Linear Wire Antennas

Transcription

1 Linear Wire Antennas Ranga Rodrigo August 4, 010 Lecture notes are fully based on Balanis?. Some diagrams and text are directly from the books. Contents 1 Infinitesimal Dipole 1 Small Dipole 7 3 Finite-Length Dipole 9 Wire antennas, linear or curved, are some of the oldest, simplest, cheapest, and in many cases the most versatile for many applications. 1 Infinitesimal Dipole An infinitesimal linear wire l λ) is positioned symmetrically at the origin of the coordinate system and oriented along the z axis. The spatial variation of the current is assumed to be constant and given by I z ) = â z I 0 where I 0 is a constant. 1

2 The source only carries an electric current I e. I m and the potential function F are zero. To find A we write Ax, y, z) = µ I e x, y, z kr e j ) 4π R dl. where x, y, z) : Observation point coordinates. x, y, z ) : Coordinates of the source. C R : The distance from any point on the source to the observation point. C : Path along the length of the source. I e x, y, z ) = â z I 0. x = 0, y = 0, z = 0, for the infinitesimal dipole. R = x x ) + y y ) + z z ) = x + y + z = r constant). dl = d z. µi l/ 0 kr Ax, y, z) = â z e j d z µi 0 l = â z 4π l/ 4π e j kr. z θ x, y, z) x, y, z ) φ y x

3 Next: H A : H A = 1 µ A, E A: H A = J + j ωɛe A. Transformation from rectangular to spherical coordinates: A x = 0, A y = 0, A z 0. A r sinθ cosφ sinθ sinφ cosθ A x A θ = cosθ cosφ cosθ sinφ sinθ A y A φ sinφ cosφ 0 A z A r = A z cosθ = µi j kr A θ = A z sinθ = µi 0l e A φ = 0. A = âr r sinθ θ A φ sinθ) φ A θ H = 1 â φ µ r r r A θ) θ A r H r = H θ = 0. H φ = j ki 0l sinθ E r = η I 0l cosθ πr E θ = j η ki 0l sinθ E φ = 0. cosθ. j kr + âθ 1 r sinθ e j kr. j kr E = E A = 1 j ωɛ H j kr e j kr j kr 1 kr ) φ A r r r A φ) + âφ r r r A θ) θ A r e j kr. The E and H components are valid everywhere except on the source itself. 3

4 z â r, E r, H r â φ, E φ, H φ θ â θ, E θ, H θ φ y x Power Density and Radiation Resistance For a lossless antenna, the real part of the input impedance is designated as the radiation resistance, that power is transferred from the guided wave to the free space wave. W = 1 E H = 1 â r E r + â θ E θ ) ). = 1 â r E θ H φ â θe r H φ W r = η I 0 l sin θ 1 8 λ r 1 j kr ) j W θ = j η k I 0l cosθ sinθ 16π r 3. 1 kr ) â φ H φ The complex power moving int eh radial direction π π P = W d s = â r W r + a θ W θ ) a r r sinθdθdφ S π π = W r r sinθdθdφ 0 0 = η π I 0 l j λ kr ) ).

5 The transverse component W θ does not contribute to the integrals. Thus P does not represent the total complex power radiated by the antenna. W θ is purely imaginary, and does not contribute to any real radiated power. It contributes to the imaginary reactive) power. The reactive power density, which is most dominant for small values of kr, has both radial and transverse components. It merely changes between outward and inward directions to form a standing wave at a rate twice per cycle. It also moves in the transverse direction. Time average power radiated is π ) p rad = η I 0 l 3 λ. For large values of kr kr 1), the reactive power diminishes. For free space η 10π, ) ) π l ) l R rad = η = 80π.. 3 λ λ Near-Field Region kr 1 E r j η I j kr 0l e πkr 3 cosθ. j kr E θ j η I 0l e 4πkr 3 E φ = H r = H θ = 0. H φ I j kr E r and E θ are in time-phase. E r and E θ are in time-phase quadrature with H φ. Therefore, there is no time-average power flow associated with them. Intermediate-Field Region kr > 1 E r η I j kr 0l e πkr cosθ. j kr E θ j η ki 4πkr E φ = H r = H θ = 0. H φ j ki j kr 5

