Study of the "Sierpinski`s Carpet" Fractal Planar Antenna by

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VOL. 2, NO. 1, JANUARY 2007
Study of the "Sierpinski’s Carpet" Fractal Planar Antenna
by the Renormalisation Method
C. LARBI *, T. BEN SALAH *, T.AGUILI *, A.BOUALLEGUE*, H.BAUDRAND**
*L.syscom, ENIT BP.37 Le Belvédère 1002, Tunis, Tunisia
Emails: [email protected]; [email protected]; [email protected]
Abstract: We are interested, in this work, with
the application of the method of renormalisation
in electromagnetism by studying the radiation of
fractal planar antennas. Our study relates, more
particularly, to the Sierpinski’s Carpet antenna,
where we combine the surface impedance model
with the renormalisation method to find out very
interesting results in particular concerning
computing time and memory resources use
compared to MoM. This new method, as far as
we know, is the only method able to study this
type of antennas on infinite scales (or iterations)
thanks to the concept of fixed point that
1. INTRODUCTION
The fractal antennas - due to their multi scale
characteristics and their auto-similarity
property - found out rapidly new application
fields, particularly for ensuring connections in
GSM, DECT and WLAN bands, which
aroused the interest of many designers and
researchers in the field. However, to analyze
this new generation of antennas, the traditional
methods - known as "full waves" - remain
limited because of important requirements in
memory resources and computing time.
Indeed, to well describe the geometry of the
fractal objects we need take into consideration
details and irregularities which do not cease
increasing from one scale to another.
However, these structures - in spite of their
apparent geometrical complexity - can be built
and studied recursively according to various
transformations from one scale to another [1],
using their interesting auto-similarity property.
Although these techniques appear very
tempting, they were applied to relatively
simple 1D electromagnetic circuits and could
not be extended to two-dimensional ad
infinitum reiterated planar structures [ 7 ], such
as the “Sierpinski’s Carpet” antenna studied in
guarantees the convergence of the results
towards this point when the scale rises
indefinitely.
Keywords: renormalisation, surface impedance,
fixed point, fractal structure
this article. To circumvent the latter difficulty,
we develop in this work a new electromagnetic
method of calculation based on the technique
of renormalisation and the surface impedance
model.
2. RENORMALISATION
Renormalisation method of was firstly
introduced by K.Wilson[2] to explain the
behaviour of ferromagnetic materials around
and at the temperature of Currie where
magnetic susceptibility presents a narrow peak
and has an infinite value. This critical point is
characterized by an infinite characteristic
length of the largest fluctuations; in other
words, the associated physics phenomena are
several scales wide.
Practically, this type of problems does not have
exact solutions, because of the significant
number of intervening parameters. Indeed, at
this particular temperature of Currie, the
number of strongly coupled sites of spins can
reach 100 generating a number of possible
configurations of 210000. This number confirms
impossibility of solving this problem using the
traditional methods.
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To circumvent this difficulty, and to calculate
the magnetic properties in the vicinity and at
the critical point, the only method providing
appreciable solution is the method of
renormalisation. The principle of this method
consists in cutting out a major problem in some
simpler to solve sub-problems. Therefore,
instead of studying all spins at the same time
(having as characteristic dimension the length
of correlation), one is limited to a super-spin.
The passage from spin to a super spin
(grouping several elementary spins) is
governed by the laws which give the key of the
problem. This process of back "zoom" allows,
using successive operations of scaling, the
establishment of the behaviour of the macrospin based on the elementary spins behaviour.
The method of renormalisation, as K.Wilson
presented it, makes up of three steps repeated
several times. The first step, which is the
decimation, consists in dividing the network
into blocks containing each one some spins.
The average spin of each block (or super-spin)
is built according to an arbitrary rule: the
direction of the super spin is that of the
majority of the spins in the block. This is the
second step. The third step consists in reducing
the scale to find initial dimensions and the
natural interactions. To let theses steps make
sense in the transformations of the
renormalisation, one must ensure the
invariance of the spin network, in other words
the physic properties of studied material must
remain unchanged during various passages or
iterations.
