Date post: | 18-Dec-2015 |
Category: |
Documents |
Upload: | jocelyn-craig |
View: | 216 times |
Download: | 0 times |
1
3.4 Basic Propagation Mechanisms & Transmission Impairments
(1) Reflection: propagating wave impinges on object with size >> • examples include ground, buildings, walls
(2) Diffraction: transmission path obstructed by objects with edges
• 2ndry waves are present throughout space (even behind object)
• gives rise to bending around obstacle (NLOS transmission path)
(3) Scattering propagating wave impinges on object with size < • number of obstacles per unit volume is large (dense)• examples include rough surfaces, foliage, street signs, lamp posts
2
Models are used to predict received power or path loss (reciprocal)based on refraction, reflection, scattering
• Large Scale Models
• Fading Models
at high frequencies diffraction & reflections depend on
• geometry of objects
• EM wave’s, amplitude, phase, & polarization at point of intersection
3
3.5 Reflection: EM wave in 1st medium impinges on 2nd medium • part of the wave is transmitted• part of the wave is reflected
(1) plane-wave incident on a perfect dielectric (non-conductor)
• part of energy is transmitted (refracted) into 2nd medium
• part of energy is transmitted (reflected) back into 1st medium
• assumes no loss of energy from absorption (not practically)
(2) plane-wave incident on a perfect conductor
• all energy is reflected back into the medium
• assumes no loss of energy from absorption (not practically)
4
(3) = Fersnel reflection coefficient relates Electric Field intensity of reflected & refracted waves to incident wave as a function of:
• material properties,
• polarization of wave
• angle of incidence
• signal frequency
boundary between dielectrics (reflecting surface)
reflected wave
refracted wave
incident wave
5
(4) Polarization: EM waves are generally polarized
• instantaneous electric field components are in orthogonal
directions in space represented as either:
(i) sum of 2 spatially orthogonal components (e.g. vertical & horizontal)
(ii) left-handed or right handed circularly polarized components
• reflected fields from a reflecting surface can be computed using superposition for any arbitrary polarizationE||
E
6
3.5.1 Reflection from Dielectrics • assume no loss of energy from absorption
EM wave with E-field incident at i with boundary between 2 dielectric media
• some energy is reflected into 1st media at r
• remaining energy is refracted into 2nd media at t
• reflections vary with the polarization of the E-field
plane of incidence
reflecting surface= boundary between dielectrics
irt
plane of incidence = plane containing incident, reflected, & refracted rays
7
Two distinct cases are used to study arbitrary directions of polarization
(1) Vertical Polarization: (Evi) E-field polarization is
• parallel to the plane of incidence • normal component to reflecting surface
(2) Horizontal Polarization: (Ehi) E-field polarization is
• perpendicular to the plane of incidence• parallel component to reflecting surface
plane of incidence
irt
EviEhi
boundary between dielectrics (reflecting surface)
8
• Ei & Hi = Incident electric and magnetic fields
• Er & Hr = Reflected electric and magnetic fields
• Et = Transmitted (penetrating) electric field
Hi Hr
Ei Er
i r
t
1,1, 1
2,2, 2
Et
Vertical Polarization: E-field in the plane of incidence
HiHrEi Er
i r
t
1,1, 1
2,2, 2
Et
Horizontal Polarization: E-field normal to plane of incidence
9
(1) EM Parameters of Materials
= permittivity (dielectric constant): measure of a materials ability to resist current flow
• = permeability: ratio of magnetic induction to magnetic field intensity
• = conductance: ability of a material to conduct electricity, measured in Ω-1
dielectric constant for perfect dielectric (e.g. perfect reflector of lossless material) given by
0 = 8.85 10-12 F/m
10
often permittivity of a material, is related to relative permittivity r
= 0 r
lossy dielectric materials will absorb power permittivity described with complex dielectric constant
(3.18) where ’ =f
2
(3.17) = 0 r -j’
highly conductive materials
r & are generally insensitive to operating frequencyr
f
0
• 0 and r are generally constant • may be sensitive to operating frequency
11
Material r /r0 f (Hz)Poor Ground 4 0.001 2.82 107 108
Typical Ground 15 0.005 3.77 107 108
Good Ground 25 0.02 9.04 107 108
Sea Water 81 5 6.97 109 108
Fresh Water 81 0.001 1.39 106 108
