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Radar 2009 a 7 radar cross section 2

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.

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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
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
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
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
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
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(
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
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
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
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
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 =
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.

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Radar 2009 a 7 radar cross section 2

  • 1.
    IEEE New HampshireSection 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 Course2 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 Course3 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 Course4 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 Course5 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 Course6 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 Course7 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 Course8 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 Course9 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 Course10 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 Course11 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 Course12 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 Course13 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 Course14 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 Course15 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 Course16 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 Course17 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 Course18 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 Course19 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 Course20 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 Course21 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 Course22 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 Course23 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 Course24 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 Course25 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 Course26 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 Course27 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 Course28 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 Course29 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 Course30 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 Course31 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
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    Radar Systems Course32 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 Course33 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 Course34 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 Course35 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 Course36 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 Course37 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 Course38 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 Course39 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 Course40 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 Course41 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 Course42 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 Course43 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 Course44 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 Course45 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 Course46 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 Course47 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.