Introduction to Antennas & Arrays

Transcription

1 Introducton to Antennas & Arrays Antenna transton regon (structure) between guded eaves (.e. coaxal cable) and free space waves. On transmsson, antenna accepts energy from TL and radates t nto space. J.D. Kraus, Antennas, McGraw Hll, 988 On recepton, antenna gathers energy from ncdent free-space wave and sends t nto TL. Recprocty theorem: almost all antenna propertes are the same on transmsson and recepton. Hence, no need to study them twce. Lecture 3 8-Jan-3 (2)

2 Smart (Adaptve) Antennas Antenna array (of smple antennas or sensors, e.g. dpole or monopole) + sgnal processng va optmum combnng. R.A. Monzngo, T.W. Mller, Introducton to Adaptve Arrays, Wley, New York, 98. Lecture 3 8-Jan-3 2 (2)

3 Antenna pattern Antenna pattern - descrbes drectonal propertes of an antenna. Assume a plane wave ncdent upon the antenna from a partcular drecton The output sgnal s j( t ( )) x( t) = A θ e ω +ϕ θ (3.) ( ) θ e( t) = Ee ω +ϕ j( t ) The ampltude pattern s Smlarly, power pattern F( θ ) = A( θ ) / A (3.2) max F 2 ( θ ) (3.3) Antenna drectvty: rato of maxmum radaton ntensty (power per unt sold angle) to the average radaton ntensty. Gan = drectvty f the antenna s lossless (assumed below). Radaton ntensty: c constant. 2 Φ( θ ) = c F ( θ ) (3.4) Recprocty: the Tx and Rx patterns are the same. θ - ncdence angle or angle-of-arrval (AoA). Lecture 3 8-Jan-3 3 (2)

4 Sold Angle and Sphercal Coordnate System J.D. Kraus, Antennas, McGraw Hll, 988. Lecture 3 8-Jan-3 4 (2)

5 Gan expressons: 2π π Φ max = c; Φ = Φ( θ, ϕ)sn θdθdϕ 4 Gan : G Φ = max = Φ θ - elevaton, ϕ - azmuth π (3.5) 2π π F 2 4π ( θ, ϕ)sn θdθdϕ Antenna gan compares max. radaton ntensty of a gven antenna wth that of an sotropc antenna for the fxed total power radated. (3.6) In receve mode, antenna gan tells us how much more power s receved by gven antenna as compared to the sotropc one Omndrectonal antenna: F( θ, ϕ ) = F( θ ) (3.7) G = π F 2 2 ( θ)sn θdθ (3.8) Isotropc antenna: F( θ, ϕ ) = (3.9) Lecture 3 8-Jan-3 5 (2)

6 Antenna Beamwdth 2 F( θ) θ θ θ Antenna beamwdth: 3dB beam wdth θ- dstance between the ponts of F( θ ) = / 2 ( 3 db) (3.) Zero-to-zero beamwdth, θ, s dstance between the ponts of F( θ ) =. Lecture 3 8-Jan-3 6 (2)

7 Antenna Pattern Components J.D. Kraus, Antennas, McGraw Hll, 988. Lecture 3 8-Jan-3 7 (2)

8 Antenna Pattern n 3-D Lecture 3 8-Jan-3 8 (2)

9 Unform Lnear Array (of sotropc elements) Plane wave: nput sgnal (feld) s a plane wave or a combnaton of plane waves (functon of space and tme) and output sgnal s a conventonal sgnal (functon of tme only). Plane wave sgnal k x(t) e( t, r) E wave ampltude ω frequency (radal) k ( ) e ω k r e( t, ) = E t r (3.) - wave vector (sometmes called "number") r - poston vector (dentfes a pont n space) k 2π ω = = ; λ wavelength λ c Drecton of k s the same as the drecton of the wave front, the surface of constant ampltude and phase s a plane (for a gven moment of tme) kr = const. Notatons: bold captal (K) matrces; bold lower case (k) vectors; lower case regular (k) scalars; k - -th column of K. Lecture 3 8-Jan-3 9 (2)

10 Why plane waves are mportant? Almost any reasonable (physcal) feld can be decomposed nto plane waves (Fourer transform n 3-D). EM feld from a dstant source (far zone) looks locally lke a plane wave. Note: f k r, then 2π ϕ = kr = r -> phase shft due to λ propagaton along dstance r. Lecture 3 8-Jan-3 (2)

11 ULA response to a plane wave nput plane wave θ A A 2 A N d + x output ( t ) x (t) = ae ω +ϕ (3.2) (t) - output sgnal of -th antenna element (assumed to be stropc) a - sgnal ampltude (the same for all elements) ϕ sgnal phase j t Drop e ω -> t s everywhere, further work wth complex ampltude, jϕ 2π A = a e, assume ϕ = : ϕ 2 = d cosθ λ 2π ϕ = d( )cos θ λ (3.3) (3.4) Lecture 3 8-Jan-3 (2)

