current distribution and impedance of monopole

8/12/2019 Current Distribution and Impedance of Monopole

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INT ELECTRONICS, 1974, VOL. 37, SO. 1 49-60

Current distribution and impedance of monopole antennaon lossy ground

A . H I Z A LElectronic and Electrical Engineering Department,University of B irmingham B 15 2TT, England[Received 20 June 19731A Hnllen a type integral equation is solved by the method of moments with aubseetionsl basis nd parobolie expansion functions. Tho effect of the ground is takeninto account by reflection-coefficient spproximation. A vsriationnl expression iaderived for the impedsnoo changes resulting from the presence of a loaey ground.Numorieal maults are presented for the current diatributian and impednnee changesof qunrtcr weve vorticnl antennas on an inhomogonaous wet ground at MHz and5 MHz as a function of thickness and elevation of antennos.

I introdu tionThe current d is t r ibut ion of a vert ically polarized dipole is significantlyaffected by the presence of a lossy ground plane. A knowledge of the t ruecurren t d is t r ibut ion i s requi red for accu rate f ield and im pedance calcula tions.Fo r certa in ranges of the pa rame ters the s inusoidal current d is t r ibut ion canbe used . Hansen 1972) and Olsen and Chang 1971) have used s inusoidalcurrent distribution for half-wave dipoles on lossy grounds for field andimpedance calculat ions, respectively. Th e measured dat a presented by Rashid1970) shows th at a t h igh frequencies 60 MH z) the curr ent d is t r ibut ion of ahal f-wave d ipole on a lossy ground i s nearly s inusoidal and a s the frequency

is decreased 15 MH z) i t is s ignif icantly d is torted . Chang and W ai t 1970)have computed th e curren t d is t r ibut ion of hal f-wave d ipoles on lossy groundsa t 1 0 0 MHz an d 600 31 .H ~ s a funct ion of e levat ion . Th e resul t s indicatetha t , apa r t f rom the p rox imi ty e f fec t t he cu r ren t a t t he l ower pa r t of t h ean ten na is h igher than th at in the upper pa rt ) which i s signi ficant when theante nna i s c lose to th e groun d, the real par t of th e cut re nt d is t r ibut ion i sclose to a s inusoidal funct ion b ut th e imaginary part i s s igni f icant ly d is torted ,especially ne ar t h e feeding point .F o r a quar t e r -wave an t en na t he c u r ren t d i s t r ibu t ion migh t be expec t edto be similar to half of a sinusoid. Although num erical results based on a nintegral equat ion so lu t ion are avai lable Mil ler el al 1072) for the radia t ionpat tern of quarter-wave antennas on lossy grounds, no extensive numericalresul t s are previously reported for the current d is t r ibut ion and impedanceof qua rter-w ave ant enn as on lossy grounds. This might be expected sincein p rac t i ce snch an t enn as a re o f ten i nco rpo ra ted wi th an ea r th ing . sys t emso th at a good image of th e ante nna i s formed. How ever, in certa in appl ica-t ions such as mul t i -beam receiv ing a rray s S tarbu ck 1969) a n eff ic ient ear th-ing system can be uneconomical. Also in such systems, i one accounts forthe changes in t h e impedance and pa t t e rn cha rac te r i s ti c s of t he a r ray i n t h epresence of a lossy ground, the necessi ty of a n earth in g system can beJ.E. I n


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e l iminated. I n thi s case, however , th e coverage a t low angles of e leva t ionneeds i mprovement by o t he r a r rangement s such a s us i ng e l eva t ed- feedant en nas o r e l eva t i ng t he a n t e nn a i ts e lf .Th e present . paper i s nlot iva ted b y th e lack of numerica l resul t s showingthe effect of lossy groun ds on th e curre nt di s t r ibut ion a n d impedan ce ofqua r t e r -wav e an t ennas . Th e ind i v i dua l e l ement s of t he formul a t i on used i nt h i s pape r a re u rell known. The refore no c l a i n ~ s made wi t h rega rd t o t h eorigina li ty of th e theore t ica l app roach used in the formu la t ion. How ever ,the way in which the wel l known techniques a re combined haa not beenpreviously reported.

