On the Accuracy of Equivalent Antenna Representations

1.8 MB·

by  Johan Malmström

MALMSTRO¨Metal.:ONTHEACCURACYOFEQUIVALENTANTENNAREPRESENTATIONS 1
On the Accuracy of Equivalent
Antenna Representations
Johan Malmstro¨m, Henrik Holter, and B. L. G. Jonsson
Abstract— and the complete platform in the same simulation, since the
Theaccuracyoftwoequivalentantennarepresentations,near- computational time and memory requirement grow with the
7 field sources and far-field sources, are evaluated for an antenna electrical size [5].
1 installedonasimplifiedplatforminaseriesofcasestudiesusing Simulations of electrically large platforms in combination
0 different configurations of equivalent antenna representations.
2 Theaccuracyisevaluatedintermsofinstalledfar-fieldsandsur- with complex antennas, can lead to extreme computational
face currents on the platform. The results show large variations time and memory requirement. In these cases, a less complex

n between configurations. The root-mean-square installed far-field antenna model can significantly reduce the overall complexity
a error is 4.4% for the most accurate equivalent representation.
of the simulation. Using an equivalent representation of the
J Whenusingfar-fieldsources,thedesignparametershavealarge
antenna is one way to reduce the complexity of the antenna
2 influence of the achieved accuracy. There is also a varying
1 accuracy depending on the type of numerical method used. model. An equivalent antenna representation also opens the
Based on the results, some recommendations on the choice of possibility to utilize different numerical methods in different

] sub-domain for the equivalent antenna representation are given. parts of the simulation domain, sometimes referred to as a
h
In industrial antenna applications, the accuracy in determining
hybrid method.
p
e.g. installed far-fields and antenna isolation on large platforms
- On large platforms, special techniques can be used to
arecritical.Equivalentrepresentationscanreducethefine-detail
s
s complexity of antennas and thus give an efficient numerical avoid limitations of platform size from memory requirements.
a descriptions to be used in large-scale simulations. The results One commonly used technique is asymptotic methods, e.g.
cl inthispapercanbeusedasaguidelinebyantennadesignersor physical optics (PO) or geometrical optics (GO), which are
. system engineers when using equivalent sources. high-frequency approximations. Another technique is domain
s
c IndexTerms—Electromagneticanalysis,Electromagneticmod- decompositionthatsplitalargesimulationintoseveralsmaller
si eling, Computational electromagnetics, Antenna feeds. sub-problems that are solved in parallel [6]. This has been
y usedine.g.[7–10].Adrawbackwithdomaindecompositionis
h
I. INTRODUCTION that the complexity is increased as compared to a full-domain
p
[ THEINCREASINGaccuracyoffull-wavelarge-scalesim- simulation, since information has to be exchanged over the
interface between adjacent sub-domains.
ulation of complex electromagnetic (EM) problems has
1 OnewayofdecomposingalargeEMproblemistoanalyze
v changed the industrial design-process of radio frequency (RF)
theantennasinisolation,andimprinttheresultsintheplatform
5 systems. The construction of scaled models for measurements
model. In a basic domain decomposition, multiple scattering
7 haswidelybeenreplacedbysimulations.Thisworkflowsaves
2 effects between structures in different domains are neglected,
both time and money as compared to building prototypes
3 which implies that there is no need for iterations over the
for measurements [1]. For simulations to be trustworthy, in
0 interface between adjacent sub-domains. In that case, the sub-
. particularinindustrialdesignprocesses,ana-prioripredictable
1 domainwiththeantennaservesasanequivalentrepresentation
accuracy is essential.
0 of the antenna, in the same way as in this work. When
The reason for the large interest in simulations within the
7
installing antennas on a platform, the antennas are often
1 electromagnetic community is that electromagnetic scattering
placed so that the scattering from the platform is minimized.
: problemsarerarelypossibletosolveanalytically.Problemsin-
v Therefore, neglecting multiple scattering is often a minor
volvingrealisticantennasarethereforeusuallysolvednumeri-
i
X approximation in antenna placement studies.
cally.Forsmall-scalesystems,adetailedmodelofthephysical
Commercial software for computational electromagnetics
r antenna serves as a good model for accurately determining
a (CEM), see e.g. [11–14], and recommendations for CEM has
associated fields, impedances, and currents with numerical
beenthoroughlydiscussedintheliteratureseee.g.[15].From
methods. This is documented by e.g. the annual benchmarks
an industrial perspective it is of major interest to get a highly
given by the EurAAP working group on software [2], where
accurate solution, but equally important to get a predictable
different simulation methods and measurements are compared
accuracy.Simulationsoftwaredocumentationcangivedesign-
[3].Thesimulationshaveshownanincreasingagreementwith
ers valuable rule-of-thumb recommendations about preferred
measurementsoverthelastfewyears[4].Forelectricallylarge
equivalent model configurations, but the recommendations
platforms, it is problematic to include models of the antenna
seldom give any information about the expected accuracy
when using the equivalent representation. Unfortunately, it
J. Malmstro¨m and B. L. G. Jonsson are with the School of Electrical is not at all obvious which configuration of the equivalent
Engineering,KTHRoyalInstituteofTechnology,Stockholm,Sweden.
antenna model that is most accurate.
J. Malmstro¨m and H. Holter are with Saab Surveillance, Stockholm,
Sweden.TheworkwassupportedbySaabSurveillance. Source modeling in electromagnetics has been of scientific
MALMSTRO¨Metal.:ONTHEACCURACYOFEQUIVALENTANTENNAREPRESENTATIONS 2
interest for a long time, where major contributions have been tangentialelectricandmagneticfieldsonthesurfaceΓ .Since
e
made by e.g. Halle´n, King and Harrington. Some of these the currents J, M produce zero fields inside the surface Γ ,
e
works are reviewed in e.g. [16]. These classical works focus the currents on Γ can be written as [25], [27]
e
mainly on antenna excitations for a specific type of antenna,
J = ˆn×H, (1)
e.g. wire antennas. Here, we do not model the feeding, but
ratheruseanequivalentmodelfortheentireantenna.Thetwo M = −ˆn×E, (2)
herestudiedequivalentrepresentations,near-fieldssourcesand
where E and H are the electric and magnetic fields on Γ
far-fieldsources,aregeneralinthesensethattheycanbeused e
and ˆn is the outward pointing normal to the surface Γ .
foranytypeofantenna.Theunderlyingtheoriesforthesetwo e
representations, the equivalence principle and far-field pattern It can be beneficial to use a near-field representation of an
generation, are both described in literature; see e.g. [17] and antennainsteadofamodelofthephysicalantenna.Onereason
the references within. They are also implemented in several isifthenumberofmeshcellsinthemodelincludingtheplat-
commercial software, see e.g. [11], [13]. Results from using formdecreases.Also,itopensupforusingdifferentnumerical
equivalentrepresentationsaredescribedinseveralarticles,e.g. methods when analyzing the antenna and the platform.
