Small Loop Antenna and Duality Theorem

Transcription

1 Small Loop Antenna and Duality Theoem Page 1 Small Loop Antenna and Duality Theoem Having studied the ideal electic dipole we now tun ou attention to an inteesting antenna closely elated to the electic dipole: the electic loop antenna. Such an antenna is shown in Figue 1. The antenna consists of a loop of thin wie wound into a cicle of adius a in the xy-plane. A cuent I flows along the wie aound the loop in the ˆφ-diection. Also shown in the figue is an equivalent magnetic dipole at the oigin, which can be ignoed fo now. It will be discussed late. Figue 1: Small loop antenna souce: C. A. Balanis, Antenna Theoy, Analysis, and Design, 2nd ed., John Wiley and Sons) The vecto magnetic potential in Catesian coodinates, fo a wie conducto 1D) is Ax, y, z) = µ ˆ Ix, y, z ) e jkr R dl. 1) We make the following assumptions: 1. The adius of the loop a is small compaed to the wavelength, a λ; C 2. The cuent along loop flows only in the ciumfeential diection, that is, I only has a ˆφ component; 3. The cuent along the loop can be teated as unifom. Collectively, these imply that the cuent on the conducting loop is I = I φ ˆφ, φ 2π. 2)

2 Small Loop Antenna and Duality Theoem Page 2 Then, in cylindical coodinates, Iρ, φ, z )dl = I φ adφ ˆφ 3) We can now evaluate A in cylindical coodinates. Since thee is only a φ-component of the cuent, we only expect a φ-component of A. A is then given by Aρ, φ, z) = ˆφ µ ˆ 2π = ˆφ µ ai φ I φ adφ ˆφ ˆφe jkr ˆ 2π R cosφ φ ) e jkr R dφ 4) whee it is impotant to note that the unit vectos ˆφ and ˆφ do not point in the same diection because we ae in a cylindical coodinate system, and hence thei dot poduct gives the cos) function seen hee. The length R can be found as R = x x ) 2 + y y ) 2 + z z ) 2. 5) In spheical coodinates, we can expess these quanties as follows, Then, The expession fo A φ becomes x = cos φ 6) y = sin φ 7) z = cos θ 8) x = a cos φ 9) y = a sin φ 1) z =. 11) R = 2 2 cos φa cos φ 2 sin φ a sin φ + a 2 A φ = µ ai φ = 2 + a 2 2a cosφ φ ). 12) ˆ 2π cosφ φ ) exp[ jk 2 + a 2 2a cosφ φ )] dφ 13) 2 + a 2 2a cosφ φ ) Since the poblem has axial symmety, A φ does not depend on φ; theefoe, we evaluate it at an abitay obsevation angle. Let s take φ = fo simplicity, A φ = µ ai φ ˆ 2π cosφ ) exp[ jk 2 + a 2 2a cosφ )] dφ 14) 2 + a 2 2a cosφ )

3 Small Loop Antenna and Duality Theoem Page 3 Integation of this expesssion is challenging. We appoximate the integand by expanding the expx)/x tem in a Taylo seies about x =, i.e., fx) = f) + f )x + 1 2! f )x n 1)! f n 1) )x n ) whee in this case fa) = exp[ jk 2 + a 2 2a cosφ )] 2 + a 2 2a cosφ ) 16) which is valid if the loop is small a ). Then, keeping only the fist two tems of the seies, Hee, fa) f) + f )a. 17) fa = ) = e, 18) and the second tem is a bit moe involved. Let us let whee we see that u) = and Then, by the chain ule, These deivatives ae and Then, u = 2 + a 2 2a cosφ ), 19) fu) = e jku u. 2) f a) = df du du da. 21) df jke jku du = e jku, 22) u u 2 du da = a 2 2a cos φ ) 1/2 2a 2 cos φ ). 23) 2 f a) a= = df du du jke da = a= Retuning to ou seies appoximation fo fa), ) fa) e jke + + e cos φ )a [ 2 1 jk e + a + 1 ) ] cos φ 2 Substituting this esult into 14), A φ µ ai φ [ˆ 2π e ) e cos φ ) 24) 2 25) jk cos φ dφ + a + 1 ) ˆ 2π ] e cos 2 φ dφ. 26) 2

