III. Spherical Waves and
Radiation
Antennas radiate spherical waves into free space
Receiving antennas, reciprocity, path gain and path loss
Noise as a limit to reception
Ray model for antennas above a plane earth and in a
street canyon
Cylindrical waves
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© 2002 by H. L. Bertoni
1
Radio Channel Encompasses Cables,
Antennas and Environment Between
Radio Channel
Transmitting
Antenna
Receiving
Antenna
Tx
Information
Rx
Cable
Cable
• Transmitter impresses information
onto the voltage of a high power
RF carrier for transmission
through the air - called modulation
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Information
• Receiver extracts the information
from the voltage of a low power
received signal - called
demodulation
© 2002 by H. L. Bertoni
2
Examples of Different Cellular Antennas
Half wave dipole
l /2
Full wave monopole
above ground plane
l /2
Dipole in corner
reflector
l /4
l /4
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3
PCS Antennas
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4
Base Station Antennas
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5
Antennas Radiate Electromagnetic Waves
Transmitting
Antenna
• EM waves have:
– Electric field E (V/m)
– Magnetic field H (A/m)
Cable
• E and H
– Perpendicular to each other and to direction
of propagation - Polarization
– Amplitude depends on direction of
propagation - Radiation Pattern
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E
H
6
Spherical Waves Radiated by Antennas
z
E
ar

r
I
For large r, localized current sources
radiate fields in the form of Spherical
Waves
H
e  jk r
E  aE ZI
f , 
r
1
H  ar  E
y
x


I : terminal Current
Z : constant with units of ohms
 : 120
ar  a E  1
Radial Power Flux
2
1
1 ZI
2

2
P  ReE  H  a r
f


,


(watt
s/m
)
2
2
2 r
Antenna pattern = f  ,   2
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7
Power Radiation Pattern
P()

• Power density radiated by antenna
P() = ExH* watts/m2
Poynting vector in the radial direction
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8
Omnidirectional Antennas
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9
Parabolic Reflector Antenna
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10
Horn Antennas
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11
Log Periodic Dipole Array
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12
Dual Polarization Patch Antenna
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13
Total Radiated Power
PT 
 P  a dA,
r
where dA  r2 sin dd
P
sphere
2
1
2
PT 
ZI 
2
0

 f ,  
2
sin  dd
dA
ar
0
r
PT is independent of r, as required by
conservation of power.
Normalization for f  ,  
2 
  f  , 
0
2
2
4r 2  area of sphere
is:
sind d  4
0
f ,  
4
2
ZI and P  a rPT
.
2
2
4r
2
T hen: PT 
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14
Antenna Gain and Radiation Resistance for
No Resistive Loss
Directive gain = g(,)= |f (,)|2 and
Antenna gain = G = Max. value of g(,
If isotropic antennas could exist, then |f (,)|2 = 1, G = 1
Radiation Resistance Rr = effective resistance seen at antenna terminals
1 2
4
I Rr  PT 
ZI
2
2
4 2
Rr 
Z
2

Z 
Rr
4
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15
Antenna Directivity, Gain, Efficiency
Maximum P ointing Vector Pm (r)
Direct ivit y=

Average P oint ing Vect or Pav (r)
Pm (r)

PT 4 r2 
Pm (r)
Gain =
includes t he effect of antenna resist ance
2
Pte rminal 4 r 
Efficiency =
PT
Pte rminal
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Gain

Direct ivit y
© 2002 by H. L. Bertoni
16
Short (Hertzian) Dipole Antenna
T he radiated field can be
z
E
L<<l
I

writt en in t he desired form
z
E  a E ZI
H
I (z)
Start ing with M axwell'
s equations,
it is found that
 jk r
LI e
E  a j 
sin 
2l r
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e
 jk r
r
sin
if
f   
3
sin 
2
Z  j
2 L
3 2l
G  f 90  3/2
2
2  L 
Rr  
3  l
2
17
Half Wave Dipole Antenna
The radiated field can be written:
z
E

I
l /4
H
r
I (z)
e  jk r
E  a E ZI
f  
r
where

cos cos 
2

f   
0.781sin
0.781
Z j

2
G  f 90  1.64
2
l /4
10logG  2.2dB
4
2
0.781
Rr 
j
  73

2
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18
Summary of Antenna Radiation
• Radiation in free space takes the form of spherical
waves
• E, H and r form a right hand system
• Field amplitudes vary as 1/r to conserve power
• Power varies as 1/r2, and varies with direction
from the antenna
• Direction dependence gives the directivity and
gain of the antenna
• Radiation resistance is the terminal representation
of the radiated power
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19
Receiving Antennas and Reciprocity
r
For a linear two-port
V1 =Z11I1 + Z12I2
V2 =Z21I1 + Z22I2
Reciprocity
Z12 = Z21
Equivalent Circuit
If I2 = 0, V2 = Z12I1 ~ 1/r
For r large,
|Z12| << |Z11|, |Z22 |
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+
I1
Z11-Z12
V1

