JOURNAL OF RESEARCH of the National Bureau of Standards-D. Radio Propagation Vol. 67D, No.3, May- June 1963

Impedances of Long Antennas In Air and In Dissipative Medial

D. W . Gooch/ C. W. Harrison l Jr. /3 R. W. P. King/2 and T. T. Wu 2

(R eceived J anuary 4, 1963)

Graphs arc pro vided for t he normali zed impcdance of ce nte r-dri ven cylind rical dipole a nLe nnas when irnm_ersed in a ir or in a dissipaLive medium. The electric half-le ngth ranges from 1 to 100 for dipoles in a ir and from 1 to 19.7 for dipoles in a diss ipaLi ve medium. Three rat ios of r adius of the antenna to waveleng t h have been used. The proper t ies of the medium a re exp ressed in terms of the ratio 01 / f3 in the r a nge from zero to one where f3 a nd 01 are, respectively, t he real a nd imaginary parts of t he complex propagatio n cons tant k.

1. Introduction

Antennas completely imm ersed 111 a dissipative medium that is characterized by the constitutive parameters u, E, and J-l have been a subj ect of interest for many years. :Most investigations were concerned primarily wi th the electromagnetic fields of idealized dipole SO Ul'ces and a very extensive li teratUl'e exists on this subject. Studios of the circuit El,nu field IJl"OP­erties of insulated antennas [Moore, 1951) have been based on a transll1ission -line approximation that is useful for media that are rather highly conducting so that ((J/WE) > >1. The distribution of CUl'l'ent and t he impedllllce ch,uact~ l'i s ti cs of center-driven bare lwtennas of half-length h and radius a have been reported specifically for the electrically shor t antenna [King, Harrison, and Denton, 1961] with {3b :::;0.3, 0:::; (U /WE ) :::; ro; and for the half-wave dipole [King and Harrison, 1960] with {3h = 7r/2, 0:::; (U/WE) :::;0.4.

" Approximate formulas have also heen obtained [King, 1962; King and Iizuka, 1963] for cylindrical antennas in the ranges {3h< 7r, O:::;(U/WE):::;ro .

Extensive investigations of the circuit properties of electrically long antennas have been concluded recently. These include a theoretical study by T. T. IVu [1961] which provides an asymptotic formula,

> derived by the Wiener-Hopf method, for the il1lped-1. ance of a long cylindrical antenna cen ter driven by a I delta-function generator under the very general con-

ditions, (a/h) < <1, (a/ }... ) < <1, {3h> l, 0:::; (U/WE):::; OJ,

1 Suppor ted in part by Joint Sen -ices Oontract Nonr 1866(32) between H an-ard University, the Office of Naval Research, the Signal Oorps of the U.S . Army, and the U.S. Air Force; National Scien ce Foundation Grant N SF-G20225; and the Sandia Oorporation.

, Oordon McKay Laboratory. H arvard Unh-ersity, Oambridge, Mass. 3 Sand ia Oorporation, Albuquerque, N. Mex.

and a complete tfl.bulation of the impedance for four values of a/}... in the range 1 :::; {3h :::; 30 when u = O. Analytical difficulties associated with the idealized generator have been discussed in detail [IVu and King, 1959; King and Hanison, 1960, and King, 1962]. A careful seri es o[ experimental measuremen ts or the impedance o[ long antennas in ail' [Iizuka, King, and Prasad, 1963J (which takes l'ull fl.ccount of tel'inin al-zone effocts and the relationship ueL,,yeeli impedances measured with an actual transmission line and those computed 1'01' a deltfl.- function gen­eJ'ator) confu'ms the aCCUffLCY o[ the theoretical results when {3h? 7r and shows them to be useful approxim fLtions e\Ten when {3h is as small as 1.

2 . Antennas in Dissipative Media

Since no complete quantitative data have been published of the impedance of bare antennas in dissipative media in tbe range 1 :::; {3h:::; ro, 0:::; (U /WE) :::; ro extensive computations have been made from Wu's formula. The purpose was to provide quantitatively useful information on the properties of antennas with a wide range of lengths and diam eters when immersed in media mnging from air (~T= l , u= O) thro ugh dry earth (ET"-' 4, u,,-,10 - 5). moist earth (Er"-'lO, u ,,-, 10- 3) to salt water (~r "-'80, u"-'4). Such information has obvious appli­cations to subsurface communication , to the study of the properties of dissipative media, and to the theory of ground systems in general. Although the results apply specifically to an infinite medium, tbe impedance of an antenna is detennined primarily by the adjacent medium especially when this is even moderately conducting. It follows that the impedance of an antenna in moist earth or salt

355

water neal' an ail' interface is well approximated by the impedance of an antenna in ~n infinite medium if the distance to the surface IS not too small . :Nleasurements to determine the effect of such an interface will be reported in another paper.

