1. Radiation from Infinitesimal Dipole (Electric-current Element)

Transcription

1 LECTUE : adiation fom Infinitesimal (Elementay) Souces (adiation fom an infinitesimal dipole. Duality of Maxwell s equations. adiation fom an infinitesimal loop. adiation zones.). adiation fom Infinitesimal Dipole (Electic-cuent Element) Definition: The infinitesimal dipole is a dipole whose length l is much smalle than the wavelength λ of the excited wave, i.e. l λ (Δl < λ / 50). The infinitesimal dipole is equivalent to a cuent element I l, whee dq I l = l. dt dl Q +Q Idl z A cuent element is best illustated by a vey shot (compaed to λ) piece of infinitesimally thin wie with cuent I. Fo simplicity, the cuent is assumed to have constant magnitude along dl. The ideal cuent element is difficult to ealize in pactice, but a vey good appoximation of it is the shot top-hat antenna. To ealize a unifom cuent distibution along the wie, capacitive plates ae used to povide enough chage stoage at the end of the wie, so that the cuent is not zeo thee... Magnetic vecto potential due to a cuent element z I I l / l / The magnetic vecto potential (VP) A due to a linea souce is (see Lectue ): e PQ A( P) = µ IQ ( ) dl Q (.) 4 π L PQ µ I e PQ A z dl. (.) 0 ( P) = ˆ l PQ If we assume that the dipole s length l is much smalle than the distance fom its cente to the obsevation point P, then PQ holds both in the exponential tem and in the denominato. Theefoe, Nikolova 06

2 e A= µ 0I l z. ˆ (.) Equation (.) gives the vecto potential due to an electic cuent element (infinitesimal dipole). This is an impotant esult because the field adiated by any complex antenna in a linea medium is a supeposition of the fields due to the cuent elements on the antenna suface. We epesent A with its spheical components. In antenna theoy, the pefeed coodinate system is the spheical one. This is because the fa field adiation is of inteest whee the field dependence on the distance fom the souce is decoupled fom its angula dependence. This angula dependence is descibed conveniently in tems of the two angles in the spheical coodinate system (SCS) and. Also, this field popagates adially (along ˆ ) when the souce is located at the oigin of the coodinate system. The tansfomation fom ectangula to spheical coodinates is given by A sincos sinsin cos Ax A = coscos cossin sin A y. (.4) A sin cos 0 Az Applying (.4) to A in (.) poduces e A = Azcos = µ 0I l cos e A = Az sin = µ 0I l sin (.5) A = 0 z A A A y x Nikolova 06

3 Note that: ) A does not depend on (due to the cylindical symmety of the dipole); ) the dependence on, e /, is sepaable fom the dependence on... Field vectos due to cuent element Next we find the field vectos H and E fom A. a) H = A (.6) µ The cul opeato is expessed in spheical coodinates to obtain A ( A ) H = ˆ µ φ. (.7) Thus, the magnetic field H has only a -component, i.e., e H = ( I l) sin +, H = H = 0. (.8) b) j E = jω jωε H = Α ωµε A (.9) In spheical coodinates, the E field components ae: e E = jηβ ( I l) cos ( β) e j E = jηβ ( I l) sin + β β π E = 0. β j ( ) 4 (.0) Notes: ) Equations (.8) and (.0) show that the EM field geneated by the cuent element is quite complicated unlike the VP A. The use of the VP instead of the field vectos is often advantageous in antenna studies. ) The field vectos contain tems, which depend on the distance fom the souce as /, / and / ; the highe-ode tems can be neglected at lage distances fom the dipole. Nikolova 06

4 ) The longitudinal ˆ -components of the field vectos decease fast as the field popagates away fom the souce (as / and / ): they ae neglected in the fa zone. 4) The non-zeo tansvese field components, E and H, ae othogonal to each othe, and they have tems that depend on the distance as /. These tems elate by the intinsic impedance η and they descibe a TEM wave. They epesent the so-called fa field which satisfies the Sommefeld adiation bounday conditions. The concept of fa field will be e-visited late, when the adiation zones ae defined... Powe density and oveall adiated powe of the infinitesimal dipole The complex vecto of Poynting P descibes the complex powe density flux. In the case of infinitesimal dipole, it is * * P= ( E H * * ) = ( Eˆ ˆ) ( ˆ) ( ˆ ˆ + E Hφ = EH EH ). (.) Substituting (.8) and (.0) into (.) yields η I l sin P = j, 8 λ ( β) (.) I l cossin P = jηβ +. 6π ( β) The oveall powe Π is calculated ove a sphee, and, theefoe, only the adial component P contibutes: Π= P ds= Pˆ+ Pˆ ˆ sindd S S ( ), (.) π I l j Π= η, W. (.4) λ ( β) The adiated powe is equal to the eal pat of the complex powe (the timeaveage of the total powe flow, see Lectue ). Theefoe, the adiated powe of an infinitesimal electic dipole is π I l Π ad = η, W. (.5) λ Nikolova 06 4