2. heory2.1. Current dislribufion

t h i s pape r SI uni t s a re used and on l y ti me dependence i a s sumed t obe ex p ( i wt) . Th e geome t ry of t he prob l em is shown i n f ig . I T h e t h e o r yis based on a metho d-of-m omen ts solut ion of a Hal len 's ty pe integra l equ at ionincorporated with the reflect ion-coefficient method discussed by hfi l ler l nl.(1972) .

Fig. 1

Geometry of the problem and variation of the ground conductivityo=10+20 exp (-IOyld,) mmh o/m, O S y S do 10 mmholm, y > ,do=20mmho lm

Miller el nl (1972) have so l ved Pockl i ng t on ' s t y pe i n t egra l eq ua t i on (Tesche1972) by invok ing th c usual thin wi re approxinla t ion. The e lec tr ic f ieldproduced b y each segm ent of th e ant en na is combined w i th th e re flec tion-coeffic ient -weighted e lec t r ic f ield produced b y th e image segment . Th ecurr ent di s t r ibut ion on a segm ent i s expressed by a t r igonom etr ic ser ies wi tht hree t e rms . Th e me t hod of Chang and W ai t (1970) is t he m os t ri gorousapproach t o t h e pre sen t p rob lem. Howeve r , an ex t ens ive am ou nt of compu t e rt ime i s reqyi red a s a resul t of numerica l integra t ions involved in obta ining th e


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Current distribution nd impedance of monopole ante nna 51Green s function for a given grou nd. I n th e reflection-coefficient metho d theGreen s funct ion is obtained f rom a s imple express ion. Chang an d W ai t(1970) hav e used also th e method of Ma a nd W alters (1969) incorpo rated withthe f ive- term Chang-King model of two coll inear ante nna s . I n this metho dthe image current dis t r ibut ion is weighted by the reflection coefficient fornorm al incidence hence it is a coarse degree of approxinlation since a singlereflection coefficient is used for the entire antenna.Appa rent ly th e solnt ion of a Hal len s ty pe integral equat ion incorporatedwith th e ref lect ion-coeff icient metho d is no t avai lable. Here we solve suchan equat ion us ing the method of mo men ts wi th sub-sectional bas is . For theexpansion functions we use parabolic functions following the method of Changan d Wa i t (1970) .A Hal len s type integral equat ion for N coupled ante nna s whose radi i aremuch less than the wavelength can be wri t ten s

where Iq(zQ ) is the unknown cnrrent dis t r ibut ion of th e qth ante nna ,G,,(r,, 2, ) i s th e Green s funct ion for the p th ante nna an d P,,(z ,) i s th eexci tat ion funct ion of the pth ante nna . According to the ref lect ion-coeff icientapproxim at ion th e G reen s funct ion is given by (hl il ler el al . 1972) : (below,is the free-space propagation coefficient)

whereR,,* = [ (d, 5dq+z, TZ, )~ b p 2 ] l l ~ 3 )

R, (~ ,,) =( co s a,,-Z,)/(cos O,,+Z,) 4 )R,(O,,) is t h e ref lec tio n coe fficie nt, 2 is th e surface impe dance of a lossygrou nd which is normalized t o the intrinsic impedanc e of free-space an d

cos OD = (d, dq z, z,,)/R,,- 5 )The exci tat ion funct ion, for a del ta- funct ion type vol tage source appl ied atzP = ,, is give n by

where B, a n d C , are unknown cons tants .The solut ion of eqn. ( I ) is facili tated by following the scheme of Chang(IOGl), Chang an d King (1968) and Chan g and W ait (1970). We divide theq th an t .enna in to Nq segments o f equa l l eng th a nd r epresen t the c ur ren t oneach segmcnt by a polynomial of degree three. Th us we set


