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IEEE New Hampshire SectionRadar Systems Course 1Review E & M 1/1/2010 IEEE AES Society
Radar Systems Engineering Lecture 2
Review of Electromagnetism
Dr. Robert M. O’DonnellIEEE New Hampshire Section
Guest Lecturer
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Radar Systems Course 2Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Reasons for Review Lecture
• A number of potential students may not have taken a 3rd year undergraduate course in electromagnetism
– Electrical/Computer Engineering Majors in the Computer Engineering Track
– Computer Science Majors– Mathematics Majors– Mechanical Engineering Majors
• If this relatively brief review is not sufficient, a formal course in advanced undergraduate course may be required.
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Radar Systems Course 3Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Outline
• Introduction– Coulomb’s Law– Gauss’s Law– Biot -
Savart Law– Ampere’s Law– Faraday’s Law
• Maxwell’s Equations
• Electromagnetic Waves
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Radar Systems Course 4Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Coulomb’s Law
• If two electric charges, and , are separated by a distance, , they experience a force, , given by:
• Two charges of opposite sign attract; and two charges of the same sign repel each other.
• The magnitude of the electric force is proportional to the magnitude of each of the two chares and inversely proportional to the distance between the two charges
• This electric force is along the line between the two charges
r1q 2q
2o
21
r4r̂qqF
επ=
rFr
1q 2qrCharles Augustin de Coulomb
(1736-1806)
= permittivity of =free space
( )2212 mN/C10x85.8 −oε
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Radar Systems Course 5Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Electric Field
• The electric field
of a charge , at a distance from the electric charge is defined as:
2o
1
r4r̂q)r(Eεπ
=r
1q r
Pr1q
P
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Radar Systems Course 6Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Electric Field
• The electric field
of a charge , at a distance from the electric charge is defined as:
• Remember, that the force on a charge located a distance due to is give by
• Linear Superposition– The total electric field at a point in space is due to a number
of point charges is the vector sum of the electric fields of each charge
• Electric field of a point charge
2o
1
r4r̂q)r(Eεπ
=r
1q r
2q r
2o
21
r4r̂qqF
επ=
r
r1q 2q
1q
q− q+
EqFrr
=
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Radar Systems Course 7Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Gauss’s Law
• Define: the “Electric Flux Density :
• Then, Gauss’s Law states that :
• Integrating the Electric Flux Density over a closed surface gives you the charge enclosed by the surface
• Using vector calculus, Gauss’s law may be cast in differential form:
(1777-1855)Carl Freidrich Gauss
ED orr
ε=
∫∫∫∫∫ ρ==⋅ dVQQSdD EnclosedEnclosedrr
VolumeChargeDensity
ρ=⋅∇ Dr
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Radar Systems Course 8Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Biot Savart Law
(1774-1862)
(1791-1841)
Jean-Baptiste Biot
Felix Savart
• Define: = Magnetic Field and = the Magnetic Flux Density
• The Biot-Savart law: – The differential magnetic field generated by a steady current
flowing through the length is:
– where is a unit vector along the line from the current element location to the measurement position of and is the distance between the current element location and the measurement position of
• For an ensemble of current elements, the magnetic field is given by:
Hr
Br
( )m/AR
R̂xld4IHd 2 ⎥
⎦
⎤⎢⎣
⎡⎥⎦⎤
⎢⎣⎡π
=r
r
ldr Hd
r
∫⎥⎦⎤
⎢⎣⎡π
=l
2RR̂xld
4IH
rr
RR̂
Hdr
Hdr
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Radar Systems Course 9Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Magnetic Flux and the Absence of Magnetic Charges
• Law stating that there are no magnetic charges:
• Integrating the Magnetic Flux Density over a closed surface gives you the magnetic charge enclosed by the surface (zero magnetic charge)
• This is “Gauss’s Law”
for magnetism– Law of non-existence of magnetic monopoles– A number of physicists have searched extensively for magnetic
monopoles Find one and you will get a Nobel Prize
• Magnetic field lines always form closed continuous paths, otherwise magnetic sources (charges) would exist
0B0SdB =⋅∇=⋅∫∫rrrr
Magnetic Fiend of the Earth
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Radar Systems Course 10Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Amperes Law
• Ampere’s law (for constant currents):
• If c is a closed contour bounded by the surface , then
• The sign convention of the closed contour is that and obey the “right hand rule”
(1775-1836)Andre-Marie Ampere
JHxISdJsdHSc
rrrrrrr=∇=⋅=⋅ ∫∫∫
S
Ir
Hr
• The line integral of around a closed path c equals the current moving through that surface bounded by the closed path
Hr
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Radar Systems Course 11Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Faraday’s Law
tBExSd
tBsdE
Sc ∂∂
−=∇⋅∂∂
−=⋅ ∫∫∫r
rrrr
rr
• A changing magnetic field induces an electric field.
