research article an effective math model for...

Research ArticleAn Effective Math Model for Eliminating Interior ResonanceProblems of EM Scattering

Zhang Yun-feng12 Zhou Zhong-shan12 Su Zhi-guo2

Wang Rong-zhu2 and Chen Ze-huang2

1Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters Nanjing University of Information Scienceamp Technology Nanjing 210044 China2Key Laboratory for Aerosol-Cloud-Precipitation of China Meteorological Administration Nanjing University of Information ScienceNo 219 Ningliu Road Nanjing 210044 China

Correspondence should be addressed to Zhang Yun-feng zhangyunfengnuisteducn

Received 22 May 2014 Revised 26 November 2014 Accepted 15 December 2014

Academic Editor Giancarlo Bartolucci

Copyright copy 2015 Zhang Yun-feng et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It is well-known that if an 119864-field integral equation or an119867-field integral equation is applied alone in analysis of EM scattering froma conducting body the solution to the equation will be either nonunique or unstable at the vicinity of a certain interior frequencyAn effective math model is presented here providing an easy way to deal with this situation At the interior resonant frequenciesthe surface current density is divided into two parts an induced surface current caused by the incident field and a resonance surfacecurrent associated with the interior resonance mode In this paper the presented model based on electric field integral equationand orthogonal modal theory is used here to filter out resonant mode therefore unique and stable solution will be obtained Theproposed method possesses the merits of clarity in concept and simplicity in computation A good agreement is achieved betweenthe calculated results and those obtained by other methods in both 2D and 3D EM scattering

1 Introduction

Electric field integral equation (EFIE) and magnetic fieldintegral equation (MFIE) have been widely employed toanalyze electromagnetic scattering of conducting bodies [1ndash3] However interior resonance phenomena exist in solvingelectromagnetic scattering problems with surface integralequations When working frequencies of conductors are near(or exactly at) to the frequencies associated with interiorresonances the single equation will become highly ill-conditioned (or singular) which makes the solution unstableor nonunique Also the interior resonance behavior hassignificant influence on the late time stability associated withtime domain EFIE and MFIE [4 5]

Several ways of dealing with this numerical problemhave been proposed Nowadays the popular combined fieldintegral equation (CFIE) technique to overcome this problemis a proper combination of the electric field integral equationand the magnetic field equation [6 7] The CFIE technique

requires the calculation of both 119864 and119867 impedance matricesand it is not suitable for aperture problems The combinedsource integral equation (CSIE) [8 9] techniquemakes up foraperture structures Also a technique has been proposed byMittra andKlein [10] involving application of the generalizedboundary condition [11] and consists of additional points inthe interior of the conductor and forces the field to be zeroat those points The problem with this technique is the factthat the chosen interior points must be carefully selected soas not to lie on nodal lines which is not too practical for largebodies of simple shape for which a slight change in frequencycan take us fromone resonance to the next of differentmodaldistribution There are also some iterative methods reportedin the works of Sarkar and Ergul [12 13] to deal with thissituationand they are used to compute the minimum normsolution (which produces the correct scattered fields but notthe true tangential fields) and to calculate the LQSR Anotherwork related to the use of extended integral equationshas been presented by Mautz and Harrington [8] which

Hindawi Publishing CorporationInternational Journal of Microwave Science and TechnologyVolume 2015 Article ID 724702 7 pageshttpdxdoiorg1011552015724702

2 International Journal of Microwave Science and Technology

involves application of the boundary element method withobservation points lying on an internal closed surface nearthe boundary of the scatterer Unfortunately the resultingmatrix equations are also ill-conditioned since the internalcontour can resonate by itself The authorsrsquo proposed schemeof allowing the internal contour to varywith thewave numberseems impractical and in reality does not solve the problemexcept in some isolated cases of electrically small simpleshapes whose resonances are known More recently Canning[14] illustrated a matrix algebra technique known as thesingular value decomposition (SVD) has been proposed formoment method calculations involving perfect conductorsSuch a technique diagonalizes the matrix equation isolatingthe resonant contribution which is then omitted in the calcu-lation We also refer to the interesting work of Yaghjian whooriginally presented his augmented electric or magnetic fieldequation [15] and more recently Tobin et al [16] pointed outtheir drawbacks for an arbitrarily shaped multiwavelengthbody and most importantly introduced a modification theso-called dual-surface integral equation which is applicableto the perfectly conducting bodies and supposedly eliminatesall the spurious solutions Finally the modal orthogonalcharacteristics [17 18] were applied to solve the same problemin two-dimensional EM scattering

Here we present an effective method to solve the scatter-ing problem of conductor bodies at or in the neighborhoodof the resonant frequencies in both 2D and 3D EM scatteringAt the resonant frequencies Inagaki modes firstly applied toanalyze the antenna array [19] are employed here to be vali-dated away from resonance with a unique and stable solutionto the equation It can both stabilize the numerical calculationand yield to reliable results of the surface current density andexterior field for conductors at interior resonances

