peak near wc, which is not present when computing the impedance for an assumed sinusoidal current distribu- tion. This resonance peak is similar to the one observed for a parallel magnetic field in the vicinity of w=wc where the antenna becomes electrically long.

C. Current Distributions for Antenna .in a Plasma with a Finite Xagnetic Field

Variational approximations to the current distribu- tions have been computed from (51) of [l] for the an- tenna parameters of Figs. 4 and 5 and are shown in Figs. 6 and 7, respectively. For o,=0.5wp and w =wc the current is decaying rapidly along the antenna for thin insulating layers in Fig. 6. For we= 1 . 5 ~ ~ and w<wc the current distributions are oscillatory (as shown in Fig. 7) which indicates that the electrical antenna length is increased. For w = w e , the current distribution is attenu- ated and it approaches a triangular shape for somewhat larger values of w.

Ament et al. [4] have suggested that the wave num- ber of the current distribution along a thin antenna which is oriented in a direction perpendicular to the static magnetic field is given by k,= [ ~ 1 ~ 3 ] ~ . ? ~ k 0 , which is multivalued. The permittivity element el of (4) of [l ] becomes large as w approaches wc and its real part is negative in the frequency range wc <w <mu. The element e3 of (6) of [1] is negative for w<w,. For w=wc the product (e1e3) is positive for w>wc if wc<wpr and for w <a, if wc is larger than up. Large real values of k , may be anticipated in both cases for frequencies in the vicin- i ty of w,, but the computations show an oscillatory cur- rent distribution only for wc>w, in Fig. 7. The wave number k , is therefore defined as

- _ k s = ~ V ' C I V' €3 ko. (55)

The imaginary parts of el and el are always positive and all the square roots of (55) can be defined in the first quadrant of the complex plane including the real and imaginary axis. Using this convention a large positive value of k, can be observed only for w,>w,, when both €1 and € 3 are positive. This explains qualitatively the difference between the current distributions shown in Figs. 6 and 7 in the vicinity of w=wC. The accuracy of (55) is further examined by computing a sine wave with a complex wave number k , and plotting it along n:ith the variational approximations to the current distribu- tions. These curves, indicated by thin lines in Figs. 6

and 7, follow the general shape of the variationally com- puted current distributions for k small (solid curves), but k, seems to provide a consistantly low estimate for the wave number. The variationally computed current distributions are either more attenuated or are more oscillatory than the simple sine waves with the wave number k,.

The previous discussion refers only to antennas with negligibly thin insulating layers. For insulating layers of larger thickness (h=X/lOO) the current distribution becomes less attenuated for w=w, in Fig. 6. However, near the resonance frequency of w=0.7wp of Fig. 4 the current distribution is oscillatory for k large, whereas i t is nearly triangular in the absence of insulation. For w, = 1 . 5 ~ ~ the insulating layer tends to make the antenna electrically short for w < w c but oscillatory current dis- tributions may occur a t larger values a t w, where the current distribution is attenuated in the absence of insu- lation. The wave number k, of ( 5 5 ) cannot be used for estimating current distribution along an insulated an- tenna.

D. Comparisons with Other Impedance Calculations The present calculations are compared in Fig. 8 with

the results shown in Figs. 14 and 15 of Ament et al. [3] for w>w,. Ament et al. [3] assume a sinusoidal current distribution with a wave number d&O. The two sets of resistance data of Fig. 8 differ by 15 percent, but larger differences are noted between the reactance curves. In the low-frequency limit, closed form approxi- mations have been computed in Section V - A which have also been compared n-ith the corresponding quasi-sta- tionary approximations of Balmain [2] and with the approximate impedance expression of Ament et al. [4]. The expressions for antenna resistance were in a reason- able agreement. Balmain [2] deduces an antenna reac- tance lvhich is capacitive and is inversely proportional to the antenna length. However, the closed form expres- sion (48) and numerical calculations shown in Figs. 2, 4, and 5 indicate an inductive reactance which is propor- tional to the antenna length. For an antenna oriented in a direction transverse to the static magnetic field, Staras [SI has computed only the antenna resistance. His numerical examples are restricted to low frequencies, and his results can be compared with the resistance com- puted from (49, as can be seen from the data shown from Table I. The calculations of Seshadri [6] are made for a current strip in a uniaxially anisotropic plasma, where the static magnetic field is in a direction perpen- dicular to the antenna axis but parallel to the surface of the strip. For high frequencies, Seshadri also obtains the antenna resistance (SO), but his results are different for frequencies w <up even in the limit of w approaching zero.

