Radar 2009 a 7 radar cross section 1

IEEE New Hampshire Section
Radar Systems Course 1
Radar Cross Section 1/1/2010 IEEE AES Society
Radar Systems Engineering
Lecture 7 – Part 1
Radar Cross Section
Dr. Robert M. O’Donnell
Guest Lecturer

Radar Cross Section 1/1/2010
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Block Diagram of Radar System
Transmitter
Waveform
Generation
Power
Amplifier
T / R
Switch
Antenna
Propagation
Medium
Target
Radar
Cross
Section
Photo Image
Courtesy of US Air Force
Used with permission.
Pulse
Compression
Receiver
Clutter Rejection
(Doppler Filtering)
A / D
Converter
General Purpose Computer
Tracking
Data
Recording
Parameter
Estimation
Detection
Signal Processor Computer
Thresholding
User Displays and Radar Control
This lecture

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Definition - Radar Cross Section (RCS or σ)
Radar Cross Section (RCS) is the hypothetical area, that would intercept the
incident power at the target, which if scattered isotropically, would produce
the same echo power at the radar, as the actual target.
Figure by MIT OCW.

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Factors Determining RCS
Figure by MIT OCW.

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Threat’s View of the Radar
Range Equation
Transmitted Pulse
Received Pulse
Target Cross Section σ
Antenna Gain G
Transmit Power PT
Figure by MIT OCW.
Distance from Radar to Target
Radar Range Equation
R
S
N
=
Pt G2 λ2 σ
(4π)3 R4 k TS Bn L
Cannot Control
Can Control
Cannot Control

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Outline
• Radar cross section (RCS) of typical targets
– Variation with frequency, type of target, etc.
• Physical scattering mechanisms and contributors to
the RCS of a target
• Prediction of a target’s radar cross section
– Measurement
– Theoretical Calculation

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Radar Cross Section of Artillery Shell
M107 Shell
for
155mm Howitzer
Aspect Angle (degrees)
0 20 40 60 80 100 120 140 160 180
RadarCrossSection(dBsm)
0
-60
-20
-30
-40
-50
-10
Typical Artillery Shell
RCS vs. Aspect Angle of an Artillery Shell
Courtesy US Marine Corps

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Radar Cross Section of Cessna 150L
S Band
V V
Polarization
Measured at RATSCAT (6585th Test Group) Holloman AFB for FAA
Courtesy of Federal Aviation Administration
0 90 180 270 360
RadarCrossSection(dBsm)
0
-20
20
40
Cessna 150L (in takeoff) Cessna 150L (in flight)
Scott Studio Photography with permissionScott Studio Photography with permission

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Aspect Angle Dependence of RCS
Cone Sphere Re-entry Vehicle (RV) Example
Figure by MIT OCW.

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Examples of Radar Cross Sections
Square meters
Conventional winged missile 0.1
Small, single engine aircraft, or jet fighter 1
Four passenger jet 2
Large fighter 6
Medium jet airliner 40
Jumbo jet 100
Helicopter 3
Small open boat 0.02
Small pleasure boat (20-30 ft) 2
Cabin cruiser (40-50 ft) 10
Ship (5,000 tons displacement, L Band) 10,000
Automobile / Small truck 100 - 200
Bicycle 2
Man 1
Birds (large -> medium) 10-2 - 10-3
Insects (locust -> fly) 10-4 - 10-5
Radar Cross Sections of Targets Span at least 50 dB
Adapted from Skolnik, Reference 2

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Outline
the RCS of a target
– Measurement

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RCS Target Contributors
• Types of RCS Contributors
– Structural (Body shape, Control surfaces, etc.)
– Avionics (Altimeter, Seeker, GPS, etc.)
– Propulsion (Engine inlets and exhausts, etc.)
Inlet
Control Surfaces
Altimeter
Seeker
Body Shape
Exhaust