'\iVu's formulas for the admittance Y = G-iB and the impedance Z = R -iX (for a time dependence e- iw l ) ftre [Wu, 1961]

1 2ik Y =-Z = - (S+ CU) mhos

WiJ. (1)

where

C- -.!. (2T - T' ) sin kh-(2S-S' ) cos kh (2) - 2 T' cos kh+ S' sin kh

(3)

The following quantities are involved in the defini­t ions of C and U:

,,/= 0.57722, ,,/' = l.6449 (4)

( \1~- JnI2+ 7ri )2J (6)

where

cr p= --'

WEOEr (16)

Note that ko is the propagation constant for free space and p is the loss tangent of the Jll ediull1. The phase constant (3 and attenuation constant a are conveniently expressed in terms of the real and imaginary parts of the function

,/1 ± ip= f(p) ± ig(p) (17) where

g(p) = sinh- l Gsinh- 1 p )=~~ (,'l + li-1). (19)

These functions are readily available in the literature [King, 1945; Gooch, Harrison, King, and Wu, 1962]. With (18) and (19) the real ftnd imaginary parts of k are

(3 = ko, /ETiJ.,f(p) , a= ko,,!ETiJ. rY(P) (20)

a= O, (3= ko. (21)

Owing to the complex intrinsic impedance

(22a)

(8) where

(9)

(10)

(ll )

(12)

The complex propagation constant of the medium in which the highly conducting antenna is immersed is given by

and

11 = ~ f(p) -V;, (22b)

which is a coefficient in the formula (1) for the ad- . mittance, it is not possible to provide uni versal curves of the impedance of antennas in dissip ative media ~ with {3h as variable and a/}-.. and a/ {3 as parameters in the manner familiar for antennas in air. The most general form in which the admittance and impedance can be presented is

Y/I1 = G/IJ. -iB/I1 , ZI1 = RI1-iXI1 (23)

where the normftlizing factor 11 is given in (22b). For antennas in air, 11 = 1; for antennas in perfect dielectrics cr = O so that p = O and f(p ) = 1 and 11 =

,I ET/ iJ. T' When cr ~ 0 the actual impedance may be obtained from the normalized value ZI1 by dividing by 11 as evaluated from (22b) with the aid of the tables [King, 1945; Gooch, Harrison, King, and vVu, 1962] off(p) .

Normalized admittances and impedances haye

356

\ J

been co mputed for antennas witll three different radii , viz,

~~=~=0.001191, 0.003175, 0.008496

where A is the wavelength in the medium. T11C electrical half-lengths in dissipative media ran ge from /3h= l to /3h = 19 .7. The properties of the medium are expressed in terms of t he ratio a//3 = g(p)/J(p) in the range from a//3 = O (air) to a//3 = l (salt water). The res ults of the computation are contained in the three families of impedance curves shown in figures 1, 2, and 3 for the three values of a/t... They are available in complete tabular form in Gooch et al. [1962]. A short list of impedances is in t he table. The effect of increasing the dimen­sionless ratio a/ /3 on the resistance and reactance of an antenna as it is made longer is brought out very clearly in these figures. Note the rapid decrease ill the antireso nant resistances near /3h= n7r with n integral as a//3 is increased . . This is shown best wiLh /3h near 7r in passing from the peak at A for a//3 = O to that at I for a//3 = l. In figure 1 it is evident Jrom

1600

1400 R6

1200

800

1000~

A . 600-B-i G

- 400

·600

-800

-1000 A

2 3

A

B

C

D

CU RVE "/13 A 0

B 0.01 C

;} ,~ 0.03

0.05

I

CURVE C</IJ C</!.~0001191

0.1

F 0.2

G 04

H 07

I 10

/1

" , 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 j3h

FIG. 1. Normali zed resistance and l'eactance oj dipole antennas in ail' and in dissipative media.

1400 -,

CURVE a/r3 CURVE a//3 a/A ~O 003175

1200 A A 0 01

R6 B B 0.01 02 C 003 G 04

1000 C D 005 H 07

I 10

FIG. 2. Normali zed resistance and reactance oj dipole antennas in air and in dissipative media.

-200

-400

G

I H

~t~! 2 3 4

a/.>. ~ 0.008496 ]

5 6 7 8 9 10 II 12 13 14 15 16 17 18 19Sh

FIG. 3. Normali zed resistance and reactance oJ dipole antennas in air and in dissipative media.

curves F that with a//3 = 0.2 , Rt1 and Xt1 are essen­tially independent of Jurther increases in length beyond /3h= 12, £rom curves G with a//3 = O.4 this independence of length is seen to begin neftr /3h= 6, from curves I-I with a//3 = 0.7 the independence begins at /3h= 3. Finally for curve I with a//3 = 1 the impedance hardly changes for lengths greater than /3h = 2. It m ay be inferred that the current is negligible in those extensions of the conductor that have no effect on the impedance.

