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An Open-Source Code for the Calculation of the Effects of Mutual Coupling in Arrays of Wires and for the ASM-MBF Method

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Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2010, Article ID 137903, 10 pagesdoi:10.1155/2010/137903
Research Article
AnOpen-SourceCodeforthe Calculationofthe EffectsofMutualCouplinginArraysofWiresand fortheASM-MBF Method
Christophe Craeye,
1
Belen Andr´esGarc´ıa,
2
EnriqueGarc´ıaMu˜noz,
2
andR ´emi Sarkis
1
1
Communications and Remote Sensing Laboratory, Universit ´e Catholique de Louvain, Place du Levant, 2,1348 Louvain-la-Neuve, Belgium
2
Department of Signal Theory and Communications, Carlos III University of Madrid, Avenida de la Universidad 30, 28911 Leganes, Spain
Correspondence should be addressed to Christophe Craeye, christophe.craeye@uclouvain.beReceived 31 October 2009; Accepted 26 February 2010Academic Editor: Hon Tat HuiCopyright © 2010 Christophe Craeye et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal work is properly cited.A general description of mutual coupling in ﬁnite and inﬁnite antenna arrays is provided and an open-source code is described forthe analysis of mutual coupling in linear arrays of parallel dipoles. The ASM-MBF method is illustrated with the help of that codeand it is shown that the method also can handle situations where eigenmodes are supported by the array: single machine precisionis achieved with four Macro Basis Functions only.
1.Introduction
Mutual coupling in arrays has been a subject of intenseresearch for several decades. Regular arrays received impor-tant attention for radar design purposes, and the subjectreceived more attention recently for civilian applications, likeMIMO communication systems and observational systems;among the latter phased arrays devoted to radio astron-omy, ground-penetrating radar and medical imaging. Thepresence of several antennas within a limited volume leadsto mutual interactions that have been described in seminalreferences like [1–3].
As for antenna arrays devoted to transmission orreception with respect to very large distances, the mostimportant characteristics of arrays propably correspond totheir embedded element patterns and to their impedancematrix, or any equivalent quantities. The purpose of thepresent paper consists of providing a general understandingof mutual coupling through the speciﬁc case of regular arraysmade of wire antennas. Beyond that, a method for the fastcalculation of those quantities, the Macro Basis Functions(MBF) approach based on the Array Scanning Method(ASM), therefore named “ASM-MBF” will be described andillustrated. We believe that this method reconciles very wellfast ﬁnite-array approaches with inﬁnite-array approxima-tions. Open-source codes [4] in Matlab
language for ﬁnitearrays, inﬁnite arrays and for the ASM-MBF approach willbe described. Examples will be given for both regular casesand for situations where the array supports eigenmodes.Although related work has been published by the authorsconsidering more complex antennas [5, 6], it is expected that
the open-source codes for the simpler case of wire antennaswill form a good way to introduce graduate students andresearchers in the ﬁeld of signal processing to the importanceof mutual coupling and that further insight will be gained inthe e
ﬀ
ects of eigenmodes in antenna arrays.The remainder of this paper is organized as follows: inSection 2, a particular point of view on mutual coupling isgiven; in Section 3, the calculation of the e
ﬀ
ects of mutualcoupling for arrays of wires with the Method of Momentsis shortly recalled and an open-source code is introduced; inSection 4,theASM-MBFmethodisillustratedanddetailsaredescribed with the help of the same code.
