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Propagation Mechanisms 1
1. Propagation Mechanisms
Contents:
• The main propagation mechanisms• Point sources in free-space• Complex representation of waves• Polarization• Electric field pattern• Antenna characteristics• Free-space propagation loss• Uniform plane wave in free-space
Propagation Mechanisms 2
1. Propagation Mechanisms
Contents (cont’d):
• Uniform plane wave in a lossy medium• Reflection and transmission• Propagation over flat earth• Scattering from rough surfaces• Diffraction over/around obstacles• The uniform geometrical theory of diffraction
Propagation Mechanisms 3
Multipath propagation:
Diffraction
Reflections
Line of sight
path
Shadowing
Scattering
Uplink
Downlink
The main propagation mechanisms
Propagation Mechanisms 4
Spherical coordinate system:
x1
x2
x3
Ωar Ω( )
S
O
ah Ω( )
φ
θ
1
Tx
av Ω( )
T Ω( ) Definitions and remarks:
• : Sphere of radius 1centred at .
• : Direction
• Azimuthal and coelevationangle, resp.
• : Outward unit normal to at
• : Tangent plane to at
Transverse plane to
• :Unit vectors pointing in thedirection of increment ofand , resp.
SO
Ω S∈φ θ,
ar Ω( ) S Ω
T Ω( ) S Ωar Ω( )
ah Ω( ) av Ω( ), T Ω( )∈
φθ
Point sources in free space
Propagation Mechanisms 5
Vertically polarized isotropic point source:
d
GT
PT
d0
av Ω( )ar Ω( )
ah Ω( )
E v Ω d t, ,( )
H h Ω d t, ,( )
d
λ
Polarization
planeΩ
Far zone
region
Sphere of
equal phase
S
Near zone
region
Point sources in free space
Propagation Mechanisms 6
Vertically polarized isotropic point source (cont’d):
Field equations:
E v Ω d t, ,( ) av Ω( )E v d( ) 2πft kd ϕv+–( ) Electric field [V/m]cos=
H h Ω d t, ,( ) ah Ω( )H h d( ) 2πft kd ϕv+–( ) Magnetic field [A/m]cos=
Point sources in free space
Propagation Mechanisms 7
Characteristics of the radiated wave:• The surfaces of equal phase are spheres
-> Spherical wave
• The wave amplitude on the equal phase surfaces is constant-> Uniform wave
• The electric and magnetic fields are orthogonal and belong to .-> Transverse electromagnetic (TEM) wave
•
T Ω( )
Z0H h d( ) E v d( )=
av Ω( )E v d( )
ar Ω( )
ah Ω( )H h d( )
Point sources in free space
Propagation Mechanisms 8
Electrical characteristics of free-space:
• Permittivity
• Permeability
• Intrinsic impedance
Wave constants:
• Frequency [Hz]
• Propagation constant
• Phase velocity [m/s]
• Wavelength [m]
ε0 8.86 1012–
[F/m]⋅≈
μ0 4π 109–
[H/m]⋅≈
Z0
μ0
ε0
-----⎝ ⎠⎛ ⎞ 1 2⁄
≡ 120π Ω[ ]≈
f
k 2πf μ0ε0( )1 2⁄≡ 2πλ
------= m1–[ ]
c μ0ε0( ) 1 2⁄–≡ 3 108
m/s[ ]⋅≈
λ c f⁄≡
Point sources in free space
Propagation Mechanisms 9
Wave’s Poynting vector:
Radiated power:
: sphere of radius .
av Ω( )E v d( )
ah Ω( )H h d( )xxxxxx
⎧ ⎨ ⎩
Sr d( )
Sr Ω d,( ) av Ω( )E v d( ) ah Ω( )H h d( )× ar Ω( )E v d( )2
Z0
------------------= =
ar Ω( )Sr d( )
PT1
2--- Sr Ω d,( ) Sdd⟨ | ⟩
Sd
∫1
2--- Sr d( )d2 Ωd
S∫ 2πd
2Sr d( ) constant= = = =
Lossless medium (✥) Sd d
Point sources in free space
Propagation Mechanisms 10
Vertically polarized isotropic point source (cont’d):
It follows from (✥) that:
andxxxx
⎧ ⎨ ⎩
E v
E v d( )Z0PT
2π-------------
1
d---⋅ 60PT
1
d---⋅ E v
1
d---⋅=≈= H h d( )
E v
Z0
-------1
d---⋅=
E v Ω d t, ,( ) av Ω( )E v
d------- 2πft kd ϕv+–( )cos=
H h Ω d t, ,( ) ah Ω( )E v
Z0d--------- 2πft kd ϕv+–( )cos=
Point sources in free space
Propagation Mechanisms 11
Horizontally polarized isotropic point source:
GT
PT
d
d0
av Ω( )ar Ω( )
ah Ω( )
d
λ
E h Ω d t, ,( )
H v Ω d t, ,( )
Polarization
plane
Far zone
region
Ω
S
Near zone
region
Point sources in free space
Propagation Mechanisms 12
Horizontally polarized isotropic point source (cont’d):
Isotropic point source:
Usually, the wave radiated by a source is the superposition of a vertically
and of a horizontally polarized spherical wave:
Henceforth, we only consider the electric field of the waves.