Brick 4.44 0.001 2.54 107 4109
Limestone 7.51 0.028 4.21 108 4109
Glass, Corning 707 4 0.00000018 5.08 103 106
Glass, Corning 707 4 0.000027 7.62 105 108
Glass, Corning 707 4 0.005 1.41 108 1010
12
• because of superposition – only 2 orthogonal polarizations need be considered to solve general reflection problem
Maxwell’s Equation boundary conditions used to derive (3.19-3.23)
Fresnel reflection coefficients for E-field polarization at reflecting surface boundary
• || represents coefficient for || E-field polarization
• represents coefficient for E-field polarization
(2) Reflections, Polarized Components & Fresnel Reflection Coefficients
13
Fersnel reflection coefficients given by
(i) E-field in plane of incidence (vertical polarization)
|| =it
it
i
r
E
E
sinsin
sinsin
12
12
(3.19)
(ii) E-field not in plane of incidence (horizontal polarization)
=ti
ti
i
r
E
E
sinsin
sinsin
12
12
(3.20)
i = intrinsic impedance of the ith medium
• ratio of electric field to magnetic field for uniform plane wave in ith medium• given by i = ii
14
velocity of an EM wave given by 1
boundary conditions at surface of incidence obey Snell’s Law
)90sin()90sin( 222111 (3.21)
i = r (3.22)
Er = Ei (3.23a)
Et = (1 + )Ei (3.23b)
is either || or depending on polarization
• | | 1 for a perfect conductor, wave is fully reflected• | | 0 for a lossy material, wave is fully refracted
)90sin(sin90
2
11it
15
• radio wave propagating in free space (1st medium is free space) • 1 = 2
|| =irir
irir
2
2
cossin
cossin
(3.24)
=iri
iri
2
2
cossin
cossin
(3.25)
Simplification of reflection coefficients for vertical and horizontal polarization assuming:
Elliptically Polarized Waves have both vertical & horizontal components
• waves can be depolarized (broken down) into vertical & horizontal E-field components
• superposition can be used to determine transmitted & reflected waves
16
(3) General Case of reflection or transmission
• horizontal & vertical axes of spatial coordinates may not coincide with || & axes of propagating waves
• for wave propagating out of the page define angle measured counter clock-wise from horizontal axes
spatial horizontal axis
spatial vertical axis
||orthogonal components
of propagating wave
17
vertical & horizontal polarized components
components perpendicular & parallel to plane of incidence
EiH , Ei
V EdH , Ed
V
• EdH , Ed
V = depolarized field components along the horizontal & vertical axes
• EiH , Ei
V = horizontal & vertical polarized components of incident wave
relationship of vertical & horizontal field components at the dielectric boundary
EdH, Ed
V EiH , Ei
V = Time Varying Components of E-field
iv
iH
CT
dv
dH
E
ERDR
E
E(3.26)
- E-field components may be represented by phasors
18
for case of reflection:
• D =
• D|| || = ||
for case of refraction (transmission): • D = 1+
• D|| || = 1+ ||
R =
cossin
sincos, = angle between two sets of axes
DC =
|| ||0
0
D
D
R = transformation matrix that maps E-field components
DC = depolarization matrix
19
1.00.80.60.40.20.0
0 10 20 30 40 50 60 70 80 90
||||
r=12
r=4
angle of incidence (i)Brewster Angle (B)
for r=12
vertical polarization (E-field in plane of incidence)
for i < B: a larger dielectric constant smaller || & smaller Er
for i > B: a larger dielectric constant larger || & larger Er
Plot of Reflection Coefficients for Parallel Polarization for r= 12, 4
20
r=12
r=4
||1.00.90.80.70.60.50.40.3
0 10 20 30 40 50 60 70 80 90angle of incidence (i)
horizontal polarization (E-field not in plane of
incidence)
for given i: larger dielectric constant larger and larger Er
Plot of Reflection Coefficients for Perpendicular Polarization for r= 12, 4
21
e.g. let medium 1 = free space & medium 2 = dielectric
• if i 0o (wave is parallel to ground)
• then independent of r, coefficients || 1 and |||| 1
|| = 1cos
cos
cossin
cossin2
2
02
2
ir
ir
irir
irir
i
= 1cos
cos
cossin
cossin2
2
02
2
ir
ir
iri
iri
i
thus, if incident wave grazes the earth• ground may be modeled as a perfect reflector with || = 1• regardless of polarization or ground dielectric properties• horizontal polarization results in 180 phase shift
22
3.5.2 Brewster Angle = B
• Brewster angle only occurs for vertical (parallel) polarization • angle at which no reflection occurs in medium of origin• occurs when incident angle i is such that || = 0 i = B
• if 1st medium = free space & 2nd medium has relative permittivity r then (3.27) can be expressed as
1
12
r
r
sin(B) = (3.28)
sin(B) = 21
1
(3.27
)B satisfies