[10], [18–21], where near-field sources are used, and e.g. [22] Whenanantennaisinstalledonaplatform,thesurrounding
that shows examples when using far-field sources. material is non-homogeneous; the platform is typically made
The results in this paper add to the practical knowledge by of conducting or dielectric material that is surrounded by e.g.
examining the accuracy of equivalent antenna representations, air or vacuum. Using a near-field source to represent the
for different parameter configurations and different numerical antenna in a situation when the homogeneous condition is
methods. In addition to the installed far-field, which is one violated introduces an error because the surface currents are
of the most important characteristic of an installed antenna, notcorrectlydescribed.Hence,usinganear-fieldsourceunder
we also evaluate the induced surface current on the platform, thiscommoncircumstanceisanapproximation.Theerrorfrom
which is important in both EMC applications and in antenna this approximation is effected by the choice of sub-domain,
placement studies [1]. The results indicates what order of and is evaluated in Section IV-B. The generation of a near-
magnitudesoferrorsthatequivalentantennamodelsintroduce. field source is described in Section III-B1.
The results also serve as a guideline for an antenna designer
inordertochoosethemostaccurateconfigurationwhenusing
B. Far-field Sources
equivalent representations of antennas.
The paper consists of the following sections. Section II Antennaradiationpatternsarecommonlyusedtocharacter-
describethetheoryusedinthearticleandSectionIIItheused ize antennas. They describes magnitude, phase and polariza-
methods and models. The results are presented in Section IV. tion of the propagating waves generated by the antenna, for
Weevaluatetheaccuracyofnear-fieldsourcesinSectionIV-B, directions ϕ,θ, and a fixed distance r. The far-field radiation
far-field sources in Section IV-C, and their combination with pattern is the leading order behavior as r →∞ [25].
different numerical methods in Section IV-D. The paper ends A far-field radiation pattern can be used as an equivalent
with a discussion and conclusions in Section V. source in electromagnetic simulations. We will then refer to
it as a far-field source (FFS). For reciprocal antennas, it also
II. EQUIVALENCETHEORY describes the receiving characteristics of the antenna. The far-
field source is characterized by its far-field radiation pattern
Two types of equivalent antenna representations are eval-
imposedasaninfinitesimallysmallsourceatagivenposition.
uated in this paper; near-field sources and far-field sources.
This position together with the approximation of the platform
Both representations are applicable to any kind of antenna or
whengeneratingthefar-fieldsource,arethedesignparameters
other radiating structure.
for far-field sources. Their impact on the accuracy of the
solution is non-trivial, and is examined in Section IV-C.
A. Near-field Sources
Far-field sources are, compared to near-field sources, more
The equivalence principle, introduced by Love [23] and re- a primitive representation, and are not expected to perform as
fined by Schelkunoff [24], implies that any radiating structure good. On the contrary, they provide very efficient represen-
can be represented by electric J and magnetic M surface tations in numerical methods, from an implementation point
currents on an fictitious surface Γ enclosing the radiating of view. The error introduced by the far-field approximation
e
structure [25]. If the material outside Γ is homogeneous, it is small when the far-field source is far from surrounding
e
can be deduced that J and M reproduce exactly the same structure,e.g.ahornantennafeedingaparabolicreflector,see
electricandmagneticfieldsoutsideΓ astheoriginalantenna, e.g. [28]. However, far-field sources have be used also near
e
while the fields inside Γ are zero [25], [26]. A near-field structures, see e.g. [22].
e
source(NFS)isasetofsurfacecurrentsJ,M onthesurface Multipoleexpansionisaproceduretoincreasetheaccuracy
Γ , a Huygens’ surface. With the presence of a platform, the of a far-field approximation close to the source point, i.e. in
e
homogeneous requirement is not fulfilled and the near-field the near field region [27]. However, with a far-field source
source only reproduces the original fields approximately. installed on a platform, multipole expansion cannot be used,
The surface currents J and M, which we hereafter refer since it requires that there is no charge or current carrying
to as currents, are directly related to the discontinuity of the structurewithinacertaindistancefromthecenterpointofthe
MALMSTRO¨Metal.:ONTHEACCURACYOFEQUIVALENTANTENNAREPRESENTATIONS 3
100 mm D D D
0 0 0
40 mm D
1
(cid:45) Γ Γ
A 20 mm e e h
2 mm
G (a) (b) (c) (d)
Fig.2. Calculationdomains:(a)ThefulldomainD0isusedwhensolvingthe
200 mm referencecase,(b)thesub-domainD1isusedwhengeneratingtheequivalent
sources. The domain D0 is used for imprinting the equivalent sources, (c)
near-field sources on the interface Γe and (d) point-shaped far-field sources
Fig.1. TheplatformgeometryGwiththemono-coneantennaAplacedon placedadistancehabovetheflatplatformtopsurface.
itsflattop.
geometry G and the mono-cone antenna A are both rotational
expansion [27]. Multipole expansion is therefore not further
symmetric, it implies that fields and currents must have the
considered in this paper.
same rotational symmetry. It is hence sufficient to evaluate
Using a far-field source, its emitted radiation will locally
symmetric quantities on a plane (r,θ) and a fixed azimuth ϕ.
be a plane wave, irrespective of the true distance between the
far-fieldsourceandanevaluationpoint.Afar-fieldsourcecan
be used e.g. in an integral equation formulation, where the B. Calculation Domains
point source is used as a field source within the computation The following domain definitions, as in Fig. 2, are used:
domain. Implementation details of the point-shaped far-field • The domain D0 that contain the platform geometry G
source depend on the type of numerical method used, see e.g. and the antenna A.