(1791-1867)Michael Faraday
• Induced electric fields are determined by:
• Magnetostatic fields are determined by :Jor
μ
tB∂∂
−r
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Radar Systems Course 12Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Outline
• Introduction
• Maxwell’s Equations– Displacement Current– Continuity Equation– Boundary Equations
• Electromagnetic Waves
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Radar Systems Course 13Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Electromagnetism (Pre Maxwell)
• Surprise! These formulae are inconsistent!
→→→→
→→→
→→
→→
→→
→→
μ=ε=
⋅=⋅
⋅∂∂
−=⋅
=⋅
ρ=⋅
∫
∫ ∫∫
∫∫
∫∫∫∫∫
HBED
SdJsdH
SdtBsdE
0SdB
dVSdD
r
Gauss’s Law
Magnetic ChargesDo Not Exist
Ampere’s Law
Faradays’s Law
JtDH
tBE
0B
D
r+
∂∂
=×∇
∂∂
−=×∇
=⋅∇
ρ=⋅∇
→→→
→→→
→
→
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Radar Systems Course 14Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
The “Pre-Maxwell Equations” Inconsistency
• Inconsistency comes about because a well known property of vectors:
• Apply this to Faraday’s law
• The left side is equal to 0, because of the above noted property of vectors
• The right side is 0, because
• If you do the same operation to Ampere’s law …..Trouble..
0)Ax( =∇⋅∇rrr
( )Btt
B)Ex(rr
rrrrr
⋅∇∂∂
−=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂−
⋅∇=∇⋅∇
0B =⋅∇rr
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Radar Systems Course 15Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
“How Displacement Current Came to Be”
• The left side is 0; but the right side is not, generally 0• If one applies Gauss’s law and the continuity equation:
• The above equation become:
• So Maxwell’s Equations become consistent, if we rewrite Ampere’s law as:
• A changing electric field induces an magnetic field
o
J)Hx(μ⋅∇
=∇⋅∇rr
rrr
0t
J =∂ρ∂
+⋅∇rr
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
ε⋅−∇=⋅∇ε∂∂
−=∂ρ∂
−=⋅∇tEE
ttJ oo
rrrrr
tDJHx∂∂
+=∇r
rrr Displacement current
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Radar Systems Course 16Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Review -
Electromagnetism
James Clerk Maxwell
Maxwell’s EquationsIntegral Form
Differential Form
→→
→→
→→→
→
→
+∂∂
=×∇
∂∂
−=×∇
=⋅∇
ρπ=⋅∇
JtDH
tBE
0B
4D
Plane Wave SolutionNo SourcesVacuumNon-Conducting Medium
( )
( ) )jwtrk(j
)jwtrk(j
eBt,rB
eEt,rE
−⋅→
−⋅→
→→
→→
=
=
o
o
r
r
→→→→
→→→
→→
→→
→→
→→
→→
μ=ε=
⋅⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+
∂∂
=⋅
⋅∂∂
−=⋅
=⋅
ρ=⋅
∫∫∫
∫ ∫∫
∫∫
∫∫∫∫∫
HBED
SdJtDsdH
SdtBsdE
0SdB
dVSdD
Electric FieldMagnetic Field
x
y
z
λ
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Radar Systems Course 17Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Boundary Equations
• In the limit, when the side surfaces approach 0, Gauss’s law reduces to:
• And from
• The scalar form of these equations is
∫∫∫∫∫ ρ=⋅→→
dVSdD
n̂ 1nD
2nD
s2n1n DD σ=−
s21 )DD(n̂ σ=−⋅rr
0SdB =⋅→→
∫∫0)BB(n̂ 21 =−⋅
rr
0BB 2n1n =−
1nD is the normal component of at the top of the pillbox
Dr
Medium 2
Medium 1
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Radar Systems Course 18Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Boundary Equations (continued)
n̂1tH
2tHMedium 2
Medium 1
1tH is the tangential component of at the top of the pillbox
Hr • In the limit, when the sides of the
rectangle approach 0, Ampere’s law reduces to:
• And from Faraday’s law
• The scalar form of these equations is
s21 J)HH(xn̂rrr
=−
0)EE(xn̂ 21 =−rr
0EE
JHH
2t1t
S2t1t
=−
=−r
At the Surface of a Perfect Conductor
0Bn̂JHxn̂
Dn̂0Exn̂
s
s
=⋅=
σ=⋅=rrr
rr
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Radar Systems Course 19Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Outline