357

Short table of the normali zed impedanre of a dipole in a dissi­pative medium Zc, = Rc, - iXt:. fo r a/'A = O.003175

2.0 2.3 2.6

a R Ll - XLl RLl - XLl RLl - XLl RLl - XLl 13 _ ._- - - - - - ------------- --

0. 0 104 -4R. 8 304 - 277 721 - 374 1210 232 .01 I II - 46. 0 313 - 263 710 -336 1130 215 . 03 124 -40. 7 327 -237 684 -271 1000 189 . 05 137 -35.7 339 -213 658 -218 905 168 . 1 168 - 24. 8 :158 - 160 597 - 120 736 132 . 4 282 - 12. 0 367 - 37. 5 412 - 11. 9 427 26.8

1. 0 266 -83. 1 2Ti -85. 4 279 -83.5 280 -81. 8

3.1416 3.5 3.8

a R Ll - XLl RLl - XLl RLl - XLl R Ll - XLl

13 - - - - ---- --- ------- -----

0. 0 674 764 337 678 144 476 87. 7 335 .01 676 703 360 646 166 464 107 330 . 03 669 599 396 .183 206 438 144 318 . 0.1 654 513 421 524 240 410 177 304 . 1 603 359 451 397 305 340 246 264 . 4 417 56. 4 402 68. 2 381 69. 7 371 62.7

1. 0 280 -80.8 280 - 80.5 279 -80.4 279 -80, .1

/lh =4.1 4.4 4.7124 5.0

a RLl - XLl RLl - XLl RLl -XLl RtJ. - XLl /l

- - ------------------ --0.0 72.2 211 83.3 87. 9 144 - 50.4 272 - 188

. 01 90.6 210 105 91. 4 164 - 38. 3 292 - 159

. 03 126 206 141 97.8 201 - 15. 2 321 - 106

. 05 158 202 173 103 232 5. 92 340 -61.0

. 1 227 186 241 111 289 46.6 364 15.6

.1 367 53.6 368 46. 1 372 41. 7 377 40. 8 1.0 279 -80. 6 279 -80. 6 279 -80. 6 279 -80. 6

/lh=5.3 5.7 6.1 6.2832

a RtJ. - X Ll RLl - XtJ. RtJ. - XtJ. RtJ. - XtJ. /l

- - - - - ------ - ---------- - -0.0 572 -270 984 238 532 631 358 .194

. 01 561 - 209 883 217 535 561 381 544

.03 .,34 - 116 740 190 525 449 408 456

. 05 507 -51. 2 645 173 506 368 419 384

.1 455 36.8 514 148 459 246 418 262

. 4 380 42.3 382 45.6 381 48. 0 380 48. 6 1. 0 279 -80. 6 279 -80.6 279 -80.6 279 -80. 6

3 . Very Long Antennas in Air

A more complete range of impedances for long antenn as in air is shown in figures 4a, b where the electrical length is ex tended to {3h ~ 100. For t he r ange {3h S 4.0 the impedan ces determined from the King-Middleton [King, 1956] theory are shown s in ce t hey ar e m ore accura te in this range than the vVu theory [Wu, 1961] which, in turn , is more ac­curate for longer an tennas. Note that for dipoles in air Ll = 1, so that the normalized and actual impedan ces coincide.

The graphs for the impedance or a dipole in ail' in figures 1, 2, and 3 (a = O) and in figures 4a, b show tha t the oscillations in both R and X , with increasing v alues of electrical length {3h, decrease only very slowly in amplitude for any given a/t... The limiting values of the admittance, as the length of t he an tenna approaches infinity at a fixed fre­quency w, are given by [King and Schmit t, 1962]

(24)

(25)

where so= 1207l' ohms. Approximate expressions when ka= 27l'ajt.. are sufficiently small so that <,

(26 )

is satisfied are given by th e leading t erms 111 (24) and (25) . They are

2 G"- 71' B"- 71'

- soQo' - 2soQ6' (27)

This formula for th e conductan ce is essen tially the same as that derived by P apas [1958] . The limiting vftlue of the impedance corresponding to the ad­mi ttance defined by (27) is

Z=Jo Qo+ i So 71' 2

(28)

so that

R=~ [In (2~a)-0.5772 ] (29)

V· So . 60 1 L l.. = -2= - 7r 0 nl1S . (30)

4 . Conclusion

In conclusion i t may be stated tha t accurate quantitative data have been provided for the im­pedance of bare cylindrical antennas over a very wide range of lengths when immersed in an arbit rary hom ogeneous and isotropic medium. Although computed specifically for a delta-function generator < at the cen ter of a dipole, the results are readily applied to an tennas cen ter driven from a two-wire line [King, 1955, 1956; Iizuka and King, 1962] or a monopole base driven over a ground screen by m eans

i

of a coaxial line [King, 1955a, b , and 1956] if proper corrections are made for the differences in the terminal zones .