[5], [29], [30] for general discussions. The generation of a • Asub-domainD1 thatcontaintheantennaandaselected
far-field source is described in Section III-B2.
part of theplatform. As the equivalent source isintended
to improve calculation time and memory use, the sub-
III. METHODS domainD shouldbemuchsmallerthanthedomainD .
1 0
Weuseaseriesofcasestudiestoinvestigatetheaccuracyof • AHuygens’surfaceΓe,i.e.afictivesurfaceenclosingthe
equivalentantennarepresentationsfordifferentconfigurations. antenna.TheHuygens’surfaceΓ iscompletelyincluded
e
For electromagnetic problems, where analytic solutions can in the sub-domain D .
1
rarely be found, the case study is an effective tool to compare
In a platform specific problem, e.g. with aircraft or ships, the
configurations. The aim is to determine how equivalent an- domain D is often electrically very large. However, in this
0
tennarepresentationsaffecttheaccuracyoffieldsandcurrents work, we use a comparatively small domain D . This allows
0
in the presence of a platform and to obtain an understanding
us to obtain a highly accurate reference solution for the fields
oftherelationbetweenconfigurationoftheequivalentantenna
andcurrentsfromtheinstalledantenna,whichissubsequently
representation and accuracy.
used as our benchmark. The equivalent sources can be used
on electrically large structures as well, so the methods and
A. Geometrical Model observationsareapplicabletoantennasone.g.shipsoraircraft.
We choose here a platform geometry G and an antenna A. Anantennainducescurrentsonsurroundingstructures.Such
The platform geometry G should have a simple and well currents also contribute to the equivalent source. By using
defined shape, but still retaining platform specific features the small domain D1 to determine the equivalent sources, we
for installed antennas. The chosen platform geometry G is implicitly ignore all currents outside D1. The larger the sub-
depicted in Fig. 1. The geometry is a cut 200 mm diameter domain D1 is, the smaller the error will be, but the computa-
half-sphere, where the diameter of the cut top is 100 mm. It tional cost to determine the equivalent source increases with
is perfectly electrically conducting (PEC). The geometry G is growing size of D1.
of electrically size about 7λ at 10 GHz and has the following 1) Work Flow for Using an Equivalent Near-field Source:
platform like properties: • In the sub-domain D1; Calculate the tangential electric
• Finite support of surface currents, and magnetic fields, ˆn×E and ˆn×H, on the surface
• weak back-scattering from its edges to the antenna, Γe generated by the antenna, see Fig. 2(b).
• radiating creeping waves on the curved surface. • In the domain D0; Remove the model of the physical
The antenna A chosen for this case-study is a mono-cone antenna1 and imprint the equivalent currents J =ˆn×H
antenna, also depicted in Fig. 1, with 20 mm top radius and and M =−ˆn×E on Γe, see Fig. 2(c).
20 mm height. The mono-cone antenna is mounted centrally • Solve the problem in the domain D0 using J, M as
ontheflattopsurfaceofthemaingeometrywitha2mmfeed equivalent sources.
gap.Itisfedwithawave-guideportviaa10mmlongcoaxial We use a rectangular box as surface Γ for the equivalent
e
cable.Amono-coneantennahasanisolatedfar-fieldradiation currents that define the near-fields source.
pattern similar to a mono-pole but is more wideband.
1Insteadofremovingtheantennamodel,itcanbereplacedbyasimplified
We are interested in determining fields and currents in the
modeltoincludesomeofthemultiplescatteringeffectsthatthemodelofthe
domain on and outside the geometries G∪A. Since the main physicalantennamodelwouldgive.
MALMSTRO¨Metal.:ONTHEACCURACYOFEQUIVALENTANTENNAREPRESENTATIONS 4
z
2) Work Flow for Using an Equivalent Far-field Source:
θ y
• In the sub-domain D1; Calculate the far-field radiation
pattern generated by the antenna, see Fig. 2(b). This is
ϕ
the equivalent far-field source.
• Choose a height h over the main geometry where the x
far-field source is imprinted. (a) (b)
• In the domain D0; Remove the model of the physical
antenna and solve the problem using the far-field source, Fig.3. Definitionof(a)thesphericalanglesθ andϕ,andtheirrelationto
see Fig. 2(d). theCartesiancoordinatesx,y,z,(b)thecurveusedforevaluationofcurrents.
Thecurveisparametrizedbythearclength(cid:96),where(cid:96)=0isthecenterof
The procedures described above in III-B1–III-B2 is general theflattop,and(cid:96)≈155mmistheperipheryofthebottomofthegeometry.
andappliestoanykindofstructure,notonlytheplatformand
the antenna used in this case study.
into components parallel and orthogonal to the azimuthal unit
vectorˆe ,
ϕ
C. Evaluation
H = ˆe ·H, (3)
ϕ ϕ
The results in e.g. [2–4] indicate that full-wave simulation
|H | = |ˆe ×H|. (4)
r,θ ϕ
results can be used as a benchmark of antenna behavior.
We evaluate the electric current and the installed far-field The propagating component Hϕ is dominant outside the
pattern by comparing simulations with equivalent antenna reactive region, so that we can make the approximation
representations against an accurate reference solution using H ≈ Hϕˆeϕ. With this approximation, the electric current J
the model of the physical antenna. can be calculated as
Inordertogetreliableestimatesoftheaccuracy,itiscrucial J = ˆn×H ≈ H (ˆn׈e ). (5)
ϕ ϕ
to have a highly accurate reference solution. We verify that
The rotational symmetry implies that ˆn׈e =−ˆe , whereˆe
the solution is accurate by solving the reference case with ϕ t t
is the outward pointing tangent unit vector along the curve in
threedifferentnumericalmethods;Finiteintegrationtechnique
Fig. 3(b). The tangential current J along the curve is
(FIT), Method of moments (MoM), Finite element method t
(FEM), see e.g. [5], [31], [32]. J = ˆe ·J ≈ −H . (6)
t t ϕ
When comparing results from simulations, one must keep
We denote the complex tangential current from the references
in mind that different settings, e.g. the mesh, will affect the
solution Jref((cid:96)) and when using an equivalent source J ((cid:96)).
results. The results in this paper are from simulations where t t
The scalar (cid:96) is the arc length along the curve in Fig. 3(b). We
the solver settings are not the main limiting factor for the
define the relative magnitude error δ |J ((cid:96))| and the phase
accuracy, but rather the representation of the antenna. rel t
error (cid:54) δJ ((cid:96)) of the complex current as
Two of the most important antenna quantities are installed t
|J ((cid:96))|−|Jref((cid:96))|
far-field patterns and isolation between antennas [1]. The δ |J ((cid:96))| = t t , (7)
installed far-field patterns are of large importance when de- rel t |Jref((cid:96))|
t
termining the functionality of a radio frequency (RF) system, δ(cid:54) J ((cid:96)) = (cid:54) J ((cid:96))−(cid:54) Jref((cid:96)), (8)
t t t
whichisdeterminede.g.bythecoverageforasensorinstalled
where (cid:54) (·) denotes the argument of a complex number.
