Far field expressions (r>λr>D)》 头 ao-员[xapk町 E(6,p) .dp)-ix x 2 2n Poynting vector Antenna's radiation vector source (antenna) P' Mme.p)=j.Pep[i小aW 上 j,≠0
E(r,θ,ϕ) ≈ − j η 2λ ˆ i r × M (θ,ϕ) × ˆ i r ⎡ ⎣ ⎤ ⎦ e− jβr r H (r,θ,ϕ) ≈ 1 η ˆ i r × E S= 1 2 E × H* = E 2 2η ˆ i r M(θ,ϕ) ≈ Ji(P') V ' ∫ exp jβ r '⋅ˆ i ( r ) ⎡ ⎣ ⎤ ⎦dV ' Far field expressions ( r >> λ r >> D ) Antenna’ s radiation vector V’ P’ P r ’ r y x z D Eo (θ,ϕ) Ji ≠ 0 M t source (antenna) Poynting vector
Polarization vector p The polarization vector defines the polarization of the field (and also of the antenna) It is strictly related to the radiation vector M 二J2沟 =e明.e以= 一=M 22r
p ˆ E E ⋅ ejχ Polarization vector The polarization vector defines the polarization of the field (and also of the antenna) It is strictly related to the radiation vector M p ˆ = E E ⋅ ejχ = − j η 2λr M t e− jβr r η 2λr M t 1 r ⋅ ejχ = M t M t e − j βr+π 2 ( ) ⋅ ejχ ⇒ p ˆ = M t M t = M ˆ t
Wavefront A wavefront is a locus where the field has constant phase e.g: E.0.)-F.(. Therefore the wavefront is defined by: arg(Eos)-Br=- If r is large and arg(E)is limited we have arg(Er)-Br≈-Br=-p→r= Therefore the wavefront is a spherical surface in our case.That's why it's called"spherical wave
Wavefront r Tx Ex (r,θ,ϕ) = E0 x (θ,ϕ) e− jβr r = E0 x r ej(arg(E0 x )−βr) A wavefront is a locus where the field has constant phase Therefore the wavefront is defined by: arg(E0 x ) − βr = −ϕ If r is large and arg(E0x) is limited we have arg(E0 x ) − βr ≈ −βr = −ϕ ⇒ r = ϕ β Therefore the wavefront is a spherical surface in our case. That’s why it’s called “spherical wave”. e.g:
Spherical and plane waves spherical wavefront Spherical wave 下=ri (.6,p)=E。(0,p), In far field a spherical wave can be locally approximated with a Plane wave E(,0,p)≈E,e with E E(6,9 k=Bi, ≈plane wavefront
Spherical and plane waves Tx spherical wavefront Spherical wave E(r,θ,ϕ) ≈ Eo (θ,ϕ) e− jβr r r = rˆ ir with Eo ' = Eo (θ,ϕ) r k = βˆ ir In far field a spherical wave can be locally approximated with a Plane wave E (r,θ,ϕ) ≈ Eo ' e− j ki r ≈ plane wavefront k e