Research Article Mixed-Potential Integral Equation Based...

Research ArticleMixed-Potential Integral Equation Based Characteristic ModeAnalysis of Microstrip Antennas

Yikai Chen Liwen Guo and Shiwen Yang

University of Electronic Science and Technology of China Chengdu 611731 China

Correspondence should be addressed to Yikai Chen ykchenuestceducn

Received 26 August 2016 Accepted 1 November 2016

Academic Editor Miguel Ferrando Bataller

Copyright copy 2016 Yikai Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A characteristic mode (CM) formulation is developed for the modal analysis of microstrip antennas It is derived from the mixed-potential integral equation (MPIE) with spatial-domain Greenrsquos functions for multilayered media where spatial-domain Greenrsquosfunctions take into account the effect of the multilayered media The resultant characteristic currents and fields are orthogonalwith each other among different orders of modes Together with the eigenvalues and their deduced indicators the CMs providedeep physical insights into the radiation mechanisms of microstrip antennas Numerical results are presented to confirm CMformulationrsquos effectiveness and accuracy in determining the resonant frequencies radiating mode currents and modal fields ofmicrostrip antennas As opposed to the very popular CM formulation for conducting bodies comparative studies are presented toshow the quite different modal analysis results by considering the multilayered media

1 Introduction

Microstrip antennas (MSAs) have gained great interest inmodern wireless communications due to their attractive fea-tures such as low profile light weight and ease of fabricationNevertheless MSAs usually have narrow bandwidths andcan only operate effectively in the vicinity of resonant fre-quencies Therefore it is important to accurately determinethe resonant frequencies of MSAs Moreover obtaining theradiating modes of MSAs is also of great importance forpractical designs A number of methods have been proposedto determine resonant frequencies of MSAs These methodscan be generally classified into analytical and numericalmethods Analytical methods such as cavity models [1 2]and transmission-line [3] provide intuitive physical expla-nations to the radiation mechanism However they areonly suitable for some canonic structures Corrected for-mulae for irregular shaped MSAs become quite complicated[4] Although electromagnetic modeling techniques like themethod of moments (MoM) are well developed for MSAanalysis feedings have to be properly designed to get the trueradiatingmodes In [5] theMoMwas employed to determine

the resonances of irregularly shaped MSAs by seeking thefrequencies at which the real part of the surface currentdensity ismaximum and the imaginary part is zero Althoughthis method was effective in finding resonant frequenciesthe modal fields (or modal currents) strongly depend on theincident plane wave used in the MoM simulation

Recently the characteristic mode (CM) theory [6] forconducting bodies which is based on the electric field inte-gral equation (EFIE) has been employed for modal analysisof MSAs In [7 8] resonant frequency of the fundamentalTM10 mode for rectangular patch antennas with air substrateis analyzed using the CM theory for conducting bodiesSimilarly U-slot and E-shaped MSAs with air substrate werealso analyzed using the CM theory for conducting bodies [9]However dielectric material with 120576119903 gt 1 is used extensivelyin MSA designs As will be demonstrated the CM theory forconducting bodies hasmany limitations in themodal analysisof MSAs

This paper develops a CM formulation from the mixed-potential integral equation (MPIE) Spatial-domain Greenrsquosfunctions (GFs) of multilayered media are used in the MPIEto account for the effect of multilayered media Numerical

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 9421050 8 pageshttpdxdoiorg10115520169421050

2 International Journal of Antennas and Propagation

results are given to show the accuracy of the proposed CMformulation Comparative studies are also given to show thelimitations of the CM theory of conducting bodies in themodal analysis of MSAs with nonair dielectric substrate

2 CM Theory for MPIE withMultilayered Media GF

The conventional CM theory was initially developed byGarbacz and Turpin [10] and then refined by Harringtonand Mautz [6] in 1971 The resultant CMs define a set oforthogonal currents that can be used to expand the inducedcurrent due to external sources Because of such attractiveproperties the CM theory has been extensively applied invarious antenna designs [11ndash13] In order to adapt the CMtheory to MSAs analysis CM formulation for MSAs withmultilayered media is developed

The MPIE with spatial-domain GFs of multilayeredmedia (infinite in transverse directions) allows accuratemod-eling surface currents in multilayered media environment[14 15] We assume that a microstrip antenna is illuminatedby an incident wave E119894 and the total tangential electric fieldmust vanish on the conducting patches It reduces to thefollowing MPIE

[1198951205961205830 ⟨119866119886 J⟩ minus 11198951205961205760nabla⟨119866119902 nabla

1015840 sdot J⟩]tan

= E119894tan (1)

where J is the unknown surface current and 119866119886 and 119866119902are the spatial-domain GFs for dyadic magnetic vector andelectric scalar potential for multilayered media respectivelyThe spatial-domain GFs can be computed from an inverseHankel transformof their spectral-domain counterparts 119886119902119866119886119902 (r r1015840)

= 14120587 int+infin

minusinfin119889119896120588120588119886119902 (119896120588 119911 1199111015840)119867(2)0 (119896120588 10038161003816100381610038161003816120588 minus 120588101584010038161003816100381610038161003816)

(2)

The spatial-domain GFs are able to account for the effects ofthe grounded multilayered media in MSAs Hence the CMformulation developed fromMPIE can better capture the res-onant frequencies and radiatingmode currents ofMSAs Dueto the computational expense for the numerical integration ofthe highly oscillatory integrand several efficient algorithmshave been developed to accelerate the numerical integrationTo date the discrete complex image method (DCIM) [1415] has mostly been used for radiating patches buried inmultilayered media The DCIM is based on extracting thequasi-static part and surface wave contributions from thespectral-domain GFs and then approximating the remainderas a series of complex exponential functions The DCIMdepends on Sommerfeldrsquos identity