on a vehicle. There can be a large difference between an
The installed electric far-field E is also separated into
antennas isolated radiation pattern, which is the radiation
its components parallel and orthogonal to the azimuthal unit
patternfortheantennainfreespace,anditsinstalledradiation
vectorˆe , so that
pattern, which is the radiation pattern when installed on a ϕ
E = ˆe ·E, (9)
ϕ ϕ
platform.Hence,asiswellknown,antennasmustbeanalyzed
|E | = |ˆe ×E|. (10)
withintheircompleteenvironment[33].Theisolationbetween r,θ ϕ
antennas is related to the risk for interference between the Due to symmetry, E = 0. On large distances from the
ϕ
antennas. It is an important quantity in antenna placement antenna, the radial component vanishes, so that
studies [1]. The isolation between antennas are mainly deter-
E ≈ E ˆe . (11)
mined by direct propagating waves and surface currents, see θ θ
e.g.[34].Theaccuracyofsurfacecurrentsisthusanimportant We denote the complex electric far-field component from the
contribution to the accuracy of isolation between antennas, in references solution Eref(ϕ,θ), and when using an equivalent
θ
particular for antennas without line-of-sight. source E (ϕ,θ). From the electric far-fields Eref(ϕ,θ) and
θ θ
The following quantities are used for evaluating the accu- E (ϕ,θ), we can evaluate both the magnitude error and the
θ
racy of the results using the equivalent sources: phase error. However, in this paper we only evaluate the
• The electric current J along the red curve in Fig. 3(b). magnitude error of the installed far-field, which is the most
• The installed far-field E(ϕ,θ) for all inclination angles commonly used quantity to classify installed antennas. We
θ =[0,180◦] and a fixed azimuth ϕ=90◦. define the relative magnitude error δrel|Eθ(ϕ,θ)| as
We use (1) to evaluate the electrical current J based on δ |E (ϕ,θ)| = |Eθ(ϕ,θ)|−|Eθref(ϕ,θ)|. (12)
the magnetic field H. The magnetic field H can be separated rel θ |Eref(ϕ,θ)|
θ
MALMSTRO¨Metal.:ONTHEACCURACYOFEQUIVALENTANTENNAREPRESENTATIONS 5
15
When calculating the current J , by using (6), we evaluate
t 10 FIT
Hϕ a small distance d>0 above the surface. The reason for m) 5 FMEoMM
this is numerical stability. The field H will be zero inside A/ 0
B
the PEC structure, and by setting d > 0 we assure that the d -5
(
evaluation points are not inside the structure. We use d = )| -10
(cid:96)
0.5 mm. By evaluating the far-field component E we will be ( -15
able to see systematic errors, which would be hθidden in the refJt -20
|
-25
normalization if e.g. the directivity was evaluated.
-30
Note that it is the field components H and E in their
ϕ θ -35
respective regions that are used as inputs to the benchmarks 0 20 40 60 80 100 120 140 160
in(7)–(8)and(12).Hence,theapproximationsin(6)and(11) Curve distance (cid:96) (mm)
do not affect the accuracy of the benchmark2. The introduced Fig.4. Referencecase:Thecurrent|Jref((cid:96))|tangentiallyalongthecurvein
t
notation,δ |J (l)|,δ(cid:54) J (l)andδ |E (ϕ,θ)|,ismotivatedby Fig.3(b),solvedwiththreedifferentnumericalmethods.Theverticallineat
rel t t rel θ
the physical relevance of these quantities. l=50mmmarkstheedgeoftheflattop.
20
D. Solvers and Simulation Settings m) 15 FFIETM
To evaluate the accuracy of the equivalent sources we need V/ MoM
B
a reference case. We use a full wave solution of the detailed (d 10
antenna model in Fig. 1 as the reference. It is thus crucial θ)| 5
that the reference solution is reliable. We here utilize two ◦,
0
9 0
observations: that commercial solvers have high accuracy [3] (
and that different numerical methods yield similar solutions. refEθ -5
|
TheequivalentsourcesaregeneratedusingFIT.Intheeval-
-10
uation of the equivalent sources, we examine their robustness, 0 20 40 60 80 100 120 140 160 180
withrespecttodesignparameters,byusingthemasexcitations Inclination angle θ (degrees)
and solve in the full domain using FIT or MoM, as well as
Fig.5. Referencecase:Installedelectricfar-fieldmagnitude|Eref(90◦,θ)|,
the asymptotic method Shooting-Bouncing-Rays (SBR). solvedwiththreedifferentnumericalmethods. θ
All simulations are performed at 10 GHz using CST Mi-
crowave Studio (MWS) [11]. The frequency is chosen so that
fields in the domain D can be calculated with full-wave best with the expected exponential decay of the current on
0
methods on a desktop computer. The size of the domains D the curved surface. Of these two solutions, with good mutual
0
and D is set so that they contain the structure of interest and agreement, the FIT solution is chosen as the reference, since
1
1.5λ of space on each side. FIT is the native solver in CST MWS.
WecanrelatesomeofthepropertiesseeninFig.4–5tothe
platform.TheslopeofsurfacecurrentinFig.4for(cid:96)>50mm
IV. RESULTS
depends on the curvature of the sphere. The double beam at
A. The Reference Solution θ ≈ 17◦, θ ≈ 78◦, and the local minimum between, of the
To examine the variability of the reference results, we use installed far-field in Fig. 5 is an effect of the flat top surface
three numerical methods, FIT, MoM and FEM, applied to the of the platform. A smaller radius of the flat top would lift the
model in Fig 1 with a model of the physical antenna. beams, i.e. decrease the inclination angle θ. The oscillations
The magnitude of the tangential electrical current, |J ((cid:96))|, for θ ∈(100◦,180◦) in Fig. 5 are caused by reflections in the
t
along the line in Fig. 3(b), is depicted in Fig. 4. The currents bottom edge of the platform G.