© 2002 by H. L. Bertoni
Z22-Z12
Z12
I2
+
V2

20
Circuit Relation for Radiation into Free Space
+
V1
-
I1
Z11-Z12
Z22-Z12
+
V2 (open circuit)
-
Z12
V2 = VOC= Z12I1
V1 = Z11I1
T ransmit ted power

PT  1/2ReV1I1*  1/2Re Z11 I1
2

 (1/2)Rr1 I1
2
whereRr1  radiation resist ance of ant enna 1
T herefore: Z11  Rr1 + jX 1
Similarily: Z22  Rr2 + jX 2
whereRr2  radiation resist ance of ant enna 2
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21
Received Power and Path Loss Ratio
I1
+
V1
-
Z11-Z12
+
Z22-Z12
I2
Z12 V

Curre nt I 1 divi des be tween branches: I 2 = -I 1
+
V2 Z22* Matched Load
Rr2 - jX2
-
Z12 +

Z12
Z 22 Z 12 + Z22
2
1
1 I 1 Z12
Rece ive d Power for Matc hed Load PR 
I Rr 2 
2 2
2 2R r 2
2
Pa th Ga in
2

 -I 1
2
 I1
2
Z12
2R r 2
2
Z12
8R r 2
2
PR
I 1 Z12 8Rr 2
Z 12
PG 


2
PT
4R r1 R r 2
I 1 R r1 2
Fi nal expre ssi on for P G is t he same if a nt enna 2 ra dia tes and a nt enna 1 re cei ve s.
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22
Effective Area of Receiving Antenna
Effective Area = Ae
Ae1
Z*11
PG 
Z*22
PT
PT
PR g2 Ae1

PT 4 r 2
T hereforeg2 Ae1  g1 Ae 2
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g , 
2 Ae
4 r
PR  P  ar Ae  PT
and by reciprocity
Ae2
PG 
PR g1 Ae 2

PT
4 r2
Ae1 Ae 2
or

 same for all ant ennas
g1
g2
© 2002 by H. L. Bertoni
23
Effective Area for a Hertzian Dipole
z
I

I (z)
L<<l
e  jk r
E  a ZI
f 
r
g   ( 3 2)sin 
E
2  L 
Rr  
3  l
2
2

VOC  LE sin 
For matched termination
+
Voc
+
Z11
Z11*
-
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or
Voc
-
RR
2
+
RR Voc/2
2
Voc
1 Voc 2
PR 

2 RR
8RR
-
© 2002 by H. L. Bertoni
24
Effective Area for a Hertzian Dipole - cont.
:
For mat ched t ermination
VOC 2
2
2
l2
E  3 2  l2
 P  a r g( )
sin 


PR 
2


4
4
 2  L   2 2
8RR
8

 3 l 
PR  P  ar Ae .
In term s of t he effective area
LE sin
l2
Comparing expressions,Ae  g 
4
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25
Path Gain and Path Loss in Free Space
For any antenna
Ae1 Ae 2  A 
l2
l2



or Ae 
g
g1
g2  g  Hertz 4
4
P ath gain in free space
PR g1 Ae 2 g2 Ae1
 l 
PG 

2 
2  g1 g2
 4 r 
PT 4 r
4 r
For isot ropic antennasg1  g2  1
 l 
PG 
 4 r 
2
2
2
P
1  4 r  1
P ath Loss T 

PR PG  l  g1 g2
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26
Path Gain in dB for Antennas in Free Space
  l 2 
PGdB  PLdB  10logg1 g2

  4 r  
For isot ropic antennas,g1  g2  1
For frequency in GHz,l  c f  0.3 f GH
PGdB  32.4  20log fGH  20logr
PGdB
-32.4
-52.4
-72.4
-92.4
r =1
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fGH= 1
Slope=20
r =10
r =100
© 2002 by H. L. Bertoni
r =1000
27
Summary of Antennas as Receivers
• Directive properties of antennas is the same for
reception and transmission
• Effective area for reception Ae = gl2/4
• For matched terminations, same power is received
no matter which antenna is the transmitter
• Path gain PG = PR/PT < 1
• Path loss PL = 1/PG > 1
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28
Noise Limit on Received Power
• Minimum power for reception set by noise and
interference
• Noise power set by temperature T, Boltzman’s
constant k and bandwidth Df of receiver: N = kTDf
• For analog system, received power PR must be at
least 10N
• For digital systems, the maximum capacity C
(bits/s) in presence of white noise is given by the
limit
 PR 
C  Df log2 1+

