DRIVING POINT IMPEDANCE OF LINEAR ANTENNAS IN THE ESSENCE OF A STRATIFIED CIELECTRIC. Janis Galejs

Transcription

1 1 AFCRL DRVNG PONT MPEDANCE OF LNEAR ANTENNAS N THE ESSENCE OF A STRATFED CELECTRC By Jans Galejs APPLED RESEARCH LABORATORY SYLVAFA ELECTRONC SYSTEMS A Dvson of Syvana Electc Poducts nc. 40 SYLVAN ROAD, WALTHAM, MASSACHUSETTS Contact No. AF19(628)-2410 Poject No Task No Scentfc Repot No. 7 Apl Je' p Pepaed fo A Foce Cambdge Reseach Laboatoes Offce of Aeospace Reseach D DC Unted States A Foce EGlEDD.n Bedfod, Massachusetts AUG pggm (m UUEOEUTTE DDC-RA E

2 Requests fo addtonal copes by agences of the Depat- ment of Defense, the contactos, o othe govenment agences should be dected to: DEFENSE DOCUMENTATON CENTER (DDC) CAMERON STATON ALEXANDRA, VRGNA Depatment of Defense contactos must be establshed fo DDC sevces o have the "need-to-know" cetfed by the cognzant mltay agency of the poject o contact. All othe pesons and oganzatons should apply to the: V Cleanghouse fo Fedeal Scentfc and Techncal nfomaton (CFST) Slls Buldng 5285 Pot Royal Road Spngfeld, Vgna 22151

3 J AFCRL DRVNG PONT MPEDANCE OF LNEAR ANTENNAS N THE PRESENCE OF A STRATFED DELECTRC By Jans Galejs APPLED RESEARCH LABORATORY SYLVANA ELECTRONC SYSTEMS A Dvson of Sylvana Electc Poducts nc. 40 SYLVAN ROAD, WALTHAM, MASSACHUSETTS Contact No. AF19(628)-2410 Poject No Task No Scentfc Repot No. 7 Apl 1965 Pepaed fo A Foce Cambdge Reseach Laboatoes Offce of Aeospace Reseach Unted States A Foce Bedfod, Massachusetts

4 Reseach Repot No. UU5 DRVNG PONT MPEDANCE OF LNEAR ANTENNAS N THE PRESENCE OF A STRATFED DELECTRC by Jans Galejs Appled Reseach Laboatoy Sylvanla Electonc Systems A Dvson of Sy^vana Electc Poducts, nc. Waltham, Massachusetts ABSTRACT A lnea flat stp antenna les between seveal delectc layes n a decton paallel to the ntefaces. ts mpedance s fomulated vaatonally as an nfnte double ntegal. Fo thn half-wave antennas n a homogeneous medum the double ntegal s evaluated analytcally and t gves standad mpedance expessons. Fo antennas of fnte wdth the ntegals ae evaluated numecally and fo a homogeneous medum the mpedance has been coelated wth the theoes of Kng and Mddleton, Wu, wth the mpedance computed fom a complementay slot antenna and wth avalable measuements. The mpedance s also computed fo antennas n a delectc laye, and the decease of the adaton effcency obseved wth nceasng electcal thckness of the laye s explaned wth the nceased amounts of suface wave powe and n pat by local delectc losses. Fo nsulated antennas located n a dsspatve medum the theoy pesented hee yelds esults n bette ageement wth measuements than the conventonal tansmsson lne theoy. The tansmsson lne theoy s shown to gve too lage a esstance and eactance at the full-wave esonant peaks of the mpedance and too low a esstance fo shot antennas.

5 RR W*5 1. DTRODÜCTOM Numeous nvestgatons have been epoted of the nput mpedance of lnea antennas n homogeneous lossy meda [Kng, 1956$ Kng and Hason, 960; Kng and zuka, 1963]. Teatments of nsulated antennas n a hghly dsspatve medum have been based on tansmsson lne theoy [Mooe, 1951, (Jhose, 960, Guy and Hassejan, 1963j. The theoy of nsulated antennas n an nfnte dsspatve medum has been futhe efned by Kng [196U] but thee ae at pesent no mpedance calculatons fo lnea antennas n the pesence of statfed meda of abtay paametes. Ths may be due n pat to the dffculty of teat*,g the! 1 cylndcal bounday of the antenna and ts nsulaton smultaneously wth the plana bounday of the delectc layes. Slot antenna mpedance fo plasma and delectc layes have been consdeed by Galejs [96U, 1965a,b] and by Vlleneuve [965] and the mpedance of an nsulated loop n a dsspatve medum has been calculated by Galejs [1965c]. A smla vaatonal mpedance fomulaton s deved heen fo lnea antennas suounded by a statfed delectc. The lnea antenna s assumed to be n the shape of a ectangula stp of zeo thckness that s excted at ts cente and s sandwched between paallel delectc layes as shown n Fg. 1. The delectc egons ae suounded by fee space, but the layes may be unsymnetcal wth espect to the antenna. Fo sem-nfnte meda one of the boundaes between the delectc and the fee space can be made to ecede to nfnty. Fo delectc laye«that may be lossy o that can suppot suface waves the powe adated nto the two lossless half-space egons wll geneally dffe fom the powe suppled to the antenna. Hence, sepaat«expessons ae deved fo the nput mpedance of the antenna and fo ts adaton esstance. The ntegals of the esultng expessons ae evaluated n closed fom fo antennas n a homogeneous medum. The well known esults fo a thn half wave antenna n fee space o n a homogeneous delectc povdes a smple check of the development, as s shown n secton secton 3-2 the pesent fomulaton s compaed wth avalable theoes fo homogeneous meda. Calculatons fo an nsulated stp antenna n a delectc laye ae epoted n Secton 3-3 Secton J.k consdes the nsulated antenna n a dsspatve meda?.. Compasons ae made wth the conventonal tansmsson lne th oy and seveal measuement«.

6 RR W ANTENNA DRVNG PONT MPEDANCE AND RADATON RESSTANCE The dvng pont mpedance of a fat ctp antenna may be confuted fom the expesson // E J^ dx dy (1) tl(x-o)] 4 whch s statonay wth espect to small changes of the suface cuent densty J o of the cuent l(x,y) about ts coect value (Hangton [1961], Secton 7-9). The suface cuent J s assumed to have only an x-component J, whch n addton, s an even functon about x«0 y«0. The anteua felds can be expessed as a supeposton of TE and TM modes and t follows fom the Appendx that Ja»» [2* (x-o)]' p» eo p )cos ux cos vy dx dy du dv (2) whee F(u,v) 1-R. aop 2^ 2 ' op 1+fl u +v '- * aop 1~R alq + 7 q 1+R aq k 14«kf 1 u op pop q U 2 +v 2 L 7 op 1_R bop 7 q 1_R bq a (3) The desgnaton of the delectc layes by subscpts s seen fom Fg. 1 and n patcula the subscpts op and q desgnate delectc layes next to z«c fo z J> 0 and z, 0 espect- vely. The othe symbols ae defned as k.. jl va&» o, J ( +lam o o,, je 7, j t J vk 2 jl -u 2 -^, whee j-0 o and J o l,2...p o l,2...q. The eflecton coeffcents of the TE modes R and of ajx the TM modes R depend on the delectc stuctue fo z > 0. Fo statfed delectc layes as shown n Fgue 1 the eflecton coeffcent R (d=* o b) n the egon of z M 1 zj zj s elated to the eflecton coeffcent R, x of the egon zj z z j by the expessons 2 Vl z f! e +R -R 2 t \z g \ a(j-l) L a(/-l) &*l '~*>T^ +R a(-l) L e27 '-^'-R a(l-l) 00 *b ^1-lN 2 f. 27!-!'*! 1 e *>\*\l \*t) L e **b(l-l) '-l e L e _R b{4-l) '"bd-d (5)

7 RR kk$ whee th* subscpt "J" has been omtted n the ght hand sdes of (h) and (5). Ft. t s deved by consdeng the scala functons (^0) to (U3) n two adjacent layes and by equng the tangental feld components (kk), (45), {kj) and (^8) to be contnuous acoss the nteface. The computatons should stat wth 1-1 whee R. /,.v R 0 n (U) and (5). A sees of computatons gves then R d1l, R,,«, R d,, and fnally R.. o R. n (3). The antenna cuent densty s assumed to be epesentable by [Oalejs 963, 965] the tal functon J x (x,y) - U sn[k(j-jx )] + B{l-cos[k(J- x )]}]f(y) (6) wth f(y) l/(2t) - const. Substtutng (6) n (2) and utlzng the statonay chaacte of the mpedance expesson fo detemnng the complex ampltudes A and B, t follows that [Galejs 1963, 1965] A 7 BB F A ' 7 A3 F B (7) * " 7 AA F B " 7 AB F A 7 AA 7 BB " 7 AB Z 7 (8) wth A ' *S 7 AA " 2F A F B 7 AB + A 7 BB (9) V *» W (10) V 1 " C0S k (11) 'm'-j- 2 ^{^^J l du F(u,v)g 1 (u)g M (u) g A (u) - -~5 (cos u - cos k) /,,) k -u %(*) " ~~z ( «n u - sn k). {lk) k -u The mpedance can be also computed fo the snusodal feld dstbuton tem by settng B - 0 n (6). Ths gves 7 AA A The powe adated nto the egon outsde of the delectc slabs can be computed as P - Rej J E x Ff* ds (16) (12)