calculated with different methods agree well, with a relative
RMSdifferencebetweenFITandMoMcurrentsof3.4%.The B. Results for the Near-field Source Configuration
difference between FIT and FEM is 9.3% and between MoM
The realization of near-field sources depend on the choice
and FEM it is 9.9%.
ofthesub-domainD andtheplacementofthesurfaceΓ ,see
The installed far-field magnitude |E (ϕ,θ)| is depicted in 1 e
θ Fig. 2, but also on the choice of ground-plane geometry. The
Fig. 5 for the azimuthal direction ϕ=90◦ and θ ∈[0,180◦].
aim here is to evaluate the freedom of choice with respect
Again,theresultsfromthenumericalmethodsagreewell,with
to these geometrical parameters. We consider six different
a relative RMS difference between the far-fields calculated
configurationsofsurfacesΓ andgroundplanes,asdefinedin
withFITandMoMof3.1%.ThedifferencebetweenFITand e
Fig. 6, with dimensions given in Table I. The configurations
FEM is 7.4% and between MoM and FEM it is 6.2%.
evaluated are motivated briefly below.
The agreement of the results from the different numerical
Configuration (a) is generated with a thin sheet PEC plate
methods indicate that the reference solutions are accurate. We
and has non-zero currents also on the lower half of Γ , as
see in Fig. 4 that the solutions using MoM and FIT conform e
can be seen in Fig. 7(a) and Fig. 8(a). The same applies to
(e), because Γ coincide with the PEC boundary. The infinite
2For a rotational symmetric problem, as in this case, the expressions (6) e
and(11)areexact. groundplanein(b)isessentiallyestimatingthewholeplatform
MALMSTRO¨Metal.:ONTHEACCURACYOFEQUIVALENTANTENNAREPRESENTATIONS 6
TABLEI
DIMENSIONSOFSURFACESΓeFOREVALUATEDCONFIGURATIONS.
Conf. x y z
NFS(a) |x|=25mm |y|=25mm |z|=25mm
(a) (b) (c)
NFS(b) |x|=25mm |y|=25mm |z|=25mm
NFS(c) |x|=25mm |y|=25mm z=0,z=25mm
NFS(d) |x|=25mm |y|=25mm |z|=25mm
NFS(e) |x|=25mm |y|=25mm |z|=25mm
NFS(f) |x|=25mm |y|=25mm z=1mm,z=25mm
(d) (e) (f)
Fig. 7. Tangential magnetic field magnitudes |ˆn×H| on Γe for the
configurationsinFig.6.Thelogarithmiccolorscalerangesfrom0to1A/m.
(a) (b) (c)
(a) (b) (c)
(d) (e) (f)
Fig.6. Theconfigurationsofgroundplane(blue)andsurfaceΓe (yellow)
when generating near-field sources; the ground-plane is (a) a thin 100 mm
diameter PEC plate, (b) a thin infinite PEC plate, (c) and (f) a thin 50×
50 mm2 PEC plate,(d) asolid 100 mm diameter,25 mm thickPEC plate, (d) (e) (f)
(e) the part of the structure G contained inside the interface Γe. Note that
tihse1bmotmtomaboovfethtehesugrrfoaucnedΓpelacnoeinic(ifd)e. withthegroundplanein(c),whileit Fig. 8. Tangential electric field magnitudes |ˆn×E| on Γe for the con-
figurationsinFig.6.Thelogarithmiccolorscalerangesfrom0to320V/m.
TABLEII
as a ground plane, whereas (a) and (d) account for the local ROOT-MEAN-SQUAREERRORSUSINGNEAR-FIELDSOURCEANDFIT.
geometry of the platform. The diameter of the circular PEC
Configu- RMS(linearscale)
plate for the configurations (a) and (d) is 100 mm (3.3λ) and ration δrel|Jt((cid:96))|, δ(cid:54) Jt((cid:96)), δrel|Eθ(90◦,θ)|,
corresponds to the flat top of the platform G, see Fig. 1. (cid:96)∈(25,155]mm (cid:96)∈(25,155]mm θ∈(0,180◦)
Configuration (b) uses an infinite ground-plane. Hence, NFS(a) 9.2% 4.0◦ 6.0%
NFS(b) 12% 7.1◦ 4.4%
there will be no discontinuities in the surface currents on the
NFS(c) 18% 7.2◦ 8.4%
groundplane.Inconfigurations(d)and(e)theground-planeis NFS(d) 8.0% 4.5◦ 5.1%
solid, resulting in a 90◦ edge at radius 50 mm on the ground NFS(e) 19% 6.9◦ 7.4%
plane. In the other configurations, (a), (c), and (f), the ground NFS(f) 17% 3.8◦ 7.3%
planes are thin sheets, resulting in sharp edges.
One of the key features in (a) and (d) is that they capture a
larger part of the platform geometry as compared to the other model.
configurations. Configurations (c), (e), (f) all take a ground The resulting near-field source representations of the an-
plane with a side length 50 mm (1.7λ) that correspond to tenna are depicted in Fig. 7–8, for each of the configurations
the horizontal size of the equivalent surface Γ . The effect used. The strong fields on the bottom surface of (c) and (f) is
e
caused by the currents on the ground plane can be observed due to the coaxial feed cable that penetrates the surface Γe.
by comparing (c) and (f), since the surface Γ coincide with The accuracy is evaluated by the tangential current errors
e
thegroundplanein(c)whileitis1mmabovethegroundplane δ |J ((cid:96))| and δ(cid:54) J ((cid:96)), according to (7) and (8). They are
rel t t
in (f). Note that the square ground plane in configurations (c), presented in Table II as RMS errors and depicted in Fig. 9
(e), (f) does not conform to the azimuthal symmetry of the for the interval (cid:96) ∈ (25,155] mm, i.e. outside the equivalent
original problem. However, the field solutions corresponding surface Γ . The installed far-field errors δ |E (90◦,θ)|, ac-
e rel θ
to configurations (c) and (e) show less asymmetry than (f), as cording to (12), are depicted in Fig. 10 and the RMS errors
canbeseeninFig.8,possiblyduetotheeffectsoftheground for θ ∈(0,180◦) are listed in Table II.
