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Computational Electromagnetics :

Summary of Integral Equation Methods

Uday Khankhoje

Electrical Engineering, IIT Madras

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Topics in this module

1 Surface v/s Volume Integral Approach

2 Finding the Radar Cross-Section (RCS)

3 Computational Considerations

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Table of Contents

1 Surface v/s Volume Integral Approach

2 Finding the Radar Cross-Section (RCS)

3 Computational Considerations

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Quick aside: Surface Integral Equations and PECs

How do we deal with scatters that are made of perfect electric conductors?

Recall boundary conditions

The original system of equations:∮[g1(r, r

′)∇φ(r) · n̂− φ(r)∇g1(r, r′) · n̂] dl = φi(r′), r′ ∈ V2∮

[g2(r, r′)∇φ(r) · n̂− φ(r)∇g2(r, r′) · n̂] dl = 0, r′ ∈ V1

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Surface v/s Volume Integral Equations

Surface approach: Volume approach:

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Surface v/s Volume Integral Equations

Surface approach:

φ(r′) = φi(r′) +

∮S [φ(r)∇g1(r, r

′)− g1(r, r′)∇φ(r)] · n̂ dl

Volume approach:

φ(r′) = φi(r′) + k20

∫V2g1(r, r

′)[εr(r)− 1]φ(r) dr

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Table of Contents

1 Surface v/s Volume Integral Approach

2 Finding the Radar Cross-Section (RCS)

3 Computational Considerations

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Definition of Radar Cross-Section (RCS): σ

σTM(θ, θi) =

limr→0

2πr |E2z (r,θ|2

|Eiz(0,0)|2

σTM(θ, φ, θi, φi) =

limr→0

4πr2 |E2z (r,θ,φ|2

|Eiz(0,0,0)|2

Mono-static and Bi-static

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Approximations in the RCS

An integral involving Green’s function : φ(r′) = φi(r′) + k20

∫V2g1(r, r

′)χ(r)φ(r) dr

In 2D:g(r, r′) = −j4 H

(2)0 (k|r − r′|)

for x� 1, H(2)0 (kρ) ≈

√2jπkρ exp(−jkρ)

In 3D: g(r, r′) = 14π|r−r′| exp(−jk|r − r

′|)

Note: RCS independent of r

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Table of Contents

1 Surface v/s Volume Integral Approach

2 Finding the Radar Cross-Section (RCS)

3 Computational Considerations

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Computational Considerations

• How fine do you discretize?

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Topics that were covered in this module

1 Surface v/s Volume Integral Approach

2 Finding the Radar Cross-Section (RCS)

3 Computational Considerations

Reference: