This document discusses various methods for calculating radar cross section (RCS), including the finite difference time domain method, method of moments, geometrical optics, physical optics, geometrical theory of diffraction, and physical theory of diffraction. It provides overviews and comparisons of each method, explaining their approaches and areas of applicability. The document also includes examples of RCS calculations and summaries of key points about specific methods.
RADAR - RAdio Detection And Ranging
This is the Part 1 of 2 of RADAR Introduction.
For comments please contact me at solo.hermelin@gmail.com.
For more presentation on different subjects visit my website at http://www.solohermelin.com.
Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
RADAR - RAdio Detection And Ranging
This is the Part 2 of 2 of RADAR Introduction.
For comments please contact me at solo.hermelin@gmail.com.
For more presentation on different subjects visit my website at http://www.solohermelin.com.
Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
Stealth Technology is totally based on the EM wave propagation and reflection.RADAR technology plays a major role in military and others.For an object to be discovered in the RADAR it must be able to reflect back the EM waves that hit it.So stealth technology is totally based on this concept.
This presentation is also useful for STEALTH TECHNOLOGY.This presentation inckludes the methods of reducing the area of an object in the RADAR i.e.,to disappear the object by not visible in the RADAR
SEMINAR CREDITS:
SANKOJU YASHWANTH
ELINT Interception and Analysis course samplerJim Jenkins
The course covers methods to intercept radar and other non-communication signals and a then how to analyze the signals to determine their functions and capabilities. Practical exercises illustrate the principles involved.
RADAR - RAdio Detection And Ranging
This is the Part 1 of 2 of RADAR Introduction.
For comments please contact me at solo.hermelin@gmail.com.
For more presentation on different subjects visit my website at http://www.solohermelin.com.
Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
RADAR - RAdio Detection And Ranging
This is the Part 2 of 2 of RADAR Introduction.
For comments please contact me at solo.hermelin@gmail.com.
For more presentation on different subjects visit my website at http://www.solohermelin.com.
Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
Stealth Technology is totally based on the EM wave propagation and reflection.RADAR technology plays a major role in military and others.For an object to be discovered in the RADAR it must be able to reflect back the EM waves that hit it.So stealth technology is totally based on this concept.
This presentation is also useful for STEALTH TECHNOLOGY.This presentation inckludes the methods of reducing the area of an object in the RADAR i.e.,to disappear the object by not visible in the RADAR
SEMINAR CREDITS:
SANKOJU YASHWANTH
ELINT Interception and Analysis course samplerJim Jenkins
The course covers methods to intercept radar and other non-communication signals and a then how to analyze the signals to determine their functions and capabilities. Practical exercises illustrate the principles involved.
Describes Pulse Compression in Radar Systems.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
Since some figures were not downloaded, I recommend to see this presentation on my website under RADAR Folder, Signal Processing subfolder.
Tutorial Content
This tutorial provides a broad-based discussion of radar system, covering the following topics:
-Introduction to Radars in Military and Commercial Applications
-Radar System Block Diagram
-Radar Antennas (slotted waveguide array, planar array), Transmitter (magnetron, solid-state), Receiver, Pedestal and Radome
-Plot Extraction, Tracking Algorithms and Display
-Radar Range Equation, Detection Performance
-Wave Propagation and Radar Cross Section
-Emerging and Advanced Radar Systems (phased-array, multi-beam, multi-mode, FMCW, solid-state)
In the discussion, practical systems, technical specifications and data will be used to enhance learning.In addition, simulation results will also be used to present findings.
The objective of the tutorial session is to equip participants with solid understanding of radar systems for system level applications and prepare them for advanced and professional radar courses, projects and research.
This tutorial is designed and developed based on the following references:
[1] G. W. Stimson, Introduction to Airborne Radar Second Edition, Scitech Publishing, 1998.
[2] L. V. Blake, A Guide to Basic Pulse-Radar Maximum-Range Calculation, NRL Report 6930, 1969.
[3] K. H. Lee, Radar Systems for Nanyang Technological University, TBSS, 2014.
Applied Physical Oceanography And ModelingJim Jenkins
This three-day course is designed for engineers, physicists, acousticians, climate scientists, and managers who wish to enhance their understanding of this discipline or become familiar with how the ocean environment can affect their individual applications. Examples of remote sensing of the ocean, in situ ocean observing systems and actual examples from recent oceanographic cruises are given.
