Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission equaion is essenial in he analysis and design of wieless communicaion sysems. I elaes he powe fed o he ansmiing anenna and he powe eceived by he eceiving anenna when he wo anennas ae sepaaed by a sufficienly lage disance ( R D max /, i.e., hey ae in each ohe s fa zones. We deive Fiis equaion nex. A ansmiing anenna poduces powe densiy W (, in he diecion (,. This powe densiy depends on he ansmiing anenna gain in he given diecion G(,, on he powe of he ansmie P fed o i, and on he disance R beween he anenna and he obsevaion poin as P P W G (, e D (,. (6.1 4R 4R Hee, e denoes he adiaion efficiency of he ansmiing anenna and D is is dieciviy. The powe P a he eminals of he eceiving anenna can be expessed via is effecive aea A and W : P A W. (6. (, R (, To include polaizaion and hea losses in he eceiving anenna, we add he adiaion efficiency of he eceiving anenna e and he PLF: P e PLF AW AW e ρ ρ, (6.3 P D (, W e ρ ρ. (6.4 4 A Nikolova 01 1
Hee, D is he dieciviy o he eceiving anenna. The polaizaion vecos of he ansmiing and eceiving anennas, ρ and ρ, ae evaluaed in hei especive coodinae sysems; his is why, one of hem has o be conjugaed when calculaing he PLF. The signal is inciden upon he eceiving anenna fom a diecion (,, which is defined in he coodinae sysem of he eceiving anenna: P P D (, e (, D e ρ ρ. (6.5 4 4R The aio of he eceived o he ansmied powe is obained as (, (, P e e ρ ρ D D. (6.6 P 4 R If he impedance-mismach loss faco is included in boh he eceiving and he ansmiing anenna sysems, he above aio becomes (1 (1 ee D (, D (, P ρ ρ. (6.7 P 4 R The above equaions ae vaiaions of Fiis ansmission equaion, which is well known in he heoy of EM wave popagaion and is widely used in he design of wieless sysems as well as he esimaion of anenna adiaion efficiency (when he anenna gain is known. Fo he case of impedance-mached and polaizaion-mached ansmiing and eceiving anennas, Fiis equaion educes o P D (, D (,. (6.8 P 4 R The faco ( / 4 R is called he fee-space loss faco. I eflecs wo effecs: (1 he decease in he powe densiy due o he spheical spead of he wave hough he em 1/ (4 R, and ( he effecive apeue dependence on he wavelengh as / (4.. Maximum ange of a wieless link Fiis ansmission equaion is fequenly used o calculae he maximum ange a which a wieless link can opeae. Fo ha, we need o know he nominal powe of he ansmie P, all he paamees of he ansmiing and W Nikolova 01
eceiving anenna sysems (such as polaizaion, gain, losses, impedance mismach, and he minimum powe a which he eceive can opeae eliably P. Then, min max P 4 P min R (1 (1 e e ρ ρ D (, D (,. (6.9 The minimum powe a which he eceive can opeae eliably is dependen on numeous facos, of which vey impoan is he signal o noise aio (SNR. Thee ae diffeen souces of noise bu we ae mosly concened wih he noise of he anenna iself. This opic is consideed in he nex lecue. 3. Rada coss-secion (RCS o echo aea The RCS is a fa-field chaaceisic of a ada age, which ceaes an echo by scaeing (eflecing he ada EM wave. The RCS of a age σ is he equivalen aea capuing ha amoun of powe, which, when scaeed isoopically, poduces a he eceive powe densiy equal o ha scaeed by he age iself: Hee, Ws Es lim 4 R lim 4 R R R W i i, m. (6.10 E R is he disance fom he age, m; W i is he inciden powe densiy, W/m ; W is he scaeed powe densiy a he eceive, W/m. s To undesand bee he above definiion, we can e-wie (6.10 in an equivalen fom: Wi lim W ( s R R 4 R. (6.11 The poduc Wi epesens some equivalen ineceped powe, which is assumed o be scaeed (e-adiaed isoopically o ceae a ficiious spheical wave, whose powe densiy in he fa zone W s deceases wih disance as 1/ R. I is hen expeced ha Wi is a quaniy independen of disance. W s mus be equal o he ue scaeed powe densiy W s poduced by he eal scaee (he ada age. Nikolova 01 3
