Antennas and Radiation - Lecture Slides | ECE 3317, Study notes of Electrical and Electronics Engineering

Download Antennas and Radiation - Lecture Slides | ECE 3317 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! ECE 3317 Prof. David R. Jackson N t 22o es Antennas and Radiation [Chapter 7] Antenna Radiation Antenna Radiation We consider here the radiation from an arbitrary antenna. z ( ), ,r r θ φ S +- y r x r → ∞ "far field" The far-field radiation acts like a plane wave going in the radial direction. Antenna Radiation (cont.) Hence we have ( )22 1ˆS E E 2r θ φ ⎛ ⎞ = + ⎜ ⎟ ⎝ ⎠0η or 2E ˆS 2 r ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ 0η⎝ ⎠ Note: in the far field, the Poynting vector is pure real (no reactive power flow). Radiation Pattern Th f fi ld l h th f ll i fe ar e a ways as e o ow ng orm: ( ) ( ) 0 E , , E , jk r Fer θ φ θ φ −⎛ ⎞ = ⎜ ⎟ ⎝ ⎠r ( )E ,F θ φ ≡ Normalized ar - field electric fieldf In dB: ( ) ( ) ( )10 E , dB , 20log F F θ φ θ φ ⎛ ⎞ ⎜ ⎟= ⎜ ⎟E ,m mθ φ⎝ ⎠ ( ),m mθ φ = direction of maximum radiation Radiation Pattern (cont.) The far-field pattern is usually shown vs. the angle θ (for a fixed angle φ) in polar coordinates. ( ) ( )E , dB 20log F θ φ θ φ ⎛ ⎞ ⎜ ⎟= ⎜ ⎟( )10 , E ,F m mθ φ⎝ ⎠ θ 30°30° 60° 60° 0φ = 0 dB θ -10 dB 120° 120° m-20 dB -30 dB 150° 150° Directivity (cont.) Th di ti it i d i t f th f fi ld tte rec v y s now expresse n erms o e ar e pa ern. ( ) ( )( )2 S , , / 4 rD r P r θ φ θ φ π ≡ → ∞ rad ( ) 2E 1F θ φ⎛ ⎞⎛ ⎞⎜ ⎟ Hence we have ( ) ( ) 2 0 2 2 2 , 2 , 4 1 F r D r rπ π η θ φ π ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠= → ∞ ∫ ∫ 0 0 0 E , sin 2 d dθ φ θ θ φ η Therefore, ( ) ( ) ( ) 2 2 2 4 E , , E sin F F D d d π π π θ φ θ φ θ φ θ θ φ = ∫ ∫ 0 0 , Directivity (cont.) Two Common Cases Short dipole wire antenna (l << λ0): D = 1.5 Resonant half-wavelength dipole wire antenna (l = λ0 / 2): D = 1.643 +h z feed 2l h= y x -h Beamwidth The beamwidth measures how narrow the beam is (The narrower the . beamwidth, the higher the directivity). HPBW = half-power beamwidth Infinitesimal Dipole (cont.) ⎡ ⎤⎛ ⎞ The exact fields of the infinitesimal dipole in spherical coordinates are 0 0 2 0 I 1 11 cos 2 jk r r lE e r jk r η θ π −= +⎢ ⎥⎜ ⎟ ⎝ ⎠ ⎣ ⎦ 0 0 2 0 0 I 1 1 1( ) 1 sin 4 ( ) jk rlE j e r jk r jk rθ ωμ θ π − ⎡ ⎤⎛ ⎞= + +⎢ ⎥⎜ ⎟ ⎝ ⎠ ⎣ ⎦ 0 0 0 I 1 1( ) 1 sin 4 jk rlH jk e r jk rφ θ π − ⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟ ⎝ ⎠ ⎣ ⎦ Infinitesimal Dipole (cont.) In the far field we have: 0 0 I ( ) sin 4 jk rl eE jθ ωμ θ −⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∼ 0 0 I ( ) sin 4 jk r r l eH jk rφ π θ π −⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∼ 0 I ( )sinF lE jθ ωμ θ= Hence, we can identify 0 4 I ( )sin 4 F lH jkφ π θ π = Infinitesimal Dipole (cont.) The radiation pattern is shown below . 0 I ( )sinF lE jθ ωμ θ= 4π θ 30°30° 60° 60° -9 -3-6 0 dB 120° 120° 150° 150° Infinitesimal Dipole (cont.) l ( ) 23, sin 2 D θ φ θ=0 I ( )sin 4 FE jθ ωμ θπ = θ 30°30° ( ), 1.0D θ φ = ( )o54 7θ 0 dB 60° 60° .