The students will be able to access educational Java applets to visualize waves and key acoustic phenomena: Click here to view
Other web-based resources include acoustic demonstration podcasts and iPod apps to conduct acoustic measurements. The student will also be armed with Internet resources for up-to-date information on sonar systems, undersea sound propagation models, and environmental databases. The student will leave with a clear understanding of how the ocean influences undersea sound propagation and scattering.
A digital revisitation_of_analog_beamforming_techniques - aiaaicssc2013_lisiMarco Lisi
"A Digital Revisitation of Analog Beam Forming Techniques", by P. Angeletti and M.Lisi: presentation at the 19th Ka and 31st AIAA ICSSC Joint Conference, Florence, October 2013
CEM Workshop Lectures (4/11): CEM of High Frequency MethodsAbhishek Jain
Above lecture can be downloaded from www.zeusnumerix.com
Computational Electromagnetics Workshop Lecture 4 of 11. The lecture focuses on the CEM methods used for high-frequency incident radiation. The methods explained are Geometric Optics and Physical Optics. PO-PTD and GO-GTD methods are mainly used for large objects where time-domain methods will be very expensive. Mathematical modeling, pitfalls and modifications to these methods are discussed.
Initial study and implementation of the convolutional Perfectly Matched Layer...Arthur Weglein
In this report, first steps and results of the implementation of the Convolutional Perfectly
Matched Layer (CPML), for the modeling of the 2D acoustic heterogeneous wave equation
are presented. We also compare the conditions to set to zero, for all angles of incidence, the
reflection coefficient at the interface between two PML media, with the analogous conditions
for the reflection coefficient at an interface between two acoustic media. A side product of the
present work for the M-OSRP is a code to create synthetic data, using Finite-Difference (FD)
methods with PML BCs.
We also provide a short description of the main stages involved in the original Reverse Time
Migration (RTM) algorithm, with focus on the 2D acoustic heterogeneous wave equation. We
include a derivation of the equations of the CPML for the backward propagation of the data,
which is part of the RTM. As far as the authors knowledge, these equations and derivations
have not been reported in the literature. The reason we include the RTM is because the present
report can be considered part of a broader research project whose objective is to compare the
RTM with PML BCs with the Green’s theorem based RTM, developed within the M-OSRP.
Initial study and implementation of the convolutional Perfectly Matched Layer...Arthur Weglein
In this report, first steps and results of the implementation of the Convolutional Perfectly
Matched Layer (CPML), for the modeling of the 2D acoustic heterogeneous wave equation
are presented. We also compare the conditions to set to zero, for all angles of incidence, the
reflection coefficient at the interface between two PML media, with the analogous conditions
for the reflection coefficient at an interface between two acoustic media. A side product of the
present work for the M-OSRP is a code to create synthetic data, using Finite-Difference (FD)
methods with PML BCs.
We also provide a short description of the main stages involved in the original Reverse Time
Migration (RTM) algorithm, with focus on the 2D acoustic heterogeneous wave equation. We
include a derivation of the equations of the CPML for the backward propagation of the data,
which is part of the RTM. As far as the authors knowledge, these equations and derivations
have not been reported in the literature. The reason we include the RTM is because the present
report can be considered part of a broader research project whose objective is to compare the
RTM with PML BCs with the Green’s theorem based RTM, developed within the M-OSRP.
Radiation patterns account of a circular microstrip antenna loaded two annularwailGodaymi1
In this paper, theoretical study of circular microstrip antenna loaded two annular (CMSAL2AR) and calculation
of the radiation pattern using principle equivalence with moment of method formulation of electromagnetic
radiation in this these based on the bodies of revolution (BoR), which are generated by revolution a planar curve
about an axis called axis of symmetry to solving the electric fields integral equation (EFIE) and magnetic field
integral equation (MFIE). To find an unknown electric current density on the conductor surface ,and both
unknowns electric and magnetic density current on the dielectric surface which are responsible for the
generation of far fields radiation in the space for the components (Eθ ,Eφ) ,the surface currents was represented
by a set of basis functions that give the Fourier series because the body has a circular symmetry property and
then select a set of weighted functions to find a linear system by using Galerkin method which requires that the
weighted functions are equal to the complex conjugate of the current ( ) * W = J .from radiation pattern
calculated the Directive gain can be utilized to the directive gain increased to (G= 21.30 dB) when
( 0.015λ 1 = g R ) for the ratio of (Rab= 5.5), and bandwidth has been better (BW%= 19.9%) when
( 0.01λ 1 = g R ) for the ratio (Rab= 6.5) .