A. Application of the renormalisation in
electromagnetism
The
application
of
this
step
in
electromagnetism, in particular to calculate the
diffraction of the electromagnetic waves, or the
radiation by two-dimensional fractal structures,
requires the choice of physical variable which
will have to be reiterated from one scale to
another. The currently studied fractal structures
[3][4][5] present a pattern repeated ad
infinitum with dimensions getting smaller and
smaller.
We know that dimension compared to
wavelength is an important parameter in
electromagnetism. In this case, whatever the
dimension of the structure is, after several
iterations, the pattern will reach quickly
sufficiently small dimensions to let us consider
them as surface impedances. This impedance
could replace a set of patterns reiterated ad
infinitum when independence from the
environment is guarantied. So, it should be
verified that the coupling between close
localized energies is negligible.
Then, the substitution of a set of patterns by
homogeneous impedances does not disturb the
remainder structure. The passage from one
scale to another will be done in terms of
impedance. At scale N the pattern has an
impedance Zn, on the scale n-1, the impedance
to be calculated is the one presented by the
structure with sub-patterns, Zn-1, as illustrated
in figure 1.
Zn
Zn
Zn
-1
Zn
Zn
Zn
-1
Fig 1 : Transformation from scale n to scale n-1
in renormalisation method: Iris of cantor
In a recent work [6], we validated this new
method by applying it to calculation of the
electromagnetic characteristics of a selfinductive iris of cantor. To show the robustness
and the interest of this method on the
numerical level, we propose in this paper
studying the "Sierpinski’s Carpet" antenna,
famous for its multi-band radiation in and its
compactness. This antenna (figure 2) is an
extension of 1D to 2D of the Cantor iris [6]
and its construction on the various scales is
similar with this last.
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determinate the radiation characteristics of the
studied antenna. This enables as to calculate
the fixed point of the relation of
renormalisation so that we simulate the
electromagnetic behavior of the antenna on the
various scales and particularly on infinite
scales.
x
3. VALIDATION
OF THE MODAL
SURFACE IMPEDANCE MODEL
Fig 2 : Sierpinski’s Carpet antenna
It is characterized primarily by its fractal
dimension calculated using the definition given
by Hausdorff Bicosovitch:
D=
Log ( N )
1
Log  
r
B. Proposed Calculation Method
N: number of copies (generating element)
R: ratio of reduction (from one scale to
another)
The application of this formula to the studied
structure gives
D=
As verification of the model, we are concerned
in this part of study in calculation of the first
resonance frequency of the Sierpinski’s
antenna on scales 2 and 3 (figures 3 and 4). We
compare the obtained results of structures
modelled by surface impedance with those
provided in the case of real structure.
Log (8)
= 1.875
Log (3)
Another descriptor (other that D) of the fractal
structure is the lacunarity. Indeed, the
measurement of the irregularity given by D
cannot inform us sufficiently about nature and
the degree of filling of.
Some configurations of this space can affect
the electromagnetic characteristics seriously.
In this work we consider the studied antenna
only characterized by its fractal dimension. We
introduce, in the next paragraph, the modal
surface impedance model and discuss its
validity field accordingly to the principle of
equivalence [8].
A’
Fig 3 : Sierpinski’s Carpet (second scale)
Then we dedicate a paragraph to the
application of the method of renormalisation to
It consists in breaking up the discontinuity
presented by the transverse plan of the antenna
into different sub-surface fields superimposed
at cross-sections of virtual guides limited by
periodic walls (figure 2). We justify the choice
of periodic walls later. On each one of the
guides, the electromagnetic field is described
on local modal bases where the weights
associated with these modes are calculated
separately. Indeed, the contribution of the
evanescent higher modes (passive modes) is
calculated locally while the lower modes are
coupled with a larger scale (active modes).
Thus the description of the boundary
conditions on the antenna is simplified by
reducing it to an equivalent impedance matrix
[Zs] translating the coupling between the
various active modes taken in an area and
depending on its various passive modes.