2 Theory

The principle is elaborated as follows The solution of anill-conditioned system of the equation will consist of (1) thecorrect physical solution to the problem and (2) the reso-nant solution which is in conjunction with Greenrsquos Theo-rem which produces the nonzero complementary resonantmodes When the orthogonal modes are used to solve theelectric field integral equation the solutions will be dividedinto the inducedmodes and the resonant modes correspond-ing to eigenvalues We can easily obtain the current densityand exterior field from the induced modes

21 Why Is the Solution Unstable Consider a PEC scattererexcited by an incident wave and the scatterer defined by thesurface 119878 in an impressed electric field 119864

119904 (see Figure 1)According to the boundary condition we get

119899 times (119864119894

+ 119864119904

) = 0 on 119878 (1)

An operator equation for the current 119869 on 119878 is

119879 (119869) = 119899 times 119864119894

on 119878 (2)

Ei

J = Ji + Jr

Es

S

Figure 1 PEC scatterer impressed by electromagnetic field

The operator 119879 is defined by [2]

119879 (119869) = 119895120596120583119899 times int119878

[119869 (1199031015840

) +1

1198962nabla1015840

sdot 119869 (1199031015840

) nabla]119866 (119903 1199031015840

) 1198891199041015840

(3)

where 119903 denotes a field point 1199031015840 denotes a source point and120596 120583 and 119896 denote angular frequency permeability and wavenumber respectively of free space 119866(119903 1199031015840) is the free spaceGreenrsquos function

119866(119903 1199031015840

) =

exp (minus119895119896 10038161003816100381610038161003816119903 minus 119903101584010038161003816100381610038161003816)

412058710038161003816100381610038161003816119903 minus 119903101584010038161003816100381610038161003816

(4)

Equation (2) is discretized into a set of simultaneous linearalgebraic equations This set of equations can be compactlyrewritten in a matrix form as

[119878] [119869] = [119884] (5)

where [119878] is the general impedance matrix [119884] denotes thegeneral voltage vector and [119869] stands for the coefficient ofthe current density to be determined Usually (noninteriorresonant condition) the current density of the conductor canbe obtained from (5) with a unique and stable solution butat the frequencies associated with interior resonances thesituation will be on the contrary

As a result of interior resonance the current density 119869

on 119878 at discrete resonant frequencies will be composed oftwo parts an induced surface caused by the incident fieldand a resonance surface current associated with the interiorresonance mode

119869 = 119869119894+ 119869119903 (6)

where 119869119903is the resonant current density which is determined

by the conducting body alone and it is independent of theincident wave 119869

119894is the induced current density which is

jointly determined by the incident field and the conductingbody itself

If the incident119864119894 is removed the electric field119864119904 producedby the resonant current 119869

119903on 119878 satisfies the following

homogeneous 119864-field integral equation

119879 (119869119903) = 0 on 119878 (7)

International Journal of Microwave Science and Technology 3

In other words the solution to the inhomogeneous EFIEis not unique since the non-zero- solution to its correspond-ing homogeneous equation exits at some discrete frequenciesNamely in addition to the induced current 119869

119894determined

jointly by the incident field 119864119894 and the conducting body

there exists also the resonant current 119869119903at some discrete

frequencies determined alone by the conducting body itselfIt follows from the nonuniqueness of the solution to EFIE119879(119869) = 119899 times 119864

119894 that the solution to its correspondingmoment matrix equation (5) will also not be unique atthe interior resonant condition The nonuniqueness of themoment matrix equation infers that the moment matrix [119878]

is singularSimilar to the behavior of a conducting cavity the

resonant current density 119869119903on a conducting body is also

not able to produce scattered field 119864119904 in space external to

119878 Therefore the scattered field and the radar cross section(RCS) external to 119878 determined by the electric field integralequation should be unique theoretically Although resonantcurrent theoretically does not change the exterior scatteredfield it does make the method of moments analysis unstableand unreliable since the homogeneous operator equation119879(119869119903) = 0 has non-zero-solution 119869

119903

22 How to Stabilize the Solution For the sake of simplicityonly the electric field integral equation is involved hereTo solve the EFIE by method of moments first of allwe should choose a set of expansion functions and a setof weighting functions Here we choose basis functionsaiming at the diagonalization of the moment matrix whichwas proposed by Inagaki and Garbacz [19] in which theexpansion functions 119906

119899 are chosen to be the eigenfunctions

of a composite Hermitian operator 119879lowast

119879 where 119879lowast is the

adjoint operator of 119879

119879lowast

119879119906119899= 120582119899119906119899 (8)

the weighting functions to be the response of them

V119899= 119879119906119899 (9)

The orthogonal property of Inagaki modes [13] leads to thesatisfaction of the moment matrix diagonalization conditionthat is