Investigation of repeated blasts at Aitik mine using waveform cross correlationIvan Kitov
We present results of signal detection from repeated events at the Aitik and Kiruna mines in Sweden as based on waveform cross correlation. Several advanced methods based on tensor Singular Value Decomposition is applied to waveforms measured at seismic array ARCES, which consists of three-component sensors.
Time Domain Modelling of Optical Add-drop filter based on Microcavity Ring Re...iosrjce
IOSR Journal of Electronics and Communication Engineering(IOSR-JECE) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of electronics and communication engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in electronics and communication engineering. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Design of a Selective Filter based on 2D Photonic Crystals Materials IJECEIAES
Two dimensional finite differences temporal domain (2D-FDTD) numerical simulations are performed in cartesian coordinate system to determine the dispersion diagrams of transverse electric (TE) of a two-dimension photonic crystal (PC) with triangular lattice. The aim of this work is to design a filter with maximum spectral response close to the frequency 1.55 μm. To achieve this frequency, selective filters PC are formed by combination of three waveguides W 1 K A wherein the air holes have of different normalized radii respectively r 1 /a=0.44, r 2 /a=0.288 and r /a= 0.3292 (a: is the periodicity of the lattice with value 0.48 μm). Best response is obtained when we insert three small cylindrical cavities (with normalized radius of 0.17) between the two half-planes of photonic crystal strong lateral confinement.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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Radar 2009 a 7 radar cross section 2
1. IEEE New Hampshire Section
Radar Systems Course 1
Radar Cross Section 1/1/2010 IEEE AES Society
Radar Systems Engineering
Lecture 7 Part 2
Radar Cross Section
Dr. Robert M. O’Donnell
IEEE New Hampshire Section
Guest Lecturer
2. Radar Systems Course 2
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
Physical Optics
(PO)
Currents Approximated by Tangent
Plane Method
Physical Theory of
Diffraction (PTD)
Physical Optics with Added Edge
Current Contribution
Geometrical Optics
(GO)
Current Contribution Assumed to Vanish
Except at Isolated Specular Points
Geometrical Theory of
Diffraction (GTD)
Geometrical Optics with Added Edge
Current Contribution
3. Radar Systems Course 3
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Electromagnetic Scattering
Incident
Plane Wave
Scattered Field
(Radiated by Induced Currents)
• Two step process to determine scattered fields
– Determine induced surface currents
– Calculate field radiated by currents
Induced
Surface
Currents
Courtesy of MIT Lincoln Laboratory
Used with permission
4. Radar Systems Course 4
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Method of Moments (MoM) Overview
• The Method of Moments calculations predict the exact
solution for the target RCS
• Method – Solve integral form of Maxwell’s Equations
– Generate a surface patch model for the target
– Transform the integral equation form of Maxwell’s equations into a
set of homogeneous linear equations
– The solution gives the surface current densities on the target
– The scattered electric field can then be calculated in a straight
forward manner from these current densities
– Knowledge of the scattered electric field then allows one to readily
calculate the radar cross section
• Significant limitations of this method
– Inversion of the matrix to solve the homogeneous linear
equations
– Matrix size can be very large at high frequencies
Patch size typically ~λ/10
Surface Patch
Model
For a Sphere
5. Radar Systems Course 5
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Standard Spherical Coordinate System
x
y
z
θ
φ
)z,y,x(
r
6. Radar Systems Course 6
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Spherical Coordinate System for MOM
Calculations
• Source currents distributed over surface