• Introduction
• Maxwell’s Equations
• Electromagnetic Waves– How they are generated– Free Space Propagation– Near Field / Far Field– Polarization– Propagation
Waveguides Coaxial Transmission Lines
– Miscellaneous Stuff
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Radar Systems Course 20Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Radiation of Electromagnetic Waves
• Radiation is created by a time-varying current, or an acceleration (or deceleration) of charge
• Two examples:– An oscillating electric dipole
Two electric charges, of opposite sign, whose separation oscillates accordingly:
– An oscillating magnetic dipole A loop of wire, which is driven by an oscillating current of the
form:
• Either of these two methods are examples of ways to generate electromagnetic waves
tsindx 0 ω=
tsinI)t(I 0 ω=
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Radar Systems Course 21Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Radiation from an Oscillating Electric Dipole
Illustration of propagation and detachment of electric field lines from the dipole
Two charges in simple harmonic motion
T21t =
T41t =
T83t =0t =
T81t =
+
-
+
-
+
-
+
-
+-
0 current
0 currentMaximumcurrent
Wave frontexpands
Field linesbreak from dipole
Wave front expands and forms closed loop
Wave front continuesto expand
Wave frontinitiates
= Period of dipole oscillationT
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Radar Systems Course 22Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
MATLAB Movies for Visualization of Antenna Radiation with Time
• Generated via Finite Difference Time Domain (FDTD) solution– We will study this method in a later lecture
• Two Cases:– Single dipole / harmonic source
– Two dipoles / harmonic sources
Electric charges are needed to create an electromagnetic wave,but are not required to sustain it
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Radar Systems Course 23Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Dipole Radiation in Free Space
Horizontal Distance (m)
Dipole*
Vert
ical
Dis
tanc
e (m
)
*driven by oscillating
source0 0.5 1 1.5 2
-1
0
-0.5
0.5
1
Courtesy of MIT Lincoln LaboratoryUsed with Permission
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Radar Systems Course 24Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Two Antennas Radiating
Horizontal Distance (m)
Vert
ical
Dis
tanc
e (m
)
0 0.5 1 1.5 2-1
0
-0.5
0.5
1
Dipole1*
Dipole2*
*driven by oscillatingsources
(in phase)Courtesy of MIT Lincoln LaboratoryUsed with Permission
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Radar Systems Course 25Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Electromagnetic Waves
Radar Frequencies
Courtesy Berkeley National Laboratory
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Radar Systems Course 26Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Why Microwaves for Radar
The microwave region of the electromagnetic spectrum (~3 MHZ to ~ 10 GHZ) is bounded by:
– One region ( > 10 GHz) with very heavy attenuation by the gaseous components of the atmosphere (except for windows at 35 & 95 GHz)
– The other region (< 3 MHz), whose frequency implies antennas too
large for most practical applications
Tran
smis
sion
(%)
0
100
50
No TransmissionThrough
Ionosphere
WindowFor
Radio andMicrowaves
Three Infrared Windows
Visible TransmissionThrough Atmosphere
HeavyAtmosphericAttenuation
Wavelength
Frequency
1 µm 1 mm 1 m 1 km
1 THz 1 GHz 1 MHz
Transmission vs. Wavelength
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Radar Systems Course 27Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Electromagnetic Wave Properties and Generation / Calculation