2.GeneralConsiderationsaboutMutualCoupling
Mutual coupling is a relatively general concept that may beunderstood in di
ﬀ
erent ways. The most general deﬁnitionof mutual coupling probably corresponds to the process
2 International Journal of Antennas and Propagationthrough which the properties of a given radiator—mainly its input impedance and radiation pattern—are modiﬁedwhen another object is brought in the near ﬁeld of theradiator. In this particular (or rather “general”) perspective,mutual coupling between several antennas is not so di
ﬀ
erentfrom mutual coupling between an antenna and a supportingelement (the circuit to which the antenna is connected, thebody of a person wearing the antenna, for instance), as issketched in Figure 1. A possible interpretation is that thesupporting element is part of the antenna, which thereforehas other radiation properties than the isolated antenna. Itis di
ﬃ
cult to say which is the minimal distance from whichmutual coupling becomes negligible. In general, one willassume distances comparable to the wavelength. The e
ﬀ
ectof objects located further away will in general be accountedforinadi
ﬀ
erentway,forinstancewiththehelpofray-tracingor multiple-scattering approaches.The object (see cup in Figure 1(b)) located in the nearﬁeld of the (original) antenna can change the radiationpattern, absorb a fraction of the radiated power and alsochange the input impedance of the radiator. These phenom-ena can be understood as resulting from the change of theelectric (and magnetic, if volume or surface equivalence areused) current distribution on the antenna itself and fromthe appearance of currents on the perturbating object. Bothchanges in current distributions can be regarded as due tothe introduction of new boundary conditions, imposed by the presence of the nearby object. In traditional antennaterminology, these changes in current distribution willimpact the antenna input impedance, its radiation patternand its radiation e
ﬃ
ciency. As for the latter quantity, whenone involves the power absorbed by the nearby object inthe estimation of the e
ﬃ
ciency of the antenna, one implicity assumes the object as part of the radiator. It seems reasonabletosaythatmutualcouplingcanbeneglected,orcouldatleastbe treated with simpliﬁed multiple-scattering approaches,when the impact of the neighboring object on the antennainput impedance is negligible.Let us consider now the case where the near-ﬁeld objectsreferred to above are made of one or several other radiators(see second antenna in Figure 1(c)). Following the samephilosophy as above, when considering the properties of agiven antenna in the array, the other antennas in the array,terminated by the impedance of their transmitter or receiver,can momentarily be considered as part of that particularantenna. The obtained radiation pattern then correspondsto the
embedded element pattern
and the input impedanceof the antenna in the array (sometimes called the
passive
impedance) can be substantially di
ﬀ
erent from that of theradiator taken in isolation. In the case of arrays of radiators,obviously, coupling coe
ﬃ
cients can also be deﬁned, in termsof impedance, admittance or S-parameter matrices. It isimportant to notice, however, that all the e
ﬀ
ects of mutualcoupling cannot in general be described with the help of such matrices only. Minimum-scattering antennas form anexception to this rule [7]. Nevertheless, several links canbe established between the radiation properties and the N-port coupling coe
ﬃ
cients. A link between
S
-parameters andradiation patterns is given in [1, 8], while in [9–11], the link
between element patterns and the active input impedance of antennas in inﬁnite periodic arrays is established.Mutual coupling in large arrays is the subject of intensee
ﬀ
orts from the numerical point of view. In the following,details are provided for the case of dipole antennas, andwe show how very high accuracy can be achieved wheninﬁnite-array solutions are incorporated in the analysis of ﬁnite arrays. All explanations are accompanied by a shortintroduction to the open-source codes provided in parallelwith this paper and allowing the reader to become familiarwith the di
ﬀ
erent aspects of mutual coupling.
3.MoMfor the ArrayofWires
This section brieﬂy recalls the Method of Moments for theanalysis of linear arrays of parallel thin wires, of radius
a
smaller than about a hundredth of the wavelength, spacedby a distance
d
. This mehod has been devised severaldecades ago [12, 13], but it is recalled here, with the help
of some details given in Appendix B, such that the readercan understand the open-source Matlab
code [4] andadapt it by himself to other situations (conﬁgurations withground planes, planar arrays
...
). This section also allowsus to introduce the concepts necessary to the understandingof the ASM-MBF method described in the next section.The traditional Pocklington method for thin wires is used.Currents are represented by triangular basis functions,
J
i
,located on the
z
axis of the wires:
J
(
z
)
M i
=
1
x
i
J
i
, wherethe
x
i
are the unknowns. Fields are tested on the externalpart of the wire, with the help of the same set of functions(displaced by the radius of the wire). The element of theMoM impedance matrix corresponding to basis function
J
j
(
z
) and testing function
J
i
(
z
) can be written as:
Z
ij
=
1
jω
G
(
z
,
z
)
k
2
J
i
(
z
)
J
j
(
z
)
−
dJ
i
(
z
)
dz dJ
j
(
z
)
dz
dz
dz
(1)where
ω
is the radian frequency,
k
is the wavenumber,
is the permittivity of surrounding space and
G
=
exp(
−
jkR
)
/
(4
πR
) is the free-space Green’s function, with
R
=
(
z
−
z
)
2
+
a
2
if basis and testing functions are locatedon the same wire, and
R
=
(
z
−
z
)
2
+
e
2
if they arelocated on di
ﬀ
erent -parallel- wires, with
e
=
pd
thedistance between centers of the wires, where
p
is integer.The excitation vector
v
, that is, the right-hand side of thesystem of equations, is zero everywhere, except for the entry corresponding to the testing function that overlaps the delta-gap source. That entry equals
−
V
, where
V
is the excitationvoltage. When the excitation is represented by a voltagesource
V
followed by a series load
Z
L
, the voltage jump atthe delta-gap level is equal to
V
1
=
V
−
Z
L
x
s
, where
x
s
is the current coe
ﬃ
cient that multiplies the basis functionoverlapping the source. Since this coe
ﬃ
cient belongs to theunknowns, the corresponding term can be moved to the left-hand side. This leads to subtraction of
Z
L
from
Z
s
,
s
. Finally,the current coe
ﬃ
cients
x
are obtained by solving the
Zx
=
v
International Journal of Antennas and Propagation 3
Z
in
∼
(a)
Z
in
The antenna
∼
(b)
Z
in
The antenna
∼
(c)
Figure
1: Mutual coupling between an antenna and an object located in its vicinity. E
ﬀ
ects on radiation pattern and input impedance: (a)isolated antenna; (b) antenna with object in its vicinity; (c) antenna and other antenna in its vicinity.