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Th e unknown cur ren t s a re Iq j , 2 , 3, ... 21 . Th e cur ren t s a t zq l= -hqand z,, tS.+l=h, , a re equated t o zero as demanded by t he b oundary condi-tions. It should be noticed that these boundary conditions depend merelyon the end geometry of the antenn u. For an clectricnlly thin an tenna or ahol low-tube with very thin walls the current a t the ends of the antenn a canbe taken a s zero. Th e unknow n co efficients a,,, s= 1, 2, 8 may be expressedin terms of the unknown currents I,), 2 , 3 ... 2Xq, hy evalua t ing eqn. (9)a t z =z,, j;l', zqjl an d , j+l . Thus,

aq2=Lq-2(41q,+1+*Iq,-l--Iqj (10 cSubs t i tu t ing eqns . (10 a -c ) in to eqn . (9) then eqn . (9) in to eqn . I ) and mani -pulat,ing with the resulting terms, one obtains

for m 2 4, .. 21\7 and

for n 3, 5 , ... Np . Above, v=z/L,.The unknown coefficients B,, a n d C,, i n eqn. 6) are de termined f rom thebound ary condi t ions a t the end s of th e antenn a . To inc lude H n d C,, i n t othe solut ion we assume tha t I,,= B,6,, and 1,. ,,;+,=C,S,,, where 6,,, is th eKronecker 's delta T h u s e q n . I I ) becomes

We se t z , , , = - h J , + ( n - I ) L , , n = l , 2 , ... N (16)as the sample poin ts to conver t e qn. (14) into a se t of l inear a lgebra ic equa-t ions . Th e nnmber of complex equat ions (A ,,) is given by

n .A ,,= 2hT,,+ )p-1


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Ctcrrenl dislribulion und impednnce o monopole anlenna 5whcre :V is t he numb er of s egment s fo r t he p t h an t e nn a , The inversion ofcqn . ( 1 4 ) gi ves t he unknown cur ren ts a t th ree po i n t s on each segmen t andalso B a n d C,,. 'I'o c a l cu la t e t he cur ren t a t any po i n t on t he q t h an t e nnaw e s u b s t i t u t e e q n s. ( I 0 n-c with I = I .vq l = nt o eqn . ( I).2.2. Impedance

Th e numerica l procedure descr ibed ub ovc yie lds the admi t tan ces . f weare conce rned wi t h an an t e nn a a r ray wi t h m ore t han few c l ement s , a con-s iderably la rge amount of computer t ime is requi red in orde r t o so l ve t hecomp le te a r ray as coupled antennas . In such applicat .ions it is more pract icalto solve fcw e lements of th e nrray a s coupled antennrrs in o rder t o obta in ag o od e s t im a t e of t h e c u r r e n t d i st r ib u t io n . T h e c o m p ~ ~ t e durrent di s t r ibut ioncan then bc used to ca lcula te th e self and m utua l impedances . For thi spurpose a var ia t ioni l l express ion bused on the induced EMF neth hod may be~ ~ s e t l .Following, t l i c prescr ipt ion of th e induced Eh IF m ethod the m utu alimpeda nce be tween tw o anten nas i s given by (Olsen an d Chang 19; I

where I a n d I a r e t h e f ee d c u r r e n t s a n d

is the kerne l of the equat ion, which a l so appears in the integra l equat ionused by hl il le r el a l . (1972) . In the i r equa t ions , however , the di ffe rent ia lope ra tor in eqn. (10) does not a pp ly to th e re flect ion coeff ic ient in eqn . (2) .Th us , the re flect ion-coeff ic ient a ppr oxim at ion used in eqn . (18) may beregar ded a s a n impro ved version of th at used by Miller el al. (1972) . Inorde r see th e e ffec t of th e lossy groun d o n t he im pedances expl ic it ly i t i sconveni en t t o sepa ra t e eqn . (18) i n t o t wo pa r t s , name l y

where Z,, is th e impedan ce of the an ten na (wi th a curre nt di s t r ibut ion inth e presence of a lossy gro un d) placed on a perfec t groun d p lane , and JZis th e impedance incremen t resul ting from t he presence of th e lossy groun d.Z,,,m is given by e qn . (18 ) with th e replac em ent of R,(R,,,) by 1.0 which impliesa perfec t image of th e ante nn a . T o obta in aZ, , we wri te RJR,, ) I - FC(8, , )where the first an d second ter m s give respectively Z,,, an d aZ,,,. Frome q n . 4) i t readi ly fol lows tha t