• A radiated
electromagnetic wave consists of electric and magnetic fields which jointly satisfy Maxwell’s Equations
• EM wave is derived by integrating source currents on antenna / target– Electric currents on metal– Magnetic currents on apertures (transverse electric fields)
• Source currents can be modeled and calculated– Distributions are often assumed for simple geometries– Numerical techniques are used for more rigorous solutions
(e.g. Method of Moments, Finite Difference-Time Domain Methods)
Electric Currenton Wire Dipole
Electric Field Distribution(~ Magnetic Current) in Slot
a
b
y
x
z
λ/4
a/2
a/2
λ/2λ
3λ/22λ
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Radar Systems Course 28Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Antenna and Radar Cross Section Analyses Use “Phasor Representation”
Harmonic Time Variation is assumed : tje ω
[ ]tje)z,y,x(E~alRe)t;z,y,x(E ω=r
α= je)z,y,x(E~ê)z,y,x(E~
)t(cos)z,y,x(E~ê)t;z,y,x(E α+ω=rInstantaneous
Harmonic Field is :
Calculate Phasor :
Any Time Variation can be Expressed as aSuperposition of Harmonic Solutions by Fourier Analysis
InstantaneousElectric Field
Phasor
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Radar Systems Course 29Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Field Regions
• All power is radiated out • Radiated wave is a plane wave• Far-field EM wave properties
– Polarization– Antenna Gain (Directivity)– Antenna Pattern– Target Radar Cross Section
(RCS)
• Energy is stored in vicinity of antenna• Near-field antenna Issues
– Input impedance– Mutual coupling
λ< 3D 62.0R
Reactive Near-Field Region Far-field (Fraunhofer) Region
λ> 2D2R
Far-Field (Fraunhofer)Region
Equiphase Wave Fronts
Plane WavePropagatesRadially Out
D
R
Reactive Near-FieldRegion
Radiating Near-Field(Fresnel) Region
r̂Er
Hr
Courtesy of MIT Lincoln LaboratoryUsed with Permission
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Radar Systems Course 30Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Far-Field EM Wave Properties
re),(E),,r(E
jkroff
−
φθ≅φθrr
ffjkr
off Er̂1r
e),(H),,r(Hrrr
×η
=φθ≅φθ−
Ω=εμ
≡η 377o
o
λπ= 2k
where is the intrinsic impedance of free space
is the wave propagation constant
Standard Spherical
Coordinate System
θ̂
φ̂
r̂
x
z
yφ
rθ
Electric FieldMagnetic Field
x
zλ
• In the far-field, a spherical wave can be approximated by a plane wave
• There are no radial field components in the far field• The electric and magnetic fields are given by:
y
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Radar Systems Course 31Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Polarization of Electromagnetic Wave
• Defined by behavior of the electric field vector as it propagates in time as observed along the direction of radiation
• Circular used for weather mitigation• Horizontal used in long range air search to obtain reinforcement
of direct radiation by ground reflection
φE
r̂
φE
θE φE
θE
– Linear–Vertical or Horizontal
–CircularTwo components are equal in amplitude,and separated in phase by 90 degRight-hand (RHCP) is CW aboveLeft-hand (LHCP) is CCW above
– Elliptical
Major AxisMinor Axis
Courtesy of MIT Lincoln LaboratoryUsed with Permission
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Radar Systems Course 32Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Polarization
rE
HorizontalLinear
(with respectto Earth)
• Defined by behavior of the electric field vector as it propagates in time
(For air surveillance looking upward)r
E
(For over-water surveillance)
VerticalLinear
(with respectto Earth)
ElectromagneticWave Electric Field
Magnetic Field
θ
φ
r
Courtesy of MIT Lincoln LaboratoryUsed with Permission