= -9 -3-6 ( ), 1.5D θ φ = 120° 150° 150° 120° The far-field pattern is shown, with the directivity labeled at various points. Wire Antenna A center-fed wire antenna is shown below . +h z I (z) feed 2l h= yI 0 x -h I⎛ ⎞ ⎡ ⎤ A good approximation to the current is: ( ) ( ) 0 0 0 I( ) sin sin z k h z k h = −⎜ ⎟ ⎣ ⎦⎜ ⎟ ⎝ ⎠ Wire Antenna (cont.) A sketch of the current is shown below. ( ) ( ) 0 0 0 II( ) sin sin z k h z k h ⎛ ⎞ ⎡ ⎤= −⎜ ⎟ ⎣ ⎦⎜ ⎟ ⎝ ⎠ +h l +h l h h resonant dipole (l = λ0 / 2, k0h = π/2) - short dipole (l <<λ0 / 2) ⎡ ⎤ - ⎛ ⎞ 0I( ) I 1 z z h ≈ −⎢ ⎥ ⎣ ⎦ 0I( ) I cos 2 z z h π = ⎜ ⎟ ⎝ ⎠ Wire Antenna (cont.) z Far-field observation point +h R ( ), ,x y z y feed rdz' x ( ) ( ) ( ) 22 2 'R x y z z= + + − -h ( )2 2 2 2 2 2 ' 2 ' ' 2 ' x y z z zz r z zz = + + + − = + − 2 2 2 '1 2z zzr r r ′⎛ ⎞ ⎛ ⎞= + −⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ '1 2z z zr r r r ′⎛ ⎞ ⎛ ⎞⎛ ⎞= + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠ Wire Antenna (cont.) z Far-field observation point +h r R ( ), ,x y z y feed dz' θ x h cos /z rθ = - ( ) ( ) ( ) 2'z z z z′ ′ ′⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 2 cos 1 2 cos 1 cos cos R r r r r r r r r z θ θ θ θ = + − ≈ − ≈ −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ′= − 1 1 2 xx+ ≈ +Note: 1x << Wire Antenna (cont.) z Far-field observation point +h R ( ), ,x y z y feed rdz' x ( ) ( ) 0 cos 0 1 ( )sin I 4 cos h jk r zeE j z dz r z θ θ ωμ θπ θ ′− − ′ ′≈ ′∫-h ( ) ( ) 0 0 cos 0 1 ( )sin I 4 1 / cos h hjk r jk z h e ej z dz r z r θ ωμ θ π θ − ′− + − ⎛ ⎞ ′ ′= ⎜ ⎟ ′−⎝ ⎠ ∫ ( ) 0 0 cos 0 1 ( )sin I 4 hjk r jk z h ej e z dz r θωμ θ π − − ′+ − ⎛ ⎞ ′ ′≈ ⎜ ⎟ ⎝ ⎠ ∫ Wire Antenna (cont.) In summary, we have ( ) 0 0 1 ( )sin jk reE j AFθ ωμ θ θ −⎛ ⎞ ≈ ⎜ ⎟ ( ) 0 0 0I cos( cos ) cos( )2 k h k hAF θθ ⎛ ⎞ ⎡ ⎤− ⎜ ⎟ ⎢ ⎥ 4 rπ ⎝ ⎠ ( ) 20 0sin sink h k θ = ⎜ ⎟ ⎣ ⎦⎝ ⎠ ⎛ ⎞ ⎡ ⎤⎛ ⎞ Thus, we have ( ) 0 0 0 0 0 0 0 I cos( cos ) cos( )1 ( ) 2 sin sin jk r k h k heE j h r k h k hθ θωμ π θ − −⎛ ⎞≈ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎣ ⎦⎝ ⎠ Wire Antenna (cont.) fFor a resonant hal -wave dipole antenna 0 / 4 / 2 h k h λ= π θ⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥ 0 π= ( ) 0 0 0 cos cos I1 2( ) 2 / 2 sin jk reE j h rθ ωμ π π θ − ⎛ ⎞⎛ ⎞⎛ ⎞ ⎝ ⎠⎢ ⎥≈ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎢ ⎥⎣ ⎦ ( )/ 2, 1.643D π φ =The directivity is 3-dB beamwidth = 90° 3-dB beamwidth = 87° 3-dB beamwidth = 78° 3-dB beamwidth = 64° 3-dB beamwidth = 47.8 Wire Antenna (cont.) 21 The radiation resistance is defined from 0P I2rad rad R= +h z 2 0 2P I rad radR = Circuit ModelI (z)feed 2l h= Z0 Zin y in in inZ R jX= + in radR R= x -h For a resonant antenna (l ≈ λ0/2), Xin = 0. Wire Antenna (cont.) The radiation resistance is now evaluated. 2PradR = 2 0I rad This yields the result [ ]2 20 00 0 0 cos( cos ) cos( )1 2 sin( ) sinrad k h k h R d k h π θη θ π θ −⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ∫ λ /2 Di l 0h k hλ π [ ]73R Ω0 po e: 0,4 2= = rad ≈ Wire Antenna (cont.) The result can be extended to the case of a monopole antenna h 0 / 4h λ≈ Feeding coax 1 2 monopole dipole in inZ Z= [ ]36.5radR ≈ Ω