A Fast Analysis of Scattering and Radiation of Large Microstrip Antenna Arrays
2218IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,VOL.51,NO.9,SEPTEMBER 2003
A Fast Analysis of Scattering and Radiation of Large
Microstrip Antenna Arrays
Ning Yuan,Tat Soon Yeo ,Fellow,IEEE ,Xiao-Chun Nie,and Le Wei Li ,Senior Member,IEEE
Abstract—An accurate and efficient method that combines the precorrected fast Fourier transform (FFT)method and the discrete complex image method (DCIM)is presented to charac-terize the scattering and radiation properties of arbitrarily shaped microstrip patch antennas.In this method,the mixed potential integral equation (MPIE)is discretized in the spatial domain by means of the discrete complex image method.The resultant system is solved iteratively using the generalized conjugate residual method (GCR)and the precorrected-FFT technique is used to speed up the matrix-vector multiplication.The precor-rected-FFT eliminates the need to generate and store the usual square impedance matrix and thus leads to a significant reduction in memory requirement and computational cost.Numerical results are presented for arbitrarily shaped microstrip antenna arrays to demonstrate the accuracy and efficiency of this technique.Index Terms—Discrete complex image method,microstrip an-tenna array,precorrected fast Fourier transform (FFT)method,scattering and radiation.
I.I NTRODUCTION
T
HE spatial-domain method of moment (MoM)has long been the corner-stone in the analysis of electromagnetic scattering of arbitrarily shaped objects.However,numerical solution of the MoM matrix equation
requires
memory to store the matrix
elements,
where
computation is needed per iteration in the latter,the
number of iteration needed could still be equal to,or larger
than,
pr e cor r e ct e d-F FTmet h od Whi t e [17],[18]t o s o l v eel beenext e ndedt o s o l v es c Ni e etal .[19].I n t h i s pap al g or i t h mt o handl e pr o bl mi x ed-p ot e nt i a li n t e gr a le t h eRao–Wi l t o n–Gl i s o n(
YUAN et al.:FAST ANALYSIS OF SCATTERING AND RADIATION OF ANTENNA ARRAYS 2219
than AIM.The P-FFT method generates the projection opera-tors between the uniform grid and irregular meshes via matching the vector and scalar potentials while AIM generates projections by equating a finite number of multipole moments of the basis functions.For AIM,the error of the matrix element caused by the use of auxiliary basis functions is controlled by the Taylor series remainder,which requires a restriction on the expansion
order
th-order polynomial within the support of
the basis function and within the expansion box.On the other hand,
increasing
,,
and
for
and
(1)
where
is the unit normal vector of the patch
surface
(2)
where
is assumed and suppressed.
B.Green’s Functions
The spatial-domain Green’s
functions
components are used in the solution of
(2).These two components are denoted here
as
and
(7)
and applying the 2-D divergence theorem,the Galerkin’s method yields a matrix
equation
(9)
2220IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,VOL.51,NO.9,SEPTEMBER 2003
where
represent the testing and basis functions,
and denote their supports respectively.The elements
of
can be computed
by
storage.
The linear system in (8)can be solved via either a direct or an iterative method.A direct scheme
requires
operations per iteration.
These requirements will render the method computationally in-tractable when the microstrip structure is electrically large.In this paper,the precorrected-FFT method is used to reduce these
computational requirements
to
and 8uniform grid.It should be
noted that the number of grid points is required to be a mul-tiple of two.The triangular elements are then sorted into cells formed by the grid,with each cell containing only a few trian-gular elements.Equivalent point sources are placed at the cell
vertices
.Assume a RWG basis
function th edge is contained within a given
cell
.We first consider the pro-jection of the currents.The vector potential produced by
the
denotes the vector potential at
the
denotes that due to the
grid currents,which can be computed by,
respectively
(13)
Here,
and
are the positions of
the
is the Dirac delta
function,
th grid point.
Substituting (12),(13)into (11)for
all
YUAN et al.:FAST ANALYSIS OF SCATTERING AND RADIATION OF ANTENNA ARRAYS
2221
Fig.2.Geometric parameters of a series-fed microstrip antenna array L= 10:08mm,W=11:79mm,L=13:4mm,L
=3:93mm,d
=2:1,f=9:42GHz.
where
are the
same for each
cell.are the mappings between
patch currents and test-point vector potentials and are given
by
(16)
.