119890minus119895119896119903119903 = minus 1198952 int

+infin

minusinfin119889119896120588120588119867(2)0 (119896120588 10038161003816100381610038161003816120588 minus 120588101584010038161003816100381610038161003816) 119890

minus119895119896119911|119911|

119896119911 (3)

Therefore if the spectral-domain GF is approximated interms of exponential functions the inverse Hankel transfor-mation in (2) can be evaluated analytically in a closed form

for each exponential term The CPU time for the numericalevaluation of the transformation in (2) is thus reducedHence theDCIM is implemented in this paper for the spatial-domain GFs evaluations

After the spatial-domain GFs is calculated the MPIEcan be easily discretized using the MoM procedure Byapplying the standard Galerkinrsquos procedure to the MPIE withusing the RWG basis functions f119898(r) the elements of theresultant impedance matrix ZMPIE can be computed from thefollowing

ΖMPIE119898119899 = 1198951205961205830 int

119879119898

int119879119899

f119898 (r) sdot f119899 (r1015840)119866119886 (r r1015840) 119889r1015840 119889r

+ 11198951205961205760

sdot int119879119898

int119879119899

nabla sdot f119898 (r) nabla1015840 sdot f119899 (r1015840)119866119902 (r r1015840) 119889r1015840 119889r

(4)

As observed from (3) the impedance matrix ZMPIE is sym-metric Following the derivations in [6] an MPIE basedgeneralized eigenvalue equation can be obtained for CManalysis

XMPIEJ119899 = 120582119899RMPIEJ119899 (5)

where RMPIE and XMPIEare the real and imaginary parts ofZMPIE Both of them are symmetric matrix Moreover RMPIE

is also a semidefinite matrix 120582119899 is the eigenvalue associatedwith each characteristic current J119899 Because of the symmetryof RMPIE and XMPIE both of J119899 and 120582119899 are real

The characteristic current is generally normalized accord-ing to ⟨J119899RMPIE sdotJ119899⟩ = 1 such that each characteristic currentJ119899 radiates unit radiation power After the normalizationthe characteristic currents satisfy the following orthogonalrelationships

⟨J119898RMPIE sdot J119899⟩ = ⟨Jlowast119898RMPIE sdot J119899⟩ = 120575119898119899⟨J119898XMPIE sdot J119899⟩ = ⟨Jlowast119898XMPIE sdot J119899⟩ = 120582119899120575119898119899

(6)

where 120575119898119899 is the Kronecker delta functionThe electric field E119899 andmagnetic fieldH119899 radiated by the

characteristic current J119899 are termed as characteristic fieldsMaking use of the complex Poynting theorem orthogonalrelationships for the characteristic fields are obtained [6]

119875 (J119898 J119899) = ⟨Jlowast119898ZMPIE sdot J119899⟩= ⟨Jlowast119898RMPIE sdot J119899⟩ + 119895 ⟨Jlowast119898XMPIE sdot J119899⟩= (1 + 119895120582119899) 120575119898119899=∯1198781015840(Elowast119898 timesH119899) sdot 119889s

+ 119895120596∭119881(120583Hlowast119898 sdotH119899 minus 120576Elowast119898 sdot E119899) 119889V

(7)

International Journal of Antennas and Propagation 3

In practical radiation problems the radiating field ofa microstrip antenna is of finite extent If we choose theradiation surface 1198781015840 in (7) to be the radiation sphere at infinitefar-field region we have

1120578∯119878infinE

lowast119898 sdot E119899119889119904 = 120575119898119899 (8)

Equation (8) illustrates that the characteristic fields form anorthogonal set in the far-field region The orthogonality ofcharacteristic currents and fields can be also observed fromthe numerical results in Section 3

Let 119898 = 119899 in (7) we can see that the complex powerdue to the characteristic current J119898 consists of two parts theradiated power (real power) and stored energy (imaginarypower)

119875real =∯119878infin

(Elowast119898 timesH119898) sdot 119889s = 1 (9)

119875imag = 119895120596∭119881(120583Hlowast119898 sdotH119898 minus 120576Elowast119898 sdot E119898) 119889V = 119895120582119898 (10)

Apparently characteristic mode with 120582119898 = 0 does notcontribute to imaginary power It indicates that such radiatingmode can be fully excited for radiation purpose Howeverfor those inefficient characteristic modes with larger |120582119898|current density with larger amplitude is required to giveunit radiating power because large amount of energy isstored within the systemTherefore for modes with the samecurrent density the radiated power (ie real power) formode120582119898 = 0 is maximized as compared with those that havelarger |120582119898| This is the most direct way to understand howthe magnitude of an eigenvalue is related to the behavior ofa mode Generally eigenvalue 120582119898 ranges from minusinfin to +infinEquation (10) clearly shows that the sign of the eigenvalueindicates whether a mode stores more electric energy thanmagnetic energy (120582119898 lt 0) or vice versa (120582119898 gt 0) 120582119898 = 0indicates that the structure is in modal resonance

Other than the eigenvalues tell how the mode can beefficiently excited for radiation purposemodal significance isalso a popular indicator in understanding resonant behaviors

MS119899 = 110038161003816100381610038161 + 1198951205821198991003816100381610038161003816 (11)