23
e.g. 1st medium = free space
Let r = 4
116
14
sin(B) = = 0.44
B = sin-1(0.44) = 26.6o
Let r = 15
115
1152
sin(B) = = 0.25
B = sin-1(0.25) = 14.5o
24
3.6 Ground Reflection – 2 Ray Model
Free Space Propagation model is inaccurate for most mobile RF channels
2 Ray Ground Reflection model considers both LOS path & ground reflected path
• based on geometric optics• reasonably accurate for predicting large scale signal strength for distances of several km
• useful for - mobile RF systems which use tall towers (> 50m)- LOS microcell channels in urban environments
Assume • maximum LOS distances d 10km • earth is flat
25
Let E0 = free space E-field (V/m) at distance d0
• Propagating Free Space E-field at distance d > d0 is given by
E(d,t) =
c
dtw
d
dEccos00 (3.33)
• E-field’s envelope at distance d from transmitter given by
|E(d,t)| = E0 d0/d
(1) Determine Total Received E-field (in V/m) ETOT
ELOS
Ei
E r = E g
i 0
d
ETOT is combination of ELOS & Eg
• ELOS = E-field of LOS component
• Eg = E-field of ground reflected component
• θi = θr
26
d’
d”
ELOS
Ei
E gi 0
d
ht h r
E-field for LOS and reflected wave relative to E0 given by:
and ETOT = ELOS + Eg
ELOS(d’,t) =
c
dtw
d
dEc
'cos
'00 (3.34)
Eg(d”,t) =
c
dtw
d
dEΓ c
"cos
"00 (3.35)
assumes LOS & reflected waves arrive at the receiver with - d’ = distance of LOS wave - d” = distance of reflected wave
27
From laws of reflection in dielectrics (section 3.5.1)
i = 0 (3.36)
Eg = Ei (3.37a)
Et = (1+) Ei (3.37b)
= reflection coefficient for ground
E g
d’
d”
ELOS
Ei
i 0
Et
28
resultant E-field is vector sum of ELOS and Eg
• total E-field Envelope is given by |ETOT| = |ELOS + Eg| (3.38)
• total electric field given by
c
dtw
d
dEc
'cos
'00
c
dtw
d
dEc
"cos
")1( 00 (3.39)ETOT(d,t) =
Assume i. perfect horizontal E-field Polarization
ii. perfect ground reflection
iii. small i (grazing incidence) ≈ -1 & Et ≈ 0
• reflected wave & incident wave have equal magnitude
• reflected wave is 180o out of phase with incident wave
• transmitted wave ≈ 0
29
• path difference = d” – d’ determined from method of images
2222dhhdhh rtrt = (3-40)
if d >> hr + ht Taylor series approximations yields (from 3-40)
d
hh rt2 (3-41)
(2) Compute Phase Difference & Delay Between Two Components
ELOS
d
d’
d”i 0
ht
h r
h
ht +
h r
Ei Eg
30
once is known we can compute
• phase difference =c
wc
2
(3-42)
• time delay d =cfc
2
(3-43)
As d becomes large = d”-d’ becomes small
• amplitudes of ELOS & Eg are nearly identical & differ only in phase
"'000000
d
dE
d
dE
d
dE (3.44)
if Δ = /n = 2π/n 0 π 2π
Δ
31
(3) Evaluate E-field when reflected path arrives at receiver
0cos"
)1('"
cos'
0000
d
dE
c
ddw
d
dEc
(3.45)ETOT(d,t)|t=d”/c =
t = d”/c reflected path arrives at receiver at
1cos00
cw
d
dEc
1cos00 d
dE=
100 d
dE=
32
(3.46)
22
200 sin1cos
d
dE=
2
2
0022
00 1 sind
dEcos
d
dE|ETOT(d)|=
=
=
2sin2 00
d
dE
cos2200
d
dE(3.47)
(3.48)
ETOT
"00
d
dE
'd
dE 00
Use phasor diagram to find resultant E-field from combined direct & ground reflected rays:
(4) Determine exact E-field for 2-ray ground model at distance d
33
As d increases ETOT(d) decreases in oscillatory manner
• local maxima 6dB > free space value
• local minima ≈ - dB (cancellation)
• once d is large enough θΔ < π & ETOT(d) falls off asymtotically with increasing d
-50
-60-70
-80
-90
-100
-110-120
-130
-140101 102 103 104 m
fc = 3GHzfc = 7GHzfc = 11GHz
Propagation Loss ht = hr = 1, Gt = Gr = 0dB
34
if d satisfies 3.50 total E-field can be approximated as:
k is a constant related to E0 ht,hr, and
radd
hh rt 3.022
2
1
2
(3.49)
d > (3.50)
rtrt hhhh 20
3
20 this implies
For phase difference, < 0.6 radians (34o) sin(0.5 )
22 00
d
dE|ETOT(d)|
e.g. at 900MHz if < 0.03m total E-field decays with d2
200 22
d
k
d
hh
d
dE rt
(3.51)ETOT(d)
V/m
35
Received Power at d is related to square of E-field by 3.2, 3.15, & 3.51
Pr(d) = (3.52b)
4120
)(
120
)( 2220 rR
eGdE
AdE
Pr(d) = 4
22
d
hhGGP rt
rtt (3.52a)
• received power falls off at 40dB/decade
• receive power & path loss become independent of frequency
rthhif d >>
36
Path Loss for 2-ray model with antenna gains is expressed as:
• for short Tx-Rx distances use (3.39) to compute total E field
• evaluate (3.42) for = (180o) d = 4hthr/ is where the ground
appears in 1st Fresnel Zone between Tx & Rx
- 1st Fresnel distance zone is useful parameter in microcell path loss models
PL(dB) = 40log d - (10logGt + 10logGr + 20log ht + 20 log hr ) (3.53)
PL = 1
4
22
d
hhGG
P
P rtrt
r
t
• 3.50 must hold