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Single and Multiple Frequency RCS
Calculations with the FD-FD Technique
• RCS Calculations for a Single Frequency
– Illuminate target with incident sinusoidal wave
– Sequentially in time, update the electric and magnetic fields, until
steady state conditions are met
– The scattered wave’s amplitude and phase can the be calculated
• RCS Calculations for a Multiple Frequencies
– Illuminate target with incident Gaussian pulse
– Calculate the transient response
– Calculate to Fourier transforms of both:
Incident Gaussian pulse, and
Transient response
– RCS at multiple frequencies is calculated from the ratios of these two
quantities

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Scattering Mechanisms
for an Arbitrary Target
Tip
Diffraction at
Aircraft Nose
Diffraction at
Corner
Return
From
Engine Cavity
Edge
Diffraction
Tip
Diffraction from
Fuel Tank
Specular
Surface
Reflection
Multiple
Reflection
Wave Echo from
Traveling Wave
Gap, Seam, or
Discontinuity
Echo
Backscatter
from
Creeping Wave
Curvature
Discontinuity
Return

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Measured RCS of C-29 Aircraft Model
Full Scale C-29
BAE Hawker 125-800
RCS(dBsm)
0 60 120 180 240 300 360
20
-20
-10
0
10
-30
1/12 Scale
Model
Measurement
X-Band
HH Polarization
Waterline Cut
Wing Leading
Edge
Fuselage
Specular
Courtesy of Arpingstone
Adapted from Atkins, Reference 5
Courtesy of MIT Lincoln Laboratory

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Outline
the RCS of a target
– Measurement

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Techniques for RCS Analysis
Scaled Model Measurements
Theoretical Prediction
Full Scale Measurements
Used with Permission

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Full Scale Measurements
Target on Support
• Foam column mounting
– Dielectric properties of Styrofoam close to those of free space
• Metal pylon mounting
– Metal pylon shaped to reduce radar reflections
– Background subtraction can be used
Derived from: http://www.af.mil/shared/media/photodb/photos/050805-F-0000S-003.jpg

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Full Scale Measurement of
Johnson Generic Aircraft Model (JGAM)
RATSCAT Outdoor Measurement
Facility at Holloman AFB

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Compact Range RCS Measurement
Radar Reflectivity Laboratory (Pt. Mugu) / AFRL Compact Range (WPAFB)
Courtesy of U. S. Navy.
Target
Main
Reflector
Feed
Antenna
Plane
Wave
Low RCS
Pylon
Sub-Reflector

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Scale Model Measurement
Subscale
Measure at frequency S x F
Scale Factor
S
(Reduced Size)
Full Scale
Measure at frequency f
MQM-107 Drone in 0.29, 0.034, and 0.01 scaled sizes

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Scaling of RCS of Targets
Scale Factor
S
Length L L´ = L / S
Wavelength λ λ´ = λ / S
Frequency f f´ = S f
Time t t´ = t / S
Permittivity ε ε´ = ε
Permeability μ μ´ = μ
Conductivity g g´ = S g
Radar Cross Section σ σ´ = σ / S2
Quantity Full Scale Subscale

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Outline
the RCS of a target
– Measurement

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Radar Cross Section Calculation Methods
• Introduction
– A look at the few simple problems
• RCS prediction
– Exact Techniques
Finite Difference- Time Domain Technique (FD-TD)
Method of Moments (MOM)
– Approximate Techniques
Geometrical Optics (GO)
Physical Optics (PO)
Geometrical Theory of Diffraction (GTD)
Physical Theory of Diffraction (PTD)
• Comparison of different methodologies

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Radar Cross Section of Sphere
Rayleigh Region
λ >> a
σ = k / λ4
Mie or Resonance
Region
Oscillations
Backscattered
wave interferes
with creeping wave
Optical Region
λ << a
σ = π a2
Surface and edge
scattering occur
Circumference/ wavelength = 2πa / λ
RadarCrossSection/πa2
10
1
10-3
10-2
10-1
0.1 0.2 0.4 0.7 1 2 4 7 10 20
Rayleigh
Region
Resonance or Mie
Region
Optical
Region
a
Higher Wavelengths Lower Wavelengths
Figure by MIT OCW.
λ >> a
λ << a