An ex pl ic i t expression for aZ can be obta ined from eqns . (18)-(21) by work-ing out the necessary di ffe rent ia t ions . It can be shown t ha t


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wherellf,,(z,, 2,) Z,t iQ2 ex p -ikR,,,-)[P(I Q Q2)+ i2QTZ

+QPU(5QT+iZT+PQPU)] (23)and whereT =k(z, +z,) (24)

l'hc second line of eqn. (29) does not arise in the reflcction-coefficient approxi-mation used by Miller l al. (1972) because the reflection coefficient is regardedas constant with respect to the differential operator in eqn (19). In practicethis can be justified in view of the fac t that t he contribution of the secondline can be much smaller than th a t of th e first line.

For p = q in eqn. (22) the self-impedance is obtained. As eqn. (18) con-tains a numerical singularity for p=q it is practical to calculate the self-impedance from eqn. (22). For this purpose we must calculate Z,,,,'. Thereal pa rt of Z,, can be determined by Poynting's vector method. Accordingto this method

where R,,, is the real part of Z,,,,'. Then the radiation resistance is givenby R,,=R,,,,'+aR,,, where aR,, is the real pa rt of aZ ,, fo rp =q in eqn. (22).The self-reactance X an d self-susceptance B,, can be obtained from Z,,=l/Y,,,, where Y,,, is the sclf-admittance. It can be easily shown that

X B,,R,,IG,, (30)where Y, ,i= G,, + iB,, and Z R,, + i X . The reason why we expressthe self-susceptance in terms of the self-conductance and self-resistance isth at th e numerical solution for the imaginary par t of the current distributiondoes not satisfactorily converge a t th e feed point (Chang and Wai t 1970).I t is possible, however, to smooth out the imaginary par t of th e feed currentartificially in order to obtain a realistic feed curren t distribution . This isequivalent to modifying the original feed geometry such that the voltagegenera tor is connected t o a finite source-gap instead of a n infinitesimal source-gap. A similar procedure has been previous applied to half-wave antennas infree-space (King el al. 1968). Using this current distr ibution in eqns. (22)-(28) for p = q , R,,, can be obtained. The susceptance and reactance valuesfor the modified feed geometry are then given by eqns. (29) and (30), respec-tively.

3. Numerical results and discussionsThe accuracy of t he numerical procedure is checked by applying it to

examples given b j Chang and Wait (1970) an d Miller el al. (1972) for the


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Current rlislribulion and impedance of monopole antenna 55half-wave antennas. Using 12 segments, a very good agreement is obtainedbetween th e results for the c urrent distributions of half-wave antennas a t100 MHz an d 600 NH z on moist ground and sea water for d,,,,=0.27A an db,,,,=0.007h (A=2n/k is th e wavelength). It is found that the agreementfor th e real par t of t he current distribution is better than th at for the imaginarypar t. Miller el al. (1972) have presented a curve for the inpu t resistance of a

Fig.

Current distributions at MHz showing the effect of nntennn-thickness2,=0.111 exp (-i20.4 ), ~,,,=61.4, o = 14.75 rnmho/rn

parasi tic ar ray of two half-wave dipoles (b,,,,=bq,=0~0005A) at 3 MHz onmoist ground as a function of separat ion. The present algorithm withN,,=N,= 12 has reproduced their curve almost exactly . A very good agree-ment between the two results is expected since both techniques use thereflection-coefficient approxirnatiou.