E h Ω d t, ,( ) ah Ω( )E h
d------- 2πft kd ϕh+–( )cos=
H v Ω d t, ,( ) av Ω( )E h
Z0d--------- 2πft kd ϕh+–( )cos–=
E Ω d t, ,( ) E h Ω d t, ,( ) E v Ω d t, ,( )+=
Point sources in free space
Propagation Mechanisms 13
Complex representation of spherical waves:
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
E h d t,( )
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
E v d t,( )
xxxxxxx
⎧ ⎪ ⎨ ⎪ ⎩
xxxxxxx
⎫ ⎪ ⎬ ⎪ ⎭
xxxxxxxxxxxxxxx
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
Eh d( )
xxxxxxxxxxxxxxxx⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
Ev d( )
E h Ω d t, ,( ) ah Ω( )ReE h
d------- jϕh( )exp jkd–( )exp j2πft( )exp⋅
⎩ ⎭⎨ ⎬⎧ ⎫
=
[Complex] electric fields
E v Ω d t, ,( ) av Ω( )ReE v
d------- jϕv( )exp jkd–( )exp j2πft( )exp⋅
⎩ ⎭⎨ ⎬⎧ ⎫
=
Time-dependent
part
Complex representation of waves
Propagation Mechanisms 14
Concise notation for spherical waves:
E h
d------- jϕh( )exp jkd–( )exp
E v
d------- jϕv( )exp jkd–( )exp
E h jϕh( )exp⋅
E v jϕv( )exp⋅
1
d--- jkd–( )exp⋅ ⋅==
E d( )Eh d( )
Ev d( )≡
E
xxxxxxxxxxx
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
and are complex 2-dim. vectorsE d( ) E
E d( ) E1
d--- jkd–( )exp⋅ ⋅=
Complex representation of waves
Propagation Mechanisms 15
Polarization of the electric field vector:
E v d t,( )
t
E h d t,( ) E Ω d t, ,( )
T Ω( )ar Ω( )
ah Ω( )
av Ω( )
Polarization
Propagation Mechanisms 16
Polarization of the electric field vector:• : linearly polarized wave
• and : circularly polarized wave
ϕh ϕv= T Ω( )
Trajectory of in
as a function ofwith fixed.
E d t,( )T Ω( ) t
d
E h
E v
E Ω d t, ,( ) ar Ω( ) E h d t,( )
E v d t,( )
ϕh ϕv– π 2⁄( ) mod π= E h E v=
CW RH ϕh, ϕv– 3π 2⁄ mod 2π( )=
T Ω( )E Ω d t, ,( )
ar Ω( )E h
E v
CCW LH ϕ, h ϕv– π 2⁄ mod 2π( )=E h d t,( )
E v d t,( )
Polarization
Propagation Mechanisms 17
Polarization of the electric field vector:
• Otherwise the wave is said to be elliptically polarized
T Ω( )
E h
E v
E Ω d t, ,( )
ar Ω( )
E v d t,( )
E h d t,( )
Polarization
Propagation Mechanisms 18
Anisotropic sources:Usually, the source does not radiate isotropically:
depend on , i.e.
•
•
[Normalized] electric field pattern of a source:
E h v ϕh v,⇒ Ω
E h v E h v Ω( )→ ϕh v, ϕh v Ω( )→
E h v ϕh v( )exp⋅ Eh v Eh v Ω( )→=
f Ω( )f h Ω( )
f v Ω( )≡
f h v Ω( )Eh v Ω( )
E h v
--------------------≡
E h v maxΩ Eh v Ω( ){ }≡⎩⎪⎪⎨⎪⎪⎧
with
f h v Ω( )
1
Electric field pattern
Propagation Mechanisms 19
Electric field pattern of a vertical -dipole antenna:λ 2⁄
f v φ θ,( )
x1
x2
x3
φ
θ
π2--- θ( )coscos
θ( )sin-----------------------------------=
f h Ω( ) 0=
f v Ω( ) f v φ θ,( )=
Electric field pattern
Propagation Mechanisms 20
Spherical wave radiated by an anisotropic source:
xxxxxxx
⎧ ⎪ ⎨ ⎪ ⎩
E Ω( )
E Ω d,( )f h Ω( )E h
f v Ω( )E v
1
d--- jkd–( )exp⋅ ⋅=
Electric field pattern
Propagation Mechanisms 21
Far-zone region:
Electric field pattern:As already discussed.
[Normalized] power pattern of a linear [polarized] wave:
d d0 2D
2
λ------≡≥
: maximal dimension ofthe antenna in meter.
D λ≥
p Ω( ) S Ω d,( )maxΩS Ω d,( )---------------------------------≡
f Ω( ) 2E 2
maxΩ f Ω( ) 2E 2{ }------------------------------------------------= f Ω( ) 2