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Radar Systems Course 33Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Circular Polarization (CP)
• Electric field components are equal in amplitude, separated in phase by 90 deg• “Handed-ness”
is defined by observation of electric field along propagation direction
• Used for discrimination, polarization diversity, rain mitigation
Propagation DirectionInto Paper
θ
φ
θ
φ
-A 0 A
-A
0
A
( ) cos( )E t A wtθ =( ) sin( )E t A wtφ =
E Aθ =2jE Ae πφ
−=
Phasors Instantaneous
Right-Hand(RHCP)
Left-Hand(LHCP)
Electric FieldCourtesy of
MIT Lincoln LaboratoryUsed with Permission
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Radar Systems Course 34Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Propagation –
Free Space
• Plane wave, free space solution to Maxwell’s Equations:– No Sources– Vacuum– Non-conducting medium
• Most electromagnetic waves are generated from localized sources and expand into free space as spherical wave.
• In the far field, when the distance from the source great, they are well approximated by plane waves when they impinge upon a target and scatter energy back to the radar
( )
( ) )trk(j
)trk(j
eBt,rB
eEt,rE
ω−⋅→
ω−⋅→
→→
→→
=
=
o
o
r
r
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Radar Systems Course 35Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Pointing Vector –
Physical Significance
• The Poynting Vector,
, is defined as:
• It is the power density (power per unit area) carried by an electromagnetic wave
• Since both and are functions of time, the average power density
is of greater interest, and is given by:
• For a plane wave in a lossless medium
HxESrrr
≡
Sr
Hr
Er
( ) AV* WHxERe21S ≡=
rrr
2E
21S
rr
η=
o
o
εμ
=ηwhere
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Radar Systems Course 36Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Modes of Transmission For Electromagnetic Waves
• Transverse electromagnetic (TEM) mode– Magnetic and electric field vectors are transverse
(perpendicular) to the direction of propagation, , and perpendicular to each other
– Examples (coaxial transmission line and free space transmission,
– TEM transmission lines have two parallel surfaces
• Transverse electric (TE) mode– Electric field, , perpendicular to – No electric field in direction
• Transverse electric (TM) mode– Magnetic field, , perpendicular to– No magnetic field in direction
• Hybrid transmission modes
k̂
k̂
k̂k̂
Er
k̂Hr
k̂
Er
Hr
TEM Mode
Used forRectangularWaveguides
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Radar Systems Course 37Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Guided Transmission of Microwave Electromagnetic Waves
• Coaxial Cable (TEM mode)– Used mostly for lower power and in low
frequency portion of microwave portion of spectrum
Smaller cross section of coaxial cable more prone to breakdown in the dielectric
Dielectric losses increase with increased frequency
• Waveguide (TE or TM mode)– Metal waveguide used for High power radar
transmission From high power amplifier in transmitter to
the antenna feed– Rectangular waveguide is most prevalent
geometry
Coaxial Cable
Rectangular Waveguide
Courtesy of Tkgd 2007Courtesy of Tkgd 2007
Courtesy of Courtesy of Cobham Sensor Systems.Cobham Sensor Systems.Used with permission.Used with permission.
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Radar Systems Course 38Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
How Is the Size of Radar Targets Characterized ?
• If the incident electric field that impinges upon a target is known and the scattered electric field is measured, then the “radar cross section”
(effective area) of the target may by calculated.