system of equations. The most important part of the work consists of computing the double integrals (1).In the
wiremom-finite.m
code, this is carried out by the inter and inner0 routines. The integrations are carriedout with
N
points over the interval of length
dl
, whichcorresponds to the width of the basis function. For a giventesting function, variables
jr
and
djr
represent the currentsand their derivatives versus
z
, while they are named
jrp
and
djrp
for the basis functions. The integrals are carried outwith the trapezoidal rule, the variables
w
and
wp
representthe corresponding weights.It is well known that, for very thin wires, the integralswill not be accurate [12], because of the nearly-singular(The function is not perﬂectly singular since
R
≥
a
.)behavior of the Green’s function. Hence, when basis andtesting functions are very close to each other, it is advised toextract the singularity from the Green’s function and to carry out separately the integrals for the singular part, 1
/
(4
π R
),of the Green’s function (in the
inter
routine,
extract
=
1indicates that extraction is to be carried out; the criterionis based on proximity between basis and testing functions).The integration is carried out numerically over the testingfunction and analytically over the basis function. In the
wiremom-finite.m
code, those operations are organizedas follows. First, the numerical integration over the testingfunction is carried out in the same
inter
routine. Theintegration over the basis function is carried out in the
inner1
routine, using the Green’s function from which thesingularity has been extracted. Besides this, for the singularpart, the analytical integration over the basis function iscarried out with the help of the
inner2
routine. Detailsregarding this integration are given in Appendix B. In the
inter
routine, the contributions from the regular andsingular parts of the inner integrals are represented by theint1 and int2 variables, respectively.Finally, it is interesting to notice that the main code of
wiremom-finite.m
makes use of the regularity of the basisfunctions over a given antenna and of the regularity of thearray by exploiting the fact that the interactions (1) only depend on the vector distances between basis and testingfunctions. For a given pair of antennas, such interactionsare ﬁrst stored in the
z
vector (Here, regularity along thewire is exploited.) and then reorganized into the
zz
matrix.Such calculations are carried out for all relative positionsbetween pairs of antennas (Here, regularity of the array isexploited.) and the
zz
matrices are accordingly organizedinto the
Ztot
impedance matrix for the whole array. Thesolution is then computed for excitation on successiveantennas to obtain, for each excitation, all the currents onthe antennas. Those currents, in turn, allow the computationof the array impedance matrix and of all embedded elementpatterns. The patterns are very easy to compute in the planeperpendicular to the antennas, where contributions fromdi
ﬀ
erent segments of a given antenna appear in phase. In thiscase, for an array of
N
a
elements, the embedded pattern isproportional to:
F
(
θ
)
=
N
a
n
=
1
M
j
=
1
x
n
,
j
e
jkd
(
n
−
1)cos
θ
(2)where
θ
is the angle with respect to the array axis and
x
n
,
j
refers to the coe
ﬃ
cient that multiplies basis function
j
onantenna
n
. Figure 2 shows embedded element patterns forelements 1, 2, 4 and 8 in arrays of 17 elements made of wiresof 1mm radius and with spacings of 15cm. The ﬁrst examplehas termination impedances of 100Ohm and the wire lengthand wavelength are 15cm and 30cm, respectively. Thesecond example has 0Ohm termination impedances, whilethe wire length and wavelength are 30cm and 66cm, respec-tively.Thelattersupportseigenmodes,whichsrcinatesfrommutual coupling between elements of the array and whichleads to complex patterns, mainly oriented along the array axis. In both cases, 21 triangular basis functions have beenused on each antenna. The impedance matrix (and fromthere also, the scattering matrix) of the array can be obtainedfrom currents calculated for all excitations. Let us denote by
Y
t
the matrix whose column
i
contains the port currentsfor excitation of port
i
with a unit voltage. Then, the array impedance matrix is given by
Z
=
Y
−
1
t
−
UZ
L
where
U
in aunit matrix and
Z
L
is the termination impedance referred to
4 International Journal of Antennas and Propagation
210240270300330030600
.