8 RR W*5 whee the ntegaton s caed out ove the outemost sufaces of the delectc (2 «z. and z..) and vhee ds t n the decton of the outwad nomal. Afte substtutng the Held expessons of the Appendx, (l6) becomes a sx fold ntegal. The x and y ntegals ae caed out fst and they gve delta functons. The two subsequent ntegatons ae tval and the emanng double ntegal becomes P - -/ du / dv Vk2- u 2-v2 (u 2 +v 2 ) (l 7 ) A * + A 00 ' lo 2 +k 2 B j 2 +k 2 o 1 oo 1 o B 10! 2 ] whee A, and B. desgnate the complex ampltudes of TE and TM modes fo z > z, (when J-O) Jo Jo " ol and fo 1 z.. (when J-). The ampltudes A and B, ae elated to the coespondng amp- '11 Jo Jo ltudes n the delectc egon nea the antenna A and B. (s-p fo J-O, s-q fo J-) by js js matchng the tangental feld components acoss the delectc ntefaces. Jo Jo s e'^'+r "VV n * R ajj e l-l 7 e j(l-l)l Z -7 J l + j(-l),z JJ! J» '»j(l-l) e _ 7.jl Z.jl W l W j(l-l)l Z Jjl " - 7 J(.1) Z J ^B 4 7 J8 Jo Ths gves (18) (19) whee R. - R,. - 0 and whee T - 1 n absence of delectc layes (s-0). A, and B, ae ajo njo Js Js elated to the ampltudes A and B of the tal functon fo the antenna cuent densty (6) by J.!«_,. 1-R 7 op m aop aop lq 1+K aq H(tt,v) (20) 7 J. B J- + tt V op J>qp t>lq op ^ op 7 q *'\ q H(u,v) (21) wth («, ). ^ (^) 2!^ja a U ga ( u) %(U )1 u +v L J (22)

9 RR U5 The mnus sgn of (21) should be used wth J-o and sap, the plus sgn wth J» and s»q. Wth P of (l6) elated to A and B by (17) to (22), the adaton esstance R may be defned as R L_- F_ (2 3 ) l(x-0) 2!AF A+ BF B 2 whee the complex ato A/B S gven by (7). Wth no losses pesent n the delectc layes the powe caed by suface waves that ae guded along the delectc ntefaces can be computed as P g - Re // E x H* ds (2k) whee the ntegaton s caed out ove the sufaces x - ± CD and y ± OD and whee s n the decton of the outwad nomal. Consdeng an antenna located at the cente of a sngle lossless delectc laye (p-q«l) of thckness 2a z - 1, the ntegal* epesentng the felds n (2k) ae sngula fo those values X u + v, whee R.- R..» 1 o R. «R -1. Ths gves t-nfak vc -»*) - ^± (Sg) v J 4T^z and. v Vc - w2 tanfak Vt -v 2 ) (26) V J c ^w2t wth 1 (w-x/k ) vt and c -t,/e -e.,/t as the equatons fo detemnng the chaacte- 0 ox o x o stc values X fo TE and TM modes espectvely. Equaton (25) has eal solutons fo the TE modes fo any thckness of the delectc laye. Equaton (26) has eal solutons fo the TM modes only fo ak Jt -1 > x/2. Fo lossless delectc thee ae no complex solutons whch satsfy the adaton condton followng Whltme [19U8]. Substtutng the feld expessons of the Appendx, (2k) becomes a sx-fold ntegal. n the ntegals ove the X"±a>(y-±ao) sufaces the y (x) ntegals gve delta func- tons, whch makes one of the v (u) ntegals tval. The two u (v) ntegals ae computed by evaluatng the esdues at the poles gven by (25) and (26). The z ntegatons ae tv- al and (2k) s educed to

10 RH»»5 P - P P j TE 8 TM (27) whee TE WbfH» 8n(-27 ola) l" -, coa*(-l7^a) "87 o coa v - 17 ol a; 1 -\- ^ ; (28) (l xl + V TM "BT 2 ol t 'oo 'ol k ol X " v(t 'oo' 7 ) 2-7 'ol 2 7 (29) al 8ln(-27 ol a) + - ^8ln 2 (-7.a) 7 3 0l 00»W 311 / dv V vlo >/5?^5 g( Vx 2 -v2) h(v) (30).., - f du N/X 2 -U 2 y u»o 5(u) h( Vx2- U 2) (3D _ - f dv N/X 7^5 x2 j V%0»o g( Vx2-v2) h(v) (32) X... u 2 g(u) h Vx 2 "^ (33) g(u) - 2X [A g A ( tt ) * B g B (u)l (34) h(v) - sn fv/tv, g A (u) and g^u) ae defneu by (13) and (k), 7 nn U negatve and 7 puely magnay. The antenna nput esstance R g assocated wth the suface wave powe s computed fom (27) by analogy wth (23). 00 s

11 1 3.0 NUMERCAL CALCUATOHS 3.1 The Fee Space mpedance of a Thn Antenna RR Wl5 The antenna mpedance (8) o (15) 1* expessed n tems of double ntegals (12), whch can be evaluated analytcally fo the antenna n a homogeneous medum. The fee space mped- ance (1?) of a thn half-wavelength antenna wll be consdeed as a smple example. Fo 1 X/k - x/(2k ), k of the tal functons (6) set equal to k» m sp c, and wth R.., - 0 o 000 OjJ» 7. A of (12) smplfes to 'AA 2 ^o - /.ntvf» ^cos * 2 ( H).&.#(&.&) Wth F. 1, substtuton of (35) n (15) gves the eal pat of the mpedance as ReZ g.-^ kdueos 2 (2g) x au cos g gj k 2 -u 2 OM. ot O x k du cos 2 ( ) ~ TFuTT JtüT * >/^u2 dv Vk2-u?-v2 Afte expandng the denomnato n patal factons and changng the vaables of ntega- ton to (k-u) and (k+u) Re Z g 60 / 2 sln2 ( x) dx - 30 [log(2x) Cl(2x)] whee C(x) s the cosne ntegal. The magnay pat of the mpedance s obtaned as la Z - s 2BU P^^H"*].2/X U\ co * ( g k } /sn «v\ 2 1 k 2 -u 2 \ " J &&& The v-ntegatons become elementay by notng that (sn cv/cv) * 1 ove the ange of v whee Vv2«H 2 «k 2 + v. (Ths apples stctly f t-»0 and u u Q - fnte.) t follows that (35) (36) (37) (38) (38)

