Download Elements of Simulation

Satellite Communications
Electromagnetic Wave
Propagation
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Overview
Electromagnetic Waves
Propagation
Polarization
Antennas
Antenna radiation patterns
Propagation Losses
Goldstone antenna at twilight, NASA
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© 2012 Raymond P. Jefferis III
1
Reference
Reference is specifically made to the
following highly recommended source:
Kraus, J. D. and Marhefka, R. J., Antennas For All
Applications, Third Edition, McGraw-Hill, 2002
from which the antenna radiation equations
used below were drawn.
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© 2012 Raymond P. Jefferis III
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Overview
• Satellite communication takes place through
the propagation of focused and directed
electromagnetic (EM) waves
• Since both received and transmitted waves
are simultaneously present at very different
power levels, in a satellite, both frequency
separation and EM field polarization are
used to decouple the channels
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© 2012 Raymond P. Jefferis III
3
Maxwell’s Equations
Maxwell’s equations in terms of free charge and current, WIKIPEDIA
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© 2012 Raymond P. Jefferis III
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Wave Equation
For scalar variable, u (E & M Fields)
u
2 2

c(u)

u
2
t
2
Solutions are sinusoids in time and space (waves)
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© 2012 Raymond P. Jefferis III
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EM Wave Propagation
Wikipedia
• Electromagnetic (EM) waves propagate energy,
contained in their electric and magnetic fields,
through space with velocity v, which is the speed
of light under the conditions of propagation.
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© 2012 Raymond P. Jefferis III
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Transverse EM (Plane) Wave Properties
• Velocity of propagation (near light speed)
• Electric field is normal to the magnetic field
• Both electric and magnetic fields are normal
to direction of propagation (plane wave)
• The relation of electric to magnetic fields is
a constant for the medium (air, vacuum)
• Waves are polarized, as determined by the
direction of the electric field orientation
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© 2012 Raymond P. Jefferis III
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Impedance
• The electric field strength E and magnetic field
intensity H in a propagating wave are related by,
H
1

E
where,



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 = magnetic permeability [Henry/meter]
 00-7 [Henry/meter] in vacuum
= dielectric constant [Farads/meter]
0 = 1/36*10-8 [Farads/meter] in vacuum
 = impedance of the medium
( 0 =376.7 Ohms in free space)
© 2012 Raymond P. Jefferis III
8
Impedance Change At Boundaries
• At a boundary between two media of differing
impedances (air and raindrops for instance), Z1
and Z2 [Ohms]
– Part of the incident wave from Medium1 is reflected
– Part of the incident wave is transmitted into Medium2
Z1 
1
1
Z2 
2
2
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© 2012 Raymond P. Jefferis III
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Wave Energy
• The electric and
magnetic energy
densities in a
plane wave are
equal. [J/m2]
• The total energy is
the sum of these
energies. [J/m2]
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1 2
wE   E
2
1
wH   H 2
2
wE  w H
wT  wE  wH
wT  12  E 2  12  H 2
© 2012 Raymond P. Jefferis III
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Wave Energy Density
• The energy density of a plane wave is the
Poynting energy, S [Watts/m2]
SRMS
 2
 EH  E 
E


1
2
1
1 2 1  2
SAv  EH 
E 
E
2
2
2 
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© 2012 Raymond P. Jefferis III
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Vertical Polarization Behavior
• Radio frequency energy at frequency, f,
propagates
• The wave propagates away from the
observer (into the paper), along the z-axis
• Energy propagates with velocity, v,
• As a function of distance, z, and time, t,
the vertical electric field is described by,
z 


E  Ey  Em cos  2 f  t   

v 

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© 2012 Raymond P. Jefferis III
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Horizontal Polarization
• Radio frequency energy at frequency, f ,
propagates
• The wave propagates away from the
observer, along the z-axis
• Energy propagates with velocity, v,
• As a function of distance, z, and time, t,
the horizontal E-field is described by,
z 