We noe ha in geneal he RCS has lile in common wih any of he cosssecions of he acual scaee. Howeve, i is epesenaive of he eflecion popeies of he age. I depends vey much on he angle of incidence, on he angle of obsevaion, on he shape and size of he scaee, on he EM popeies of he maeials ha i is buil of, and on he wavelengh. The RCS of ages is simila o he concep of effecive apeue of anennas. Lage RCSs esul fom lage meal conen in he sucue of he objec (e.g., ucks and jumbo je ailines have lage RCS, 100 m. The RCS inceases also due o shap meallic o dielecic edges and cones. The educion of RCS is desied fo sealh miliay aicaf mean o be invisible o adas. This is achieved by caeful shaping and coaing (wih special maeials of he oue suface of he aiplane. The maeials ae mosly designed o absob elecomagneic waves a he ada fequencies (usually S and X bands. Layeed sucues can also cancel he backscae in a paicula bandwidh. Shaping aims mosly a diecing he backscaeed wave a a diecion diffeen fom he diecion of incidence. Thus, in he case of a monosaic ada sysem, he scaeed wave is dieced away fom he eceive. The sealh aicaf has 4 RCS smalle han 10 m, which makes i compaable o smalle han he RCS of a penny. 4. Rada ange equaion The ada ange equaion (RRE gives he aio of he ansmied powe (fed o he ansmiing anenna o he eceived powe, afe i has been scaeed (e-adiaed by a age of coss-secion. In he geneal ada scaeing poblem, hee is a ansmiing and a eceiving anenna, and hey may be locaed a diffeen posiions as i is shown in he figue below. This is called bisaic scaeing. Ofen, one anenna is used o ansmi an EM pulse and o eceive he echo fom he age. This case is efeed o as monosaic scaeing o backscaeing. Bea in mind ha he RCS of a age may consideably diffe as he locaion of he ansmiing and eceiving anennas change. Assume he powe densiy of he ansmied wave a he age locaion is PG (, Pe D (, W 4R 4R, W/m. (6.1 Nikolova 01 4
R (, (, R The age is epesened by is RCS, which is used o calculae he capued powe Pc W (W, which when scaeed isoopically gives he powe densiy a he eceiving anenna ha is acually due o he age. The densiy of he e-adiaed (scaeed powe a he eceiving anenna is Pc W PD (, W e. (6.13 4 R 4 R (4 R R The powe ansfeed o he eceive is PD (, P e A W e D (, e 4. (6.14 (4 RR Re-aanging and including impedance mismach losses as well as polaizaion losses, yields he complee ada ange equaion: (1 (1 ee 4R R 4 P D (, D (, ρ ρ.(6.15 P Fo polaizaion mached loss-fee anennas aligned fo maximum diecional adiaion and ecepion: D 0D 0 P P 4R R. (6.16 4 The ada ange equaion is ofen used o calculae he maximum ange of a ada sysem. As in he case of Fiis ansmission equaion, we need o know Nikolova 01 5
all paamees of boh he ansmiing and he eceiving anennas, as well as he minimum eceived powe a which he eceive opeaes eliably. Then, ( R R e e (1 (1 ρ ρ max (6.17 P D (, D (,. P min 4 4 Finally, we noe ha he above RCS and ada-ange calculaions ae only basic. The subjecs of ada sysem design and elecomagneic scaeing ae huge eseach aeas hemselves and ae no going o be consideed in deail in his couse. Nikolova 01 6