5 Hee, we intoduce the concept of adiation esistance, which descibes the powe loss due to adiation in the equivalent cicuit of the antenna: Π Π= I = (.6) I id π l = η λ, Ω. (.7) Note that (.7) holds only fo an infinitesimal dipole, i.e., when the cuent is assumed constant ove the length l of the dipole.. Duality in Maxwell s Equations Duality in electomagnetics means that the EM field is descibed by two sets of quantities, which coespond to each othe in such a manne that substituting the quantities fom one set with the espective quantities fom the othe set in any given equation poduces a valid equation (the dual of the given one). We deduce these dual sets by simple compaison of the equations descibing two dual fields: the field of electic souces and the field of magnetic souces. Note that duality exists even if thee ae no souces pesent in the egion of inteest. Tables. and. summaize the duality of the EM equations and quantities. TABLE.. DUALITY IN ELECTOMAGNETIC EQUATIONS Electic souces ( J 0, M = 0) Magnetic souces( J = 0, M 0) E= jωµ H H = jωε E H = jωε E+ J E= jωµ H M D =ρ B =ρ m B =0 D =0 J = jωρ M = jωρ m A+ β A= µ J F+ βf= εm e A= µ J e dv F= εm dv V H = µ V A E= ε F E= jωa+ ( jωµε ) A H = jωf+ ( jωµε ) F Nikolova 06 5

6 TABLE.. DUAL QUANTITIES IN ELECTOMAGNETICS given E H J M A F ε µ η /η β dual H E M J F A µ ε /η η β. adiation fom Infinitesimal Magnetic Dipole (Electic-cuent Loop).. The vecto potential and the field vectos of a magnetic dipole (magnetic cuent element) Im l Using the duality theoem, the field of a magnetic dipole I m l is eadily found by a simple substitution of the dual quantities in equations (.5), (.8) and (.0) accoding to Table.. We denote the magnetic cuent, which is the dual of the electic cuent I, by I m (measued in volts). (a) the electic vecto potential e F = Fzcos = ε0 ( Im l) cos e F = Fzsin = ε0 ( Im l) sin (.8) F = 0 (b) the electic field of the magnetic dipole E = ( Im l) sin + E = E = 0 e (.9) (c) the magnetic field of the magnetic dipole ( I ) cos j m l e β H = η + H H ( m ) I l sin e = η + β β = 0 j (.0) Nikolova 06 6

7 .. Equivalence between a magnetic dipole (magnetic cuent element) and an electic cuent loop Fist, we pove the equivalence of the fields excited by paticula configuations of electic and magnetic cuent densities. We wite Maxwell s equations fo the two cases: (a) electic cuent density E = jωµ H (.) H = jωε E + J E ω µεe = ωµ J (.) j (b) magnetic cuent density E = jωµ H + M (.) H = jωε E E ω µεe = M (.4) If the bounday conditions (BCs) fo E in (.) ae the same as the BCs fo E in (.4), and the excitations of both fields fulfill jωµ J = M, (.5) then both fields ae identical, i.e., E E and H H. Conside a loop [L] of electic cuent I. Equation (.5) can be witten in its integal fom as jωµ J ds= M dl. (.6) S [ C ] C Q m I m = 0 [ C] I = ρ v l + S[ C ] I m l + [ L] A[ L ] + Q m Nikolova 06 7

8 The integal on the left side is the electic cuent I. M is assumed non-zeo and constant only at the section l, which is nomal to the loop s plane and passes though the loop s cente. Then, jωµ I = M l. (.7) The magnetic cuent I m coesponding to the loop [L] is obtained by multiplying the magnetic cuent density M by the aea of the loop A [ L], which yields jωµ IA[ L] = Im l. (.8) Thus, we show that a small loop of electic cuent I and of aea A [L] ceates EM field equivalent to that of a small magnetic dipole (magnetic cuent element) Im l, such that (.8) holds. Hee, it was assumed that the electic cuent is constant along the loop, which is tue only fo vey small loops ( a < 0.λ, whee a is the loop s adius and the loop has only tun). If the loop is lage, the field expessions below ae inaccuate and othe solutions should be used. We will discuss the loop antennas in moe detail in a dedicated lectue... Field vectos of an infinitesimal loop antenna The expessions below ae deived by substituting (.8) into (.9)-(.0): e E = ηβ ( IA) sin +, (.9) e H = j β( IA) cos +, (.0) e H = β ( IA) sin + β, (.) E = E = H = 0. (.) The fa-field tems (/ dependence on the distance fom the souce) show the same behaviou as in the case of an infinitesimal dipole antenna: () the electic field E is othogonal to the magnetic field H ; () E and H elate though η ; () the longitudinal ˆ components have no fa-field tems. The dependence of the Poynting vecto and the complex powe on the distance is the same as in the case of an infinitesimal electic dipole. The adiated powe can be found to be Π = ηβ 4 IA / π. (.) ad ( ) Nikolova 06 8