E  Ex  Em cos  2 f  t   

v 

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© 2012 Raymond P. Jefferis III
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Manipulated Variable Example
Run mCos example:
• Vary the frequency and observe the results
• Pick a position (say z = 0.5), and change the
z-variable to see how the wave propagates
past the selected location
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© 2012 Raymond P. Jefferis III
Lect 00 - 14
Antennas
• Electromagnetic circuits comparable in size
to the wavelength of an alternating current
• Have alternating electric and magnetic fields
resulting in Electromagnetic (EM) radiation
• Have a polarization specified by the electric field
direction (horizontal or vertical)
• Radiation pattern is affected by the shape of the
current-carrying conductor(s)
• The EM radiation propagates in space
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© 2012 Raymond P. Jefferis III
Lect 00 - 15
Vertically Polarized Antenna
• Total antenna length typically  /2
• Electric field shown normal to the plane
of the earth (vertical)
• Oscillating electric fields produce
accelerating and decelerating conduction
electrons, with consequent radiation of
EM-energy
• A magnetic field surrounds the currentcarrying wire
• The phases of the electric and magnetic
fields differ by 90 degrees
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© 2012 Raymond P. Jefferis III
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Horizontally Polarized Antenna
• Total antenna length typically  /2
where λ = c/f
• Electric field shown parallel to the plane
of the earth (horizontal)
• Oscillating electric fields produce
accelerating and decelerating conduction
electrons, with consequent radiation of
EM-energy
• A magnetic field surrounds the currentcarrying wire
• The phases of the electric and magnetic
fields differ by 90 degrees
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© 2012 Raymond P. Jefferis III
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Polarization Match Angles
• A match angle,  M, is defined as the angular
polarization difference between a transmitting and
a receiving antenna
• Smaller match angles result in greater coupling
between transmitting and receiving antennas
• If the antennas are at opposite polarizations
(vertical - horizontal) the received power will be
zero, theoretically.
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© 2012 Raymond P. Jefferis III
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Circular Polarization
• Radio frequency energy at frequency, f,
propagates as an EM wave, away from
the observer, along the z-axis (into the
paper)
• The energy propagates with velocity, v
• The electric and magnetic fields rotate in
time (space) according to,


Ex  Em cos  2 f  t 


z 

v  
z  


Ey  Em cos  2 f  t    

v 2 

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© 2012 Raymond P. Jefferis III
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Circularly Polarized Antenna
Circular Polarization, Wikipedia
Note the spiral net electric field resolves into
time-varying Ex and Ey components.
Conductor (black); Ex => Green; Ey => Red
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© 2012 Raymond P. Jefferis III
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The Isotropic (Ideal) Antenna
• The gains of antennas can be stated relative to an
isotropic ideal antenna as G [dBi], where G > 0.
• This antenna is a (theoretical) point source of EM
energy
• It radiates uniformly in all directions
• A sphere centered on this antenna would exhibit
constant energy per unit area over its surface
• The gain of an isotropic antenna is 0 dBi
Lect 05
© 2012 Raymond P. Jefferis III
Lect 00 - 21
Radiation Patterns of Antennas
• Electric field intensity is a function of the radial
distance and the angle from the antenna
• A radiation pattern can be plotted to show field
strength (shown as a radial distance) vs angle
• The angle between half-power points (denoted as
HPBW) is a measure of the focusing (Gain) of the
antenna. [Note: Half-power = 3 dB]
• Note: Antenna Gain is with respect to an ideal
isotropic antenna (Gain = 1.0 or 0.0 dBi)
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© 2012 Raymond P. Jefferis III
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Antenna Gain Calculation
• G = PA/PI
where,
PI is the power per unit area radiated by an
isotropic antenna, and
PA is the antenna power per unit area radiated by a
non-isotropic antenna,
G is the amount by which the isotropic power
would be multiplied to give the same power per
unit area as the gain antenna exhibits in the chosen
direction
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© 2012 Raymond P. Jefferis III
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Antenna Gain Calculation
•
•
•
•
Pr = radiated power per unit area
W = total applied power
Rr = antenna radiation resistance
Im = maximum value of antenna current
4 r 2 Pr
G
W
2
 Im 
W 
Rr

 2
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© 2012 Raymond P. Jefferis III
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Antenna Gain and Aperture Calculations
G
4 Ae

Ae   A
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2
G = antenna gain
Ae = effective aperture area
 = carrier wavelength
η = aperture efficiency
A = aperture area (r2)
© 2012 Raymond P. Jefferis III
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Half-Wave Dipole Power



cos
cos


 
Im 
2

E  60 
r 
sin 



15I m 2
Pr 
r2
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


cos
cos


 

2


sin 




θ is the angle normal
to the antenna
2
© 2012 Raymond P. Jefferis III
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Dipole Radiation Patterns
• Two dipole lengths shown:
 L =  /2 (half wave dipole)
HPBW = 78˚
Gain = 2.15 dBi
L =  (full wave dipole)
HPBW = 47˚
Gain = 3.8 dBi
• The longer antenna focuses the
energy into a more narrow beam
and thus has higher Gain.