119904119898119899

= ⟨V119898 119879 119906119899⟩ = 120582119899120575119898119899

(10)

namely

[119878] =

[[[

[

1205821

0

sdot sdot sdot

0

0

1205822

sdot sdot sdot

0

sdot sdot sdot

sdot sdot sdot

sdot sdot sdot

sdot sdot sdot

0

0

sdot sdot sdot

120582119873

]]]

]

(11)

So the modal solution to the operator equation becomes

119869 = 119880119879

[119878]minus1

119884 (12)

Here 119880 is the expansion function vector

119880 = [1199061 1199062 119906

119873] (13)

where 119884 is a column excitation vector given by

119910119899= ⟨119879119906

119899 119899 times 119864

119894

⟩ (14)

Then method of moments is employed here for computationof the modes To do so functions 119891

1 1198912 119891

119873are taken

as both expansion functions and weighting functions for amoment method analysis to the operator eigenvalue equa-tion giving

119906119899=

119873

sum

119895=1

119887(119899)

119895119891119895 119899 = 1 2 119873

119873

sum

119895=1

119887(119899)

119895119879lowast

119879119891119895= 120582119899

119873

sum

119895=1

119887(119899)

119895119891119895

119873

sum

119895=1

119887(119899)

119895⟨119891119894 119879lowast

119879119891119895⟩ = 120582

119899

119873

sum

119895=1

119887(119899)

119895⟨119891119894 119891119895⟩

(15)

whose matrix form is

[119876] 119899= 120582119899[119875] 119899 119899 = 1 2 119873 (16)

In thismatrix eigenvalue equation both [119875] and [119876] are119873times119873

matrices with their elements respectively as

119901119894119895= ⟨119891119894 119891119895⟩

119902119894119895= ⟨119891119894 119879lowast

119879119891119895⟩ = ⟨119879119891

119894 119879119891119895⟩

(17)

And 119899is a column vector composed of the unknown

coefficients 119887(119899)119895

as

119899= [119887(119899)

1 119887(119899)

2 119887

(119899)

119873]119879

(18)

Thematrix [119876] is simply themomentmethodmatrix [119878] onlyif a least-squaresmethod ofmoments is applied to the generaloperator equation in which 119891

1 1198912 119891

119873are taken to be the

expansion functions and their responses 1198791198911 1198791198912 119879119891

119873

the weighting functionsAt the interior resonances the surface current is com-

posed of the resonant current and the induced currentThe resonant current is the solution to the correspondinghomogeneous 119864-field integral equation It follows from thenonuniqueness of the solution to 119864-field integral equationthat the solution to its corresponding moment matrix equa-tion will also not be unique at the interior resonance con-dition The nonuniqueness of the moment matrix equationinfers that the moment matrix [119878] is singular

Actually the moment matrix will not be exactly singularbut rather it will be highly ill-conditioned since the round-off and the truncated error exist during the computation ofthe matrix elements According to the orthogonal propertyof Inagaki modes [13] the surface current of conductors iscomposed of serials of modal currents corresponding to theeigenvalue 120582

119899

119869 =

119873

sum

119899=1

119869119899=

119873

sum

119899=1

⟨119879119906119899 119899 times 119864

119894

⟩119906119899

120582119899

(19)


At the interior resonances when eigenvalue 120582119898is much

smaller than the other eigenvalue the corresponding modalcurrent is resonant current and the other is the inducedcurrent These modal currents are mutually orthogonal soto eliminate the interior resonant mode in the Inagaki modesolution what we need to do is just to replace the number1120582119898in [119878]minus1 by zeroThe incident current 119869

119894can be acquired

from the following equation

119869119894=

119873

sum

119899=1

⟨119879119906119899 119899 times 119864

119894

⟩119906119899

120582119899

minus

⟨119879119906119898 119899 times 119864

119894

⟩119906119898

120582119898

(20)

Theoretically the scattered field and the radar crosssection external to 119878 determined by the electric field integralequation should be unique However due to the ill-conditionof the moment matrix the exterior fields from conductorswill be unstable and unreliable After filtering out the reso-nant current the exterior field obtained from the inducedcurrent is stable and reliable

In fact the resonance current does not really exist on thesurface of the conductors The interior resonance problemis just caused by the deficiency of the selected mathematicalmodel After getting rid of the virtual resonance current wecan obtain the stable and reliable current density and exteriorfield of conductors at interior resonances

3 Numerical Results

Here we confine our attention to not only two-dimensionalstructures but also three-dimensional structures The pre-sented method is applied here to get the surface currentdensity and exterior field of conductors when they are at (ornear) the interior resonant frequencies

31 An Infinitely Long Circular Cylinder As the first casefor testing the approach described above scattering from acircular conducting cylinder was examined here It was foundthat when the TMplanewave normally illuminates an infinitecircular cylinder the interior resonance takesplace if 119896119886 =