9 4. adiation Zones Intoduction The space suounding the antenna is divided into thee egions accoding to the dominant field behaviou. The boundaies between the egions ae not distinct and the field behaviou changes gadually as these boundaies ae cossed. In this couse, we ae mostly concened with the fa-field chaacteistics of the antennas. Next, we illustate the thee adiation zones though the field of the small electic dipole. 4.. eactive nea-field egion This is the egion immediately suounding the antenna, whee the eactive field dominates and the angula field distibution is diffeent at diffeent distances fom the antenna. Fo most antennas, it is assumed that this egion is a sphee with the antenna at its cente, and with a adius NF 0.6 D / λ, (.4) whee D is the lagest dimension of the antenna, and λ is the wavelength of the adiatiation. The above expession will be deived in Section 5. It must be noted that this limit is most appopiate fo wie and waveguide apetue antennas while it is not valid fo electically lage eflecto antennas. At this point, we discuss the geneal field behaviou making use of ou knowledge of the infinitesimal electic-dipole field. When (.4) is tue, is sufficiently small so that β (note that D λ fo the infinitesimal dipole). Then, the most significant tems in the field expessions (.8) and (.0) ae ( I l j ) e β H sin I l e E E ( ) β ( ) H = H = E = 0 jη I l e jη πβ sin cos, β. (.5) This appoximated field is puely eactive (H and E ae in phase quadatue). Since e j β we see that: () H has the distibution of the magnetostatic Nikolova 06 9

10 field of a cuent filament I l (emembe Bio-Savat s law); () E and E have the distibution of the electostatic field of a dipole. That the field is almost puely eactive in the nea zone is obvious fom the powe equation (.4). Its imaginay pat is Im π I l Π = η λ β { } Im{ Π } dominates ove the adiated powe, ( ) Nikolova (.6) π I l Π ad = e{ Π } = η λ, (.7) when 0 because β and Π ad does not depend on. The adial component of the nea-field Poynting vecto P has negative imaginay value and deceases as / 5 : nea η I l sin P = j. (.8) 8 λ β5 The nea-field P component is also imaginay and has the same ode of dependence on but it is positive: o P nea P nea ( I l) cossin = jηβ (.9) 6π η I l = j 8 λ 4.. adiating nea-field (Fesnel) egion ( β) sin( ). (.40) β5 This is an intemediate egion between the eactive nea-field egion and the fa-field egion, whee the adiation field is moe significant but the angula field distibution is still dependent on the distance fom the antenna. In this egion, β. Fo most antennas, it is assumed that the Fesnel egion is enclosed between two spheical sufaces: 0.6 D D. (.4) λ λ Hee, D is the lagest dimension of the antenna. This egion is called the Fesnel egion because its field expessions educe to Fesnel integals.

11 The fields of an infinitesimal dipole in the Fesnel egion ae obtained by neglecting the highe-ode (/β) n -tems, n, in (.8) and (.0): ( I l) e H sin β ( I l) e E η cos πβ, β. (.4) β ( I l) e E jη H = H = E = 0 sin The adial component E is not negligible yet but the tansvese components E and H ae dominant. 4.. Fa-field (Faunhofe) egion Only the tems / ae consideed when β. The angula field distibution does not depend on the distance fom the souce any moe, i.e., the fa-field patten is aleady well established. The field is a tansvese EM wave. Fo most antennas, the fa-field egion is defined as D / λ. (.4) The fa-field of the infinitesimal dipole is obtained as ( I l) e H sin β j ( I l) e β E jη sin, β. (.44) E 0 H = H = E = 0 The featues of the fa field ae summaized below: ) no adial components; ) the angula field distibution is independent of ; ) E H; 4) η = Z0 = E / H; 5) P= ( E H* ) / = ˆ0.5 E / η = ˆ0.5 η H. (.45) Nikolova 06