The modal significance represents the contribution of aparticularmode to the total radiation responseswhen an idealexternal source is applied to fully excite the mode In partic-ular MS119899 = 1 indicates that the structure is in resonance atthe calculated frequency In addition characteristic angle isalso of great importance for antenna designs where multiplemodes need to be excited for certain radiation performance(eg circular polarization) [9] It is defined as follows

120572119899 = 180∘ minus tanminus1120582119899 (12)

It denotes the phase delay angle between the characteristiccurrent and its associated characteristic field on the radiatingsurface

3 Numerical Results and Discussion

To confirm the validity of the MPIE based CM theory MSAswith single layered and three-layered dielectric substrates areinvestigated Comparative study is performed to intuitivelyshow the differences between the CM theory of conductingbodies and the MPIE based CM theory for multilayeredmedia

31 Rectangular Patch Antenna A rectangular MSA with di-mensions of 119871 times 119882 = 76 times 50mm2 was analyzed from08GHz to 22GHz The patch is printed on a groundeddielectric substrate with relative permittivity 120576119903 = 338 andthickness ℎ = 1524mm In [2] the resonant frequencies ofthe first fourmodes are computed using the cavitymodelThemodes TM10 TM01 TM11 and TM20 resonate at 1075GHz1605GHz 1955GHz and 2145GHz respectively Thesemodes have proved to be the natural radiatingmodes that canbe physically excited at their own resonant frequencies

To clearly demonstrate the effectiveness of the proposedapproach a comparison study among three different modelsis carried out In the first model only an isolated rectangularpatch in free space is considered This model has been usedfor CM analysis of MSA in [11] to get some physical insightsofMSAs Figure 1(a) shows themodal significances solved forthis model by using the CM theory of conducting bodies Ascan be seen the patch modes exhibit very wide bandwidthalthough such wide bandwidth is generally not achievable inpractical MSA designsThis is mainly due to the ignorance ofthe ground plane in practice

To investigate the effect of the ground plane the secondmodel which consists of a rectangular patch and an infiniteground plane is considered for CM analysis The patch sizeis the same as that in the first model The space betweenthe patch and the ground plane is filled with air and thedistance between them is 1524mm Figure 1(b) shows thesolvedmodal significance of this model As can be seen thereare four resonant frequencies found in the frequency rangeof 08ndash40GHz However they are quite different from thosepredicted in [2]Thediscrepancy in resonant frequenciesmaybe due to the ignorance of the dielectric material in the CManalysis

To fully consider the true configurations of the MSAthe dielectric material (120576119903 = 338) between the patch andthe ground plane is included in the third model and theproposed MPIE based CM formulation is implemented tosolve the resonant frequencies Figure 1(c) shows the modalsignificances of the third model The CMs are sorted overthe whole frequency band by using the modal trackingtechnique in [16] Regarding the physical meaning of themodal significance four resonances are observed over thefrequency band They are located at 108GHz 160GHz195GHz and 213GHz respectively Evidently they agreewell with the resonant frequencies reported in [2]

However rectangular MSAs with grounded dielectricsubstrates were previously analyzed using the CM formula-tion of conducting bodies [6 11] where either the groundplane or the dielectric material is ignored in CM analysisAlthough one can observe some radiating modes from such

4 International Journal of Antennas and PropagationM

odel

sign

ifica

nce

1E minus 3

001

01

1

10 12 14 16 18 20 2208

Frequency (GHz)

1st mode2nd mode3rd mode

4th mode5th mode6th mode

(a) Modal significances for isolated rectangular patch in free space

Frequency (GHz)10 15 20 25 30 35 40

High-ordermodes

188GHz 281GHz 339GHz 378GHz

1st mode2nd mode

3rd mode4th mode

1E minus 3

001

01

1

Mod

el si

gnifi

canc

e

(b) Modal significances for rectangular patch with ground planeThe spacebetween the patch and the ground plane is filled with air (120576119903 = 1)


10 12 14 16 18 20 2208

Frequency (GHz)

modesHigh-order

1E minus 3

001

01

1

Mod

el si

gnifi

canc

e

1st mode2nd mode

3rd mode4th mode

(c) Modal significances for rectangular patch with grounded dielectricsubstrate (120576119903 = 338)

Figure 1 Modal significances for the same rectangular patch but with different dielectric substrate and ground plane configurations

CM analysis there are many physical explanations that havenot been fully addressed This example clearly shows theimportance of considering the grounded dielectric substratein the CM analysis of MSAs

Figure 2 gives the characteristic currents and fields (attheir own resonant frequencies) solved from the MPIE basedCM formulation for the third model First we observe thatthe fourmodes are orthogonal with each other in terms of thesurface current polarizations and magnitudes Second theyagree well with the physically excited current distributionsobtained by putting a coaxial probe near the corner of thepatch [2] Third because CMs are excitation independentwhich means they are inherent to geometry and material

properties hence it is quite attractive to get these modalsolutions before feeding designs The characteristic currentsin turn provide useful guidelines on how to design feedingsfor the excitation of each radiating mode

For comparison purpose the characteristic currents andfields solved from the first model (an isolated rectangularpatch in free space) are plotted in Figure 3 As Figure 1(a)shows that the resonances of the isolated patch are not asevident as in Figures 1(b) and 1(c) all the CMs in Figure 3 areplotted at the center frequency 15 GHz As can be seen themodal fields of the 1st 2nd 4th and 5th modes in Figure 3are very close to the radiating modes TM10 TM01 TM11 andTM20 respectively This is the reason that the CM theory of


0 01 02 03 04 05 06 07 08 09 1

(a) TM10 (1st mode)