6 E r and E θ approach time-phase quadrature. They form a rotating vector whose tip traces and ellipse in a plane parallel to the direction of propagation: cross field. Far Field kr 1 The ratio of E θ to H φ is equal to E θ j η ki j kr 4πkr E r E φ = H r = H θ = 0. H φ j ki j kr Z w = E θ H φ η. where Z w is the wave impedance and η is the intrinsic impedance πΩ for free-space.) E- and H-field components are perpendicular to each other, transverse to the direction of propagation, and r variations are separable from those of θ and φ. This relationship is applicable in the far-field region of all antennas of finite dimensions. Directivity The average power density The radiation intensity W av = 1 Re E H ) 1 = â r η E θ η = â r ki 0 l sin θ 4π r. U = r W av = η ki 0 l 4π sin θ = r Eθ r,θ,φ). η The maximum value occurs at θ = π/: U max = η ki 0 l 4π. D 0 = 4π U max P rad = 3. 6

7 The maximum effective aperture λ A em = )D 0 = 3λ 4π 8π. P rad = 1 I 0 R rad. ) l. R rad = η π 3 λ Small Dipole The current distribution of the infinitesimal dipole l < λ/50) is I 0, a constant. For a small dipole λ/50 l λ/10) the triangular current distribution approximation must be used. z I 0 I The current distribution is I e x, y, z ) = Ax, y, z) = µ 0 â z I ) e j kr 4π l/ l z R {â z I 0 1 l z ), 0 z l, â z I l z ), l z 0. d z + â z l/ 0 I 0 1 ) e j kr l z R d z. Because the overall length of the dipole is small, the value of R for different values of z along the length of the wire are not much different from r. 7

8 z l/ d z z θ θ R r Pr,θ,φ) l/ φ = φ y x Maximum phase error due to the assumption R r is kl = π 10 = 18 for λ/10. j kr 1 µi0 e A = â z A z = â z, which is one half of that obtained in the previous section for the infinitesimal dipole. Far-Zone Fields, kr 1 E θ j η ki j kr 8πkr E r E φ = H r = H θ = 0. H φ j ki j kr 8πr Directivity and the maximum effective area are the same as for the infinitesimal dipole. R rad = P ) rad l I 0 = 0π λ which is 1/4 of the value for the infinitesimal dipole. 8

9 3 Finite-Length Dipole We can analyze the radiation characteristics of a dipole with any length using magnetic vector potential A. For a thin, center-fed finite-length dipole l λ/10,d λ), the approximate current distribution can be written as I e x = 0, y = 0, z â z I 0 sin k l ) = z ), 0 z l, â z I 0 sin k l + z ), l z 0. In the far field, we have, z r, θ θ. For amplitude: R r. For phase: R r z cosθ. z z l/ d z z θ θ R r Pr,θ,φ) d z z θ θ R r Pr,θ,φ) l/ x φ = φ y x φ = φ y Maximum phase error due to the assumption R r is kl = π 10 = 18 for λ/10. Ax, y, z) = µ 4π C kr µe j Ax, y, z) = C I e x, y, z kr e j ) R dl. I e x, y, z )e j kz cosθ d z. The finite dipole antenna is subdivided into a number of infinitesimal dipoles of length δz. For an infinitesimal dipole of length d z positioned along the z-axis at z de θ j ηki ex, y, z j kr )e sinθd z. 4πR de r de φ = d H r = d H θ = 0 d H φ j ki ex, y, z j kr )e sinθd z. 4πR 9

10 Using the far field approximation de θ j ηki ex, y, z j kr )e sinθe j kz cosθ d z. Summing the contribution from all the infinitesimal elements E θ = l/ l/ kr ke j l/ de θ = j η sinθ I e x, y, z )e j kz cosθ d z l/ Simplifying E θ j η I j kr 0e πr H φ j I j kr 0e πr cos cos kl cosθ ) cos sinθ kl cosθ ) cos sinθ kl kl ) ).. Power Density, Radiation Intensity, and Radiation Resistance W av = 1 Re E H = 1 Re â θ E θ â φ H φ = 1 Re â θ E θ â φ E θ /η 10