Since the collocation in(14)is linear in the patch and grid cur-
rent distributions,the contributions of
the
to,
given
by
(17)
where denotes
the in-
dicates the generalized Moore–Penrose inverse
of.Since
these matrices are small and are the same for each cell,the rel-
ative computational cost for this operation is insignificant.By
using the
vectors
.For any patch cur-
rent
are generated by summing over all the cur-
rents in the cell.Patch currents outside
cell
(20)
Here,
is the
mapping between patch charges and test point scalar potentials.
With the projection operator in(18),we can project the element
charges unto a uniform grid of point
charges.
(a)
(b)
Fig.3.E plane radiation patterns of a series-fed microstrip antenna array:
(a)L=14:6mm.
The accuracy of the above projection scheme depends on the
proper selection of the test
points
,we
choose9points uniformly located on a circle containing the cell
as test points.
2)Computation of Grid Potentials Using FFT:Once the
currents and charges have been projected onto a rectangular
grid,the relationship between the vector/scalar potential at each
grid point and the grid currents/charges is in fact a convolution.
The convolution can be rapidly calculated by using the discrete
FFT.So the vector and scalar potentials at the grid points can
be computed
by
2222IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,VOL.51,NO.9,SEPTEMBER 2003
TABLE
I
TABLE
II
It should be noted
that
to the vector and scalar potentials
are then given
by
(23)
for
each
cell
(27)
where
,which are
inaccurate.
is the in-dices of the set of cells which are “close”to
cell
.Because for each
cell is a small set
and each
matrix
,
and
.
The requirement of the FFT
is
,
where
because the overlaid grid can be
much coarser than the original mesh.But for very large objects which are not
flat,
YUAN et al.:FAST ANALYSIS OF SCATTERING AND RADIATION OF ANTENNA ARRAYS
2223
Fig.4.Geometry of a 824microstrip corporate-fed antenna array,L =
10:08mm,W =11:79mm,d
=3:93mm,L =18:48mm,D =22:40
mm.
Fig.5.Radiation patterns of the 824microstrip corporate-fed antenna array.
A.Example A
As the first example,we consider a four-element series-fed microstrip antenna array,which is fed at the left end.The an-tenna was proposed by Wu et al.[27]in 1991.The geometric parameters are taken from [13]and reproduced in Fig.2.In this example,the delta voltage source is applied at the excitation port so that the right-hand side
vector of (8)is zero everywhere,except at the excitation edge.The E plane radiation pattern at
9.42
grid spacing
and
a
=2:17,d =1:58mm,
f =3:7GHz.(a)323array,(b)727array,(c)11211array.
tion of the P-FFT system remain essentially unchanged from the original MoM system.This is of critical importance for fast iterative solutions since an increase in the iteration count could annul the faster computation of the matrix-vector product.Even though the geometries in this example are still rather small to demonstrate the full impact of the P-FFT,we can already see the efficiency of this fast algorithm.B.Example B
The next example is a
8
and
thickness
2224IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,VOL.51,NO.9,SEPTEMBER 2003
TABLE III
R ESOURCE R EQUIREMENTS OF THE C ONVENTIONAL
MoM
TABLE IV
R ESOURCE R EQUIREMENTS OF THE
P-FFT
MoM,the memory requirement is about 600MB and the CPU time per iteration is 22.2seconds on a Pentium 1G PC.However,the P-FFT uses only 15MB of memory and takes 5.71seconds per iteration when a grid spacing
of
directions is 55.517mm.The geometry is shown in Fig.6.The
monostatic
RCS
for a
311array comprised of the same
elements at 3.7GHz is shown in Fig.7(c).Good agreement is observed between our results and those obtained by King and Bow [29].
The resource requirements of the conventional MoM and pre-corrected-FFT method of these examples are listed in Tables III and IV .A grid spacing
of
7array which
has 10878unknowns,the MoM is estimated to need 902.79MB memory and require 66.7hours to produce the solution for 90incident angles.On the other hand,the P-FFT needs only 2.7%of the MoM memory and takes 12.8hours of computation time.For the
11
3array when a threshold distance
of
are used.It is observed that even a relatively large
grid
spacing
and the grid
spacing varies
from
,
,
and
YUAN et al.:FAST ANALYSIS OF SCATTERING AND RADIATION OF ANTENNA ARRAYS2225
(a)
(b)
Fig.8.Investigation of the impact of the grid spacing and threshold distance
(a)RCS(
2226IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,VOL.51,NO.9,SEPTEMBER 2003
[26]N.G.Alexopoulos and D.R.Jackson,“Fundamental superstrate (over)
effects on printed circuit antennas,”IEEE Trans.Antennas Propagat.,vol.32,pp.807–816,Aug.1984.