0 01 02 03 04 05 06 07 08 09 1

(b) TM01 (2nd mode)

0 01 02 03 04 05 06 07 08 09 1

(c) TM11 (3rd mode)

0 01 02 03 04 05 06 07 08 09 1

(d) TM20 (4th mode)

Figure 2 The characteristic currents and fields of the rectangular patch with grounded dielectric substrate (120576119903 = 338)

conducting bodies can be used to roughly predict the modalbehavior of MSAs On the other hand the 3rd and 6thmodesdo not appear in the MPIE based CM analysis results Theyare defined as the inductive modes in [11] because the charac-teristic currents form closed loops over the two-dimensionalplate and exhibit special inductive property In [11] Cabedo-Fabres et al have also demonstrated that such modes didnot resonate and presented an inductive contribution at eachfrequency [11] However they are not present in the case ofa patch with ground plane This comparison study clearlyshows the impact of the grounded dielectric substrate and thenecessity to consider the grounded dielectric substrate in CManalysis through using the MPIE formulation especially inthe case that we need to get all the true radiating modes of aMSA

32 Dual-Band Circularly Polarized Antenna Although CMtheory for PEC bodies [6] has been used to investigatethe circular polarization potential of an isolated triangularpatch without dielectric substrate and ground practicalcircularly polarized MSA with grounded substrate has notbeen discussed before in the framework of characteristicmode theory The mechanism of dual-band MSA has alsonot been discussed through characteristic mode analysis Tofully explore the application potential of the MPIE based CMtheory in more complicated microstrip antenna designs thissection focuses on the CM analysis of a dual-band circularlypolarized MSA with three-layered dielectric substrates usingthe MPIE based CM formulation The dual-band circularly

polarized MSA was originally reported in [17] and theconfiguration and dimensions can be also found in the insetof Figure 4(b) A coaxial probe directly feeds the top patchthrough a via hole on the bottom patch Because CMs areexcitation independent it is able to get modal solutions ofMSAs without considering any specific feeding structuresHence the coaxial probe and the via hole are neglected in theCM analysis

Figures 4(a) and 4(b) show the modal significances andcharacteristic angles respectively As can be seen there arefour resonances over the frequency band of 18ndash31 GHzThe first two modes coresonate in the 19 GHz band andthe 3rd and 4th modes coresonate in the 28GHz bandThe vertical dash lines in Figures 4(a) and 4(b) indicatethe frequency at which the characteristic angle differencesbetween modes 1 and 2 and modes 3 and 4 exactly equal90∘ As can be seen the first two dash lines center at about19 GHz and the 3rd and 4th dash lines center at about28GHz band Moreover the modal significances of mode 1and mode 2 between the first two dash lines are comparablewith each other The same observation is clearly seen inthe 28GHz band Considering the physical meanings of themodal significance and the characteristic angle if the firsttwo modes were excited simultaneously they will providecomparable radiation power and generate far fields with aphase difference around 90∘ Clearly these are the radiationmechanisms of this dual-band CP antenna

Figure 5 shows the characteristic currents and charac-teristic fields of the four CMs In Figures 5(a)ndash5(d) the left


0 01 02 03 04 05 06 07 08 09 1

(a) 1st mode

0 01 02 03 04 05 06 07 08 09 1

(b) 2nd mode

0 01 02 03 04 05 06 07 08 09 1

(c) 3rd mode

0 01 02 03 04 05 06 07 08 09 1

(d) 4th mode

0 01 02 03 04 05 06 07 08 09 1

(e) 5th mode

0 01 02 03 04 05 06 07 08 09 1

(f) 6th mode

Figure 3 The characteristic currents and fields of the rectangular patch in free space

figure shows the characteristic currents on the top patchwhile the right figure shows the characteristic currents onthe bottom patch As can be seen the first two modes haveorthogonal radiating currents on the bottom patch whereasthe currents on the top patch are very weak Hence thebottom patch dominates the radiation in the lower bandTheorthogonality of the first twomodes ensures the CP radiationThe characteristic currents and characteristic fields of the3rd and 4th modes are also orthogonal with each other Thecharacteristic currents on the bottom and top patches havecomparable current magnitudes which indicate that both ofthe two patches resonate in the higher band and contributetogether to the total CP radiation in the 28GHz band

In summary the modal significance characteristic angleand characteristic currents clearly illustrate that CP radiationcan be realized by properly exciting and combining coreso-nant CMs Dual-band performance is also observed from theCM analysis results

4 Conclusion

The CM theory based on the MPIE with spatial-domain GFsfor multilayered media is proposed for the modal analysisof microstrip antennas with multilayered media Againstthe conventional CM theory in [6] which was formulated


Mod

el si

gnifi

canc

e

Frequency (GHz)18 20 22 24 26 28 30

High-ordermodes

1E minus 3

001

01

1

1st mode2nd mode

3rd mode4th mode

(a)

Top patch

Bottom patch

Feed for LHCP

Ground plane

Frequency (GHz)18 20 22 24 26 28 30

b2

b1

d 45∘

45∘

a

a

t

x

x

y

y

h1

h2

1205761

120576

80

100

120

140

160

180

200

220

240

260

280

Char

acte

ristic

angl

e (de

gree

)

1st mode2nd mode

3rd mode4th mode

90∘ difference

(b)

Figure 4 Modal significances (a) and characteristic angles (b) for the dual-band CP antenna 1205761 = 1205762 = 44 ℎ1 = 16mm ℎ2 = 08mm119905 = 05mm 119886 = 2186mm 1198871119886 = 095 and 1198872119886 = 096 The grayed dash lines indicate the frequency at which the characteristic angledifferences exactly equal 90∘