In this paper we confine th e examples to isolated monopole ante nna onwet ground. The purpose of the cnrves presented is to show the effect ofvarious parameters on t he shape and magnitude of t he curr ent distribution.The parameters are the ground constants, the radius of the antenna and theelevation of t he antenn a above th e ground. The ground parameters at twodiffere nt frequencies correspond t o a n inhomogeneous ground with a n exponen-tial decrease of conductivity and dielectric constant. The effective surfaceimpedance and t he material properties are computed by a numerical procedurein which the wave equation for the Hertz potential is numerically integratedafter converting it int o two coupled first-order linear equations. Such a


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Current distributions a t M z showing the effect of antenna-thickness.Z,= 0 ,062 exp ( - i 3 5 ) , r,lf=88.9, o,,,= 13.58 mmholmprocedure has been used by Alostafavi and Mittra 1972) to calculate thereflection coefficient of an inhomogeneous medium.

Figures 2 and 3 show the variatiou of curr ent distribution as a func tion ofthe antenna-radius at 5 M z and MHz espectively. It is assumed thatthere is n infinitesimal gap between the ground and the lower end of theantenna. Figure 4 shows th e effect of elevation of the ant enna on the cur ren tdistribution. In the computations A T , , = X , = 2 0 segments are used. Th econvergence of th e results are checked by increasing th e numb er of segmentsup to 30. I t is found t ha t t he solutions converge satisfactorily except a t th efeed point for th e imaginary current distribution. This behaviour is due tothe assumed delta-function generator feeding the antenna. It is observedth at t he number of segments required increases when t he radius of the ant ennaincreases and/or t he surface impedance of th e ground decreases. If the radius


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Current dislribulion nd impedance of monopole anlennuBig. 4

Current distributions at M z nd 5 M H z showing the effect of antenna-thicknessand antenna-elevation . The ground con stants are as in figs. 3 and 2.of th e an ten na i s increased more charge is accumu la ted a t t he ends of t hean t enn a and hence t he s lope of t he cur ren t d i s tr i bu t ion i nc rease s . Al t e rna -t ive ly if th e ante nn a is placed on a ground w i th a smal le r surface impedan ceth e charge on th e lower par t of th e anten na increases which gives r i se to ave ry s t eep s lope. Consequent l y i n o rde r t o repre sen t t he cur ren t nea r t heend s of th e anten na espec ial ly ne ar the lower end more segments a re needed.A non-uni form d is t r ibu t i on of t h e number of s egment s a l ong t h e an t en nawould be su i t ab le for t h i s purpose. In t h i s pape r t h e segment s a re un iformlydi s t r i bu t ed . Th e r ipp l es seen in t he curves nea r t he ends of t h e an t enn a a redue to th e insuffic iency of th e num ber of segments a t these points. As thenum ber of segmen ts a re increased these r ipples a re f il t e red an average curvepassi ng t h roug h t he se r i pp l es i s observed t o be l i tt l e af fec t ed by t he num berof seg men ts for N = A ,= 20 It shoul d a l so be no t i ced t ha t a s t he an t enna