• Field observation point located at
• Point on surface is
S′
x
y
z
θ
θ′
φ
φ′
)z,y,x( ′′′
)z,y,x(
Target Surface S′
r′
R
r
rrR
rrr
′−=
S′ )z,y,x( ′′′
)z,y,x(
7. Radar Systems Course 7
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Method of Moments
• Maxwell’s Equations transform to the Stratton and Chu
Equations using the vector Green’s Theorem and yield:
• Free space Green’s function is an spherical wave falling of
as:
• Also, note:
( ) ( ) ( )[ ]
( ) ( ) ( )[ ]
rrR
R4
e
SdHnˆxHxnˆExnˆiH
SdEnˆxExnˆHxnˆiE
ikR
'S
S
'S
S
′−==⎥
⎦
⎤
⎢
⎣
⎡
π
=ψ
′ψ∇⋅−ψ∇−ψεω+=
′ψ∇⋅+ψ∇+ψμω+=
+
∫∫
∫∫
rrrrrr
rrrrrr
Free Space
Green’s Function
SI
SI
HHH
EEE
rrr
rrr
+=
+=
R/1
8. Radar Systems Course 8
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Method of Moments (continued once)
• On the surface of the perfectly conducting target these
equations become:
– Total tangential electric field zero at surface
– No magnetic sources of currents or charges as source of
scattered fields
• Electric Field Integral Equation (EFIE)
• Magnetic Field Integral Equation (MFIE)
• Causes of scattered fields
– Scattered electric field – electric currents and charges
– Scattered magnetic field – electric currents
∫∫∫∫ ′′
′ψ∇=′ψ∇=
SS
S SdxJSdx)Hxnˆ(H
rrr
[ ] Sd
1
JiSd)Enˆ()Hxnˆ(iE
SS
S
′⎥
⎦
⎤
⎢
⎣
⎡
ψ∇ρ
ε
+ψμω+=′ψ∇⋅+ψμω+= ∫∫∫∫ ′′
rrr
9. Radar Systems Course 9
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Method of Moments (continued twice)
• Applying the boundary conditions for Maxwell’s Equations
and the Continuity Equation to free space yields:
• Procedure to calculate the scattered electric field:
– Convert the integral equation into a set of algebraic equations
– Solve for induced current density using matrix algebra
– With the current density known, the calculation of the
scattered electric field, , is reasonably straightforward and
the cross section can be calculated:
SdxJxnˆ
2
J
Hxnˆ
SdJ
i
JixnˆExnˆExnˆ
S
I
S
SI
′ψ∇−=
′⎥
⎦
⎤
⎢
⎣
⎡
ψ∇⋅∇
εω
+
+ψμω+=−=
∫∫
∫∫
′
′
r
r
r
rrrr
S
E
r
2I
2S
2
E
E
R4 π=σ
10. Radar Systems Course 10
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Method of Moments (continued again)
• Break up the target into a set of N discrete patches
– 7 to 10 patches per wavelength
• Expand the surface current density as a set of known
basis functions
• Define the “Magnetic Field Operator”, , as
• Insert the series expansion of currents and bringing
the sum out of the operator, we get:
Surface Patch Model
For Sphere∑=
=
N
1n
nn )r(BI)r(J
rrrr
∑=
==
N
1n
I
nHnH Hxnˆ))r(B(LI)J(L
rrrr
∫∫′
′ψ∇−≡
S
H SdxJxnˆ
2
J
)J(L
r
r
r
)J(LH
r
11. Radar Systems Course 11
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Method of Moments (one last time)
• Multiply by the weighting vector, , and integrating over
the surface:
– Point Testing
– Galerkin’s Method
• This is a set of N equations in N unknowns (current
coefficients, ) of the form:
• The only difficulty is inversion of a very large matrix
( )[ ] 0dSSd))r(B(LWiIdSHxnˆ)r(W n
S
m
'S
N
1n
n
S
I
=′⋅μω−⋅ ∫∫∫∫∑∫∫ =
rrrrr
mW
r
VIZ
vrrr
=
)r(BW mm
rrr
=
)rr(W mm
rrr
−δ=
mI
VZI 1
vrrr −
=
N...,3,2,1m =
12. Radar Systems Course 12
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Monostatic RCS of a Square Plate
• 15 cm x 15 cm Plate 6.0 GHz HH Polarization
Aspect Angle (degrees)
-90 -60 -30 0 30 60 90
RadarCrossSection(dBsm)
-30
-20
-10
0
10
20
Measurement
13. Radar Systems Course 13
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Monostatic RCS of a Square Plate
• 15 cm x 15 cm Plate 6.0 GHz HH Polarization
Aspect Angle (degrees)
-90 -60 -30 0 30 60 90
RadarCrossSection(dBsm)
-30
-20
-10
0
10
20
Measurement
Method of Moments
Patch
Size
λ/4
N = 12
14. Radar Systems Course 14
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Surface Patch Model of JGAM for
Method of Moments RCS Calculation
• 1.0 GHz 1350 unknowns
Top View
Side View
Photo of JGAM on Pylon
Courtesy of MIT Lincoln Laboratory
Used with Permission
15. Radar Systems Course 15
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Summary - Method of Moments
• Method of moments solution is exact
– Patch size must be small enough