0 01 02 03 04 05 06 07 08 09 1

(a)

0 01 02 03 04 05 06 07 08 09 1

(b)

0 01 02 03 04 05 06 07 08 09 1

(c)

0 01 02 03 04 05 06 07 08 09 1

(d)

Figure 5 Characteristic currents and fields for (a) 1st mode (b) 2nd mode (c) 3rd mode and (d) 4th mode In figures (a)ndash(d) the left figureshows the characteristic currents on the top patch while the right figure shows the characteristic currents on the bottom patch

from EFIE of PEC bodies and then was borrowed for modalanalysis of MSAs the present approach fully describes thereal physics of the MSA geometry Owing to this essentialdifference this method accurately solves the resonant fre-quencies and nearfar modal fields for dominant mode as

well as higher order modes These modal analysis results arequite attractive and will systematically guide practical MSAdesigns Examples are presented to show the accuracy andvalidity MSAs designs with the proposed CM theory will befurther addressed in our follow-up efforts


Competing Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

This work is supported by the Natural Science Foundation ofChina under Grant no 61671127

References

[1] K-F Lee K-M Luk and J S Dahele ldquoCharacteristics of theequilateral triangular patch antennardquo IEEE Transactions onAntennas and Propagation vol 36 no 11 pp 1510ndash1518 1988

[2] Z N Chen and M Chia Broadband Planar Antennas DesignandApplications JohnWileyamp Sons NewYork NYUSA 2005

[3] M El Yazidi M Himdi and J P Daniel ldquoTransmissionline analysis of nonlinear slot coupled microstrip antennardquoElectronics Letters vol 28 no 15 pp 1406ndash1408 1992

[4] X Gang ldquoOn the resonant frequencies of microstrip antennasrdquoIEEE Transactions on Antennas and Propagation vol 37 no 2pp 245ndash247 1989

[5] M D Deshpande D G Shively and C R Cockrell ldquoResonantfrequencies of irregularly shaped microstrip antennas usingmethod of momentsrdquo NASA Technical Paper 3386 1993

[6] R Harrington and J Mautz ldquoTheory of characteristic modesfor conducting bodiesrdquo IEEE Transactions on Antennas andPropagation vol 19 no 5 pp 622ndash628 1971

[7] J Eichler P Hazdra M Capek and M Mazanek ldquoModalresonant frequencies and radiation quality factors of microstripantennasrdquo International Journal of Antennas and Propagationvol 2012 Article ID 490327 9 pages 2012

[8] C Van Niekerk and J T Bernhard ldquoCharacteristic mode anal-ysis of a shorted microstrip patch antennardquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium (APSURSI rsquo12) pp 1ndash2 Chicago Ill USA July 2012

[9] Y Chen and C-F Wang ldquoCharacteristic-mode-based improve-ment of circularly polarized U-slot and E-shaped patch anten-nasrdquo IEEEAntennas andWireless Propagation Letters vol 11 pp1474ndash1477 2012

[10] R Garbacz and R Turpin ldquoA generalized expansion for radi-ated and scattered fieldsrdquo IEEE Transactions on Antennas andPropagation vol 19 no 3 pp 348ndash358 1971

[11] M Cabedo-Fabres E Antonino-Daviu A Valero-NogueiraandM F Bataller ldquoThe theory of characteristicmodes revisiteda contribution to the design of antennas for modern applica-tionsrdquo IEEE Antennas and Propagation Magazine vol 49 no 5pp 52ndash68 2007

[12] Y Chen and C-F Wang ldquoHF band shipboard antenna designusing characteristic modesrdquo Institute of Electrical and Electron-ics Engineers Transactions onAntennas andPropagation vol 63no 3 pp 1004ndash1013 2015

[13] Y Chen and C-F Wang ldquoElectrically small UAV antennadesign using characteristic modesrdquo IEEE Transactions onAntennas and Propagation vol 62 no 2 pp 535ndash545 2014

[14] D G Fang J J Yang and G Y Delisle ldquoDiscrete image theoryfor horizontal electric dipoles in a multilayered mediumrdquo IEEProceedings H Microwaves Antennas and Propagation vol 135no 5 pp 297ndash303 1988

[15] F Ling C-F Wang and J-M Jin ldquoAn efficient algorithm foranalyzing large-scale microstrip structures using adaptive inte-gral method combined with discrete complex-image methodrdquoIEEE Transactions onMicrowaveTheory and Techniques vol 48no 5 pp 832ndash839 2000

[16] B D Raines and R G Rojas ldquoWideband characteristic modetrackingrdquo IEEE Transactions on Antennas and Propagation vol60 no 7 pp 3537ndash3541 2012

[17] J-Y Jan and K-L Wong ldquoDual-band circularly polarizedstacked elliptic microstrip antennardquo Microwave and OpticalTechnology Letters vol 24 no 5 pp 354ndash357 2000

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results are given to show the accuracy of the proposed CMformulation Comparative studies are also given to show thelimitations of the CM theory of conducting bodies in themodal analysis of MSAs with nonair dielectric substrate

2 CM Theory for MPIE withMultilayered Media GF

The conventional CM theory was initially developed byGarbacz and Turpin [10] and then refined by Harringtonand Mautz [6] in 1971 The resultant CMs define a set oforthogonal currents that can be used to expand the inducedcurrent due to external sources Because of such attractiveproperties the CM theory has been extensively applied invarious antenna designs [11ndash13] In order to adapt the CMtheory to MSAs analysis CM formulation for MSAs withmultilayered media is developed