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Radar Cross Section Calculation Issues
• Three regions of wavelength
Rayleigh (λ >> a)
Mie / Resonance (λ ~ a)
Optical (λ << a)
• Other simple shapes
– Examples: Cylinders, Flat Plates, Rods, Cones, Ogives
– Some amenable to relatively straightforward solutions in some
wavelength regions
• Complex targets:
– Examples: Aircraft, Missiles, Ships)
– RCS changes significantly with very small changes in frequency
and / or viewing angle
See Ref. 6 (Levanon), problem 2-1 or Ref. 2 (Skolnik) page 57
• We will spend the rest of the lecture studying the different
basic methods of calculating radar cross sections

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High Frequency RCS Approximations
(Simple Scattering Features)
Scattering Feature Orientation Approximate RCS
Corner Reflector Axis of symmetry along LOS
Flat Plate Surface perpendicular to LOS
Singly Curved Surface Surface perpendicular to LOS
Doubly Curved Surface Surface perpendicular to LOS
Straight Edge Edge perpendicular to LOS
Curved Edge Edge element perpendicular to LOS
Cone Tip Axial incidence
Where:
22
eff /A4 λπ
22
/A4 λπ
22
/A4 λπ
21 aaπ
πλ /2
2/a λ
)2/(sin42
αλ
Adapted from Knott is Skolnik Reference 3

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Radar Cross Section Calculation Issues
• Three regions of wavelength
Rayleigh (λ >> a)
Mie / Resonance (λ ~ a)
Optical (λ << a)
• Other simple shapes
– Examples: Cylinders, Flat Plates, Rods, Cones, Ogives
– Some amenable to relatively straightforward solutions in some
wavelength regions
• Complex targets:
– Examples: Aircraft, Missiles, Ships)
– RCS changes significantly with very small changes in frequency
and / or viewing angle
See Ref. 6 (Levanon), problem 2-1 or Ref. 2 (Skolnik) page 57
• We will spend the rest of the lecture studying the different
basic methods of calculating radar cross sections

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RCS Calculation - Overview
• Electromagnetism Problem
– A plane wave with electric field, , impinges on the target of
interest and some of the energy scatters back to the radar
antenna
– Since, the radar cross section is given by:
– All we need to do is use Maxwell’s Equations to calculate the
scattered electric field
– That’s easier said that done
– Before we examine in detail these different techniques, let’s
review briefly the necessary electromagnetism concepts and
formulae, in the next few viewgraphs
IE
r
2
I
2
S2
r
E
E
r4lim r
r
π=σ
∞→
SE
r

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Maxwell’s Equations
• Source free region of space:
• Free space constitutive relations:
0)t,r(B
0)t,r(D
t
)t,r(D
)t,r(H
t
)t,r(B
)t,r(E
=⋅∇
=⋅∇
∂
∂
=×∇
∂
∂
−=×∇
→
→
rr
rr
rr
rr
rr
rr
)t,r(H)t,r(B
)t,r(E)t,r(D
o
o
rrrr
rrrr
μ=
ε= = Free space permittivity
= Free space permeability
oε
oμ

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Maxwell’s Equations in Time-Harmonic Form
• Source free region:
• Time dependence
0)r(B
0)r(D
)r(Di)r(H
)r(Bi)r(E
=⋅∇
=⋅∇
ω−=×∇
ω=×∇
→
→
rr
rr
rrrr
rrrr
{ }
{ }ti
ti
e)r(HRe)t,r(H
e)r(ERe)t,r(E
ω−
ω−
=
=
rrrr
rrrr

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Boundary Conditions
• Tangential components of and are continuous:
• For surfaces that are perfect conductors:
• Radiation condition:
– As
E
r
H
r
21
21
HxnˆHxnˆ
ExnˆExnˆ
rr
rr
=
=
0Exnˆ =
r
r
1
)r(Er ∝∞→
rr
2222
1111
HE
HE
rr
rr
εμ
εμ nˆ
Surface
BoundaryMedium 1
Medium 2

