ANTNNAS Vecto and Scala Potentials Maxwell's quations jωb J + jωd D ρ B (M) (M) (M3) (M4) D ε B Fo a linea, homogeneous, isotopic medium and ε ae contant. Since B, thee exists a vecto A such that B A and ( A). A is called the magnetic vecto potential. Thee ae infinitely many vectos A that satisfy B A thus we late need to specify A. (M) can be witten as -jω A ( + jωa) () Vecto Vecto (some scala) () + jωa - ϕ ϕ : scala potential (3) - jωa - ϕ (4) (M) can be witten as B J + jωε (5) A J + ( jωa ϕ) (6) A + ( A) J + ω εa ϕ (7)
To simplify the equation, choose A as A - jωεϕ (8) This is called the Loentz Condition. This leads to A + ω εa - J D'Alembet's quation (9) The poblem now consists of finding the vecto potential A due to a souce J. Fom the knowledge of A, and can be detemined. Review spheical coodinates, gadient, divegence, cul, and laplacian in spheical coodinates. (Textbook, Appendix A pp. 689-69). In spheical coodinates, the solution fo the vecto potential A() is given by A() V jβ J(') e ' 4π ' ' dv' () In the above equation, V' is the volume defining the souce ove which the integation is pefomed. is the vecto fom the oigin of the coodinate system, O, to the obseve and ' is the vecto fom O to a souce point within V'. ' -' obseve O V' souce Figue Fom A, use Maxwell's quations to deive and. B A () ()
3 etzian Dipole Assume a souce point infinitely small in size located at the oigin of the coodinate system with elemental length dl, and diven by a cuent with stength I o in the +z diection. The equation fo the cuent density of such a system is given by Upon substitution in () J(') i z I o dlδ(x')δ(y')δ(z') (3) A() i z I odl exp(-jβ) (4) In spheical coodinates: i z i cos - i sin A() (i cos - i sin) I odl exp(-jβ) (5) A I odl cos exp(-jβ) (6) A - 4π I odl sin exp(-jβ) (7) Calculate and fields. A (8) A A ( sin ) sin (9) A A ( ) sin () A A ( ) () I odl sin exp(-jβ) jβ(+ jβ ) ()
4 Using, we can deive the components of the electic field. ( sin ) sin (3) cosidl o jβ { e } jβ + jβ (4) ( ) (5) jβi dl o jβ sin e jβ (6) jβ β { } + In summay, the exact solutions fo the fields of an infinitesimal antenna ae given by jβ e cos + jβ ( jβ) ε (7) jβ e sin + + jβ ( jβ) ε (8) jβ e sin + (9) jβ In most pactical cases, the obseve is located seveal wavelengths away fom the souce. This defines a fa field egion which is the egion whee the distance fom the souce to the obseve is much lage than the wavelength λ π β. In this case >> λ so that β >>; consequently, the tems vaying as / and / 3 can be neglected. The fa-field solutions fo the infinitely small antenna thus become (3) ε e jβ sin (3)
5 e jβ sin (3) Note that the atio / is the chaacteistic impedance η of the popagation medium. Ove a small egion, the fa field solution is a plane-wave solution since the electic and magnetic fields ae in phase, pependicula to each othe and thei atio is the intinsic impedance η, and ae pependicula to the diection of popagation. oweve, unlike plane waves, the fa field solution is a function of the elevation angle, and does not have constant magnitude (/ dependence). Radiation Pattens The gaph that descibes the fa-field stength vesus the elevation angle at a fixed distance is called the adiation patten of the antenna. In geneal, adiation pattens vay with and. The distance fom the dipole to a point on the adiation patten is popotional to the field intensity o powe density obseved in that diection. Figue shows the -field and powe density adiation pattens of a etzian dipole. As can be veified these pattens ae based on the sin and sin dependence of the -field and powe density espectively. z z (a) (b) Figue. (a) Radiation patten fo field (b) Radiation patten fo powe density Time Aveage Powe in Radiation Zone In ode to calculate the powe adiated in the fa field, we need to detemine the timeaveage Poynting vecto o powe density <P>.
6 i < P> Real * [ ] Real (33) ε < P > i η βidl Real o π sin (34) 4 The total powe adiated P T at a distance is by definition obtained by integating the Poynting vecto ove a sphee of adius. π Total Powe PT < P> ds (35) ds is the elemental suface of adius and is given by so that π ds sindd (36) i P T π π η βiodl 3 sin dd (37) dl PT η βio 4π π π sin d (38) 3 P T 4πη βiodl 3 4π dl π η 3 λ I o (39) The diective gain is a figue of meit defined as Diective Gain Poynting powe density AveagePoynting powe density ove aea of sphee of adius o Diective Gain Fo an infinitesimal antenna, we get < P > P π T /4 (4)
7 Diective Gain η βiodl sin 4π 4πη βiodl 3 4π 3 sin (4) The diectivity is the diective gain in the diection of its maximum value. Fo an infinitesimal antenna, the diection of maximum value is fo π/ and the diectivity is.5. Radiation Resistance The adiation esistance of an antenna is defined as the value of the esisto that would dissipate an equal amount of powe than the powe adiated fo the same value of cuent. Using P T R ad I o (4) We get R ad P T I o (43) Fo an infinitesimal antenna, we get R ad 4π βiodl η I 3 4π o (44) Using β π λ, and η π ohms, we obtain R dl ad π λ 8 Ω (45)