plane that coincide with Γ in (c) and (e), but not in (f). An The best behavior in terms of δ |J ((cid:96))| is given by con-
e rel t
advantageofsettingtheground-planesizeequaltothesizeof figurations (d) and (a) with 8.0% and 9.2% relative error,
the surface Γ , as in (c), (e), and (f), is that the sub-domain respectively, whereas (c), (e), (f) are the worst, with about
e
D is minimal, leading to shorter simulation times. twice as large relative errors as (a) and (d). The correct local
1
TheworkflowdescribedinSectionIII-B1isfollowedwhen geometry of the ground-planes in (a) and (d) is important for
generatingandusingthenear-fieldsources.Sincethereference the accuracy of the surface currents.
solution was solved with FIT, we use FIT again to solve the For the installed far-field in Fig. 10, we see that the
problem with the near-field sources imprinted in the platform variation is smaller between the different configurations, as
MALMSTRO¨Metal.:ONTHEACCURACYOFEQUIVALENTANTENNAREPRESENTATIONS 7
30
region range from 4◦ with (b) to 18◦ with (c) and (f). We see
20 that the edge of the ground-plane seems to play an important
role for the accuracy in this region, where (b) performs best
10
(no discontinuity), and the configuration with a square thin
0
sheet ground-plane, (c) and (f), give the least accurate results.
%) -10 Comparing (a) and (d), we see that (d) with a solid ground-
)|( plane(90◦ edge)ismoreaccuratethan(a)withathinground-
(cid:96) -20
( plane (sharp edge). We see the same pattern when comparing
Jt NFS(a),FIT
δ|rel -30 NNFFSS((bc)),,FFIITT (thc)inwsihthee(te)g;rothuends-oplliadnegrowuinthd-aplasnhearppeerfdogrem.sTbheettseirzethaonf tthhee
NFS(d),FIT
-40 NFS(e),FIT ground plane also plays a role. A small ground plane gives
NFS(f),FIT
-50 a lifting effect of the pattern from the horizontal plane (as
0 20 40 60 80 100 120 140 160
discussed in Sec. IV-A). The large errors for (c), (e), (f) is
Curve distance (cid:96) (mm)
partly caused by this effect, where the beam maximum is
15
shifted from θ ≈78◦ to θ ≈72◦. The computationally simple
configuration (b) with an infinite ground-plane gives accurate
10
results, especially for the installed far-fields.
The expectedhigher sensitivity of thecurrents ascompared
5
es) with the far-field is clearly observed with a difference of
e
gr a factor of about two for RMS errors. Compared with the
e 0
d
( estimated RMS uncertainty in the reference solutions in Sec-
)
((cid:96) -5 tion IV-A (3.4% for the current magnitude and 3.1% for the
Jt NNFFSS((ab)),,FFIITT installedfar-fieldmagnitudes),weseeinTableIIthatthenear-
δ(cid:54) -10 NNFFSS((cd)),,FFIITT fieldsourcesincreasethecurrentmagnitudeuncertaintywitha
NFS(e),FIT factor of 2.3–5.6 and the far-field magnitude uncertainty with
NFS(f),FIT
-15 a factor of 1.4–2.7 .
0 20 40 60 80 100 120 140 160
To conclude this section, we note that the RMS errors of
Curve distance (cid:96) (mm)
the currents, is about 9% for the best case (see Table II),
Fig. 9. The relative surface current magnitude error δrel|Jt((cid:96))| (top) and withmax-normdeviationsupto±20%for(cid:96)∈(25,150]mm,
surfacecurrentphaseerrorδ(cid:54) Jt((cid:96))(bottom),fordifferentnear-fieldsource rising up to ±50% close to the bottom platform edge at (cid:96)=
configurations,seeFig.6.TheerrorsareevaluatedalongthecurveinFig.3(b).
155 mm (see Fig. 9). Similarly, the phase has about 5◦ RMS
Theverticallineat(cid:96)=50mmmarkstheedgeoftheflattopofthegeometry.
errorforthebestconfigurations(seeTableII),withmax-norm
deviation up to ±20◦ (see Fig. 9). For the installed far-fields,
25
NFS(a),FIT we note that the RMS errors are about 5% for the best cases,
20 NFS(b),FIT
NFS(c),FIT with corresponding max-norm deviations up to ±10% over
NFS(d),FIT
15 NFS(e),FIT θ ∈ [0,160◦]. We note that the RMS errors of the far-field
NFS(f),FIT
%) 10 vary with a factor of two between the most and least accurate
( configurations.
)| 5
θ
◦0, 0
9 C. Results for Far-field Source Configuration
(
θ -5
E
| When generating the far-field sources, we consider three
δrel -10 configurations, all defined in Fig. 11. From these configu-
-15 rations, we use FIT to calculate their far-field patterns. The
-20 results, which are used as far-field sources, are depicted in
0 20 40 60 80 100 120 140 160 180 Fig. 12(a) for the vertical cut ϕ = ±90◦ and Fig. 12(b)
Inclination angle θ (degrees) for the horizontal cut θ = 70◦. The inclination θ = 70◦
Fig.10. Relativeinstalledelectricfar-fielderrorsδrel|Eθ(90◦,θ)|usingthe is depicted since the three far-field patterns have similar
equivalentnear-fieldsourcesillustratedinFig.6. magnitude making it easy to compare the curves and identify
asymmetries. Note that configuration (b), with an infinite
ground plane, results in a far-field pattern that is identical to
compared to the current error in Fig. 9. This is expected due zero in the lower hemisphere, θ >90◦. Configuration (c) will
to the smoothing effect for far-fields. The smallest RMS error not preserve the symmetry in ϕ of the original problem, but
δ |E (90◦,θ)| are for (b) with 4.4% and (d) with 5.1%, the effect is small, as can be seen in Fig. 12(b).