1901201501800
.
05(a)210240270300330030600
.
6901201501800
.
40
.
2(b)
Figure
2: Radiation patterns of elements 1(-), 2(:), 4(.-) and 8(- -) in arrays of 17 elements. (a) without eigenmodes, (b) with eigenmodes.The horizontal axis corresponds to the array axis.
above (When terminations are not all the same,
UZ
L
shouldbe replaced by a diagonal matrix with the
i
th diagonal entry equal to the
i
th termination impedance.).The validation of the code, illustrated in Appendix A,has been carried out in two di
ﬀ
erent ways. At the elementlevel, the input admittance has been compared with resultsprovided by the NEC2 free software as provided by [14], aswell as with results shown in [15]. At the array level, thee
ﬀ
ects of mutual coupling have been veriﬁed by establishingthe balance between power accepted by a given port and thesum of radiated and dissipated powers. The radiated power isobtained through integration over the unit sphere of the 3-Dpower density pattern. The dissipated power corresponds topower delivered to the loads terminating the other elementsof the array; in dense arrays, it can correspond to a largefraction of the delivered power. In all cases, the relative errorwas less than 0.1 percent.
4.FastCalculationswiththe ASM-MBF Method
4.1. Inﬁnite-Array Simulations.
For inﬁnite regular arrayswith uniform amplitude and linear phase excitation, themethod above can be extended such that calculations canbe limited to a unit cell of the array, while involving all thee
ﬀ
ects of mutual coupling. This is done with the help of the introduction of an inﬁnite-array Green’s function. Fora linear array, with inter-element phase shifts equal to
ψ
, itreads:
G
(
R
)
⇒
∞
n
=−∞
exp
−
jkR
n
4
πR
n
e
−
jnψ
(3)where
R
n
=
(
a
+
d n
)
2
+ (
z
−
z
)
2
. For
n /
=
0, in view of the thin wire approximation, the term
a
can be omitted aslong as
d
a
. Other formulations for this Green’s function,with faster convergence, can be found in the literature [16,17]. Indeed, as such, formulation (3) converges very slowly.
However, here, the extra computation time required for theintroduction of the inﬁnite-array Green’s function is mademarginal by (
i
) using series accelerators, like the Levin-Taccelerator [18] and (
ii
) by tabulating the Green’s functionbefore ﬁlling the MoM impedance matrix. In the case of arraysofparallelwires,thetabulationcanbelimitedtoaone-dimensional table, since the inﬁnite-array Green’s functiondepends only on the
z
−
z
coordinate. To be more precise, itisinterestingtonoticethat,intheGreen’sfunctionabove,the
n
=
0termcorrespondstothecaseoftheisolateddipole. Thecorresponding contribution to the inﬁnite-array impedancematrix can hence be recuperated from the code referred to inthe previous section. The remainder is then computed usingthe Green’s function (3) made non-singular by withdrawingthe
n
=
0 term from it. The latter point also greatly facilitatesthe tabulation and interpolation steps.Inﬁnite-array simulations can be carried out with thehelp of the
wiremom-infinite.m
code [4]. With respectto the
wiremom-finite.m
code, the di
ﬀ
erences are thefollowing. First, the solution is looked for in the unit cell only and the result depends on phase shift
ψ
between consecutiveelements. Second, the inﬁnite-array Green’s function istabulated in subroutine
table1
. It is computed as theexplicit summation of contributions from successive dipoles.The summation is accelerated with the help of the Levin-T accelarator [18]. It is important to notice that, for thelatter to be e
ﬃ
cient, the summation must be carried outindependently for the two semi-inﬁnite arrays that composethe inﬁnite array. The contribution from the closest dipole isexcluded and added when necessary in the inner0 and inner1routines. As in the code for ﬁnite arrays, those routinescalculate the interactions over basis functions, for the casewhen singularity is extracted and for the case where it is notextracted, respectively. The Green’s function is interpolatedfrom the table using a simple ﬁrst-order approach.