12 m» - «du au CO«cojf (» c) - (39) 2 2*0 dy log ysnjg.. Sl(2j ).. ^ * 4> *-y 2 whee S(x) 1«the sne ntegal. The numecal values (37) and (39) ae ecognzed as stan- dad esults [Kaus, 1950, Sec. 10.3]. Consdeng the pevous example of fee space and a thn half-wavelength antenna, the adaton esstance R (23) smplfes unde smla assumptons to Re Z of (36) and (37). s 3.2 Antennas n a Homogeneous Delectc mpedance calculatons fo an antenna of a fnte wdth ae fst caed out fo homo- geneous meda whee othe solutons ae aleady avalable. The fee space mpedance of a x/2 antenna s shown n Fgue 2 fo vaous values of the wdth c o of the paamete 0 2ln(Ul/t). Also shown n Fgue 2 ae the mpedance of a cylndcal antenna of adus a c/2 (Table.30. of Kng [ 1956]), and the mpedance Z computed fom the admttance 7 of a complementay slot antenna [Galejs, 190k] usng the fom- ula Z p Y/l2c(l+ tan 6)]. The assumpton of a snusodal cuent dstbuton s shown to gve an ncoect esstance fo antennas of fnte wdth. Fo A 10 the eactance computed by Kng [1956J, s nealy the same as fo the assumed snusodal cuent dstbuton and t dffes fom the othe computatons. Thee s good ageement between the vaous computatons fo antennas of 0 > 11. The admttance of the stp antenna n a lossy medum s compaed n Fgue 3 wth the admttance of a complementay slot antenna [Oalejs, 96*], wth measuements of zuka and Kng [Fg. 10, 1962] and wth the theoy of Wu [1961, Oooch, et al 1963]. The antenna ad- mttance calculated fom Kng [1996] fo a lossless delectc s also ndcated n Fgue 3. The antenna length s eq-aal to one-half wave length fo low delectc losses and c - e./«. «. o changes fom 78 to 69 wth nceasng losses as ndcated by zuka and Kng [1962]. Thee s good ageement between the vaous theoetcal cuves fo tan The measued con- ductance appeas too low fo a low loss delectc. The theoy of Wu gves lowe susceptance fgues fo hgh loss tangents. The admttance of the complementay slot has been computed 8

13 RR 1^5 usng a complex value of k-k vt-/«vl-m. tan 6. n the tal functons (6), whle the ado a o a mttance of the stp antenna was computed usng eal values of k-k vct/c. Ths may account fo some of the dffeences between the two sets of calculatons fo lage values of tan Antenna n Delectc Layes The mpedance of a lnea antenna located ethe n the cente o on the suface of delectc layes of vaous thcknesses 2d s shown n Fgues k-9 along wth the coespondng effcency fgues. All the calculatons of these fgues ae made fo antennas of fl» 8 and the heavy hozontal lnes ndcated on the sdes of the fgues denote the mpedance fo d-0 o ao computed fom a complementay slot antenna. The adaton effcency s computed as a ato between the powe adated nomally though the delectc suface as ndcated n (l6) and the powe suppled to the antenna. The suface wave effcency denotes the ato between the total powe caed by the suface wave fo a lossless delectc n the decton paallel to the delectc suface and the powe suppled to the antenna. Fo a lossy delectc that pat of suface wave powe whch flows outsde of the delectc laye wll be eventually dsspated nsde the delectc and ths powe wll not appea as pat of the powe adated outsde the lossy delectc n (l6). The sum of the suface wave effcency and the adatng effcency ae a lttle less than unty fo a low loss delectc (tan 0. >0.03) n fgues k and 5, and appecably less than unty fo hghe delectc losses (tan and l) n fgue 6, when sgnfcant dsspaton occus n mmedate vcnty antenna. Fo antennas located n the cente of the delectc laye the adaton effcency s deceased by nceasng the delectc constant of the laye (fgue 5), but t s unchanged fo the seveal antenna lengths shown n fgue k. Fo antennas located on the suface of the delectc laye (fgues 7 and 8) the adaton effcency s pactcally unaffected by changes of the antenna length shown n fgue 7, and t s also deceased fo hghe values of c.. The calculatons of suface wave effcency have not been caed out n the case of antennas located on the suface of the delectc laye. Fo delectc layes of «/«1 (appoxmaton of a plasma laye by an sotopc d- electc) the mpedance cuves of fgue 9 ae much smoothe than fo e./e > 1, when the CL O layes could suppot suface waves. Also the adaton effcency s deceased moe gadually