The differences between the present calculations and the results of Seshadri [6] and Balmain [2] may indi- cate that the impedance is quite sensitive to changes in the antenna shape for w<w,.

746 IEEE TFLWSACTIONS O N AWl'ENNAS AND PROPAGATION NOVEMBER

.Y -I

= o 0.2 0.4 0.6 0.8 1.0

DISTANCE ~'1

Y cz 0 0.2 0.4 0.6 0.8 1.0

DISTANCE - d!

A Y 0 0.2 0.4 0.6 0.8 1.0

DISTANCE - Z'L

. 2 Y c 0 0.2 0.4 0.6 0.8 1.0

DISTANCE - d! h = IO-'h - - -h = 10-2h sin [i<JZ- zj]/rin (k? 1

ks Y J m k o

Fig. 6. Variational approximations to the current distributions antenna of Fig. 4 with W,=O.jW,.

DISTANCE - &

1.6 I I I I I

DISTANCE - 4 DISTANCE - d - h = IO-'r. --- h = IO-'h - s i n P J 4 - d]/s in ( k t )

ks = 4 m k o

Fig. 7. Variational approximations to the current distributions antenna of Fig. 5 with we= 1.5w,.

1966 GALEJS: ANTENNA IN MAGhETIC FIELDS 747

10K u = 1.1130 I I

R

I' 4

0.4 RELATIVE AMENNA LENGTH~/A

0.5 0.6

Fig. 8. Antenna impedance for w>w, . (G) Present formulation. (A) Calculation by Ament et al.

TABLE I LOW-FREQUEKCP ANTESK'A RESISTANCE

f= 5 X lo3 c / s ~=0.555X10-~0,

w,=o.1550p e = 0.06 m

1 I 3 I 30 1 300

R (Staras) 94 14 3

R [from (64)] 1 122 1 19.8 1- 2 .77

quency wz. For gyro-frequencies wc>wp there are further resonances for w<wc, but these resonances are not ob- served if w,<wp. Numerically computed current dis- tributions show that k, = v'&d/E3 ko represents a first- order low estimate of the wave number for the current distributions along the antenna if the antenna insulation is of negligible thickness. This estimate does not apply to insulated antennas. The antenna resistance of the present paper can be compared with resistance calcula- tions of other authors [2]- [5] but some of the differences in the reactance calculations for w<wp are probably caused by differences in the antenna shapes.

APPENDIX THE h I P L I T U D E RATIO x= Ez-/EZ+

The Fourier transforms of the field components E,j and Hzj have possibly different values at the two plasma boundaries a=lz and -h and they are denoted as ??ip(Tjp) and Ejn(77i*), respectively. The same sub- scripts ( p or n) will be used for distinguishing the values of Yi, X, Ra, and R b at the two boundaries. The ampli- tudes A and B of (14) and (15) and also the field com- ponents (16) to (19) which refer to 2 2 0 and z 5 0 are denoted similarly by subscripts p and n.

A unique determination of the reflection coefficients Ra and R b requires a knowledge of the various mode amplitudes ??jp and Ejn which can be derived after examining the boundary conditions near the antenna a t z = 0. The boundary conditions of E,, = E,,, Egp = Egn lead to

E. Summary of C o d u s i o n s For an x-directed linear current density J , of the an- tenna, the boundary conditions Hzp = Hz,, Hun = Hup+ J , lead to This paper has presented impedance calculations over

the complete frequency range. For a uniaxial medium there is an impedance peak at w = wp. The closed form Ap(1 - R a p ) + A n ( 1 - R a n ) = (58) expressions for short antennas compare favorably with the more accurate numerical calculations. For finite magnetic fields and negligibly thin insulating layers, the where a and /3 are defined by (27) and (28). Using (21) antenna impedance exhibits peaks at the plasma fre- and (23) for A and B and applying (12), the boundary quency up and near the upper hybrid resonance fre- conditions (56) to (59) can be expressed as