Knowing that reactive energy is concentrated
on the edges of the various slits constituting
the studied antenna, we show in the following
how to isolate this stored energy and to
validate the surface impedance model Zs by
describing it by a unique active mode.
Classically
the
calculation
of
the
characteristics of the considered antenna is
brought back to a description of the electric
fields on areas with dimensions α3A (figure 4)
(α : fractalisation coefficient taken equal to 1/3
in our study; A: antenna dimension)
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The dimensions ratio on these scales is very
important, which can cause problems of
numerical convergence particularly when α is
low.
y mn : mode admittance in the virtual guide
The factor 2 translated the contribution of both
half spaces on both sides of discontinuity.
From the equivalent diagram of figure 6, one
obtains the following equation:
E   0 1   J 0 
 J  = − 1 Yˆ   E 
  
 e 
E e = ∑ Ven g n
y
n
Gn : test functions
Ven : associated amplitude
x
Fig 4 : Sierpinski’s
Carpet (scale 3)
Fig 5 : Sierpinski’s Carpet
(scale 3) modelled by
surface impedances
By applying the method of Galerkin to this
problem, we obtain the following matrix
equation
V   0
0 = 
  − M
M t I 0 
 
Yˆ  Ve 
[]
To circumvent this disadvantage, using the
technique of scaling, one can reduce the
complexity of the problem by introducing an
intermediate description on the scale α2A
(figure 5).
[Yˆ ] = g mYˆg n
Thus the boundary problems in the virtual
guide described locally, can be represented by
the following equivalent diagram (figure 6)
f0 standardized function representing the
fundamental mode in the virtual guide.
J
]
m =1.. N
n =1.. N
Then impedance Z s = M t [Y ] M =
−1
^
Y
E
0
e
Fig. 6 : Equivalent diagram of the studied structure
(Sierpinski’s Carpet)
On this diagram (figure 6), there is J0 the
excitation in magnetic field with a phase of π/2
imposed on the fundamental mode in the
virtual guide, Ee the induced field on the slit
)
domain and Y the operator representing the
reaction of the environment; it describes the
contribution of the localized modes. The
)
operator Y who is an alternative of the Green
operator in the spectral field can be represented
by a modal base {fmn n є IN*} of the virtual
guide:
)
Y = ∑ f mn 2 y mn f mn
M = [ f0 , gn
]
Ve = [Ven ]
J
E
[
[11] [12]
V
I0
The Zs impedance calculated is actually the
impedance seen by the fundamental mode of
the virtual guide. We limit ourselves in this
work to a representation of this last as a dipole.
We justify the validity of this modeling by
imposing restrictions on dimensions of these
guides.
Indeed by considering dimensions of these
elementary guides small in front of the
wavelength, we can suppose that the higher
modes are still localized. Indeed, in a former
work [6], we showed that the electromagnetic
coupling between close guides is practically
ensured by the fundamental modes excited in
these guides (active modes). Moreover, (by
considering an extension of calculation 1D in
[6] to 2D) for a reduced dimension k0.a=0.1
and a polarization according to ox , the
variation observed between this coupling and
other modes can reach 70db. It is the case of
the Sierpinski antenna observed on scale 3
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(figure 4) where a whole of sub-structure of
this antenna is replaced by the surface
Partie réelle de l'impédance d'entrée
20
1.0
15
0.8
0.6
10
0.4
5
0.2
0.0
Re(Zin) : Modèle de Zs
Re(Zin) : Structure réelle
1.2
The weak error obtained in this calculation
case (0.3%) does not depend only on the
criterion imposed on dimensions of the virtual
guide. Indeed, for the case k0A' =0.44
(dimension more important than in the
preceding case), the error can reach the 6%,
limit of validation of the model suggested
(figure 8). Moreover, the calculation precision
of Zs also plays an important role in the
precision of the final result.
0
0.5
1.0
1.5
In the same way, we notice that the real values
of the part of modeled Zin are more important
than those of the real antenna. The latter seems
not to be adapted to this frequency of
resonance (figure 12), contrary to the modeled
antenna where Zs is purely reactive.