14 RR U%5 wth nceasng laye thckness. Valatlonal appoxmatons to the cuent dstbutons of the antenna confguatons shown n fgues k-9 ae computed fon equatons (6) and (7) and ae shewn n fgues 10 and U. Thee ae sgnfcant devatons fom the usually assumed half wave snusodal cuent dstbuton and these devatons become moe sgnfcant wth nceasng thckness of the delectc laye d, the antenna length 1 o the delectc constant c.. Thck layes of a lossy delectc make the cuent dstbuton appoach a tangula one. A smla change s obseved wth nceasng laye thckness d f c. c and wth the deceasng e, n fgue U when the antenna tends to become electcally shote. The above appoxmate cuent dstbutons ae the best possble fts to the actual cu«ent dstbuton usng the tal functon (6) and should be ndcatve of the tends obseved n the actual cuent dstbutons wth changng antenna paametes. 3.k nsulated Antenna n Dsspatve Medum The calculated admttance of an nsulated stp antenna n a dsspatve medum s com- paed wth admttance measuements by zuka [1963] n Fgue 12. A cylndcal we of ad- us a nsulated by a cylndcal delectc shell of oute adus b fom the suoundng ds- spatve medum s compaed wth a stp lne of wdth Ua layng n the cente of a delectc laye of thckness 2(b-a). n the lmt of a hghly conductng oute medum the chaactes- tc mpedance of the lne Z n the cylndcal geomety of b/a - l.k to 10 appoxmates Z of o o the coespondng plana geomety wth a 10 pecent accuacy [Refeence data, 1996] whch may ndcate an appoxmate equvalence of the two geometes fo tan 6.» 1. The compasons of the calculated and measued admttance data ndcates a qualtatve ageement. n Fgue 13 the mpedance of an nsulated cylndcal we measued by Guy [Fgues 3.2k and 3.25, 1962} s compaed wth tansmsson lne calculatons followng Guy and HasseJan [1963] and wth calculatons fo the appoxmately equvalent model of the stp antenna. The calculated esstance data agee vey well wth measuements but thee s only fa ageement wth the measued eactance. The tansmsson lne theoy gves too low esstance fgues fo small antenna lengths and t also poduces too lage esonance peaks n esstance and eact- ance. These obsevatons ae patculaly ponounced n the pesent example whee tan 5 va- 8 es fom 2.k to 0.3 n the fequency ange ndcated n Fgue 13* The same basc dffeences 10

15 RR kk$ between tansmsson lne theoy and the measued data can be seen fom othe measuements wth hghe loss tangents (Fgues 3.12 to 3.29 of Ouy [19&] o Fgues 3 and 5, Fenwlck and Weeks [1962]). 'The mpedance of an nsulated stp antenna located nea the nteface between fee space and a lossy delectc half space (gound) s shown n Fgue k fo seveal values of the loss tan 6. The tansmsson lne theoy s appled to an appoxmately equal geomety wth a cylndcal conducto n Fgue 15«Fo tan thee s fa ageement between the two sets of calculatons, but fo tan 6 10 the tansmsson lne theoy gves an ncease n the sze of the esonant peaks, whle they ae defntely lowe fo the calculatons shown n Fgue k. Also thee s a sgnfcant ncease of the esstance fo shot antenna lengths and thee s a elatvely small vaaton of the esstance component wth the changng an- tenna length fo tan 6 1. The hgh values of esstance fo a small antenna lengths n- dcate sgnfcant powe dsspaton n the vcnty of the antenna. Fo a dpole n dect contact wth the gound, the esstance should appoach nfnty, but nsulaton and the fnte antenna length have the effect of keepng ths esstance fnte. The adaton ef- fcency should be mpoved by deceasng values of tan 8, but fo shot antenna lengths and fo tan 8 10 the effcency s nealy the same as fo tan 8=1, due to the vey hgh e- 8 E sstance components shown n Fgue k fo tan 8 1. The esstance components shown fo a shot antenna n Fgue k wth *J* 6 ae hghe than those of Fgue 13 wth «J«80 fo the lowe fequences, because n the latte case thee s a lage dffeence n delec- tc constant between the delectc and gound, and as a consequence, less of the nea felds s stoed n the gound wth smalle attendant ohmc losses. 3.5 Conclusons n the pesent valatonal fomulaton the antenna mpedance s expessed as a double ntegal that s evaluated numecally, except fo the tval case of a thn half-wave antenna. Fo an antenna n a homogeneous medum the pesent fomulaton gves easonable ageement wth othe theoes [Kng, 1956; Wu, 961] and t has been coelated also wth mpedances computed fom complementay slot antennas [Qalejs, 196k). The antenna mpedances and ada- ton effcences have been also computed fo antennas on o wthn delectc layes And adaton effcences of 60 pecent o moe can be acheved wth delectc layes of mod- eately delectc constants (*J* 3) and wth laye thckness of less than 0.01 X. 11

16 RR 1*5 Fo nsulated antenna wthn an nfnte daslpatlve medum the calculatons have been coelated vth measuements of zuka [1963] and Guy [1962] and also wth tansmsson lne theoy. Futhe compasons have been made between the pesent fomulaton and the tansmsson lne theoy fo nsulated antenna nea the nteface between fee space and a semnfnte lossy delectc. The tansmsson lne theoy s shown to gve too lage esonance peaks of the nput esstance and eactance. Also t gves too low esstance fo shote antennas. k.o ACKNOWLEDGEMENT Appecaton s expessed to M. M. Clcca fo computng pogammng. Ths eseach was suppoted n pat by AFCRL, Offce of Aeospace Reseach, unde Contact Ho. AF 19(628)-21*10. 12