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Scattering Matrix
• For a linear polarization basis
• The incident field polarization is related to the scattered field
polarization by this Scattering Matrix - S
• For and a reciprocal medium and for monostatic radar cross
section:
• For a circular polarization basis
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=⎥
⎦
⎤
⎢
⎣
⎡
=
HI
VI
HHHV
VHVV
ikr
HS
VS
S
E
E
SS
SS
r
e
E
E
E
r
2
VHVH
2
HHHH
2
VVVV
S4
S4
S4
π=σ
π=σ
π=σ
RLLLRR ,, σσσ
HVVH σ=σ
Scattering Matrix - S

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Radar Cross Section Calculation Methods
• Introduction
– A look at the few simple problems
• RCS prediction
– Exact Techniques
Finite Difference- Time Domain Technique (FD-TD)
Method of Moments (MOM)
– Approximate Techniques
Geometrical Optics (GO)
Physical Optics (PO)
Geometrical Theory of Diffraction (GTD)
Physical Theory of Diffraction (PTD)
• Comparison of different methodologies

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Methods of Radar Cross Section
Calculation
RCS Method
Approach to Determine
Surface Currents
Finite Difference-
Time Domain (FD-TD)
Solve Differential Form of Maxwell’s
Equation’s for Exact Fields
Method of Moments
(MoM)
Solve Integral Form of Maxwell’s
Equation’s for Exact Currents
Geometrical Optics
(GO)
Current Contribution Assumed to Vanish
Except at Isolated Specular Points
Physical Optics
(PO)
Currents Approximated by Tangent
Plane Method
Geometrical Theory of
Diffraction (GTD)
Geometrical Optics with Added Edge
Current Contribution
Physical Theory of
Diffraction (PTD)
Physical Optics with Added Edge
Current Contribution

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Finite Difference- Time Domain (FD-TD)
Overview
• Exact method for calculation radar cross section
• Solve differential form of Maxwell’s equations
– The change in the E field, in time, is dependent on the change in the H
field, across space, and visa versa
• The differential equations are transformed to difference equations
– These difference equations are used to sequentially calculate the E
field at one time and the use those E field calculations to calculate H
field at an incrementally greater time; etc. etc.
Called “Marching in Time”
• These time stepped E and H field calculations avoid the necessity
of solving simultaneous equations
• Good approach for structures with varying electric and magnetic
properties and for cavities

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Maxwell’s Equations in
Rectangular Coordinates
• Examine 2 D problem – no dependence:
• Equations decouple into H-field polarization and E-field
polarization
ZoXY
YoZX
XoYZ
E
t
H
y
H
x
H
t
E
x
E
z
E
t
H
z
H
y
∂
∂
ε=
∂
∂
−
∂
∂
∂
∂
μ−=
∂
∂
−
∂
∂
∂
∂
ε=
∂
∂
−
∂
∂
ZoXY
YoZX
XoYZ
H
t
E
y
E
x
E
t
H
x
H
z
H
t
E
z
E
y
∂
∂
μ−=
∂
∂
−
∂
∂
∂
∂
ε=
∂
∂
−
∂
∂
∂
∂
μ−=
∂
∂
−
∂
∂
0
y
=
∂
∂
• H-field polarization • E-field polarization
ZXY EEH ZXY HHE
y

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Maxwell’s Equations in
Rectangular Coordinates
• Examine 2 D problem – no dependence:
• Equations decouple into H-field polarization and E-field
polarization
0
y
=
∂
∂
• H-field polarization
ZXY EEH
• E-field polarization
ZXY HHE
ZoXY
YoZX
XoYZ
E
t
H
y
H
x
H
t
E
x
E
z
E
t
H
z
H
y
∂
∂
ε=
∂
∂
−
∂
∂
∂
∂
μ−=
∂
∂
−
∂
∂
∂
∂
ε=
∂
∂
−
∂
∂
ZoXY
YoZX
XoYZ
H
t
E
y
E
x
E
t
H
x
H
z
H
t
E
z
E
y
∂
∂
μ−=
∂
∂
−
∂
∂
∂
∂
ε=
∂
∂
−
∂
∂
∂
∂
μ−=
∂
∂
−
∂
∂
= 0
= 0
= 0
= 0
y

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• H-field polarization:
• Discrete form:
• Electric and magnetic fields are calculated alternately by the
marching in time method
Discrete Form of Maxwell’s Equations
( ) ( )
( )t,y,xE
x
t,y,xE
z
t,y,xH
t
Z
XYo
∂
∂
−
∂
∂
=
∂
∂
μ−
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛ Δ
+−⎟
⎠
⎞
⎜
⎝
⎛ Δ
+Δ+
Δ
−
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛ Δ
+−⎟
⎠
⎞
⎜
⎝
⎛
Δ+
Δ
+
Δ
=
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛ Δ
−
Δ
+
Δ
+−⎟
⎠
⎞
⎜
⎝
⎛ Δ
+
Δ
+
Δ
+
Δ
μ
−
o
Z
ooZo
Z
oXoZ
X
oo
X
oXoZo
X
oX
Z
T
o
Z
o
X
oY
T
o
Z
o
X
oY
T
o
t,
2
z,xEt,
2
z,xE
1
t,z,
2
xEt,z,
2
xE
1
2
t,
2
z,
2
xH
2
t,
2
z,
2
xH
z
xy
Ex
EZ
HY Yee’s
Lattice
(2-D)

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FD-TD Calculations and Absorbing
Boundary Conditions (ABC)
• Absorbing Boundary Condition (ABC) Used to Limit Computational Domain
– Reflections at exterior boundary are minimized
– Traditional ABC’s model field as outgoing wave to estimate field quantities outside
domain
– More recent perfectly matched layer (PML) model uses non-physical layer, that
absorbs waves
Ex
EZ
HY
Perfect ConductorLayer Perfectly Matched
0H
x2
1
tc
1
txc
1
y2
2
2
2
2
2
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
−
∂
∂
+
∂∂
∂
2nd Order ABC
1st Order ABC 0H
tc
1
xz2
1
y =⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
+
∂
∂
Total Field
Target
Domain of Computation
Absorbing Boundary
0ETAN =
Scattered Field

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RCS Calculations Using the FD-TD Method
• Single frequency RCS calculations
– Excite with sinusoidal incident wave
– Run computation until steady state is reached
– Calculate amplitude and phase of scattered wave
• Multiple frequency RCS calculations
– Excite with Gaussian pulse incident wave
– Calculate transient response
– Take Fourier transform of incident pulse and transient
response
– Calculate ratios of these transforms to obtain RCS at multiple
frequencies
From Atkins, Reference 5

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Description of Scattering Cases on Video
Finite Difference Time Domain (FDTD) Simulations
Ei Hi
Hi
Ei
15 deg 15 deg
Hi
Ei
Case 1 – Plate I Case 2 – Plate II Case 3 – Plate III
Case 4 – Cylinder I Case 5 – Cylinder II Case 6 – Cavity
Hi
Ei
Hi
EiEi Hi
Courtesy of MIT Lincoln Laboratory Used with Permission

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FD-TD Simulation of Scattering by Strip
0.5 m
Ey
4 m
• Gaussian pulse plane wave incidence
• E-field polarization (Ey plotted)
• Phenomena: specular reflection
Case 1
Courtesy of
MIT Lincoln Laboratory

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Case 1
Courtesy of

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FD-TD Simulation of Scattering by Strip
Case 1
Courtesy of

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FD-TD Simulation of Scattering by Cylinder
0.5 m
Hy
2 m
• Gaussian pulse plane wave incidence
• H-field polarization (Hy plotted)
• Phenomena: creeping waveCase 5
Courtesy of

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Case 5
Courtesy of

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FD-TD Simulation of Scattering by Cylinder
Case 5
Courtesy of

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Backscatter of Short Pulse from Sphere
Creeping Wave Return
Specular
Return
Radius of sphere is equal
to the radar wavelength
Figure by MIT OCW.

Radar 2009 a 7 radar cross section 1

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