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Current diatribulion nd impedance of monopole ant en na 59be c ome s th inne r th e c ur re n t d i s t r ibu t ion be c ome s more smooth , th e r ipp le sd i sa ppe a r a nd the d i s t r ibu t ion be come s a lmos t s innsoida l ; the los sy g roundbecomes ineffec tive on th e shap e of th e curves . This behaviou r becomesmore ma rke d whe n th e a n te n na is r ai se d a bove the g round . Also th e p rox imi tyeffec t is a lmost indis t inguishable for e leva ted and/or thin anlennas .Fina l ly we present a t a b le whic h shows the a dmi t ta nc e a nd impe da nc eva lue s ob ta ine d by nume r ic a l so lu t ion a nd by induc e d EMF-Poynt ing ' sve c to r me thod . Th e f i r s t two c o lumns ( Y a n d 2 show re spe c tive ly t he inpu ta dm i t ta nc e a nd impe da nc e va lue s ob ta ine d by the nume ric n l so lu tion . Th et h i r d c o l u m n ( R ) i s c om p u t e d f ro m R = a R + H whe re a R a nd R a re r e spe c -t ive ly ob ta ine d f rom e qns . (22) a nd (28) . Th e fourth a n d f i fth c o lumns Ba n d S r e re spec t ive ly c ompute d f rom c qns . (29) a nd (30) . Th e los t twoc o lumns show th e mntua l impe da nc es ob ta ine d f rom e qn . (18) . I n the va r ia -t iona l e xpre ss ions in e qns . (18) a nd (22) a nd in e qn . (28) th e c ur re n t d i s t r i -bution given in figs. 2-4 are used. In each case the imaginary par t of thefee d c ur re n t i s smoothe d by f i t t ing to the c urve s a pa ra bo la in t he v ic in i tyof th e feed point . T he parabola passes through t h e points on th e curvesde te rmine d by z/h= . 55 a nd 15 a nd a po in t de te rmine d by th e a bsc i ssaz h = -0 .35 and the ordi na te 1 .1 t ime s g re a te r tha n the ma ximum of theord ina te s of t he p re v ious two po int s . Th e ta b le shows th a t a s the r a d ius i sde c re a sed the inp u t a dm i t ta nc e ra p id ly de c rea se s, the i np u t r e s is tanc e s l ight lydecreases an d inp ut reac ta nce rapidly increuses. Th e e ffec t of othe r para-me te r s on the a dmi t ta nc e s a nd impe da nc e s c a n be c le a r ly s e e n in the t a b le .Th e a gre e me n t be twe e n t he s e l f - impe da nc e va lue s ob ta ine d by th e nume r ica lme thod a n d induc e d EXF -Poynt in g ' s ve c to r me thod re vea l s c onf ide nc e onth e va l idi ty of t he num erica l resul ts .4 Conclusions

The presence of a lossy ground has a s ignif icant e ffec t on the currentdis tr ibut ion of a th ick ant enn a if no ear th ing sys tem is used. This e ffec t ,howe ve r , c an be c ons ide rab ly r e duce d b y de c rea s ing th e r a d ius of th e a n te nn aa nd /or e le va t ing th e a n t e nn a s l igh t ly . Nume r ic al r e su l t s p re se n ted show th a tth e e ffec t of lossy ground on t he shap e of th e current dis t r ibut ion of a veryth in a n t e n na i s sma l l a n d th e d i s tr ibu t ion i s close to a s inuso idal va r ia t ion .ACKNOWLEDOMENTS

T h e a u t h o r t h a n k s t h e c o m r n u n ic a ti o n s gr o u p o f t h e E l e ct r o n ic s a nd E l e ct r ic a lEngineer ing D epa r tm en t of th e Univ ers i ty of Birm ingham for giving him t heoppo r tun i ty t o c a r ry o u t th i s re se arc h . Th e f ilm a na lys i s g roup of th eDe par tm ent of Phys ics of th e sam e Unive rs i ty is acknowledge for providingth e c omp ut ing fa ci l it i es .R E F E R E N C E S

Cfraro, D. C., 1967, Radw Sci . 2, 1043.C H A N O , . C., and \Varr J . R., 1970, I . E . E . E . Tmns. Anlenrm Propag. , 18, 182.H A N S E N ,. M., 1972, I . E . E . E . Trans. Antennas I ropag., 20, 766.K ~ N o , . W. P., MACK,R . B., a n d S A N D L E R ,. S., 1968. A r r a y of CylindricnlDipoles (Cambridge University Press), Chap. 2..MA hl T . , a n d W ~ L T E R S ,. C., 1969, ESSA Tech. Re pt . IER 104, ITS 74.


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fi0 Current dis tr ib~rl ion nd impedance of monopole antennaMILLER K . P O G G IO . J . B U R K E . J. and S E L D E N . S. 197 Ca n. J . Phys.

50 879.MOSTAFAVII . nlld ~ I I T T RA R . 1972, Radio Sci. 7 1105.OLSEN t. G., and CHAPC . C. 1971 I . E . E . E. Trans . Antenma Propug. 19 685.RASHID . F. 1Ri0, I . E . E . E . T r a n s. 4nLeanas Propug. 18 22.STARRUCK.T . 1969, Radio Electron. Engng 37 N o . 4T E S C I I E .At. 1972, I . E .E .E . Truns. Antennas Propag. 20 210.

current distribution and impedance of monopole

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