17 RR Ho. kk5 APPENDX FELD COMPONENTS The f> ds excted n the geomety of Fg. 1 wll be expessed as a supeposton of TE and TM nodes that ae deved fom scala functons T and espectvely. The egon J J of 2 > o next to the antenna s desgnated by a subscpt o, the egon of z o - by a subscpt. Fo an exp(-ut) tme dependence of the felds the scala functons ae of the fom» o - ff A O (U,»)." lux.-^(.v R^."' 0 *) du av ka) o ff B c (u "' ) '~ ±UX e " 1Vy «V + \o''' l l * * W * " j/ V U ' v) e " UX e " Vy (e * + R a e A ) du dv (42) t - ff B^U.T) e" ux e- 1^ (e" V R^e' 1 *) du dv *3) F whee R and R^ ae eflecton coeffcents. The feld components ae elated to * and * by J J P E xj " *j + E yj * - 5 *) + otox" # j (W) 3T3y *«$) öx > c öy' H xj _ [ s^ s T j + *j s *j ] o # j *3> (*6) (*7> H yj" ST [ asy- *j - k j s j] # H *j"-%(7 + *0 (W) *9) 13

18 RR No. kk$ The cuent of a lnea densty J s assumed to flow only n the x decton along the antenna and the feld components satsfy the bounday condtons E - E, E» E, AX AU J X y^ H"H,H«H +J x, foz-o. Afte substtutng (1*0) to (U3) n (WO and (U5) f and applyng the E and E bounday condtons t follows that x y V^ao* * A (l+r a ) (50) W 14^ B ( - 1+R b ) (51) The H and H bounday condtons lead to x y * - 'o B o "t,> L ^^ ^^ J (52) > 2 JF j x«lux «vy *** [ uu u o 2 A 2 u +v *--ao> \K,^m...^ WU V o 2^ 2 u +v & ) / x «lux *" * * The ampltudes A. and B. ae elated to the lnea cuent densty of the antenna J J J A (53) by (50) to (53) The scala functons T and and the feld components (kk) to (U9) ae thus unquely elated to J. [

19 RR No. 445 Refeences 1. R. C. Fenvck and V. L. Weeks, (»fay 1963), "Submeged Antenna Chaactestcs," EEE Tans, on Antennas and Popagaton, Vol. AP-11, No. 3, pp. 296 to J. Oalejs, (Mach 1963), "Admttance of a Rectangula Slot whch s backed by a Rectangula Cavty," EEE Tans, on Ant. and Pop., Vol. 11, pp J. Galejs, (Novembe 1964) "Slot Antenna mpedance fo Plasma Layes," EEE Tans. on Antennas and Popagaton, Vol. AF-12, No J. Oalejs, (Januay 1965a), "Admttance of a Wavegude Radatng nto Statfed Plasma," EEE Tans, on Ant. and Pop., Vol. AP j. Oalejs (Febuay 1965b), "Self and Mutual Admttances of Wavegudes Radatng nto Plasma Layes," RADO SCENCE, J. Res. NBS, Vol. 69D, No J. Oalejs, (Mach 1965c), "Admttance of nsulated loop antennas n a dsspatve medum," EEE Tans, on Ant. and Pop., Vol. AP R. N. Ghose, (960), "The adato-to-medum couplng n an undegound communcaton system," Poc. Nat. Elec. Confj Vol. 16, pp D.W. Oooch, C.W. Hason,J., R.W.P. Kng and T.T. Wu (May-June 1963), "mpedances of Long Antennas n A and Dsspatve Meda," RADO SCENCE, NBS J. Res., Vol. 67D, No. 3, PP O, Also. Tech. Rept. No. 353, Cuft Laboatoy, Havad Unvesty. 9. A.W. Ouy (1962), "Expemental data on bued antennas," Repot No. D , The Boeng Company, Seattle, Washngton. 10. A.W. Ouy and G. HasseJan (May 1963), "mpedance popetes of lage subsuface antenna aays," EEE Tans, on Ant. and Pop., Vol. AP-11, No. 3, pp R.F. Hangton (l96l), "Tme-hamonc electomagnetc felds," McGav-Hll Book Co., New Yok. 12. K. zuka (Septembe 1963)1 "An expemental study of the nsulated dpole antenna mmesed n a conductng medum," EEE Tans, on Ant. and Pop., Vol. AP-11, pp K. zuka and R.W.P. Kng (July 1962), "The Dpole Antenna mmesed n a Homogeneous Conductng Medum," RE Tans, on Ant. and Pop. Vol. AP-10, No. 4, pp R.W.P. Kng (1956), "Theoy of Lnea Antennas," Havad U. Pess, Cambdge, Mass. 15. R.W.P. Kng (May 1964), "Theoy of the temnated nsulated antenna n a conductng medum," EEE Tans, on Ant. and Pop., Vol. AP-12, pp R.W.P. Kng and C.W. Hason, J. (July-August, 960), "Half-wave cylndcal antenna n a dsspatve medum: cuent and mpedance," J. Res. NBS, Vol. 64D, p