2.0
Frequency
---- : Zs Model ( fr =1.188 GHz)
___ : Real Structure ( fr = 1.264 GHz)
Fig 8 : Input Impedance for k0A’=0.4
impedance (Zs=0.507 ; k0A’=0.143 ; k0=2π/λ ;
convergence obtained for Nb = 300 in the
virtual guide and a number of test functions of
nφ=4 in the opening). The calculation of Zs is
obtained using the method of the generalized
equivalent circuits associated to the method of
Galerkin, mentioned above. A complete
analysis of this method was presented in [11].
This value of Zs injected into a simulation by
the ADS Momentum program, enables us to
model the various characteristics of the studied
antenna.
The first simulated frequency of resonance of
the
modeled
antenna
on
scale
3
(frm=1.227GHz) is not too different from that
of the real antenna (fr=1.223GHz) which
corresponds to its greater dimension.
The error recorded between these frequencies
is about 0.3%. It is considered weak as long as
the obtained frequency frm is contained in the
band-width of the antenna as figure 7 shows it.
Let us not forget, moreover, that an arbitrary
choice of the isolation walls (boundaries of the
virtual guides) can lead to divergences in
calculation, or require great dimension bases of
approximation, as it is the case of the electric
or magnetic walls. This is why, and thanks to
the pseudo periodicity presented by the studied
fractal structure it self (characteristic deriving
from auto-similarity property) for calculation
of Zs, we considered virtual guides limited by
periodic walls as it is indicated on figure 2. We
thus avoid imposing particularly null values on
the electromagnetic field and taking into
consideration the various interactions between
the selected pattern and these walls, while
ensuring a good isolation between the various
patterns (of the modeled structure).
Partie réelle de l'impédance d'entrée
20
0.8
15
0.6
10
0.4
5
0.2
0.0
Re(Zin) : Modèle Zs
Re(Zin) : Structure réelle
1.0
0
0.5
1.0
1.5
2.0
Frequency
---- Zs Model : fr =1.227 GHz
___ Real Structure : fr = 1.223
GHz
Real Structure (scale 3) at resonance
Fig 7 : Re(Zin ) of the structure
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Surface Impedance Model (scale 3) at resonance
Figure 9 : Surface Current : comparison between real
and surface impedance modeled Structure
distributions of the real and the modelled
antennas (scale 3) at the first resonance
frequency. We notice, by observing these
results, that the openings of smaller
dimensions contribute little in the calculation
of the characteristics of radiation of this
antenna considering the similarity of the
diagrams of radiation. In the same way we note
that the current concentration is more marked
around the center opening. This seems to be
more adapted than the antenna characterized
by its great dimension which is indeed in
resonance as figure 9 shows it; which justifies
the explanation concerning the results
represented in figure 7.
In addition, we notice that if the structure
division introduces very unfavorable boundary
conditions into the description of the
electromagnetic field, a solution to cure it
consists in using the covering technique [8].
S11
0
Mag. [dB]
-5
-10
-15
-20
0
1
2
3
4
5
6
Frequency
figure 12 : S11 module
fr1 = 1.223 GHz for real structure
(a) Real structure (scale 3) at resonance
To explain this phenomenon, a study of
discontinuity between the feeder and the
antenna as well as the adaptation of the latter
could be the subject of another work.
4. APPLICATION TO THE CALCULATION
OF THE RADIATION CHARACTERISTICS
(b) Modeled structure (scale 3) at resonance
Figures 10 and 11
(10) Radiation diagram of the real structure at f=1.223
Ghz
(11) Radiation Diagram of the surface impedance
modelled structure at fr' 1=1.227 GHz
When speaking about renormalisation, we talk
all about relation of renormalisation.
Accordingly to an earlier work we published in
[ 5 ], we can model the passage from a scale to
another by the equivalent parallel diagram
(figure 13b) translating the boundary
conditions of the structure represented by the
figure 13a.
By
taking
into
consideration
these
observations, we represented figures 9 and 10
showing the radiation diagrams and the current
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(b)
J0
^
Y
(a)
Zk
Zk
Ee
Fig 13 : Zk , impedance seen at scale k relatively
to dimension αKA.