12 ULA Pattern (factor) Total sgnal ampltude at the output, Nψ sn y = A = a e = a 2 e sn 2 ( N ) N N j j( ) ψ 2 ψ = = (3.5) 2π ψ = d cosθ λ Array pattern = normalzed response to a plane wave versus AoA, F( θ ) = Assumng array s located along oz, Nψ sn 2 ψ N sn 2 ψ (3.6) (3.7) N sn( d kz ) F( k ) = 2 (3.8) d k N sn( z ) 2 Note: 2 π 2 π ψ max = d ; ψ mn = d (3.9) λ λ We started wth sotropc elements and managed to obtan a hghly-selectve pattern! Lecture 3 8-Jan-3 2 (2)

13 ULA Pattern ULA Pattern d = λ / N=2 π θ = θ N=5 2. N= nput plane wave θ A A 2 A N d + output Lecture 3 8-Jan-3 3 (2)

14 ULA Pattern Parameters 3 db beamwdth -> For large N (N>>), 2 F( θ ) =.5 -> no closed-form soluton λ θ 5 ; L = Nd = array aperture (length) (3.2) L Null-to-null beamwdth λ λ θ = 2 [ rad] = 4 [deg]; (3.2) L L Frst sdelobe = -3.2dB λ Gan (drectvty): f d = n, G = N 2 n = nteger number. Lecture 3 8-Jan-3 4 (2)

15 Generc Representaton of Array Response Array elements spatally sample ncomng wave where [ ] x( t) e( t, p ) e( t, p )... e( t, p ) (3.22) p = N-... p N- Usng plane-wave model of poston vectors of array elements T ( kp) e( t,p) = Ee ωt, T ωt j ( ) j j t E e kp kp kp x = e e... e N (3.23) Array manfold vector s ( ) jkp jkp jkp v k = e e... e N (3.24) It gves us a phase at each array element. Array response to a plane wave can be expressed as N = x t (3.25) = y( t) ( ) Array pattern (factor) s N ωt = E e v ( k ) (3.26) k= N k= F( k) ~ v ( k ) (3.27) T. Lecture 3 8-Jan-3 5 (2)

16 Generc case: assume that each elements has mpulse response of the form h ( τ ), ntroduce vector mpulse response of the array h ( τ) = [h ( τ) h ( τ)... h ( τ)] T (3.28) N- e( t, p ) may be a generc wave, the array output can be expressed as where t h T d (3.29) y( t) = (t - τ) ( τ) τ e( τ ) = [ e( τ, p) e( τ, p)... e( τ, p N )] T (3.3) Very smlar to conventonal response (convoluton ntegral) of an LTI system. Fourer transform to frequency doman, jωt T y( ω ) = y( t) e dt = h( ω) e ( ω) (3.3) { } FT { } h( ω ) = FT h(t), e( ω ) = e (t) (3.32) h ( ω) = vector frequency response of the array, e ( ω) = spectrum of ncomng feld at element locatons, e( ω ) = e( ω) v( k ) (3.33) Lecture 3 8-Jan-3 6 (2)

17 Array scalar frequency response s T h( ω ) = y( ω) / e( ω ) = h( ω) v( k ) (3.34) It depends on the frequency response of ndvdual elements ( h ( ω) ) and on the array geometry ( v( k )) Varyng h ( ω) may perform beamformng, e.g. to steer the beam n desred drecton. Example: delay-and-sum beamformer h ( τ ) = δ( τ + τ ) (3.35) τ = k p ω jωτ j (3.36) h ( ω ) = e = e k p (3.37) and the array pattern (factor) N F T ( ) ( ) e j e j k = h v k = k p kp (3.38) N N N = = = v ( k - k ) (3.39) N Lecture 3 8-Jan-3 7 (2)

18 Block Dagram of Antenna Array Broadsde drecton endfre drecton F k ( k ) T F( k) = h v( k) N N h h h N j j = e k p e kp N = N = k = steerng drecton. Σ out = v ( k - k ) (3.4) N Lecture 3 8-Jan-3 8 (2)

19 Beam steerng: the beam s drected towards k! In terms of angles: N( ψ ψ) sn F( θ ) = 2 ψ ψ N sn 2 2 π where d cos, 2 π ψ = θ ψ = d cosθ λ λ Half-power beamwdth, (3.4) π π θ = θ, θ <, 2 2 θ 5 λ Nd cos θ (3.42) ( ) F θ ULA Pattern: beam steerng tet= tet=3 tet=6. θ, deg. Lecture 3 8-Jan-3 9 (2)

20 Introducton to antenna arrays. Summary Antenna pattern. Man beam and sdelobes. Nulls. Half-power beamwdth and zero-to-zero beamwdth. Sdelobe level. Antenna gan (drectvty). Unform lnear array. Array pattern, beamwdth, sdelobe level. Beam steerng. Pattern parameters. References H.L. Van Trees, Optmum Array Processng, Wley, New York, 22. R.C. Hansen, Phased Array Antennas, Wley, New York, 998. A.W. Rudge et al (Eds.), The Handbook of Antenna Desgn, IEE & Peter Peregrnus, Norflok, 986. R.A. Monzngo, T.W. Mller, Introducton to Adaptve Arrays, Wley, New York, 98. Lecture 3 8-Jan-3 2 (2)