~

This paper was prepared a t Harvard University by Dilla W. Gooch, R . W. P. King, and T . T . Wu. The computations upon which the graphical r epre­sentations are based were carried out at the Sandia Corporation under the direction of C. W . Harrison, I

Jr.

358

~ } =t

0: 1600

w 0

1400 z

'" 1200

if>

iJ 1000 0:

800 -if>

" 600 I 0

400

200

00

600

400

200 w 0 z

'" -200 0

::l -400 0:

if> -600

" I 0 -800

-1000

-12000

A --~"

8-+

;:~~\ A--·.,.

~~!~ ;,;;~ \\::!

;l

~=p;ff.} 8-~' ;

A---"":';

Fig. 4a . ---, \I' U

, , , , , , , CURVE ~ CURVE olX

.001191 E .003969

.001588 .004763

.002381 .006350 .008496

.;

--'------'-----~~

26 28 kh

Resistance and reactance oj cylindrical antenna.

- - - - . K i ll g-"\ i id<lictoll 2d-or<l0r theory

~

30

30

is ' _1- -L .l....----L.....--L -.J

H m ~ • n D ~ n ~ • ~ ~ ~ M ~ e m ~ ~ kh

Fig. 4b . Resistance and reactance oj long cylindrical antenna, W u theory. a /A ~ O.OOll91 (A), O.003175(D ), O.00849G(II)

359

5 . References

Gooch. D . W ., C. W. H arrison, Jr., R. W. P. King, and T . T. Wu (J an. 1962), I mpedances and admittances of long antennas in a ir and in dissipative media with tables of the funct ions f(p) ± ig (p) = 1 ± ip, Cruft Laboratory Tech. Rpt. No. 353.

Iizuka, K , R . W. P . ICing, and S. Prasad (1963), The ad­mittance of a very long cylindrical antenna, Proc. lEE, 110, No.2, 303- 309.

Ii zuka, K ., and R. W . P . King (1962), T erminal-zone correc­tions for a dipole driven by a two-wire line, J. Res. NBS 66D (R adio Prop.) No. 6, 775-781.

King, R . W . P. (1945), Electromagnetic Engineering, [1] appendix 2 (McGraw-Hill Book Co.) . A Dover edi t!on of t his book will be available early in 1963 under t he tItle Fundamental Electromagnetic Theory.

King, R. W. P . (1955a), Transmission Line Theory, pp. 405-411, 430-437 (McGraw-Hill Book Co., New York, N.Y.).

King, R. W . P . (April 1955b), End correct ion for a coaxial line when driven over a ground screen, Trans. IRE AP-3 , 6&-72.

King, R . W. P . (1956), Theory of linear antennas, ch. II (H arvard U niversity Press).

King, R . W . P ., and C. W. H a rrison, Jr. (1960), H alf-wave cylindrical antenna in d issipative m edium: Current and impedance, J . R es. NBS 6<lD (Radio Prop.) No. 4, 365-380.

King, R . W . P ., C. W. Harrison, Jr. , and D avid H. D enton, J r. (1961), The electrically short antenna as a probe for m easuring electron densit ies and collision freq uencies in an ionized r egion, J . Res. NBS 65D (Radi o Prop.) No.4, 371-384.

King, R. W . P . (1962), Dipoles in dissipative media, pp. 199-241 in R . E. Langer, E lectromagnetic waves (University of Wisconsin Press).

King, R. W . P ., and H. J. Schmitt (May 1962), Tra nsient r esponse of linear antennas and loops, Trans. IRE AP- 10, 222-228.

K ing, R. W . P ., and K. Iizuka (1963) , T he co mplete electro­magnetic fi eld of dipoles in dissipat ive media, Trans. IRE AP- 11.

Moore, R. K . (June 1951), Theory of radio communication betwcen submerged submarines, Thesis, Cornell University.

Papas, C. H. (Sept. 1958), On the infinitely long cylindrical antenn a, Cruft Laboratory Tcch . Rpt. No. 58.

Wu, T. T ., and R. W. P. King (Jan. H)59) , Driving point and input admittance of linear antennas, J . Appl. Phys. 30,74- 76.

Wu, T . T . (1961). Theory of the d ipole antenna and t he two­wire transmission line, J . M at h. Phys. 2, 550-574.

(Paper 67D3- 269)

360