4 Small Loop Antenna and Duality Theoem Page 4 The fist integal is zeo, while the second one integates to π, so A φ µ a 2 I φ jk e ). 27) 2 Next, we find the magnetic field fom [ H = 1 A = 1 ˆ µ µ θ A φ ) ˆθ ] A φ) 28) which gives and H = jka2 I φ 2 H θ = ka)2 I φ 4 1 j ) cos θ e k 2 29) jk 1 ) e. 3) k) 2 Finally, the electic field is found as E = 1 jωɛ H = ηka)2 I φ 4 ) e jk ˆφ 31) Let us compae these fields to those poduced by an ideal dipole, H = I z jk ) e jk 32) E = I z 1 2π η j ) e cos θ ˆ + I zjωµ k 2 jk 1 ) e k) 2 33) Notice the exteme similaity between the loop magnetic field 29), 3) and the dipole electic field 33)! The same is also tue of the loop electic field 31) and the dipole magnetic field 32). In fact, if we take the dipole fields ceated fom an ideal dipole with cuent stength I m and length l, and make the following substitutions, E H 34) H E 35) ɛ µ 36) µ ɛ 37) η 1 η 1 η 38) η 39)

5 Small Loop Antenna and Duality Theoem Page 5 then the following fields esult, and If we set jk E = jki ml H = I ml 1 η2π H θ = jki ml η j k 2 ) e 4) ) cos θ e 41) jk 1 ) e. 42) k) 2 I m l = jηkπa 2 I φ = jωµsi φ, 43) whee S = πa 2 is the aea of the loop, then these fields ae exactly equal to the loop fields 29), 3), and 31)! This is not a coincidence. The elations 34) - 39) fom what is known as the Duality Theoy and it is a vey useful theoem associated with Maxwell s equations. Conside the souce-fee cul equations, which ae E = jωµh 44) H = jωɛe 45) Notice the symmety in this equations, simila to what we obseved when compaing the E and H fields of the loop and dipole. Using the Duality Theoem, we can actually synthesize the second cul equation fom the fist, without even needing to know the second equation. O vice vesa. Now intoduce souces to the equations. Ampee s Law becomes If we apply the Duality Theoy to this equation, the dual would be H = jωɛe + J 46) E = jωµh + M 47) whee M is the dual of the electic cuent souce in the fist equation, J M. 48) Such a souce is called a magnetic cuent souce. While magnetic cuent souces do not exist in natue, they can be useful mathematical tools because intoducing M in Faaday s Law makes it a pefect dual of Ampee s Law. How does this apply to the loop antenna we have deived? We see that by applying duality to the electic dipole poblem, we can aive at the fields of a loop antenna. Equivalently, this is the same as finding the fields fom a so-called magnetic dipole whose cuent is I m and length is l, as we did above in 4)-42). This is a lot simple than the long pocedue poposed initially to detemine the fields fom the loop diectly fom the electic cuent. Knowing the elationship

6 Small Loop Antenna and Duality Theoem Page 6 between the magnetic dipole stength I m and the loop antenna paametes given by 43), the fields can be elated to the geomety of the loop. We can obseve that the loop antenna / magnetic dipole antenna ceates the same fields as an electic dipole but with the oles of the E and H fields evesed. So this time, is is E that has a ˆφ-oiented componented, instead of H. Similaly, this time H has both ˆ- and ˆθ-components, instead of E. Essentially, the polaization of the antenna is othogonal to that of an electic dipole. Hence the co-location of an electic and magnetic dipole can be used to ceate two othogonally-polaized fields, which can be useful fo ceating CP o EP waves, fo example.