rel θ
which are comparable with the variations of the reference TheworkflowdescribedinSectionIII-B2isusedforgener-
solutions on 3.1%. In Fig. 10 the difference between con- atingandusingthefar-fieldsources.Becauseoftherotational
figuration is particularly large in the region θ ∈ [30◦,80◦], symmetry of the platform geometry G, the far-field source is
motivating a closer study. The max-norm deviations in this placed on the symmetry axis x = 0, y = 0. In contrast, the
MALMSTRO¨Metal.:ONTHEACCURACYOFEQUIVALENTANTENNAREPRESENTATIONS 8
FFS(a),h=0mm,MoM
350 FFS(b),h=0mm,MoM
FFS(c),h=0mm,MoM
300 FFS(a),h=2mm,MoM
(a) (b) (c) FFS(c),h=2mm,MoM
FFS(a),h=4mm,MoM
250 FFS(c),h=4mm,MoM
Fig. 11. The configurations of ground plane when generating far-field %) FFS(c),h=10mm,MoM
sources;(a)athin100mmdiameterPECplate,(b)athininfinitePECplate, ( 200
and(c)athin50×50mm2 PECplate. (cid:96))| 150
(
Jt
FFS(a) | 100
θ=0 FFS(b) ϕ=0 δrel
30 30 FFS(c) 330 30 50
0
60 60 300 60
10 0−10 5 0 -50
−20 −5 -100
90 90 270 90
0 20 40 60 80 100 120 140 160
Curve distance (cid:96) (mm)
120 120 240 120 200
150 150 210 150 150
(a) 180 (b) 180 es) 100
e
Fig.12. Realizedgainoffar-fieldsourcesfortheconfigurationsillustrated gr
inFig.11,for(a)azimuthsϕ=±90◦ andinclinationsθ∈[0,180◦],and de 50
(b)azimuthsϕ=∈[0,180◦]andinclinationθ=70◦. (cid:96))( 0
(
Jt
δ(cid:54) -50 FFS(a),h=0mm,MoM
positionhontheverticalaxisisnottrivial.Comparedtoe.g.a FFS(b),h=0mm,MoM
-100 FFS(c),h=0mm,MoM
geometrical theory of diffraction (GTD) formulation [35], the FFS(a),h=2mm,MoM
FFS(c),h=2mm,MoM
source, in that case a dipole moment, can be placed both on -150 FFS(a),h=4mm,MoM
FFS(c),h=4mm,MoM
the conducting surface or above it. We investigate four cases FFS(c),h=10mm,MoM
-200
of the design parameter h=(0,2,4,10)mm, i.e. the distance 0 20 40 60 80 100 120 140 160
above the flat platform top, see Fig. 2. The resulting problem Curve distance (cid:96) (mm)
issolvedwithMoM,sinceFITinCSTMicrowaveStudio[11]
cannot handle far-field sources. Fig.13. Therelativecurrentmagnitudeerrorsδrel|Jt((cid:96))|(top)andcurrent
phase errors δ(cid:54) Jt((cid:96)) (bottom) evaluated along the curve in Fig. 3(b) using
The accuracy of the far-field sources are evaluated with the thefar-fieldsourcesinFig.11,installedonheighthabovetheflattopofG.
currenterrorsδ |J ((cid:96))|andδ(cid:54) J ((cid:96)),accordingto(7)and(8).
rel t t
These errors are depicted in Fig. 13 and listed as RMS errors
in Table III for the investigated configurations. The far-field
FFS(a),h=0mm,MoM
errorsδrel|Eθ(ϕ,θ)|,accordingto(12),aredepictedinFig.14. 200 FFFFSS((bc)),,hh==00mmmm,,MMooMM
Thegroundplanein(c)issmallerthantheflatsurfaceofG. FFS(a),h=2mm,MoM
150 FFS(c),h=2mm,MoM
Despitethat,asdepictedinFig.12,far-fieldsource(c)radiates FFS(a),h=4mm,MoM
FFS(c),h=4mm,MoM
lessinthelowerhemisphere,ascomparedwith(a)and(b).We %) 100 FFS(c),h=10mm,MoM
noteinTableIIIthatthefar-fielderrorδ |E (ϕ,θ)|issmallest (
with (c), especially for θ >90◦. This isrelsomθewhat surprising, ◦,θ)| 50
since (a) captures the local geometry of the platform better. 0
9
(
However, the asymmetry of (c) makes it less attractive to use. θ
E 0
If (b) is installed on a height h>0 mm, there are no fields δ|rel
impinging on the platform, resulting in zero currents on the -50
platform and also zero field for θ >90◦ (which is the reason
fortheomittednumbersinTableIII).Hence,whengenerating -100
0 20 40 60 80 100 120 140 160 180
a far-field source using an infinite ground plane, the resulting
Inclination angle θ (degrees)
far-field source should be installed on the platform surface,
i.e. h=0. For the other far-field sources, i.e. (a) and (c), it is Fig. 14. The relative installed far-field errors δrel|Eθ(90◦,θ)| using the
hard to give any recommendations for the value of h. far-fieldsourcesinFig.11installedonheighthabovetheflattopofG.
Compared with the estimated RMS uncertainty in the
reference solutions in Section IV-A (3.4% for the current
magnitudeand3.1%fortheinstalledfar-fieldmagnitudes),we
see in Table III that the far-field sources increase the current Itisnotable,however,thatalsotheinstalledfar-fieldsdepicted
magnitude uncertainty by a factor of 11–48 and the far-field in Fig. 14 are more inaccurate, as compared to using NFS. It
magnitudeuncertaintybyafactorof13–43(forθ ∈(0,180◦)). is also notable that the placement h of the far-field source has
Since the near-field behavior is not captured with far-field such a strong impact, see Fig. 13–14. Its influence is in same
sources, it is expected that the surface currents are inaccurate. order of magnitude as the choice of configuration.
MALMSTRO¨Metal.:ONTHEACCURACYOFEQUIVALENTANTENNAREPRESENTATIONS 9
TABLEIII TABLEIV
ROOT-MEAN-SQUAREERRORSUSINGFAR-FIELDSOURCEANDMOM. ROOT-MEAN-SQUAREERRORSFORDIFFERENTNUMERICALMETHODS
ANDEQUIVALENTSOURCES.
Configu- RMS(linearscale)
ration δrel|Jt((cid:96))|, δ(cid:54) Jt((cid:96)), δrel|Eθ|, δrel|Eθ|, RMS(linearscale)
(cid:96)∈ (cid:96)∈ θ∈ θ∈ Configuration δrel|Eθ(90◦,θ)|,θ∈(0,180◦)
h= (3,155]mm (3,155]mm (0,180◦) (0,90◦)
FIT MoM SBR
FFS(a) m 89% 52◦ 133% 44% NFS(b) 4.4% 23% 56%
FFS(b) m 163% 44◦ 77% 57% NFS(d) 5.1% 16% 57%
FFS(c) 0 70% 69◦ 46% 41% NFS(f) 7.3% 18% 65%
FFS(a) m 112% 40◦ 133% 47% FFS(a),2mm – 133% 195%
FFS(b) m – – 74% 27% FFS(b),4mm – 74% 74%
FFS(c) 2 79% 52◦ 54% 41% FFS(c),4mm – 45% 88%
FFS(a) m 105% 45◦ 127% 30% FFS(c),10mm – 40% 91%
FFS(b) m – – 74% 27%
FFS(c) 4 66% 44◦ 45% 14%
FFS(a) m 61% 61◦ 115% 31%
FFS(b) m – – 74% 27% RMS errors are 3 times higher with MoM and 9 times higher
FFS(c) 10 39% 58◦ 40% 25% with SBR, as compared with FIT.