38214 where 119896 is the wave number and 119886 is the radius whichis the numerical resonant frequency point of E

11mode of the

same surface circular cylinder cavityFrom Figures 2 and 3 we can see that the magnitude

of induced current is much smaller than that of resonantcurrent so the interior resonant current has completelymasked the true current responsible for the scattered fieldThe true surface current computed by the presentedmethodagrees well with the exact analytical solutionThe bistatic RCSof the circular cylinder computed from the induced currentis compared with the exact solution It is clear that there areexcellent agreements between the calculated results and theanalytic results as it is shown in Figure 4

32 An Infinitely Long Square Cylinder Also an infinitelylong square conducting cylinder was considered here witha TM wave incident along the axis of the cylinder The firstand the second resonant frequency of the square cylinder arerespectively at 119889120582 = radic22 and at 119889120582 = radic52 theoretically

A ADCB

2a

A

D

C

B

90

80

70

60

50

40

30

20

10

0

|JzH

i |

Ei

k

Figure 2 Resonant current on a circular cylinder 119896119886 = 38214

A

D

C

B

Ei

k

2

18

16

14

12

1

08

06

04

02

0

Presented methodAnalytical results

|JzH

i |

A ADCB

2a

Figure 3 Induced current on a circular cylinder 119896119886 = 38214

We can get the first numerical resonant point at 119889120582 =

070775 using the presented method as shown in Figure 5Here only the first resonant frequency is considered At thefirst resonant frequency the resonant current and inducedcurrent are shown respectively in Figures 6 and 7 FromFigure 7 we can see that the induced current is comparedwellwith that obtained by CFIE technique

In Figure 8 the bistatic RCS of the square cylinderobtained through three different methods is depicted Dueto the ill-conditioned equation the RCS of square cylindercomputed by single EFIE (dash line) obviates the true resultAfter we filtered out the resonant modal current the RCS(dot-dash line) produced by the induced current coincideswith that obtained by CFIE method (solid line)

33 Two Conducting Spheres In the end two conductingspheres respectively at the 119896119886 = 2768 and 119896119886 = 4518 areanalyzed by the presentedmethodwhere 119886denotes the radius



120590(d

B)

12

10

8

6

4

2

00

50 100 150 200 250 300 350

Ei

k

2a

120601 (deg)

120601 = 0∘

Figure 4 Bistatic RCS of a circular cylinder 119896119886 = 38214

A

D C

B

Mag

nitu

de o

f min

imal

eige

nval

ue

160

140

120

100

80

60

40

20

020

161

d

070775

d120582

Ei

k

Figure 5 Minimal eigenvalue of square cylinder

A ADCB

140

120

100

80

60

40

20

0

A

D C

B

d

Ei

k

|JzH

i |

Figure 6 Resonant current on a square cylinder 119889120582 = 070775

A

D C

B

d

Ei

k

Presented methodCFIE

A ADCB

|JzH

i |

4

35

3

25

2

15

1

05

0

Figure 7 Induced current on a square cylinder 119889120582 = 070775

10

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

A

D C

B

d

Ei

k

120590(d

B)

minus2

minus4

minus6

minus8

120601 (deg)

120601 = 0∘

Figure 8 Bistatic RCS on square cylinder 119889120582 = 070775 obtainedby presented method (labeled IM) CFIE method and EFIE

of the sphere At the interior resonant frequency the bistaticRCS obtained by the single EFIE method (dash line) is awayfrom the right value After being corrected by the presentedmethod (dot-dash line) the bistatic RCS coincides with thatcalculated by CFIE technique (solid line) as shown in Figures9 and 10

4 Conclusion

A new scheme for eliminating interior resonance problemsassociated with surface integral equation is presented Theorthogonal property of Inagaki modes is used here to isolatethe resonantmode which is then omitted in the computationto obtain the right property of the conductors at the interior


8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

120590120601120601120582

2(d

B)

120579 (deg)

(a)

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

minus6

minus8

1205822

(dB)

120579 (deg)

120590120579120579

(b)

Figure 9 Bistatic RCS of a resonant sphere 119896119886 = 2768 (a) 120601120601-polarization (b) 120579120579-polarization obtained by presented method (labeled IM)CFIE method and EFIE

18

16

14

12

10

8

6

4

2

00 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

120590120601120601120582

2(d

B)

120579 (deg)

(a)

20

15

10

5

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus10

minus5

1205822

(dB)

120579 (deg)

120590120579120579

(b)


resonances This simple technique has been proven to beeffective in attenuating the resonant modes and gettingthe unique and stable solution to EFIE when conductorsare at (or near) the frequencies associated with interiorresonances Excellent numerical results away from resonanceproblems have been obtained for some shapes not only intwo dimensions but also in three dimensions Comparedwithother techniques the advantages of this approach are onlyinvolved EFIE equation and easy to get the right results but

due to the determination of eigenvalues and eigenvectorsthere is a bit time-consuming for property computation at orin the interior resonance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