The MPIE with spatial-domain GFs of multilayeredmedia (infinite in transverse directions) allows accuratemod-eling surface currents in multilayered media environment[14 15] We assume that a microstrip antenna is illuminatedby an incident wave E119894 and the total tangential electric fieldmust vanish on the conducting patches It reduces to thefollowing MPIE

[1198951205961205830 ⟨119866119886 J⟩ minus 11198951205961205760nabla⟨119866119902 nabla

1015840 sdot J⟩]tan

= E119894tan (1)

where J is the unknown surface current and 119866119886 and 119866119902are the spatial-domain GFs for dyadic magnetic vector andelectric scalar potential for multilayered media respectivelyThe spatial-domain GFs can be computed from an inverseHankel transformof their spectral-domain counterparts 119886119902119866119886119902 (r r1015840)

= 14120587 int+infin

minusinfin119889119896120588120588119886119902 (119896120588 119911 1199111015840)119867(2)0 (119896120588 10038161003816100381610038161003816120588 minus 120588101584010038161003816100381610038161003816)

(2)

The spatial-domain GFs are able to account for the effects ofthe grounded multilayered media in MSAs Hence the CMformulation developed fromMPIE can better capture the res-onant frequencies and radiatingmode currents ofMSAs Dueto the computational expense for the numerical integration ofthe highly oscillatory integrand several efficient algorithmshave been developed to accelerate the numerical integrationTo date the discrete complex image method (DCIM) [1415] has mostly been used for radiating patches buried inmultilayered media The DCIM is based on extracting thequasi-static part and surface wave contributions from thespectral-domain GFs and then approximating the remainderas a series of complex exponential functions The DCIMdepends on Sommerfeldrsquos identity

119890minus119895119896119903119903 = minus 1198952 int

+infin

minusinfin119889119896120588120588119867(2)0 (119896120588 10038161003816100381610038161003816120588 minus 120588101584010038161003816100381610038161003816) 119890

minus119895119896119911|119911|

119896119911 (3)

Therefore if the spectral-domain GF is approximated interms of exponential functions the inverse Hankel transfor-mation in (2) can be evaluated analytically in a closed form

for each exponential term The CPU time for the numericalevaluation of the transformation in (2) is thus reducedHence theDCIM is implemented in this paper for the spatial-domain GFs evaluations

After the spatial-domain GFs is calculated the MPIEcan be easily discretized using the MoM procedure Byapplying the standard Galerkinrsquos procedure to the MPIE withusing the RWG basis functions f119898(r) the elements of theresultant impedance matrix ZMPIE can be computed from thefollowing

ΖMPIE119898119899 = 1198951205961205830 int

119879119898

int119879119899

f119898 (r) sdot f119899 (r1015840)119866119886 (r r1015840) 119889r1015840 119889r

+ 11198951205961205760

sdot int119879119898

int119879119899

nabla sdot f119898 (r) nabla1015840 sdot f119899 (r1015840)119866119902 (r r1015840) 119889r1015840 119889r

(4)

As observed from (3) the impedance matrix ZMPIE is sym-metric Following the derivations in [6] an MPIE basedgeneralized eigenvalue equation can be obtained for CManalysis

XMPIEJ119899 = 120582119899RMPIEJ119899 (5)

where RMPIE and XMPIEare the real and imaginary parts ofZMPIE Both of them are symmetric matrix Moreover RMPIE

is also a semidefinite matrix 120582119899 is the eigenvalue associatedwith each characteristic current J119899 Because of the symmetryof RMPIE and XMPIE both of J119899 and 120582119899 are real

The characteristic current is generally normalized accord-ing to ⟨J119899RMPIE sdotJ119899⟩ = 1 such that each characteristic currentJ119899 radiates unit radiation power After the normalizationthe characteristic currents satisfy the following orthogonalrelationships

⟨J119898RMPIE sdot J119899⟩ = ⟨Jlowast119898RMPIE sdot J119899⟩ = 120575119898119899⟨J119898XMPIE sdot J119899⟩ = ⟨Jlowast119898XMPIE sdot J119899⟩ = 120582119899120575119898119899

(6)

where 120575119898119899 is the Kronecker delta functionThe electric field E119899 andmagnetic fieldH119899 radiated by the

characteristic current J119899 are termed as characteristic fieldsMaking use of the complex Poynting theorem orthogonalrelationships for the characteristic fields are obtained [6]

119875 (J119898 J119899) = ⟨Jlowast119898ZMPIE sdot J119899⟩= ⟨Jlowast119898RMPIE sdot J119899⟩ + 119895 ⟨Jlowast119898XMPIE sdot J119899⟩= (1 + 119895120582119899) 120575119898119899=∯1198781015840(Elowast119898 timesH119899) sdot 119889s

+ 119895120596∭119881(120583Hlowast119898 sdotH119899 minus 120576Elowast119898 sdot E119899) 119889V

(7)



1120578∯119878infinE

lowast119898 sdot E119899119889119904 = 120575119898119899 (8)



119875real =∯119878infin





MS119899 = 110038161003816100381610038161 + 1198951205821198991003816100381610038161003816 (11)


120572119899 = 180∘ minus tanminus1120582119899 (12)










odel

sign

ifica

nce

1E minus 3

001

01

1

10 12 14 16 18 20 2208

Frequency (GHz)