12 5. egion Sepaation and Accuacy of the Appoximations In most pactical cases, a closed fom solution of the adiation integal (the VP integal) does not exist. Fo the evaluation of the fa fields o the fields in the Faunhofe egion, standad appoximations ae applied, fom which the boundaies of these egions ae deived. Conside the VP integal fo a linea cuent souce: µ e A= Il ( ) d l, (.46) L whee = ( x x') + ( y y') + ( z z'). The obsevation point is at Pxyz, (,, ) and the souce point is at Qx ( ', y', z '), which belongs to the integation contou L. So fa, we have analyzed the infinitesimal dipole whose cuent is constant along L. In pactical antennas, the cuent distibution is not constant and the solution of (.46) can be vey complicated depending on the vecto function ( ') I l d l. Besides, because of the infinitesimal size of this souce, the distance between the integation point and the obsevation point was consideed constant and equal to the distance fom the cente of the dipole, = ( x + y + z) /. Howeve, if D max (the maximum dimension of the antenna) is lage and commensuate with the wavelength λ, the eo, especially in the phase tem β, due to the above assumption fo would be unacceptable. Let us divide the integal kenel e / into two factos: () the amplitude facto ( / ), and () the phase facto e j β. The amplitude facto is not vey sensitive to eos in. In both, the Fesnel and the Faunhofe egions, the appoximation / / (.47) is acceptable, povided >> Dmax. The above appoximation, howeve, is unacceptable in the phase tem. To keep the phase tem eo low enough, the maximum eo in ( β ) must be kept below π / 8 =.5. Neglect the antenna dimensions along the x- and y-axes (infinitesimally thin wie). Then, ( ) ( ' ') ( ' ' cos ) x' = y' = 0 = x + y + z z', (.48) = x + y + z + z zz = + z z. (.49) Nikolova 06

13 Using the binomial expansion, is expanded as ( ) / / / ( ) ( ) ( ) ( = + z ' z 'cos + ) ( z ' z 'cos ) +, z' z' cos = z'cos + + z cossin + O. (.50) O denotes tems of the ode (/ ) and highe. Neglecting these tems and simplifying futhe leads to the appoximation z 'cos + ' sin cos sin z + z. (.5) This expansion is used below to mathematically define the eactive nea-field egion, the adiating nea-field egion and the fa-field egion. (a) Fa-field appoximation Only the fist two tems in the expansion (.5) ae taken into account: z'cos. (.5) z ' P (,, ) dz Qz ( ') x D/ 0 = ' y (a) z-oiented dipole of length D n n n nn ( ) n nn ( )( n ) n a + b = a + na b + a b + a b +!! ( ) Nikolova 06

14 z ' = P (,, ) dz Qz ( ') x D/ 0 z'cos = ' y (b) z-oiented dipole: fa-field appoximation The most significant eo tem in that was neglected in (.5) is ( z ') e ( ) = sin, which has its maximum at = π / and z = zmax = D/: ( z max ) emax () =. (.5) The minimum, at which the phase eo ( β ) is below π /8, is deived fom: ( z max ) π β. 8 Thus, the smallest distance fom the antenna cente, at which the phase eo is acceptable is fa min = D / λ. (.54) This is the fa-zone limit defined in (.4). As a wod of caution, sometimes equation (.54) poduces too small values, which ae in conflict with the assumptions made befoe. Fo example, in ode the amplitude-tem appoximation / / to hold, the atio of the maximum antenna dimension D and the distance must fulfill D/. Othewise, the fist-ode appoximation based on the binomial expansion is vey inaccuate. Besides, in ode to neglect all field components except the fa-field ones, the condition λ must hold, too. Theefoe, in addition to (.54), the Nikolova 06 4

15 calculated inne bounday of the fa-field egion should comply with two moe conditions: D and λ. (.55) Finally, we can genealize the fa-zone limit as fa min = max ( D / λ, 0 λ, 0D). (.56) (b) adiating nea-field (Fesnel egion) appoximation This egion is adjacent to the Faunhofe egion, so its uppe bounday is specified by fa min = D / λ. (.57) When the obsevation point belongs to this egion, we must take one moe tem in the expansion of as given by (.5) to educe sufficiently the phase eo. The appoximation this time is z'cos + z' sin. (.58) The most significant eo tem is z ' e = cossin. (.59) The angles o must be found, at which e has its extema: e z' = sin ( sin + cos ) = 0. (.60) The oots of (.60) ae () o = 0 min, (.6) (),() = actan ± ± 54.7 max. o ( ) Following a pocedue simila to case (a), we obtain: π z' () () π z' βemax ( ) = cos sin, o o = λ λ π D π βemax ( ) =, note: z = D/ λ 8 D D 0.6 λ = λ. (.6) Equation (.6) states the lowe bounday of the Fesnel egion (fo wie antennas) and is identical to the left-hand side of (.4). Nikolova 06 5