2
I
2
S
R E
Elim r
r
∞→=σ=
RadarCross
Section
By MIT OCW
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Radar Systems Course 39Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Units-
dB vs. Scientific Notation
Signal-to-noise ratio (dB) = 10 log 10Signal PowerNoise Power
ScientificFactor of:
Notation
dB10
101
10100
102
201000
103
30...1,000,000
106
60
Example:
The relative value of two quantities (in power units), measured on a logarithmic scale, is often expressed in deciBel’s (dB)
0 dB
=
factor of 1
-10 dB
=
factor of 1/10-20 dB
=
factor of 1/100
3 dB
=
factor of 2-3 dB
= factor of 1/2
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Radar Systems Course 40Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Summary
• This lecture has presented a very brief review of those electromagnetism topics that will be used in this radar course
• It is not meant to replace a one term course on advanced undergraduate electromagnetism that physics and electrical engineering students normally take in their 3rd
year of undergraduate studies
• Viewers of the course may verify (or brush up on) their skills in the area by doing the suggested review problems in Griffith’s (see reference 1) and / or Ulaby’s (see reference 2) textbooks
-
Radar Systems Course 41Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Acknowledgements
• Prof. Kent Chamberlin, ECE Department, University of New Hampshire
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Radar Systems Course 42Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
References
1. Griffiths, D. J., Introduction to Electrodynamics, Prentice Hall, New Jersey, 1999
2. Ulaby. F. T., Fundamentals of Applied Electromagnetics, Prentice Hall, New Jersey,5th
Ed., 2007 3. Skolnik, M., Introduction to Radar Systems,McGraw-Hill,
New York, 3rd
Ed., 20014. Jackson, J. D., Classical Electrodynamics, Wiley, New
Jersey, 19995. Balanis, C. A., Advanced Engineering Electromagnetics,
Wiley, New Jersey, 19896. Pozar, D. M., Microwave Engineering, Wiley, New York, 3rd
Ed., 2005
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Radar Systems Course 43Review E & M 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Homework Problems
• Griffiths (Reference 1)– Problems 7-34, 7-35, 7-38, 7-39, 9-9, 9-10, 9-11, 9-33
• Ulaby (Reference 2)– Problems 7-1, 7-2, 7-10, 7-11, 7-25, 7-26
• It is important that persons, who view these lectures, be knowledgeable in vector calculus and phasor notation. This next problem will verify that knowledge
– Problem-
Take Maxwell’s Equations and the continuity equation, in integral form, and, using vector calculus theorems, transform these two sets of equations to the differential form and then transform Maxwell’s equations from the differential form to their phasor form.
Radar Systems Engineering�Lecture 2�Review of ElectromagnetismReasons for Review LectureOutlineCoulomb’s LawElectric FieldElectric FieldGauss’s LawBiot Savart LawMagnetic Flux and the Absence of Magnetic ChargesAmperes LawFaraday’s LawOutlineElectromagnetism (Pre Maxwell)The “Pre-Maxwell Equations” Inconsistency“How Displacement Current Came to Be”Review - ElectromagnetismBoundary EquationsBoundary Equations (continued)OutlineRadiation of Electromagnetic WavesRadiation from �an Oscillating Electric DipoleMATLAB Movies for Visualization�of Antenna Radiation with TimeDipole Radiation in Free SpaceTwo Antennas RadiatingElectromagnetic WavesWhy Microwaves for RadarElectromagnetic Wave Properties and Generation / CalculationAntenna and Radar Cross Section Analyses Use “Phasor Representation”Field RegionsFar-Field EM Wave PropertiesPolarization of Electromagnetic WavePolarizationCircular Polarization (CP)Propagation – Free SpacePointing Vector – Physical SignificanceModes of Transmission For Electromagnetic WavesGuided Transmission of Microwave Electromagnetic WavesHow Is the Size of Radar Targets Characterized ? Units- dB vs. Scientific NotationSummaryAcknowledgementsReferencesHomework Problems