References

[1] R F Harrington Field Computation by Moment MethodsMacmillan New York NY USA 1968

[2] R F Harrington Time Harmonic Electromagnetic FieldMcGraw-Hill New York NY USA 1968

[3] W C Chew and J M Song ldquoGedanken experiments tounderstand the internal resonance problems of electromagneticscatteringrdquo Electromagnetics vol 27 no 8 pp 457ndash471 2007

[4] H Lianrong G Tian J Fang and G Xiao ldquoThe behavior ofMFIE and EFIE at interior resonances and its impact in MOTlate time stabilityrdquo American Journal of Electromagnetics andApplications vol 1 no 2 pp 30ndash37 2013

[5] F P Andriulli K Cools F Olyslager and E Michielssen ldquoTimedomain Calderon Identities and their application to the integralequation analysis of scattering by PEC objects part II stabilityrdquoIEEE Transactions on Antennas and Propagation vol 57 no 8pp 2365ndash2375 2009

[6] J R Mautz and R F Harrington ldquoH-field E-field andcombined-field solutions for conducting bodies of revolutionrdquoAEU vol 32 pp 157ndash164 1978

[7] F P Andriulli and EMichielssen ldquoA regularized combined fieldintegral equation for scattering from 2-D perfect electricallyconducting objectsrdquo IEEE Transactions on Antennas and Prop-agation vol 55 no 9 pp 2522ndash2529 2007

[8] J R Mautz and R F Harrington ldquoA combined-source solutionfor radiation and scattering from a perfectly conducting bodyrdquoIEEE Transactions on Antennas and Propagation vol 27 no 4pp 445ndash454 1979

[9] K F A Hussein ldquoEffect of internal resonance on the radar crosssection and shield effectiveness of open spherical enclosuresrdquoProgress in Electromagnetics Research vol 70 pp 225ndash246 2007

[10] RMittra andCA Klein ldquoStability and convergence ofmomentmethod solutionsrdquo in Numerical and Asymptotic Techniques inElectromagnetics vol 3 of Topics in Applied Physics pp 129ndash163Springer Berlin Germany 1975

[11] P C Waterman ldquoMatrix formulation of electromagnetic scat-teringrdquo Proceedings of the IEEE vol 53 no 8 pp 805ndash812 1965

[12] T K Sarkar and S M Rao ldquoA simple technique for solving119864-Field integral equations for conducting bodies at internalresonancesrdquo IEEE Transactions on Antennas and Propagationvol 30 no 6 pp 1250ndash1254 1982

[13] O Ergul and L Gurel ldquoEfficient solution of the electric-fieldintegral equation using the iterative LSQR algorithmrdquo IEEEAntennas and Wireless Propagation Letters vol 7 pp 36ndash392008

[14] F X Canning ldquoSingular value decomposition of integral equa-tions of EM and applications to the cavity resonance problemrdquoIEEE Transactions on Antennas and Propagation vol 37 no 9pp 1156ndash1163 1989

[15] A D Yaghjian ldquoAugmented electric and magnetic field equa-tionsrdquo Radio Science vol 16 no 6 pp 987ndash1001 1981

[16] A R Tobin A D Yaghjian and M M Bell ldquoSurface integralequations for multiwavelength arbitrarily shaped perfectlyconducting bodiesrdquo in Proceedings of the URSI National RadioScience Meeting Digest pp 12ndash15 Boulder Colo USA January1987

[17] W Cao and J Chen ldquoThe application of bi-orthogonal modeanalysis approach to in electromagnetic scattering of conduct-ing bodies at interior resonancesrdquo Chinese Journal of RadioScience vol 10 pp 16ndash22 1995

[18] Y-F Zhang C-G Jiang and W Cao ldquoInagaki mode approachto electromagnetic scattering of conducting bodies at interiorresonancesrdquo Journal of Electronics and Information Technologyvol 28 no 9 pp 1735ndash1739 2006

[19] N Inagaki and R J Garbacz ldquoEigenfunctions of compositeHermitian operator with application to discrete and continuousradiating systemsrdquo IEEE Transactions on Antennas and Propa-gation vol 30 no 4 pp 571ndash575 1982

