Electromagnetic Induction (EMI) Response from Conducting and Permeable Spheroidal Shells

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1 Electromagnetic Induction EMI Response from Conducting and Permeable Spheroidal Shells Benjamin E. Barrowes, Kevin O Neill, Chi O. Ao, and J. A. Kong Department of Electrical Engineering and Computer Science Massachusetts Institute of Technolog PIERS Progress In Electromagnetics Research Smposium Jul,

2 Presentation Outline Introduction Motivation Eact Solution Formulation Boundar conditions Truncated solution Results and limiting cases for small and moderate sie parameter Conclusion

EMI is promising in the detection of Uneploded

permits ignoring ground effects Spheroidal geometr

replacements Needles Can discriminate between

3 EMI is promising in the detection of Uneploded Ordnance UXO Ver low frequencies 3H-3KH Low fequenc permits ignoring ground effects Spheroidal geometr allows models of comple shapes Spheres Plates PSfrag replacements Needles Can discriminate between different shapes and compositions Introduction Motivation PSfrag replacements

4 & ' PSfrag replacements Spheroidal shell EMI response conducting and permeable shell Geometr and Formulation time-harmonic ecitation quasi-magnetostatic regime Region Region Region ' " *

5 " & " Eact Solution Regions and Epand and in terms of vector spheroidal wave functions. where primar field strength, spheroidal coordinates spheroidal sie parameter, interfocal distance, and epansion coefficients

6 " " and Eact Solution Region Epand potentials where as

7 and Solutions in Terms of Spheroidal Wavefunctions Unknowns Scalar spheroidal wave function of the kind is where spheroidal angle function spheroidal radial function spheroidal aimuthal function Sum of Associated Legendre functions Sum of Spherical Bessel functions Vector spheroidal wave functions are generated from the scalar wave function

8 . : / ', ' 9 *& 9 ',. ' / : 9 *& * ' - ' " : 4 " " 5 "4 5 " ' 1 3 " & ; " 54 5 ' - ' -, ' " " " & Boundar Conditions Outer Matching After Testing 8 " : " " boundar 6 boundar Inner Matching After Testing

9 & & & * * & * & * & * * 6 are matrices, respectivel " are matrices and & Truncated Solution Truncate infinite series at Recast into matri equations where

10 Far-Field Dipole Response Far-field epression for secondar field Aial Transverse

11 Results

12 cements Normalied EMI response Prolate Spheroid, Aial primar field elongation e=1.1 sie ratio d β /d α =.1 µ r1 =µ r =1 wavenumber ratio k 3 /k 1 =1 Re{analtic shell} Im{analtic shell} Re{analtic solid} Im{analtic solid} Re{SPA solid} Im{SPA solid} Induction number logk1 a

13 cements Normalied EMI response Prolate Spheroid, Aial primar field elongation e=1.1 sie ratio d β /d α =.1 µ r1 =µ r =1 wavenumber ratio k 3 /k 1 =.1 Re{analtic shell} Im{analtic shell} Re{analtic solid} Im{analtic solid} Re{SPA solid} Im{SPA solid} Induction number logk1 a

14 cements Normalied EMI response Prolate Spheroid, Aial primar field elongation e=1.1 sie ratio d β /d α =.1 µ r1 =µ r =1 wavenumber ratio k 3 /k 1 =.1 Re{analtic shell} Im{analtic shell} Re{analtic solid} Im{analtic solid} Re{SPA solid} Im{SPA solid} Induction number logk1 a

15 cements Normalied EMI response Prolate Spheroid, Aial primar field elongation e=3 sie ratio d β /d α =.1 µ r1 =µ r =1 wavenumber ratio k 3 /k 1 =.1 Re{analtic shell} Im{analtic shell} Re{analtic solid} Im{analtic solid} Re{SPA solid} Im{SPA solid} Induction number logk1 a

16 cements Normalied EMI response Prolate Spheroid, Aial primar field elongation e=1.1 sie ratio d β /d α =.4 µ r1 =µ r =1 wavenumber ratio k 3 /k 1 =1 Re{analtic shell} Im{analtic shell} Re{analtic solid} Im{analtic solid} Re{SPA solid} Im{SPA solid} Induction number logk1 a

17 cements Normalied EMI response Prolate Spheroid, Aial primar field elongation e=1.1 sie ratio d β /d α =.4 µ r1 =µ r =1 wavenumber ratio k 3 /k 1 =.5 Re{analtic shell} Im{analtic shell} Re{analtic solid} Im{analtic solid} Re{SPA solid} Im{SPA solid} Induction number logk1 a

18 cements Normalied EMI response Prolate Spheroid, Aial primar field elongation e=1.1 sie ratio d β /d α =.8 µ r1 =µ r =1 wavenumber ratio k 3 /k 1 =.5 Re{analtic shell} Im{analtic shell} Re{analtic solid} Im{analtic solid} Re{SPA solid} Im{SPA solid} Induction number logk1 a

19 cements Normalied EMI response Prolate Spheroid, Aial primar field elongation e=1.1 sie ratio d β /d α =.4 µ r1 =µ r =1 wavenumber ratio k 3 /k 1 =.5 Re{analtic shell} Im{analtic shell} Re{analtic solid} Im{analtic solid} Re{SPA solid} Im{SPA solid} Induction number logk1 a

20 cements Normalied EMI response Prolate Spheroid, Aial primar field elongation e=3 sie ratio d β /d α =.4 µ r1 =µ r =1 wavenumber ratio k 3 /k 1 =.5 Re{analtic shell} Im{analtic shell} Re{analtic solid} Im{analtic solid} Re{SPA solid} Im{SPA solid} Induction number logk1 a

21 Conclusion Full analtic solution for the Elecromagnetic Induction EMI response of a conducting and permeabel spheroidal shell demonstrated Spheroidal shells provide fleible model for various canonical shapes modeling UXOs including solid and hollow spheres, needles, and disks. Method becomes unstable/ill-conditioned for high large sie parameter High frequenc techniques such as the Small Perturbation Approimation SPA and Asmptotic approimations for this limitation. and ma help overcome Acknowledgement This work was supported b a grant from the Cold Regions Research Laborator and through a NSF Graduate Fellowship.

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