– 7 to 10 samples per wavelength
• Well suited for small targets at long wavelengths
– Example - Artillery shell at L-Band (23 cm)
• Aircraft size targets result in extremely large matrices to be
inverted
– JGAM (~ 5m length)
1350 unknowns at 1.0 GHz
– Typical Fighter aircraft (~ 5m length)
A very difficult computation problem at S-Band (10 cm
wavelength)
16. Radar Systems Course 16
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Comparison of MoM and FD-TD Techniques
• For Single Frequency RCS Predictions (perfect conductors)
• 2-Dimensional Calculation
• 3-Dimensional Calculation
Method of Moments
(MoM)
Finite Difference-
Time Domain (FD-TD)
Method of
Calculation
Integral Equation
Frequency Domain
Differential Equation
Time Domain
No. of
Unknowns
N (2-D) N2
(3-D) N2
(2-D) N3
(3-D)
Memory
Requirement
Matrix Decomposition
N3
(2-D) N6
(3-D)
Time Steps
N3
(2-D) N4
(3-D)
Computer
Time
N2
(2-D) N4
(3-D) N2
(2-D) N3
(3-D)
Accuracy Exact Exact
17. Radar Systems Course 17
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
18. Radar Systems Course 18
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Geometrical Optics (GO) - Overview
• Geometrical Optics (GO) is an approximate method for RCS
calculation
– Valid in the “optical” region (target size >> λ)
• Based upon ray tracing from the radar to “specular points” on the
surface of the target
– “Specular points” are those points, whose normal vector points back to
the radar.
• The amount of reflected energy depends on the principal radii of
curvature at the surface reflection point
1ρ
2ρ
21 ρρπ=σ
nˆ
• Geometrical optics (GO) RCS calculations
are reasonably accurate to 10 – 15% for
radii of curvature of 2 λ to 3λ
• The GO approximation breaks down for
flat plates, cylinders and other objects
that have infinite radii of curvature; and at
edges of these targets
19. Radar Systems Course 19
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Geometric Optics
• Power Density Ratio =
• Radar Cross Section of Sphere =
• Radar Cross Section of an Arbitrary Specular Point =
– Where radii of curvature at specular point =
Ω
Ω
===
dR
d
4
a
A
A
A
1
A
1
S
S
2
2
S
I
I
S
INC
SCAT
r
r
2
2
2
2
INC
SCAT2
a
R4
a
R4
S
S
R4 π=π=π
21 ρρπ
2,1 ρρ
a2/aΩd
Image
Point
Incident Rays
Incident
Rays
Specular
Point
20. Radar Systems Course 20
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Single and Double Reflections
• RCS Calculation for Single Reflection
– Identify all specular points and add contributions
– Phase calculated from distance to and from specular point
– Local radii of curvature used to determine amplitude of
backscatter
• RCS Calculation for Double Reflection
– Identify all pairs of specular points
– At each reflection use single reflection methodology to
calculate amplitude and phase
(Geometrical Optics Method)
Perfectly
Conducting
Sphere
Perfectly
Conducting
Sphere
Two Modes of Double Reflection
Single Reflection
Single Reflection
21. Radar Systems Course 21
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
22. Radar Systems Course 22
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Physical Optics (PO) Overview
• Physical Optics (PO) is an approximate method for RCS
calculation
– Valid in the “optical” region (target size >> λ)
• Method - Physical Optics (PO) calculation
– Modify the Stratton-Chu integral equation form of Maxwell’s Equations,
assuming that the target is in the far field
– Assume that the total fields, at any point, on the surface of the target are
those that would be there if the target were flat
Called “Tangent plane approximation”
– Assume perfectly conducting target
– Resulting equation for the scattered electric field may be readily calculated
– RCS is easily calculated from the scattered electric field
• Physical Optics RCS calculations:
– Give excellent results for normal (or nearly normal) incidence (< 30°)
– Poor results for shallow grazing angles and near surface edges
e.g. leading and trailing edges of wings or edges of flat plates
23. Radar Systems Course 23
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Physical Optics
• For an incident plane wave :