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involves application of the boundary element method withobservation points lying on an internal closed surface nearthe boundary of the scatterer Unfortunately the resultingmatrix equations are also ill-conditioned since the internalcontour can resonate by itself The authorsrsquo proposed schemeof allowing the internal contour to varywith thewave numberseems impractical and in reality does not solve the problemexcept in some isolated cases of electrically small simpleshapes whose resonances are known More recently Canning[14] illustrated a matrix algebra technique known as thesingular value decomposition (SVD) has been proposed formoment method calculations involving perfect conductorsSuch a technique diagonalizes the matrix equation isolatingthe resonant contribution which is then omitted in the calcu-lation We also refer to the interesting work of Yaghjian whooriginally presented his augmented electric or magnetic fieldequation [15] and more recently Tobin et al [16] pointed outtheir drawbacks for an arbitrarily shaped multiwavelengthbody and most importantly introduced a modification theso-called dual-surface integral equation which is applicableto the perfectly conducting bodies and supposedly eliminatesall the spurious solutions Finally the modal orthogonalcharacteristics [17 18] were applied to solve the same problemin two-dimensional EM scattering

Here we present an effective method to solve the scatter-ing problem of conductor bodies at or in the neighborhoodof the resonant frequencies in both 2D and 3D EM scatteringAt the resonant frequencies Inagaki modes firstly applied toanalyze the antenna array [19] are employed here to be vali-dated away from resonance with a unique and stable solutionto the equation It can both stabilize the numerical calculationand yield to reliable results of the surface current density andexterior field for conductors at interior resonances

2 Theory

The principle is elaborated as follows The solution of anill-conditioned system of the equation will consist of (1) thecorrect physical solution to the problem and (2) the reso-nant solution which is in conjunction with Greenrsquos Theo-rem which produces the nonzero complementary resonantmodes When the orthogonal modes are used to solve theelectric field integral equation the solutions will be dividedinto the inducedmodes and the resonant modes correspond-ing to eigenvalues We can easily obtain the current densityand exterior field from the induced modes

21 Why Is the Solution Unstable Consider a PEC scattererexcited by an incident wave and the scatterer defined by thesurface 119878 in an impressed electric field 119864

119904 (see Figure 1)According to the boundary condition we get

119899 times (119864119894

+ 119864119904

) = 0 on 119878 (1)

An operator equation for the current 119869 on 119878 is

119879 (119869) = 119899 times 119864119894

on 119878 (2)

Ei

J = Ji + Jr

Es

S

Figure 1 PEC scatterer impressed by electromagnetic field

The operator 119879 is defined by [2]

119879 (119869) = 119895120596120583119899 times int119878

[119869 (1199031015840

) +1

1198962nabla1015840

sdot 119869 (1199031015840

) nabla]119866 (119903 1199031015840

) 1198891199041015840

(3)

where 119903 denotes a field point 1199031015840 denotes a source point and120596 120583 and 119896 denote angular frequency permeability and wavenumber respectively of free space 119866(119903 1199031015840) is the free spaceGreenrsquos function

119866(119903 1199031015840

) =

exp (minus119895119896 10038161003816100381610038161003816119903 minus 119903101584010038161003816100381610038161003816)

412058710038161003816100381610038161003816119903 minus 119903101584010038161003816100381610038161003816

(4)

Equation (2) is discretized into a set of simultaneous linearalgebraic equations This set of equations can be compactlyrewritten in a matrix form as

[119878] [119869] = [119884] (5)

where [119878] is the general impedance matrix [119884] denotes thegeneral voltage vector and [119869] stands for the coefficient ofthe current density to be determined Usually (noninteriorresonant condition) the current density of the conductor canbe obtained from (5) with a unique and stable solution butat the frequencies associated with interior resonances thesituation will be on the contrary

As a result of interior resonance the current density 119869

on 119878 at discrete resonant frequencies will be composed oftwo parts an induced surface caused by the incident fieldand a resonance surface current associated with the interiorresonance mode

119869 = 119869119894+ 119869119903 (6)

where 119869119903is the resonant current density which is determined

by the conducting body alone and it is independent of theincident wave 119869

119894is the induced current density which is

jointly determined by the incident field and the conductingbody itself

If the incident119864119894 is removed the electric field119864119904 producedby the resonant current 119869

119903on 119878 satisfies the following

homogeneous 119864-field integral equation

119879 (119869119903) = 0 on 119878 (7)



119894determined









119903






119879lowast

119879119906119899= 120582119899119906119899 (8)


V119899= 119879119906119899 (9)


119904119898119899

= ⟨V119898 119879 119906119899⟩ = 120582119899120575119898119899

(10)

namely

[119878] =

[[[

[

1205821

0

sdot sdot sdot

0

0

1205822

sdot sdot sdot

0

sdot sdot sdot

sdot sdot sdot

sdot sdot sdot

sdot sdot sdot

0

0

sdot sdot sdot

120582119873

]]]

]

(11)


119869 = 119880119879

[119878]minus1

119884 (12)


119880 = [1199061 1199062 119906

119873] (13)


119910119899= ⟨119879119906

119899 119899 times 119864

119894

⟩ (14)


1 1198912 119891

119873are taken


119906119899=

119873

sum

119895=1

119887(119899)

119895119891119895 119899 = 1 2 119873

119873

sum

119895=1

119887(119899)