N
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29
Sources of Thermal Noise
Sky Temp ~5o -150o K
Physical Temp
of Line = TL
TA
Temp of
Receiver TR
Physical Temp of
Antenna TAP
Ground Temp ~300o K
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30
Thermal Noise Power N
N  kTsDf
Boltsman’s constant = k =1.38x10-23 watts/(Hz oK)
System temperature = TS oK
Bandwidth = Df Hz
For TS = 300o K and Df = 30x103 Hz
• N = 1.24x10-16 watts
• (N)dB = -159.1 dBw = -129.1 dBm
– Noise figure of receiver amplifier F ~ 5 dB
– Effective noise = N + F
• For the example, N + F = -124.1 dBm
–
–
–
–
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31
WalkAbout Phones
Frequency band
450 MHz
Band width
12.5 kHz
Thermal noise 4x10-18 mW /Hz
5x10-14 mW
l = 0.667 m
-133 dBm
Receiver noise figure
5 dB typical
SNR for reception
10 dB for FM
Minimum received power
2x10-12 mW
-118 dBm
Transmitted power
500 mW
27 dBm
Maximum allowed path loss
(PTr)dB - (PRec)dB
145 dB
Minimum path gain
PRec /PTr = 10-14.5
3.2x10-15
Antenna gain / Antenna height
Assume 0 dB
1.6 m
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32
Maximum Range WalkAbouts in Free
Space
2
2
 l   l 
PG  G1G2

 3.2  1015  32 1016
 4R   4 R 
or
R
l
4
1
5

9.4
10
m  940 km or 563 miles
8
32  10
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33
Summary of Noise
• Noise and interference set the limit on the
minimum received power for signal detection
• Thermal noise can be generated in all parts of the
communications system
• Miracle of radio is that signals ~ 10-12 mW can be
detected
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34
Ground and Buildings Influence Radio
Propagation
• Reflection and transmission at ground, walls
• Diffraction at building corners and edges
Diffraction Path
Transmission
Reflection
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35
Two Ray Model for Antennas Over Flat Earth
(Antennas are Assumed to be Isotropic)
E1
Antenna
r1
E2
h1
a
r2 
h2
R

Image
2
Pr  Pt
  
 l 1
1
exp jkr1  +   exp jkr2 
 4  r1
r2
2
cos  a  r  sin 2 
cos + a r  sin2 
where   90  a and a  1  r for vertical (T M ) polarization,
or
a  1 horizontal (T E) polarizat ion
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36
Reflection Coefficients at Plane Earth
Vertical (TM) and Horizontal (TE) Polarizations
1
0.9
Horiz. Pol. r=15-j0.1
0.8
 
0.7
0.6
0.5
0.4
Vert. Pol. r=15-j0.1
0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
80
90
Incident Angle , degree
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37
Path Gain vs. Antenna Separation
(h1 = 8.7 m and h2 = 1.8 m)
-40
Brewster’s angle
Path Gain (dB)
-50
-60
-70
-80
-90
-100
-110 0
10
Vertical pol.
Horizontal pol
  1
10
f = 900MHz
1
10
2
10
3
Distance (m)
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38
Sherman Island/Rural
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39
Sherman Island Measurements vs. Theory
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40
Flat Earth Path Loss Dependence for Large R
If R  h1 and h2 then
r1,2 
R2 + h1
h2   R +
2
and ( )  -1
1
2
1 2
h1h2
2
h
h

R
+
h
+
h
 1 2

1
2
2R
2R
R
2
 l  1
1
Received power Pr  Pt
exp jkr1  +    exp jkr2 
 4  r1
r2
2
2
 l 
hh
 hh

is approximately Pr  Pt
exp jk 1 2   exp  jk 1 2 
 4 R 


R 
R 
2
or
2
2
 l 
 l 
 hh
 hh
Pr  Pt
2sin k 1 2   Pt
2sin 2 1 2 
 4 R 
 4 R 
R
lR
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2
2
41
Path Gain of Two Ray Model
2
PG 
 l 
 hh 
2sin 2 1 2
 4 R 

lR 
At t he break point,R 
 l 
PG  4
 4 R 
4h1h2
l
2
t he pat h gain has a local maximum
2
P ast t he break point
2
2
 l   2 h1h2 
h12 h22
PG 
2
 4
 4 R   l R 
R
P ast t he break point,
P G is:
Independent of frequency
Varies as 1 R 4 instead of1 R 2 .
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42
Maximum Range for WalkAbouts
on Flat Earth
For h1  h2  1.6 m ,
RB 
4h1 h2
l
2
4(1.6)