Bp(1 + R b p ) - Ba(1 + R b n ) = P ( 5 9)

748 IEEE TRhVSACTIONS ON AN“ENNAS AND PROPAGATION

where j = + or - . This is a system of four equations in four unknowns, E+,, E+,, E-,, and E-,,, where the re- flection coefficients R,, and &* depend on Ejpr and Ran and hn depend on E j , as indicated by (12), (24), and (25) . After considering E+,, E+, and X, and X, as the variables, (60) and (61) can be used for eliminating (E+JE+,). Also E+, and then E+, can be eliminated by a manipulation of (62) and (63). This gives two equa- tions in the unknowns X, and X,. I t follows from the symmetry of the resulting equations that X, = X,. Be- cause of the complexity of (60) to (63), this procedure is rather involved algebraically. I t is simpler to start out with an assumption of X,=X, and to show that this leads to no contradictions. With this assumption Rap =Ran= R, and &,= &,= &. The sign of yj \\-as re- versed by going from z>O to z < O and it follows from (13) that Yj,= - Yj,. Equations (60) and (63) yield

E+p + E+n = - ( E , + E-J (64)

and

Y+,(E+, + E+,) + Y-,(E-, + E-J = 0 (65)

or

( Y+, - u - p ) (E+, + E+,) = 0 ; (66)

hence, - E . = - E . -

I , J P (6 ’i)

and X,=Xn as postulated. Equations (62) and (63) can be simplified to

Equations (68) and (69) are combined into a homogene-

ous equation of two unknowns E+, and E-,. After divid- ing by E+, i t follows that

dentical equation for X, is obtained by substituting (67) into (68) and (69), which again shows tha t X, = X,-. Substituting R, and & of (24) and (25) into (70) and applying (12) i t follows that the ratio X is given as indi- cated in (26).

ACKKOWLEDGMENT The author wishes to thank N. Ciccia for computer

programming. REFERENCES

[l] J. Galejs, “Impedance of a finite insulated cylindrical antenna in a cold plasma with a longitudinal magnetic field,” this issue, pp.

[2] K. G. Balmain, “The impedance of a short dipole antenna in a

.AP-12, pp. 605-617, September 1964. magnetoplasma,” IEEE Trans. on Awtentzas and Propagation, vol.

[3] W . S. Ament, J. C. Katzin, M. Katzin, and B. Y . C. Koo, “Im-

tribution in a homogeneous anisotropic ionosphere,” J . Res. pedance of a cylindrical dipole having a sinusoidal current dis-

XBS/USMC-URSI (Radio Scietzce), vol. 68D, pp. 3 7 9 4 5 ,

141 W . S. Ament, ?VI. Katzin, J . R. NIcLaughlin, and W. W. Zachary, April 1964.

“Satellite antenna radiation properties a t L‘LF in the ionosphere,” Final Report on Contract Nonr4250(00)(X), Electromagnetic

[j] H. Staras, The impedance of an electric dipole in a magneto- Research Cyporation, College Park, iLId., .%pril 1965.

ionic medium,’ IEEE Traxs. otz Antenms and Propagation, vol. XP-12, pp. 695-702, Sovember 1964.

[6] S. R. Seshadri, “Radiation from a current strip in a uniaxially anisotropic plasma medium,” Can. J . Phps., vol. 44, pp. 207-

[7] J . Galejs, Driving point impedance of linear antennas in the 217, Januay 1966.

presence of a stratified dielectric,” IEEE Trans. on Atztennas and Propagatiim, vol. AP-13, pp. 725-737, September 1965.

[8] H. Hodara, “Radiation from a gyro-plasma sheathed aperture,”

January 1963. IEEE Trans. o?z Antennas and Propagetion, vol. AP-11, pp. 2-12,

(91 E. Arbel and L. B. Felsen, “Theory of radiation from sources in anisotropic media,” in Electvonzugnetic Waves, E. C. Jordan, Ed., Kew York: Pergamon, 1963, pt. I , pp. 391-459.

727-736.