Imperfections and Errors (Chapter 6) EC4630 Radar and Laser Cross Section Fall 011 Prof. D. Jenn jenn@nps.navy.mil www.nps.navy.mil/jenn AY011 1
Imperfections and Errors Imperfections and errors are deviations from the ideal assumptions. 1. Non-exact geometry: o surface finishes can be rough o surface shapes deviate from ideal whatever it might be (flat, spherical, etc.). Imperfect materials: o homogeneous constitutive parameters are usually assumed in analysis o dielectrics are specified with some nominal ( ) r,tan ε δ ± a tolerance If the errors are known exactly, the standard analysis methods can be used (MoM, FEM, FDTD, etc.) In practice, there are a large number of errors that appear to have random characteristics. Thus probabilistic methods are used. We are interested in calculating the average RCS of a target: o Usually the average is taken outside of large specular lobes o If we manufacture a large number of targets with errors and imperfections, what is the average RCS? AY011
Error Sources Some examples of manufacturing and assembly errors are shown for an array of surface tiles. Depending on the method of assembly, the errors can be independent from tile to tile, or correlated. 1. Uncorrelated errors give rise to diffuse scattering (isotropic scattering). Correlated errors give rise to periodic scattering lobes (Bragg type scattering) Tiles on poles: errors are independent Tiles on rails: errors in rows are correlated Butted tiles: errors accumulate AY011 3
Average RCS The average RCS (denoted σ or σ ave) depends on the mean scattered power pattern: * E s Es σ = 4πR E where is expected value. Two approaches to finding E s: 1. Monte Carlo simulation: method of repeated trials. Statistical analysis: assume a probability density function (PDF) for the errors and find the expected value of E s mathematically Example 6.15: A discrete array of independent scattering tiles. Assume each tile is an isotropic scatterer with location ( x,0, z ), n= 1,..., N. The tiles are randomly displaced in n the z direction by a zero mean Gaussian random variable δ with variance z θ n d AY011 4 i DISPLACED IDEAL δ N 1 3 4 N x δ
Monte Carlo Simulation The scattered field pattern (i.e., neglecting constants) is N N j kxu ( ) [( 1) sin cos ~ n + zw n j k n d θ+ δ E e e n θ = ] θ n= 1 n= 1 Assuming TM polarization, monostatic RCS, and φ = 0 plane. 1. Using a random number generator choose N random numbers δ n with the desired PDF. Compute E θ and then the resulting RCS σ 1 (the subscript 1 denotes the first trial) 3. Repeat the first two steps M 1 times, for a total of M trials 4. Average the RCS to get σ Example: N = 50, d = 0.4λ AY011 5
Statistical Analysis Scattered field average power pattern * N jkxu [ ] ~ cos n + δnw N j kx [ mu+ δmw P EE e e ] = θ θ θ n= 1 m= 1 cos N N θ n= 1m= 1 e j k( x x ) u j k( δ δ ) w e n m n m Define a new random variable that is a scaled difference of our previous random variable δ k( δ δ ) w with variance δ ( kw). Note that j j k( δn δm) w e = e = cos + j sin e, m n = 1, m= n The maximum error-free power pattern occurs when there are no errors n N N j k( xn xm) u N N max o = max n= 1m= 1 at u= 0 n= 1m= 1 P ~ e P ~ (1) N o m * AY011 6
RCS of Random Scatterers Mean power pattern P= Pe o + Ncos θ 1 e Normalize by and assume small errors Po max = + Pnorm P o norm cos N θ A ( ) Then the RCS if each scatter has effective area e A NA e 4 π A 4 A cos P π norm P θ σ = = o + norm λ λ N Comments: 1. The second term represents a nearly isotropic spatial noise that increases with the mean squared phase error. It is the average value that we are interested in.. From the top equation, as errors increase, energy is removed from P o and is spread over wide angles. 3. The first term increases as N, but the error term increases as N. AY011 7
Comparison of Monte Carlo and Closed Form Red: Monte Carlo simulation of five different linear arrays Blue: closed form expression 50 elements, d = 0.4 λ Gaussian random errors with σ = 0.03 λ Rayleigh distribution: 98 % of sildelobes are less than the average plus 6 db AY011 8
Result for a Two-dimensional Array Comparison of patterns with and without errors. Note that the peak main lobe value decreases as the error increases AY011 9
Rough Solid Surfaces Solid rough surfaces are characterized by (1) mean height of the deviations and () correlation interval Surface deviations and localized Gaussian dent approximation Illustration of large and small correlation intervals ROUGH SURFACE LARGE CORRELATION INTERVAL (CHANGES SLOWLY WITH x) δ(x) GAUSSIAN APPROXIMATION TO A SURFACE IRREGULARITY k ˆn ˆi ψ ROUGH SURFACE h SMALL CORRELATION INTERVAL (CHANGES RAPIDLY WITH x) x x AY011 10
Final Result for Continuous Rough Surface 4 π A 4k δ cos 4c π k δ e P exp c π sin θ σ = θ o + norm A λ λ c = correlation interval: distance on average that the Gaussian dents become uncorrelated δ = mean squared surface error (zero mean, Gaussian) Comments: The second term represents the increase in sidelobe level due roughness phase error. A small correlation interval results in lower more uniformly dispersed scattering than does a large c. The first term increases as while the second one increases as A. A Example of a rough surface generated by random displacement of triangle mesh nodes in the z (vertical) direction AY011 11
Sample RCS Result Results using MoM, 3λ 3λ plate Flat (no errors) With errors uniformly distributed with limits [0,0.1 λ ] AY011 1
Periodic Errors Periodic scattering arrangements result in Bragg type of scattering. Examples include: Patterns of rivet and screw heads ο Large number of small scatterers can have a large coherent RCS spike Regularly spaced ordinance and pylons ο Large individual (element) RCS ο Very large spacing = many Bragg lobes in space Surface sag between supports ο The hungry horse effect ο Large correlation interval Example of Bragg lobes due to periodic displacement errors in the z direction AY011 13