• Substituting this surface current yields (for the monostatic case)
Tangent Plane Approximation
( ) ( ) rdeHxnˆxrˆxrˆ
r4
e
i2rE rrˆki2
o
ikr
S
′
π
ωμ−= ′⋅−
∫
rrrr r
( ) rrˆki
oS eHxnˆ2rJ
rrrr ′⋅−
=′
· · ·· ·· ·· ·
··
Infinite Perfectly Conducting Plane
(Exact Solution)
Arbitrary Conducting Surface
(Approximate Solution)
·
·
·
Tangent Plane
oE
r oE
r
oH
r
oH
r
24. Radar Systems Course 24
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Normal and Oblique Incidence
• Physical Optics
contribution adds
constructively (in phase)
• For large plates, the edge
contribution is a small
part of the total current
• Except near the edges,
Physical Optics gives
accurate results
Normally Incident
Plane Wave
Obliquely Incident
Plane Wave
Edge
Current
Physical Optics
Current
Physical Optics Valid Perfectly Conducting
Plate
• Except near the edges, Physical
Optics gives accurate results
• Fresnel Zones of alternating
phase caused by phase delay
across plate
• In the backscatter direction, the
Physical Optics contribution is
predominantly cancelled
• The most significant part of total
current due to edge effects
Perfectly Conducting
Plate
Specular Scattering
Direction
Edge
Current
Physical Optics
Current
Fresnel
Zones
25. Radar Systems Course 25
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Monostatic RCS of a Square Plate
• 15 cm x 15 cm Plate 10.0 GHz HH Polarization
Aspect Angle (degrees)
-90 -60 -30 0 30 60 90
RadarCrossSection(dBsm)
-30
-20
-10
0
10
20
Measurement
Physical Optics (PO)
Approximation
2
2
MAX
A
4
λ
π=σ
26. Radar Systems Course 26
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
27. Radar Systems Course 27
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Geometrical Theory of Diffraction (GTD)
Overview
• Geometrical Theory of Diffraction (GTD) a ray tracing method of
calculating the diffracted fields at surface edges / discontinuities
– Assumption: When ray impinges on an edge, a cone (see Keller (1957)
Cone below) of diffracted rays are generated
– Half angle of cone is equal to the angle, , between the edge and the
incident ray.
In backscatter case the cone becomes a disk
– Diffracted electric field proportional to “diffraction coefficients”, and
and a “divergence factor, , and given by:
• Diffraction coefficients
– – when parallel to edge
– + when parallel to edge
• Divergence factor reduces amplitude
as rays diverge from scattering point
and accounts for curves edges
Conducting
Wedge
Edge Diffracted Rays
On
Keller Cone
IE
r
( )YX
ks2sin
ee
E
4/iiks
DIF m
r
πβ
Γ
=
π
YX
Γ
β
IE
r
IH
r
28. Radar Systems Course 28
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Geometrical Theory of Diffraction (GTD)
Ray Tracing (With Creeping Waves and Diffraction)
• Advantages
– Easy to Understand
– Multiple Interactions
• Disadvantages
– Implementation difficult for complex targets
– Requires more accurate description than PTD
COREDGEREFLSCAT EEEE
rrrr
++=CWDIFFSRERSCAT EEEEE
rrrrr
+++=
Cylinder Plate
Creeping Wave
Side Reflection
Endcap ReflectionDiffraction
Edge
Diffraction
Corner
Diffraction
Reflection
29. Radar Systems Course 29
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
30. Radar Systems Course 30
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Physical Theory of Diffraction (PTD)
Overview
• Approach: Integrate surface current obtained from local tangent
plane approximation (plus edge current)
• Advantages: Reduced computational requirements and applicable
to arbitrary complex geometries
• Disadvantages: Neglects multiple interactions or shadowing
IE
r
IE
r
IE
r
Diffraction
Specular
Reflection
POJ
r
nˆ
DIFPO JJJ
rr
+=
Edge Current
Tangent
Plane
Courtesy of MIT Lincoln Laboratory
Used with permission
31. Radar Systems Course 31
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Physical Theory of Diffraction (PTD)
• In 1896, Sommerfeld developed a method to find the total
scattered field for an the infinite, perfectly conducting wedge.
• In 1957, Ufimtsev obtained the edge current contributions by
subtracting the physical optics contributions from the total
scattered field.