Electric field intensity, half-wave dipole
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© 2012 Raymond P. Jefferis III
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Half-Wave Dipole Radiation
The radiated field and power
of a half-wave dipole
antenna are expressed by:
  

cos   cos 
 2


E
sin 
P : E2
Radiated power pattern, half-wave dipole
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© 2012 Raymond P. Jefferis III
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Half-Wave Dipole Radiation Pattern
zro = 0.000001;
e0 = 1.0;
e1 = Cos[p/2*Cos[theta]]/Sin[theta];
e2 = e1^2;
PolarPlot[{e2}, {theta, zro, Pi}, PlotStyle ->
{Directive[Thick, Black]},
PlotRange -> Automatic]
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© 2012 Raymond P. Jefferis III
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Half-Power Beam Width
• The Half-Power Beam Width (HPBW) is
defined as the included angle between the
half-power points on the radiation pattern.
The power is down by 3 dB at these points.
• For a half-wave dipole antenna this is
calculated as shown on the Mathematica®
notebook output that continues below.
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© 2012 Raymond P. Jefferis III
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Half-Wave Dipole HPBW Calculation
r1 = FindRoot[e1^2 - 0.5 == 0.0, {theta,
60.0 Degree}];
Print[r1]
w1 = theta /. r1
Print[w1/Degree]
r2 = FindRoot[e1^2 - 0.5 == 0.0, {theta,
120.0 Degree}];
Print[r2]
w2 = theta /. R2
Print[w2/Degree]
Print[(w2 - w1)/Degree]
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© 2012 Raymond P. Jefferis III
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HPBW for Half-Wave Dipole
• From the foregoing notebook, the HalfPower Beam Width is found to be:
HPBW = 78.0777 degrees
• At the outer edges of the beam (HPBW), the
power will be 70.7% of the maximum
power value.
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© 2012 Raymond P. Jefferis III
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Full-Wave Dipole Radiation
The radiated field and power
of a full-wave dipole antenna
are expressed, as a function of
angle, by:
cos [ cos  1
E
sin 
2
P: E
Power pattern, full-wave dipole
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© 2012 Raymond P. Jefferis III
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Full-Wave Dipole Radiation Pattern
zro = 0.000001;
e0 = 1.0;
en = 2.0;
e1 = (Cos[p*Cos[theta]] + 1)/(Sin[theta]*en);
e2 = e1^2;
PolarPlot[{e2}, {theta, zro, p}, PlotStyle ->
{Directive[Thick, Black]}]
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© 2012 Raymond P. Jefferis III
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Half-Power Beam Width
• The Half-Power Beam Width (HPBW) is
defined as the included angle between halfpower points on the radiation pattern. The
power is down by 3 dB at these points.
• For a full-wave dipole antenna this is
calculated as shown on the Mathematica®
notebook output that continues below.
LECT 04
© 2012 Raymond P. Jefferis III
35
Full-Wave HPBW Calculation
r1 = FindRoot[e1^2 - 0.5 == 0.0, {theta,
60.0 Degree}];
Print[r1]
w1 = theta /. r1
Print[w1/Degree]
r2 = FindRoot[e1^2 - 0.5 == 0.0, {theta,
120.0 Degree}];
Print[r2]
w2 = theta /. r2
Print[w2/Degree]
Print[(w2 - w1)/Degree]
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© 2012 Raymond P. Jefferis III
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HPBW for Full-Wave Dipole
• From the foregoing notebook, the HalfPower Beam Width is found to be:
HPBW = 47.8351 degrees
• At the outer edges of the beam (HPBW), the
power will be 70.7% of the full value.
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© 2012 Raymond P. Jefferis III
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Circular Aperture Antenna
• The electric field of a circular aperture
antenna can be calculated from:
2 J1[( D /  )sin ]
E[ ] 
D
sin
where, D/ gives the aperture diameter in
wavelengths and ϕ is the angle relative to the
normal to the plane of the aperture.
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© 2012 Raymond P. Jefferis III
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Radiated E-Field of Aperture Antenna
0.03
0.02
0.01
0.00
- 0.01
- 0.02
- 0.03
0.2
0.4
0.6
0.8
1.0
E-field for aperture with D/ = 10
The Mathematica® notebook follows, for D/ = 10:
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© 2012 Raymond P. Jefferis III
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Radiation Pattern of Aperture Antenna
Dlam = 10;
e2 = (2.0/p*Dlam)*(BesselJ[1,
p*Dlam*Sin[theta]])/Sin[theta];
PolarPlot[Abs[e2]/100, {theta, -p/6, p/6},
PlotStyle -> {Directive[Thick, Black]}]
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© 2012 Raymond P. Jefferis III
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Radiated Power from an Aperture
• The normalized radiated power can be
found from E2[] as shown below:
0.04
0.02
0.00
- 0.02
- 0.04
0.2
0.4
0.6
0.8
1.0
Normalized radiated power for aperture with D/ =
10
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© 2012 Raymond P. Jefferis III
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Radiated Power Calculation
Dlam = 10;
e2 = (2.0/p*Dlam)*(BesselJ[1,
p*Dlam*Sin[theta]])/Sin[theta];
PolarPlot[Abs[e2/100], {theta, -p/6, p/6},
PlotStyle -> {Directive[Thick, Black]},
PlotRange -> {{0, 1}, {-0.04, 0.04}}]
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© 2012 Raymond P. Jefferis III
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Half Power Beam Width
• The HPBW of an aperture having D/ = 10
is calculated to be:
5.89831 Degrees
• The Mathematica® notebook for this
calculation follows:
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© 2012 Raymond P. Jefferis III
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Aperture HPBW Calculation
p20 =((2.0/p*Dlam)*
(BesselJ[1,p*Dlam*Sin[0.00001]])/Sin[0.00001])^2
p2 = ((2.0/p*Dlam)*
(BesselJ[1,p*Dlam*Sin[theta]])/Sin[theta])^2/p20;
r1 = FindRoot[p2 - 0.5 == 0.0, {theta,1 Degree}];
w1 = theta /. r1;
Print[w1/Degree]
r2 = FindRoot[p2 - 0.5 == 0.0,{theta,-1 Degree}];
w2 = theta /. r2
Print[w2/Degree]
Print[Abs[(w2 - w1)]/Degree]
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© 2012 Raymond P. Jefferis III
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Workshop 04 - Antenna HPBW
• A circular aperture antenna has D/ = 20.
Plot the radiation pattern of this antenna and
calculate its Half Power Beam Width.
• What can you say about the aiming
requirements for such an antenna mounted
on a satellite?
LECT 04
© 2012 Raymond P. Jefferis III
Lect 00 - 45
Transmission Losses
Transmitted electromagnetic energy from a
satellite is lost on its way to the receiving
station due to a number of factors, including:
– Antenna efficiency
– Antenna aperture gain
– Path loss
LECT 04
– Rain/Cloud loss
– Atmospheric loss
– Diffraction loss
© 2012 Raymond P. Jefferis III
46
Antenna Gain and Link Losses
Pt = transmitted power
Pr = received power
At = transmit antenna aperture
Ar = receive antenna aperture
Lp = path loss
La = atmospheric attenuation loss
Ld = diffraction losses
Antenna Gain (t or r):
Gt/r = 4Ae t/r/ 2
Combined Antenna Gain (t + r):
G = GtGr
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© 2012 Raymond P. Jefferis III
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Antenna Gain
Ae   A (d / 2)
4
G  2 Ae