When using far-field sources, MoM gives more accurate
results than SBR, as seen in Table IV. None of the numerical
300 NNFFSS((df)),,FFIITT methods give accurate results for θ > 90◦ with far-field
NFS(d),MoM
250 NFS(f),MoM sources, see Fig. 15. With the combination of FFS (b), h>0
NFS(d),SBR
NFS(f),SBR mm and SBR, there are no fields impinging on the platform,
200 FFS(a),2mm,MoM
%) FFS(c),4mm,MoM resulting in a zero field for θ >90◦. In Fig. 15, it is clear that
FFS(a),2mm,SBR
( 150 FFS(b),4mm,SBR thenear-fieldsourcesareanorderofmagnitudemoreaccurate
)|
◦,θ 100 than the far-field sources. Similar effects are observed in the
90 currents as seen by comparing Table II with Table III.
( 50
θ
E
δ|rel 0 V. DISCUSSIONANDCONCLUSIONS
-50 Electromagnetic simulations of antennas installed on large
platforms are challenging problems. The often complex an-
-100
0 20 40 60 80 100 120 140 160 180 tenna in combination with an electrically large platform leads
Inclination angle θ (degrees) to very high memory requirements and long simulation times.
One way to reduce the complexity is to represent the antenna
Fig.15. Therelativeinstalledfar-fielderrorsδrel|Eθ(90◦,θ)|usingthree with an equivalent model that is more effective to use in
differentnumericalmethods;FIT,MoM,andSBR.
simulations.
This paper presents one of the first accuracy studies of
equivalent sources on platforms. The presented work aims
D. Results on the Impact of the Numerical Method
toward estimating the accuracy when using two different
The above discussed results evaluate the accuracy of two equivalent representations of antennas, near-field sources and
different types of equivalent antenna representations. To de- far-field sources, installed on a simplified platform. Several
termine how the estimated accuracy depends on the choice of different configurations has been considered, with respect to
numerical method, we use the best near-field configurations, theapproximationoftheplatformandgeometricalparameters
seeFig.10,andthebestfar-fieldsource,seeFig.14,withdif- associated with the generation of the equivalent sources. The
ferent numerical methods. For near-field sources, we compare determineddeviationfromthereferencesolutionarepresented
the accuracy of FIT with the accuracy of MoM and SBR. For for each of the examined configuration of the equivalent
the far-field sources we compare the accuracy of MoM with sources. The results can be used as recommendations for
SBR. We do not consider FIT for far-field sources since it is antenna designers and engineers to choose the most accurate
not implemented in the current version of CST. equivalent source configuration.
We use a subset of the equivalent sources from previ- In agreement with previous knowledge, near-field sources
ous sections; three near-field sources; NFS (b) with best perform significantly better than far-field sources for all con-
δrel|Eθ(90◦,θ)|,NFS(d)withbestδrel|Jt((cid:96))|,andNFS(f)with figurations considered. The results with near-field sources are
best δ(cid:54) Jt((cid:96)), and three far-field sources, FFS (a) h = 2 mm oftherightorderofmagnitudeforallconfigurationsevaluated,
with best δ(cid:54) Jt((cid:96)), FFS (c) h = 4 mm with best δ|Eθ|, while the errors with far-field sources are unexpectedly large
and FFS (c) h = 10 mm with best δrel|Jt((cid:96))|. The relative for some configurations.
installed far-field errors δrel|Eθ(90◦,θ)|, defined in (12), from We observe that near-field sources, in the presence of a
these equivalent sources are calculated with different numer- platform,werecomparablyrobust,withrespecttolocationand
ical methods, FIT, MoM, and SBR. The resulting errors are size of the equivalent surface. The resulting RMS accuracies
depicted in Fig. 15 and also listed as RMS errors in Table IV. ofthebestcasesevaluatedareabout9%and4◦ forthesurface
WeseeinTableIVthat,whenusingnear-fieldsources,FIT current magnitude and phase, respectively, and about 4% for
performssignificantlybetterthanMoMandSBR.Onaverage, the installed far-field magnitude.
MALMSTRO¨Metal.:ONTHEACCURACYOFEQUIVALENTANTENNAREPRESENTATIONS 10
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technologies.Hehasauthoredover35journalandconferencepublicationsin
[14] COMSOL,“COMSOLMultiphysics,”2016. theareaofarrayantennas.Dr.HolterwastherecipientoftheIEEEAntennas
[15] T. Weiland, M. Timm, and I. Munteanu, “A Practical Guide to 3-D andPropagationSociety2003R.W.P.KingAward.
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pp.7–23,1992.
[17] ——, Antenna Theory Analysis and Design, 3rd ed. John Wiley &
Sons,2005. B.L.G.JonssonreceivedtheM.Sc.degreeinengineeringphysicsfromUmea˚
[18] L.J.Foged,L.Scialacqua,F.Saccardi,F.Mioc,D.Tallini,E.Leroux, University,Umea˚,Sweden,in1995,andthePh.D.degreeinelectromagnetic
U.Becker,J.L.AraqueQuijano,andG.Vecchi,“Innovativerepresen- theoryin2001fromKTHRoyalInstituteofTechnology,Stockholm,Sweden.
tationofantennameasuredsourcesfornumericalsimulations,”inIEEE He was a postdoctoral fellow at University of Toronto, Canada and a
AntennasPropag.Soc.AP-SInt.Symp.,2014,pp.2014–2015. Wissenschaftlicher Mitarbeiter (postdoc) at ETH Zu¨rich, Switzerland. Since
[19] H. Wang, V. Khilkevich, Y. J. Zhang, and J. Fan, “Estimating radio- 2006heiswiththeElectromagneticEngineeringLaboratoryatKTHwherehe
frequency interference to an antenna due to near-field coupling using isprofessorsince2015.Hisresearchinterestsincludeelectromagnetictheory
decompositionmethodbasedonreciprocity,”IEEETrans.Electromagn. inawidesense,includingscattering,antennatheoryandnon-lineardynamics.
Compat.,vol.55,no.6,pp.1125–1131,2013.