• The current for finite length structures may be obtained by
truncating the edge current from that of the infinite structure
Uniform Current
(Physical Optics)
Infinite Perfectly
Conducting
Wedge
Non-Uniform
Edge Current
Incident Plane WaveScattered
Field
Scattered
Field
32. Radar Systems Course 32
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Normal and Oblique Diffraction
Diffraction
Perpendicular to Edge
Oblique Diffraction
Keller Cone
Incident
Ray
Perfectly Conducting
Wedge
Perfectly Conducting
Wedge
Scattering
Perpendicular
to Edge
• Constructive addition from edge
current contribution along entire
edge results in strong
perpendicular backscatter
• Small contribution from corner
edge current
• Perpendicular to edge, scattering
is strong in all directions
• Edge current contribution
interferes destructively in direction
of backscatter
• For near grazing angles, corner
current may be significant
• Strong scattering along “Keller
Cone”
Edge CurrentEdge Current
Corner
Current
Corner
Current
Incident
Ray
33. Radar Systems Course 33
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Trailing / Leading Edge Diffraction
• Tangential component of
electric field equals zero
along the conductor.
• Diffracted electric field is
produced by current
induced to cancel
incident electric field.
• No diffraction at back
edge because electric
field is close to zero.
• Negligible scattering at
front edge – Electric field
normal and continuous
• Traveling waves; above
and below plate develop
a relative phase delay.
• Required continuity of
electric field at back edge
causes induced edge
current, and thus a
diffracted electric field.
Trailing Edge Diffraction
Leading Edge Diffraction
0Exn =
r
Incident
Electric
Field
iE
r
Diffracted Field
Induced
Edge
Current
0E ≈
r
iE
r
Induced Edge
Current
Diffracted
Field
Surface Field
Out of Phase
34. Radar Systems Course 34
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
FD-TD Simulation of Scattering by Strip
0.5 m
Ey
4 m
• Gaussian pulse plane wave incidence
• E-field polarization (Ey plotted)
15 deg
• Phenomena: leading edge diffraction
Case 2
Courtesy of MIT Lincoln Laboratory
Used with permission
35. Radar Systems Course 35
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
FD-TD Simulation of Scattering by Strip
Case 2
Courtesy of MIT Lincoln Laboratory
Used with permission
36. Radar Systems Course 36
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
FD-TD Simulation of Scattering by Strip
0.5 m
Hy
4 m
• Gaussian pulse plane wave incidence
• H-field polarization (Hy plotted)
15 deg
• Phenomena: trailing edge diffractionCase 3
Courtesy of MIT Lincoln Laboratory
Used with permission
37. Radar Systems Course 37
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
FD-TD Simulation of Scattering by Strip
Case 3
Courtesy of MIT Lincoln Laboratory
Used with permission
38. Radar Systems Course 38
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Monostatic RCS of a Square Plate
• 15 cm x 15 cm Plate 10.0 GHz HH Polarization
Aspect Angle (degrees)
-90 -60 -30 0 30 60 90
RadarCrossSection(dBsm)
-30
-20
-10
0
10
20
Measurement
Physical Optics (PO)
Approximation
Physical Theory
Of Diffraction (PTD)
Courtesy of MIT Lincoln Laboratory
Used with permission
39. Radar Systems Course 39
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Measured and Predicted RCS of JGAM
–180 –150 –120 –90 –60 –30 0 30 60 90 120 150 180
Target Aspect Angle (deg)
RCS(dBsm)
30
20
10
0
–10
–20
–30
–40
End Cap
Fuselage Specular
Cone Specular
Wing Leading Edge
Wing Trailing Edge
Tail TailNose BroadsideBroadside
• VV polarization
• Elevation = 7°
• 9.67 GHz
RATSCAT Measurement
PTD Prediction
Johnson Generic Aircraft Model (JGAM) at
RATSCAT Outdoor Measurement Facility
Courtesy of MIT Lincoln Laboratory
Used with permission
40. Radar Systems Course 40
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Radar Cross Section Calculation Methods
• Introduction
– A look at the few simple problems
• RCS prediction
– Exact Techniques
Finite Difference- Finite Time Technique (FD-FT)
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
41. Radar Systems Course 41
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
RCS Prediction Techniques Family Tree
Exact Techniques Approximate Techniques
• Limited Geometry
• All Phenomena
• Limited Phenomena
• Computationally Speedy
• Valid for High Frequencies
Classical
Solutions
Hybrid
Methods
Numerical