International Journal of Antennas and Propagation 5
5
.
45
.
65
.
866
.
26
.
46
.
6
×
10
−
3
0 5 10 15 20
R e a l p a r t o f p o r t c u r r e n t
Element index (a)
−
4
−
2024681012
×
10
−
3
0 5 10 15 20
R e a l p a r t o f p o r t c u r r e n t
Element index (b)
Figure
3: Real part of port currents in array of 17 dipoles excited in phase. Horizontal lines: inﬁnite-array simulations. (a) withouteigenmodes; (b) with eigenmodes.
Table
1: For two types of arrays, with 8 elements each, powerdelivered to the array at port 1, radiated power, power disspiated inloads of other elements and sum of radiated and dissipated powers.Units are milliWatts.Accepted Radiated Dissipated Rad. + Diss.Type 1 1.2129 1.1542 0.0586 1.2128Type 2 14.154 14.142 0 14.142
A simulation example is provided in Figure 3 for thesame two conﬁgurations as in Section 3, the second of which supports eigenmodes. All elements are excited, withconstant amplitudes and constant phases. What is shownis the real part of port currents. Important di
ﬀ
erences areobserved between ﬁnite and inﬁnite-array solutions. This isespecially true for the array supporting eigenmodes. In thelatter case, it is interesting to notice that eigenmodes a
ﬀ
ectthe solution of the ﬁnite array, even if, in the inﬁnite array,the eigenmode solution appears only for other inter-elementphase shifts [19]. This is because the ﬁnite-array solutionactuallyinvolves a broad spectrum (in terms of inter-elementphaseshifts)ofinﬁnite-arraysolutions.Thisfactisexplicitely exploited in the ASM-MBF method described below.
4.2. Array Scanning Method.
The Array Scanning Method(ASM) allows the determination of the behavior of anactive antenna (or any single-source excitation) in an inﬁnitepassively terminated array from results obtained for fully periodic array excitations, while accounting for all the e
ﬀ
ectsof mutual coupling. The ASM is connected with the DiscreteFourier Transform and has ﬁrst been proposed by Munk and Burrell [20], although DFT-type links between single-element excitation and periodic excitation were already reported in [2]. It will be recalled here for the case of linear arrays. We consider an array with periodicity
d
alongthe
x
direction with periodic excitation at the feed-pointslevel, that is, with constant amplitudes and linear phaseprogression; the inter-element phase shift is given by
ψ
. If wedenote a given ﬁeld in the reference unit cell, which containsthe coordinate
x
=
0, by
−→
P
∞
(
−→
r
,
ψ
), then the ﬁeld obtainedin cell
m
when only the source in the reference unit cell isexcited is given by:
−→
P
0
m
−→
r
=
12
π
2
π
0
−→
P
∞
−→
r
,
ψ
e
−
jmψ
dψ
(4)where
−→
r
are coordinates relative to a reference pointconnected to each antenna. A discretized version of thisintegral is:
−→
P
0
m
−→
r
1
N
N
−
1
n
=
0
−→
P
∞
−→
r
,
ψ
n
e
−
jmψ
n
for 0
≤
m
≤
N
−
1(5)with
ψ
n
=
n/N
2
π
. It is interesting to notice that thissummation exactly has the same form as the FFT, as deﬁnedin the Matlab
language for instance. The result of thediscretization of integral (4) is that the obtained ﬁeldsactually correspond to those excited by sources located every
N
cells along the array. In other words,
−→
P
0
m
(
−→
r
) becomesperiodic along
m
with period
N
, as illustrated in Figure 5. Toavoid this aliasing problem, the solution may consist of usingadaptive integration techniques [21]. Another possiblity consists of artiﬁcally increasing the number of samplingpoints by increasing
N
and by interpolating them where thefunction
−→
P
∞
(
−→
r
,
ψ
) can be assumed smooth. Among others,particular attention (and hence, many explicitly computedvalues of ﬁelds
−→
P
∞
) will be given to the estimation of
−→
P
∞
near the limits of visible space (
ψ
= ±
kd
) and near somepossible values of
ψ
where the impedance matrix becomes

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