119895119879lowast

119879119891119895= 120582119899

119873

sum

119895=1

119887(119899)

119895119891119895

119873

sum

119895=1

119887(119899)

119895⟨119891119894 119879lowast

119879119891119895⟩ = 120582

119899

119873

sum

119895=1

119887(119899)

119895⟨119891119894 119891119895⟩

(15)


[119876] 119899= 120582119899[119875] 119899 119899 = 1 2 119873 (16)



119901119894119895= ⟨119891119894 119891119895⟩

119902119894119895= ⟨119891119894 119879lowast

119879119891119895⟩ = ⟨119879119891

119894 119879119891119895⟩

(17)


coefficients 119887(119899)119895

as

119899= [119887(119899)

1 119887(119899)

2 119887

(119899)

119873]119879

(18)


1 1198912 119891



119873




119899

119869 =

119873

sum

119899=1

119869119899=

119873

sum

119899=1

⟨119879119906119899 119899 times 119864

119894

⟩119906119899

120582119899

(19)






119869119894=

119873

sum

119899=1

⟨119879119906119899 119899 times 119864

119894

⟩119906119899

120582119899

minus

⟨119879119906119898 119899 times 119864

119894

⟩119906119898

120582119898

(20)



3 Numerical Results




11mode of the




A ADCB

2a

A

D

C

B

90

80

70

60

50

40

30

20

10

0

|JzH

i |

Ei

k


A

D

C

B

Ei

k

2

18

16

14

12

1

08

06

04

02

0


|JzH

i |

A ADCB

2a








120590(d

B)

12

10

8

6

4

2

00

50 100 150 200 250 300 350

Ei

k

2a

120601 (deg)

120601 = 0∘


A

D C

B

Mag

nitu

de o

f min

imal

eige

nval

ue

160

140

120

100

80

60

40

20

020

161

d

070775

d120582

Ei

k


A ADCB

140

120

100

80

60

40

20

0

A

D C

B

d

Ei

k

|JzH

i |


A

D C

B

d

Ei

k


A ADCB

|JzH

i |

4

35

3

25

2

15

1

05

0


10

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

A

D C

B

d

Ei

k

120590(d

B)

minus2

minus4

minus6

minus8

120601 (deg)

120601 = 0∘



4 Conclusion



8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

120590120601120601120582

2(d

B)

120579 (deg)

(a)

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

minus6

minus8

1205822

(dB)

120579 (deg)

120590120579120579

(b)


18

16

14

12

10

8

6

4

2

00 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

120590120601120601120582

2(d

B)

120579 (deg)

(a)

20

15

10

5

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus10

minus5

1205822

(dB)

120579 (deg)

120590120579120579

(b)







References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894determined









119903






119879lowast

119879119906119899= 120582119899119906119899 (8)


V119899= 119879119906119899 (9)


119904119898119899

= ⟨V119898 119879 119906119899⟩ = 120582119899120575119898119899

(10)

namely

[119878] =

[[[

[

1205821

0

sdot sdot sdot

0

0

1205822

sdot sdot sdot

0

sdot sdot sdot

sdot sdot sdot

sdot sdot sdot

sdot sdot sdot

0

0

sdot sdot sdot

120582119873

]]]

]

(11)


119869 = 119880119879

[119878]minus1

119884 (12)


119880 = [1199061 1199062 119906

119873] (13)


119910119899= ⟨119879119906

119899 119899 times 119864

119894

⟩ (14)


1 1198912 119891

119873are taken


119906119899=

119873

sum

119895=1

119887(119899)

119895119891119895 119899 = 1 2 119873

119873

sum

119895=1

119887(119899)

119895119879lowast

119879119891119895= 120582119899

119873

sum

119895=1

119887(119899)

119895119891119895

119873

sum

119895=1

119887(119899)

119895⟨119891119894 119879lowast

119879119891119895⟩ = 120582

119899

119873

sum

119895=1

119887(119899)

119895⟨119891119894 119891119895⟩

(15)


[119876] 119899= 120582119899[119875] 119899 119899 = 1 2 119873 (16)



119901119894119895= ⟨119891119894 119891119895⟩

119902119894119895= ⟨119891119894 119879lowast

119879119891119895⟩ = ⟨119879119891

119894 119879119891119895⟩

(17)


coefficients 119887(119899)119895

as

119899= [119887(119899)

1 119887(119899)

2 119887

(119899)

119873]119879

(18)


1 1198912 119891



119873




119899

119869 =

119873

sum

119899=1

119869119899=

119873

sum

119899=1

⟨119879119906119899 119899 times 119864

119894

⟩119906119899

120582119899

(19)






119869119894=

119873

sum

119899=1

⟨119879119906119899 119899 times 119864

119894

⟩119906119899

120582119899

minus

⟨119879119906119898 119899 times 119864

119894

⟩119906119898

120582119898

(20)