20 HF Mo. W»5 17. R.W.P. Kng and K, suka, (May, 1963), "The complete electomagnetc feld of dlpoles n dsspatlve meda," RE Tan», on Antennas and Popagaton, Vol. AP-11, pp J.D. Kaua (1950), "Antennas," McQaw-Hll Book Co., nc. Hew Yok, Hew Yok. 19. R.K. Mooe (June, 1951), "Theoy of Rado Communcaton between Submeged Submanes," Ph. D. Thess, Conell Unvesty, thaca, Hew Yok. 20. A. T. Vlleneuve (Jan. 965), "Admttance of Wavegude Radatng nto Plasma Envonment," EEE Tans, on Ant. and Pop., Vol. AP "Refeence Data fo Rado Engnees," (1956), nt. Tel and Tel. Cop., Hew Yok, H.Y. Uth Ed. pp. 588 and R.M. Whtme, (Septembe 19^8), "Felds n Hon-metallc Wavegudes," Poc. RE, Vol. 36, Ho. 9, PP. O5-O T.T. Wu, (July-August 1961), "Theoy of the Dpole Antenna and the Two-We Tansmsson Lne," J. of Math. Physcs, Vol. 2, Ho. k, pp U. J 1 3 f f [ 16

21 V ^o «o V ff o 'o2' M o' ff o2 1 Z 01 '02 *o3' H o' ff o3 / HF> e 3' M o' ff 3 / 2/ ^ n f A z 03 ' 3 "03 e 2' M o' ff 2 z. \2 1 1 O 1 o o z., Fgue. Antenna Geometey, 17

22 110 T 1!, STRP ANTENNA - 2 TERM TRAL FCN SNUSODAL DSTR. 5 x O COMPLEMENTARY SLOT THEORY BY KNG u z UJ 70 - J L '5 f- 2 n(4 / ) V T 1 " " x O 40 - x u z t // 30 - " // 2«V l Q n(4 /c) 20 Fgue 2. Fee Space Antenna mpedance Z R +!X. 18

23 o CO SOHWW '9 3DNVd3DSns 'O 3DNVlDnQN0D 19 Z Ü z o E D ' v 9) 1 o o c o c c c O c o E CO w O)

24 T a o /\' 0.3 TonS, =0.03 d RELATVE LAYER THCKNESS - d/'a T 1» T ! 0.8 u 0.6 UJ Ü u RADATON EFF. SURFACE WAVE EFF l l l j ' l l ' RELATVE LAYER THCKNESS - d/ A Fgue *. mpedonce ond Effcency of Antennas n the Cente of o Delectc Laye Effects of Antennas Length. 20

25 1000 (-X) /A * 0.2 TAN 6 J * 0.03 d C3 2 O 2 y Z 300 / * RELATVE LAYER THCKNESS - V'A Fgue 5. RELATVE LAYER THCKNESS mpedonce and Effcency of Antennas n the Cente of a Delectc Laye. Effects of the Delectc Constant, 21 4'V T - MM

26 1000 " ! ~ T X~l -TT R X -X> A =2 d o t/x x O 300 TonSj '"0.1 _. x^«j >0_ RELATVE LAYER THCKNESS - d/x J l_»_x RADATON EFF. SURFACE WAVE EFF l t RELATVE LAYER THCKNESS - d/x 0.3 l.o Fgue 6. mpedunce ond Effcency of Antenno n the Cente of a Delectc Laye. Effect» of Delectc Lowe. 22 MMb

27 F n 1 1 MM R X t-x) \ Ü? »A»0.2 A-0.3 ''o.3 N " M. *./* =2 do Ton6, = 0.03 a LÄ. ' l 11 / LL_ ' RELATVE LAYER THCKNESS - d/x & TT L J. J J Fgut 7. RELATVE LAYER THCKNESS - ;/.\ lmp«doncc and Effcency of on Antenna on fh-. Suface of a Delectc Love Effects of Antenna Length. 23

28 5 O u z z ll 1 1 1! R X (-X) n = 0.2 K Too6 d = 0.03! J L \,,,l..'.\ / J J RELATVE LAYER THCKNESS - d/* Fgu«8. RELATVE LAYER THCKNESS -,'\ mpedonc«and Effcency of Antennas on the Sufoc«of a Dtltcc Loyo. Eff«et«of»K«Delectc Constant

29 * T 1 ' ( R X - (-X)./ =0.1 a o / / / 300 / / X \ Ton.8 =0.03 a 0, J, - L_U! t RELATVE LAYER THCKNESS -!/>> Z RELATVE LAYER THCKNESS - /k Fgu«9. mp«danct and Effct^cy of Anetnos n th«cente of a Plasma Laye. 25