Frequency (GHz)10 15 20 25 30 35 40

High-ordermodes


1st mode2nd mode

3rd mode4th mode

1E minus 3

001

01

1

Mod

el si

gnifi

canc

e



10 12 14 16 18 20 2208

Frequency (GHz)

modesHigh-order

1E minus 3

001

01

1

Mod

el si

gnifi

canc

e

1st mode2nd mode

3rd mode4th mode








0 01 02 03 04 05 06 07 08 09 1

(a) TM10 (1st mode)

0 01 02 03 04 05 06 07 08 09 1

(b) TM01 (2nd mode)

0 01 02 03 04 05 06 07 08 09 1

(c) TM11 (3rd mode)

0 01 02 03 04 05 06 07 08 09 1

(d) TM20 (4th mode)








0 01 02 03 04 05 06 07 08 09 1

(a) 1st mode

0 01 02 03 04 05 06 07 08 09 1

(b) 2nd mode

0 01 02 03 04 05 06 07 08 09 1

(c) 3rd mode

0 01 02 03 04 05 06 07 08 09 1

(d) 4th mode

0 01 02 03 04 05 06 07 08 09 1

(e) 5th mode

0 01 02 03 04 05 06 07 08 09 1

(f) 6th mode




4 Conclusion



Mod

el si

gnifi

canc

e

Frequency (GHz)18 20 22 24 26 28 30

High-ordermodes

1E minus 3

001

01

1

1st mode2nd mode

3rd mode4th mode

(a)

Top patch

Bottom patch

Feed for LHCP

Ground plane

Frequency (GHz)18 20 22 24 26 28 30

b2

b1

d 45∘

45∘

a

a

t

x

x

y

y

h1

h2

1205761

120576

80

100

120

140

160

180

200

220

240

260

280

Char

acte

ristic

angl

e (de

gree

)

1st mode2nd mode

3rd mode4th mode

90∘ difference

(b)


0 01 02 03 04 05 06 07 08 09 1

(a)

0 01 02 03 04 05 06 07 08 09 1

(b)

0 01 02 03 04 05 06 07 08 09 1

(c)

0 01 02 03 04 05 06 07 08 09 1

(d)





Competing Interests


Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1120578∯119878infinE

lowast119898 sdot E119899119889119904 = 120575119898119899 (8)



119875real =∯119878infin





MS119899 = 110038161003816100381610038161 + 1198951205821198991003816100381610038161003816 (11)


120572119899 = 180∘ minus tanminus1120582119899 (12)










odel

sign

ifica

nce

1E minus 3

001

01

1

10 12 14 16 18 20 2208

Frequency (GHz)




Frequency (GHz)10 15 20 25 30 35 40

High-ordermodes


1st mode2nd mode

3rd mode4th mode

1E minus 3

001

01

1

Mod

el si

gnifi

canc

e



10 12 14 16 18 20 2208

Frequency (GHz)

modesHigh-order

1E minus 3

001

01

1

Mod

el si

gnifi

canc

e

1st mode2nd mode

3rd mode4th mode








0 01 02 03 04 05 06 07 08 09 1

(a) TM10 (1st mode)

0 01 02 03 04 05 06 07 08 09 1

(b) TM01 (2nd mode)

0 01 02 03 04 05 06 07 08 09 1

(c) TM11 (3rd mode)

0 01 02 03 04 05 06 07 08 09 1

(d) TM20 (4th mode)








0 01 02 03 04 05 06 07 08 09 1

(a) 1st mode

0 01 02 03 04 05 06 07 08 09 1

(b) 2nd mode

0 01 02 03 04 05 06 07 08 09 1

(c) 3rd mode

0 01 02 03 04 05 06 07 08 09 1

(d) 4th mode

0 01 02 03 04 05 06 07 08 09 1

(e) 5th mode

0 01 02 03 04 05 06 07 08 09 1

(f) 6th mode




4 Conclusion



Mod

el si

gnifi

canc

e

Frequency (GHz)18 20 22 24 26 28 30

High-ordermodes

1E minus 3

001

01

1

1st mode2nd mode

3rd mode4th mode

(a)

Top patch

Bottom patch

Feed for LHCP

Ground plane

Frequency (GHz)18 20 22 24 26 28 30

b2

b1

d 45∘

45∘

a

a

t

x

x

y

y

h1

h2

1205761

120576

80

100

120

140

160

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200

220

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260

280

Char

acte

ristic

angl

e (de

gree

)

1st mode2nd mode

3rd mode4th mode

90∘ difference

(b)


0 01 02 03 04 05 06 07 08 09 1

(a)

0 01 02 03 04 05 06 07 08 09 1

(b)

0 01 02 03 04 05 06 07 08 09 1

(c)

0 01 02 03 04 05 06 07 08 09 1

(d)





Competing Interests


Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










odel

sign

ifica

nce

1E minus 3

001

01

1

10 12 14 16 18 20 2208

Frequency (GHz)




Frequency (GHz)10 15 20 25 30 35 40

High-ordermodes


1st mode2nd mode

3rd mode4th mode

1E minus 3

001

01

1

Mod

el si

gnifi

canc

e



10 12 14 16 18 20 2208

Frequency (GHz)

modesHigh-order

1E minus 3

001

01

1

Mod

el si

gnifi

canc

e

1st mode2nd mode

3rd mode4th mode








0 01 02 03 04 05 06 07 08 09 1

(a) TM10 (1st mode)

0 01 02 03 04 05 06 07 08 09 1

(b) TM01 (2nd mode)

0 01 02 03 04 05 06 07 08 09 1

(c) TM11 (3rd mode)

0 01 02 03 04 05 06 07 08 09 1

(d) TM20 (4th mode)