 d 
G  A 
  
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2
2
Ae = effective antenna aperture
G = 4Ae/ 2 (Antenna Gain)
d = antenna diameter
λ = wavelength
 = aperture efficiency
© 2012 Raymond P. Jefferis III
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Compensating for Link Losses
• Increase antenna gain
• Increase power input to antenna
• Net effect: increase EIRP (Equivalent
Isotropically Radiated Power)
- Make sure tracking of beam is accurate
(target on beam axis).
LECT 04
© 2012 Raymond P. Jefferis III
Lect 00 - 49
EIRP
• Equivalent Isotropic Radiated Power
• – the equivalent power input that would be
needed for an isotropic antenna to radiate
the same power over the angles of interest
LECT 04
© 2012 Raymond P. Jefferis III
Lect 00 - 50
Path Loss Calculation
• Effective Aperture (transmit or receive):
Ae =  A
• Effective Radiated Power:
EIRP = PtGt = Pt tAt
• Path Loss (for path length R):
Lp = (4R/ 2
• Received Power:
Pr = EIRP*Gr/Lp
where,
Gt = 4Aet/ 2
Gr = 4Aer/ 2
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© 2012 Raymond P. Jefferis III
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Decibel (dB) Scale Definition
• PdB = 10 log10 Pt/Pr
• Logarithmic scale changes division and
multiplication into subtraction and addition
• dBW refers to power with respect to 1 Watt.
• Received power (Pratt & Bostian, Eq. 4.11):
• Pr = EIRP + Gr - Lp [dBW]
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© 2012 Raymond P. Jefferis III
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Received Power - dB Model
• (Pratt & Bostian, Eq. 4.11)
Pr = EIRP + Gr - Lp - La - Lt - Lr [dBW]
–
–
–
–
–
–
LECT 04
EIRP => Effective radiated power
Gr => Receiving antenna gain
Lp => Path loss
La => Atmospheric attenuation loss
Lt => Transmitting antenna losses
Lr => Receiving antenna losses
© 2012 Raymond P. Jefferis III
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End
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© 2012 Raymond P. Jefferis III
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