3 Numerical Results




11mode of the




A ADCB

2a

A

D

C

B

90

80

70

60

50

40

30

20

10

0

|JzH

i |

Ei

k


A

D

C

B

Ei

k

2

18

16

14

12

1

08

06

04

02

0


|JzH

i |

A ADCB

2a








120590(d

B)

12

10

8

6

4

2

00

50 100 150 200 250 300 350

Ei

k

2a

120601 (deg)

120601 = 0∘


A

D C

B

Mag

nitu

de o

f min

imal

eige

nval

ue

160

140

120

100

80

60

40

20

020

161

d

070775

d120582

Ei

k


A ADCB

140

120

100

80

60

40

20

0

A

D C

B

d

Ei

k

|JzH

i |


A

D C

B

d

Ei

k


A ADCB

|JzH

i |

4

35

3

25

2

15

1

05

0


10

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

A

D C

B

d

Ei

k

120590(d

B)

minus2

minus4

minus6

minus8

120601 (deg)

120601 = 0∘



4 Conclusion



8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

120590120601120601120582

2(d

B)

120579 (deg)

(a)

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

minus6

minus8

1205822

(dB)

120579 (deg)

120590120579120579

(b)


18

16

14

12

10

8

6

4

2

00 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

120590120601120601120582

2(d

B)

120579 (deg)

(a)

20

15

10

5

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus10

minus5

1205822

(dB)

120579 (deg)

120590120579120579

(b)







References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119869119894=

119873

sum

119899=1

⟨119879119906119899 119899 times 119864

119894

⟩119906119899

120582119899

minus

⟨119879119906119898 119899 times 119864

119894

⟩119906119898

120582119898

(20)



3 Numerical Results




11mode of the




A ADCB

2a

A

D

C

B

90

80

70

60

50

40

30

20

10

0

|JzH

i |

Ei

k


A

D

C

B

Ei

k

2

18

16

14

12

1

08

06

04

02

0


|JzH

i |

A ADCB

2a








120590(d

B)

12

10

8

6

4

2

00

50 100 150 200 250 300 350

Ei

k

2a

120601 (deg)

120601 = 0∘


A

D C

B

Mag

nitu

de o

f min

imal

eige

nval

ue

160

140

120

100

80

60

40

20

020

161

d

070775

d120582

Ei

k


A ADCB

140

120

100

80

60

40

20

0

A

D C

B

d

Ei

k

|JzH

i |


A

D C

B

d

Ei

k


A ADCB

|JzH

i |

4

35

3

25

2

15

1

05

0


10

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

A

D C

B

d

Ei

k

120590(d

B)

minus2

minus4

minus6

minus8

120601 (deg)

120601 = 0∘



4 Conclusion



8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

120590120601120601120582

2(d

B)

120579 (deg)

(a)

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

minus6

minus8

1205822

(dB)

120579 (deg)

120590120579120579

(b)


18

16

14

12

10

8

6

4

2

00 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

120590120601120601120582

2(d

B)

120579 (deg)

(a)

20

15

10

5

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus10

minus5

1205822

(dB)

120579 (deg)

120590120579120579

(b)







References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120590(d

B)

12

10

8

6

4

2

00

50 100 150 200 250 300 350

Ei

k

2a

120601 (deg)

120601 = 0∘


A

D C

B

Mag

nitu

de o

f min

imal

eige

nval

ue

160

140

120

100

80

60

40

20

020

161

d

070775

d120582

Ei

k


A ADCB

140

120

100

80

60

40

20

0

A

D C

B

d

Ei

k

|JzH

i |


A

D C

B

d

Ei

k


A ADCB

|JzH

i |

4

35

3

25

2

15

1

05

0


10

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

A

D C

B

d

Ei

k

120590(d

B)

minus2

minus4

minus6

minus8

120601 (deg)

120601 = 0∘



4 Conclusion



8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

120590120601120601120582

2(d

B)

120579 (deg)

(a)

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

minus6

minus8

1205822

(dB)

120579 (deg)

120590120579120579

(b)


18

16

14

12

10

8

6

4

2

00 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

120590120601120601120582

2(d

B)

120579 (deg)

(a)

20

15

10

5

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus10

minus5

1205822

(dB)

120579 (deg)

120590120579120579

(b)







References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

120590120601120601120582

2(d

B)

120579 (deg)

(a)

8

6

4

2

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus2

minus4

minus6

minus8

1205822

(dB)

120579 (deg)

120590120579120579

(b)


18

16

14

12

10

8

6

4

2

00 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

120590120601120601120582

2(d

B)

120579 (deg)

(a)

20

15

10

5

0

0 20 40 60 80 100 120 140 160 180


EFIE

z

xk

Hi

minus10

minus5

1205822

(dB)

120579 (deg)

120590120579120579

(b)







References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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