[27]K.L.Wu,M.Spenuk,J.Litva,and D.G.Fang,“Theoretical and experi-mental study of feed network effects on the radiation pattern of series-fed microstrip antenna arrays,”Proc.Inst.Elect.Eng.Microwave,Antennas Propagation ,pt.H,vol.138,no.3,pp.238–242,June 1991.
[28] F.Ling and J.M.Jin,“Scattering and radiation analysis of microstrip an-tennas using discrete complex image method and reciprocity theorem,”Microwave Opt.Technol.Lett.,vol.16,no.4,pp.212–216,Nov.1997.[29] A.S.King and W.J.Bow,“Scattering from a finite array of microstrip
patches,”IEEE Trans.Antennas Propagat.,vol.40,pp.770–774,July
1992.
Ning Yuan received the B.Eng.and M.Eng.degrees in electrical engineering from University of Elec-tronic Science and Technology of China,Chengdu,China,in 1993and 1996,respectively,and the Ph.D degree in electrical engineering from Xidian University,in 1999.
From September 1999to July 2000,she worked as a Postdoctoral Fellow in Telecommunications and Industrial Physics,CSIRO,Sydney,Australia.Since August 2000,she has been working as a Research Fellow in the Department of Electrical and Computer
Engineering,National University of Singapore.Her main research interests in-cludes numerical method in scattering,microwave circuits and
antennas.
Tat Soon Yeo (M’79–SM’93–F’03)received the B.Eng.(Hons I)degree from the University of Singapore,in 1979,the M.Eng.degree from the National University of Singapore,in 1981,and the Ph.D.degree from the University of Canterbury,New Zealand,in 1985under a Colombo Plan Scholarship.
Currently,he is a Professor and Director of Radar and Signal Processing Laboratory,Director of An-tennas and Propagation Laboratory in the Department
of Electrical Engineering,National University of Singapore (NUS).He is also the Director of Temasek Defence Systems Institute,a teaching institute estab-lished jointly by NUS and US Naval Postgraduate School (NPS).His current research interests are:scattering analysis,synthetic aperture radar,antenna and propagation study,numerical methods in electromagnetics,and electromagnetic compatibility.
Dr.Yeo was the recipient of the Ministry of Defense—National University of Singapore 1997Joint R&D Award and the IEEE Millennium Medal in 2000.He is the past-Chairman and a current Executive Committee Member of the MTT/AP and EMC Chapters,IEEE Singapore Section,and the Chairman of Singapore EMC Technical
Committee.
Xiao-Chun Nie received the B.Eng.,M.Eng.,and Ph.D.degrees in electrical engineering from Xi’an Jiaotong University,Xi’an,China,the University of Electronic Science and Technology of China,and Xidian University,Xi’an,China,in 1988,1993,and 2000,respectively.
From 1993to 1997,he was a Lecturer in the University of Electronic Science and Technology of China,Chengdu,China.Since September 2000,he has been working as a Research Fellow in Singapore-MIT Alliance,National University of
Singapore.His main research interests include numerical analysis of scattering,radiation problems,microwave circuits and antennas.
Le Wei Li (S’91–M’92–SM’96)received the B.Sc.degree in physics from Xuzhou Normal University,Xuzhou,China,the M.Eng.Sc.degree in electrical engineering from China Research Institute of Radiowave Propagation (CRIRP),Xinxiang,China,and the Ph.D.degree in electrical engineering from Monash University,Melbourne,Australia,in 1984,1987,and 1992,respectively.
In 1992,he worked at La Trobe University (jointly with Monash University),Melbourne,Australia,as a Research Fellow.Since 1992,He has been with the Department of Electrical Engineering,National University of Singapore,where he is currently an Associate Professor.Since 1999,he has also worked part time at the High Performance Computation for Engineered Systems (HPCES)Programme of Singapore-MIT Alliance (SMA)as an SMA Fellow.His current research interests include electromagnetic theory,radio wave propagation and scattering in various media,microwave propagation and scattering in tropical environment,and analysis and design of antennas.