0 01 02 03 04 05 06 07 08 09 1

(a) 1st mode

0 01 02 03 04 05 06 07 08 09 1

(b) 2nd mode

0 01 02 03 04 05 06 07 08 09 1

(c) 3rd mode

0 01 02 03 04 05 06 07 08 09 1

(d) 4th mode

0 01 02 03 04 05 06 07 08 09 1

(e) 5th mode

0 01 02 03 04 05 06 07 08 09 1

(f) 6th mode




4 Conclusion



Mod

el si

gnifi

canc

e

Frequency (GHz)18 20 22 24 26 28 30

High-ordermodes

1E minus 3

001

01

1

1st mode2nd mode

3rd mode4th mode

(a)

Top patch

Bottom patch

Feed for LHCP

Ground plane

Frequency (GHz)18 20 22 24 26 28 30

b2

b1

d 45∘

45∘

a

a

t

x

x

y

y

h1

h2

1205761

120576

80

100

120

140

160

180

200

220

240

260

280

Char

acte

ristic

angl

e (de

gree

)

1st mode2nd mode

3rd mode4th mode

90∘ difference

(b)


0 01 02 03 04 05 06 07 08 09 1

(a)

0 01 02 03 04 05 06 07 08 09 1

(b)

0 01 02 03 04 05 06 07 08 09 1

(c)

0 01 02 03 04 05 06 07 08 09 1

(d)





Competing Interests


Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 01 02 03 04 05 06 07 08 09 1

(a) TM10 (1st mode)

0 01 02 03 04 05 06 07 08 09 1

(b) TM01 (2nd mode)

0 01 02 03 04 05 06 07 08 09 1

(c) TM11 (3rd mode)

0 01 02 03 04 05 06 07 08 09 1

(d) TM20 (4th mode)








0 01 02 03 04 05 06 07 08 09 1

(a) 1st mode

0 01 02 03 04 05 06 07 08 09 1

(b) 2nd mode

0 01 02 03 04 05 06 07 08 09 1

(c) 3rd mode

0 01 02 03 04 05 06 07 08 09 1

(d) 4th mode

0 01 02 03 04 05 06 07 08 09 1

(e) 5th mode

0 01 02 03 04 05 06 07 08 09 1

(f) 6th mode




4 Conclusion



Mod

el si

gnifi

canc

e

Frequency (GHz)18 20 22 24 26 28 30

High-ordermodes

1E minus 3

001

01

1

1st mode2nd mode

3rd mode4th mode

(a)

Top patch

Bottom patch

Feed for LHCP

Ground plane

Frequency (GHz)18 20 22 24 26 28 30

b2

b1

d 45∘

45∘

a

a

t

x

x

y

y

h1

h2

1205761

120576

80

100

120

140

160

180

200

220

240

260

280

Char

acte

ristic

angl

e (de

gree

)

1st mode2nd mode

3rd mode4th mode

90∘ difference

(b)


0 01 02 03 04 05 06 07 08 09 1

(a)

0 01 02 03 04 05 06 07 08 09 1

(b)

0 01 02 03 04 05 06 07 08 09 1

(c)

0 01 02 03 04 05 06 07 08 09 1

(d)





Competing Interests


Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 01 02 03 04 05 06 07 08 09 1

(a) 1st mode

0 01 02 03 04 05 06 07 08 09 1

(b) 2nd mode

0 01 02 03 04 05 06 07 08 09 1

(c) 3rd mode

0 01 02 03 04 05 06 07 08 09 1

(d) 4th mode

0 01 02 03 04 05 06 07 08 09 1

(e) 5th mode

0 01 02 03 04 05 06 07 08 09 1

(f) 6th mode




4 Conclusion



Mod

el si

gnifi

canc

e

Frequency (GHz)18 20 22 24 26 28 30

High-ordermodes

1E minus 3

001

01

1

1st mode2nd mode

3rd mode4th mode

(a)

Top patch

Bottom patch

Feed for LHCP

Ground plane

Frequency (GHz)18 20 22 24 26 28 30

b2

b1

d 45∘

45∘

a

a

t

x

x

y

y

h1

h2

1205761

120576

80

100

120

140

160

180

200

220

240

260

280

Char

acte

ristic

angl

e (de

gree

)

1st mode2nd mode

3rd mode4th mode

90∘ difference

(b)


0 01 02 03 04 05 06 07 08 09 1

(a)

0 01 02 03 04 05 06 07 08 09 1

(b)

0 01 02 03 04 05 06 07 08 09 1

(c)

0 01 02 03 04 05 06 07 08 09 1

(d)





Competing Interests


Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Mod

el si

gnifi

canc

e

Frequency (GHz)18 20 22 24 26 28 30

High-ordermodes

1E minus 3

001

01

1

1st mode2nd mode

3rd mode4th mode

(a)

Top patch

Bottom patch

Feed for LHCP

Ground plane

Frequency (GHz)18 20 22 24 26 28 30

b2

b1

d 45∘

45∘

a

a

t

x

x

y

y

h1

h2

1205761

120576

80

100

120

140

160

180

200

220

240

260

280

Char

acte

ristic

angl

e (de

gree

)

1st mode2nd mode

3rd mode4th mode

90∘ difference

(b)


0 01 02 03 04 05 06 07 08 09 1

(a)

0 01 02 03 04 05 06 07 08 09 1

(b)

0 01 02 03 04 05 06 07 08 09 1

(c)

0 01 02 03 04 05 06 07 08 09 1

(d)





Competing Interests


Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Competing Interests


Acknowledgments


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









Research Article Mixed-Potential Integral Equation Based...

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