30 5 n -J 1 T 1 T- 1 ' f T, O 1! cat C O -o s 5 1 1, - T ~T~ «* o 5 _ O o 5 _ "* - v. > 'M " - o, 2 1 n f f / - "". ol /* -. > o a * " * _l l... 1 _3» _1_ «l es o o o o o c - * ( )l/(*)l - Nämno 3A1V13P 1 1 l T JSm _ 0 -o w o o -o / - n / 4. \ 5a * 1 1 \ o / M / * - O G ft s 2 ft 1 Q «sf s => u. - s 0 /* s! s 8 8 8* ( )l/(x; - 1N3!QTO 3A1V13» T" o o o (o)l/(«ll - NSüüflD 3AJ.V13S ^ ^ o *!, CD o O o o ( M/(») - ln3)j«q 3A1V13» o o o o (»»l/l«)! - N3»)jnD 3A1V13» 2 -. sc S > U 5 1 matte

31 2-1 1 ] >» fj/ f =2 6' o ^ ec D U 0.4 UJ > LU Tan6. = 1 a ^V^OT 5^^ fc/x = 0.2 ^^*^c^. ^^^ " n 1 1 1,, 1 _.. J DSTANCE - x/ j -? DSTANCE - x/l 1.0 Z UJ ac ec D U UJ > 4/A = 0.1 Tan,6 = 0.03 d &A = DSTANCE - x/l Fgu«11. Voatonal Appoxmatons to the Cuent Dstbutons ot Antennas n the Cente of a Dslectc Laye of Thckness 2d. 27

32 9 8 S SOHWW - O 3DNV13nONOD SOHWW - 9 3DNVd3Dsns 28 F E

33 MEASUREMENTS BY GUY / 0.326m o O.C9cm b cm «A" 2-47 «/'o 80 o ^0.22 MHO/m 1 _J FREQUENCY - Mc/j X (-X) / \ /' STRP ANTENNA / '. / TRANSMSSON / LNE THEORY S >^_ V "»«a -Jl P/^V ">. 30 N ' \'/ \ \ \ ~ 10 V/V MEASUREMENTS BY GUY \ \ \ " JL FREQUENCY - Mc/ Fgu* 13. mpedance of on nsulated Antenna n o Dttpotve Medum. Antfna m' Length / Above a Conductng Gound Plane. 29

34 5K 3K VM 0 J J 2a * 0.01m 2b m K f» 3 x 10 c/s *»0.3m K x O x LU u S ANTENNA LENGTH //k ANTENNA LENGTH J /X Fgue M. mpedance Z R + X of a Bued Stp Antenna and ts Radaton Effcency. 30

35 lok 1 *«*«, 1 1 2a = 0.01m 2b = 0.035m 'A' 2 ' 26 / o' 6 CO X O X UJ U z u DC UJ U z lo UJ , ANTENNA LENGTH J/\ Fgue 15. mpedance of a Bued Antenna Z = R + X Tansmsson Lne Theoy, 31

36 1 Unclassfed Secuty Classfcaton DOCUMENT CONTROL DATA R40 (Secuty claefcalo* of ttle, body of abstact and adetag ssaoftfmm «wl be cateed when the oveall epot t clattfed). oronatno ACTVTY fgwptvw««naojaj»pled RESEARCH LAB. SYLVANA ELECTRONC SYSTEMS, A Dvson of Sylvana Electc Poducts nc., 40 Sylvan PH. WalMatn, Mftflwahmfttfc« REPORT TTLE 4»a. REPORT SECURTY CLASSFCATON Unclassfed K GROUP Dvng Pont mpedance of Lnea Antennas n the Pesence of a Statfed Delectc SSffttfS Wo^U^T.^levew Pape S. AUTHOR/S/ (Lost name, ftt name, ntal) Galej8, Jans D. (P) S. REPORT DATE Apl, 1965»O. CONTRACT OR GRANT NO. AF19(828) _ TOTAL NO. Of PAOBS 7- NO. OP REFS to. ORGNATOR'S REPORT NUMBER'S/ RR No. 445 S b element s. OT y *y^(af **""»fc»»***"my6e DOD 8ubelement AFCRL «, AVALA..L.TVL.-.TATON NOTCE, Qualfed eque8to8 _* obtan COpeS Of ths epot M fom DDC. Othe pesons o oganzatons should apply to he * -* 'eal.~ n *"""* * ~ "' " - - -»Kfffm.^Mg 1 j_g_ J_^^. SUPPLEMENTARY NOTES None A Foce Cambdge Reseach Lab. Offce of Aeospace Reseach (CRf) USAF, Bedfod, Massachusetts u. ABSTRACT A nea f at: gtp antenna les between seveal delect: layes n a decton paallel to the ntefaces. ts mpedance s fomulated vaatonally as an nfnte double ntegal. Fo thn half-wave antennas n a homogeneous medum the double ntegal s evaluated analytcally and t gves standad mpedance expessons. Fo antennas of fnte wdth the ntegals ae evaluated numecally and fo a homogeneous medum the mpedance has been coelated wth the theoes of Kng and Mddle ton, Wu, wth the mpedance computed fom a complementay slot antenna and wth avalable measuements. The mpedance s also computed fo anten nas n a delectc laye, and the decease of the adaton effcency obseved wth nceasng electcal thckness of the laye s explaned wth the nceased amounts of suface wave powe and n pat by local delectc losses. Fo nsulated antennas located n a dsspatve medum the theoy pesented hee yelds esults nn FORM ym V " JAN 64 w * (contnued next page) Unclassfed Secuty Classfcaton