 15.3 m
0.667
For R  RB
(h1h2 )2
15
PG 

3.2
10
4
R
Solving t he inequalit y for
R
2
(1.6
1.6)
15
R4 

0.8
10
15
3.2 10
or
R  5.3 103 m = 5.3 km or 3.2 miles
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43
Fresnel Zone Gives Region of Propagation
r2- r1= l /2
Fresnel zone is ellipsoid about ray connecting source and receiver
and such that r2-r1 =l/2
– Ray fields propagates within Fresnel zone
– Objects placed outside Fresnel zone generate new rays, but
have only small effect on direct ray fields
– Objects placed inside Fresnel zone change field of direct ray
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44
Fresnel Zone Interpretation of Break Point
r1
Fresnel zone
(r2- r1= l /2
r2
RB
Fresnel zone definition
: l 2  r2  r1
Horizont al antenna separat ion
RB for Fresnel zone to touch t he ground
l 2  r2  r1  RB2 + (h1 + h2 )2  RB2 + (h1  h2 )2 
or
RB 
2h1h2
RB
4h1h2
l
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45
Regression Fits to the 2-Ray Model on Either
Side of the Break Point
-50
Path Gain (dB)
-60
n1=1.3
-70
-80
-90
-100
-110
-120
n2=3.6
f=1850MHz
h1=8.7
h2=1.6
Model: 2ray,
r=15
100
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101
102
Distance (m) RB
© 2002 by H. L. Bertoni
103
46
Six Ray Model to Account for Reflections
From Buildings Along the Street
R0
zT
ap
Ray lengt hs:
Rb
Ra
As seen from above
zR
w
Top view of street canyon showing relevant rays
Each ray seen from above represents two rays
when viewed from the side:
1. Ray propagating directly from Tx to Rx
2. Ray reflected from ground
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R0 
x 2 + z T  z R 
Ra 
x 2 + w + z T + z R 
Rb 
x 2 + w  z T  z R 
2
2
2
In 3D
rn1, 2 
Rn2 + h1
h2 
2
47
Six Ray Model of the Street Canyon
For x  h1 ,h2 polarizat ion coupling at walls can be neglected.
 Rn 

Angle of incidence on ground n  arctan
 h1 + h2 
For each ray pair (vert ical polarizat ion)
e  jk rn1
e jk rn 2
Vn 
+ H  n 
rn1
rn2
Wall reflection angle  a, b

x



 arctan
 w  zT + zR 
P ath Gain of six rays
2
 l 
2
PG 
V + E  a Va + E b Vb
 4  0
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© 2002 by H. L. Bertoni
48
Six Ray Model for Street Canyon
f = 900 MHz, h1= 10 m, h2= 1.8 m, w = 30 m, zT = zR = 8 m
-40
Received Power (dBW)
-50
-60
6 ray model
-70
-80
-90
-100
2 ray model
-110
-120
-130
-140
101
Polytechnic University
102
Distance (m)
103
© 2002 by H. L. Bertoni
104
49
Received Signal on LOS Route
f = 1937 MHz, hBS= 3.2 m, hm = 1.6 m
Telesis Technology Laboratories, Experimental License Progress Report to the FCC, August, 1991.
Polytechnic University
© 2002 by H. L. Bertoni
50
Summary of Ray Models for
Line-of-Sight (LOS) Conditions
• Ray models describes ground reflection for
antennas above the earth
• Presence of earth changes the range dependence
from 1/R2 to 1/R4
• Propagation in a street canyon causes fluctuations
on top of the two ray model
• Fresnel zone identifies the region in space through
which fields propagate
Polytechnic University
© 2002 by H. L. Bertoni
51
Cylindrical Waves Due to Line Source
T he concept of a cylindrical wave will
be useful for discussing diffract ion
Line
Source
P hase is const ant over t he surface
y
of a cylinder
H
For   l radiat ed fields are
E  a E ZI
H
1

e
 jk


f  
a  E
E

x
z
Field amplitudes vary as1/ 
t o conserve power.
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© 2002 by H. L. Bertoni
52