Methods•Few Geometries
•Rigorous, Exact •Computationally Slow
•Low Frequency
Surface
Integral
Techniques
Ray
Tracing
Techniques
•Computationally Slow
•All Geometries
•Computationally Slow
•All Geometries
Differential
Equation
Solutions
Hybrid
Methods
Integral Equation
Techniques
Series Solutions
MoM / UTD
MoM / PO
MoM / GO
Physical Optics (PO)
Physical Theory
of Diffraction (PTD)
Geometrical Optics (GO)
Geometrical Theory
of Diffraction (GTD)
Universal Theory
of Diffraction (UTD)
Shooting and Bouncing
Rays (SBR)
Finite Element
Finite Difference-Time Domain (FT-TD)
Finite Difference-Frequency Domain (FD-FD)
Method of Moments (MoM)
Other Integral Techniques
42. Radar Systems Course 42
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Comparison of Different RCS
Calculation Techniques
Methods of Calculation
FT-TD MOM GO - GTD PO-PTD
Calculation
Of
Current
Exact
Solve Partial
Differential
Equation
Exact
(Solve Integral
Equation)
Specular Point
Reflections
(Edge Currents)
Tangent Plane
Approximation
(Edge Currents)
Physical
Phenomena
Considered
All All Ray Tracing
Reflections
(Single & Double)
Diffraction
Main
Computational
Requirement
Time
Stepping
Matrix
Inversion
Multiple
Reflection
Diffraction
Surface Integration
-
Shadowing
Advantages
Exact
Visualization Aids
Physical Insight
Exact
- Simple
Formulation
- Good Insight
into Physical
Phenomena
Easiest Computationally
- Good Insight into Physical
Phenomena
Limitations
And/or
Disadvantages
- Low Frequency
Only
- Complex
Geometries Difficult
- Single Incident
Angle
- Low Frequency
Only
- Formulation
Difficult
(Materials)
- Single Frequency
- High Frequency
Only
- Canonical
Geometries Only
- Caustics
- High Frequency Only
- Many
Phenomena Neglected
43. Radar Systems Course 43
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Corner Reflectors
• Give a large reflection, , over a wide range of angles
– Used as test targets and for radar calibration
• Different shapes
– Dihedral
– Trihedral
Square, triangular, and circular
σ
Ray Trace for a
Dihedral Corner Reflector
(Side view)
=EFA Area of projected aperture
On the incident ray
2
2
EFA4
λ
π
=σ
RCS of Dihedral Corner Reflector
(Broadside Incidence)
Physical Optics Model
Sailboat Based
Circular Trihedral Corner Reflector
Courtesy of dalydaly
44. Radar Systems Course 44
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Summary
• Target RCS depends on its characteristics and the radar
parameters
– Target : size, shape, material, orientation
– Radar : frequency, polarization, range, viewing angles, etc
• The target RCS is due to many different scattering centers
– Structural, Propulsion, and Avionics
• Many RCS calculation tools are available
– Take into account the many different electromagnetic scattering
mechanisms present
• Measurements and predictions are synergistic
– Measurements anchor predictions
– Predictions validate measurements
45. Radar Systems Course 45
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
References
1. Atkins, R., Radar Cross Section Tutorial, 1999 IEEE National
Radar Conference, 22 April 1999.
2. Skolnik, M., Introduction to Radar Systems, New York,
McGraw-Hill, 3rd Edition, 2001.
3. Skolnik, M., Radar Handbook, New York, NY, McGraw-Hill,
3rd Edition, 2008 (Chapter 14 authored by E. Knott)
4. Ruck, et al., Radar Cross Section Handbook, Plenum Press,
New York, 1970, 2 vols.
5. Knott et al., Radar Cross Section, Massachusetts, Artech
House, Norwood, MA, 1993.
6. Bhattacharyya, A. K. and Sengupta, D. L., Radar Cross
Section Analysis and Control, Artech House, Norwood, MA,
1991.
7. Levanon, N., Radar Principles, Wiley, New York, 1988
46. Radar Systems Course 46
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Acknowledgement
• Dr. Robert T-I. Shin
• Dr. Robert K. Atkins
• Dr. Hsiu C. Han
• Dr. Audrey J. Dumanian
• Dr. Seth D. Kosowsky
47. Radar Systems Course 47
Radar Cross Section 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Homework Problems
• From Skolnik (Reference 2)
– Problems 2-10, 2-11, 2-12, and 2-13
• From Levanon (Reference 6)
– Problems 2-1 and 2-5
• For an ellipsoid of revolution, (semi major axis, a ,aligned
with the x-axis, semi minor axis, b, aligned with the y axis,
and axis of rotation is the x-axis; what are the radar cross
sections (far field) looking down the x, y, and z axes, if the
radar has wavelength λ and a >> λ and b >> λ?
• Extra credit: Solve the last problem assuming a << λ and b
<< λ.