Zk-1 is the impedance seen by the fundamental
mode for dimension α(k-1)A (α factor of
fractalisation and A dimension of the reason,
k scale). By applying the method of Galerkin
to the integral equations deduced from this
last diagram one can write:
Z k −1 = A11 + B T C −1 B = f (Z k ) (1)
A11 represents the surface impedance for the
scale k
B : matrix expressing the coupling between the
fundamental mode (taken with horizontal
polarization) and the various electric field test
functions defined in the opening.
C : matrix expressing the coupling between the
test functions and the various higher modes
excited in the discontinuity plan.
The relation (1) is the renormalisation relation
which we consider in this part of study. It
makes it possible to determine the fixed point
of this transformation which verifies the
invariance property translated by f(Zk)=Zk.
This last constitutes an essential step of this
method with the decimation, the choice of the
variable to be reiterated and the scaling.
Indeed, theoretically, this fixed point ensures
equivalence
between
the
various
transformations applied to the studied structure
and physically it corresponds to the input
impedance of the ad infinitum reiterated
structure. As far as know, today, this method
remains the only one to study this type of
fractal antennas. We show in the continuation
the interest and the robustness of this method.
The determination of this fixed point (called
Zp1) rests on the calculation of the deriving
fixed point of the diffraction study of the
structure forming the plan of discontinuity of
the studied antenna and which we call from
now on Zpa.
Fig 14 : Determination of the fixed point
The fixed point calculation is represented on
figure 14 (f(Zk)=Zk). Calculation can be also
carried out by an iterative method applied to
the relation Zk-1=f(Zk). The same result is
obtained whatever are the initial values (10-4
and 104).
To validate the point fixes Zp1, we compare it
with Z∞ impedance calculated at convergence
when the iteration count of scale tends towards
the infinite one (in our case it is equal to 5 i.e.
4700 openings corresponding to the number of
used test functions). Various simulations are
carried out by the software Momentum ADS
for k0.A=0.1. All the results are gathered in the
table represented in table 1. The various
impedances considered are reduced compared
to the characteristic impedance of the feeder of
the antenna.
Amplitude of Zin
(Momentum)
k=1
k=2
k=3
k=4
k=5 (Z∞)
Zp1 (fixed point)
0.6097807
0.7586882
6.7859281
6.6678778
6.6678778
6.7397799
Table 1: comparison of the impedance Z∞obtained
by the method of moments with that of the fixed
point
The relative error recorded from the
comparison between the impedance calculated
by the fixed point and that given by the method
of moments is estimated at 1.06%, which
makes it possible to validate the used method
of renormalisation.
To highlight the numerical interest of the
suggested method, we give an estimation of the
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requirements in computing time and memory
resources. The table represented by table 2
provides some results concerning the
calculation of the Sierpinski’s antenna on scale
3.
Real
Structure
Surface
impedance
model
3654
1596
Matrix size
152.91Mb
56.47Mb
Memory
resources
0h10m42s
0h01m32s
Computation
time
Table 2 : Computing time and memory usage to
calculate only one frequency (for structure figure 4)
interest of the method. Indeed, this condition is
quickly checked as of the second or third
iteration of the scale of observation. For the
extension of this method to the analysis of
structures of very large dimension, we can
consider the multi ports model in the place of
the surface impedance [9]. Furthermore, this
method can be applied easily to the calculation
of networks of antennas.
6. REFERENCES
References:
[1]
[2]
Furthermore, the computing time of the fixed
point is about 11 minutes 56 seconds while that
for Z∞ is higher than one week (7*24 hours);
various simulations were performed on a PC
Pentium 4 at 2.6GHz equipped with a RAM
memory of 2Go.
[3]
The memory resources consumed by the
calculation of the point fixes are about 1Mo
while for Z∞, calculation requires 30Mo.
[6]
This estimation shows well the advantage of
the surface impedance model compared to the
real structure analysis in term of computing
time and memory resources.
[4]
[5]
[7]
[8]
5. CONCLUSION
In this study, we applied the method of
renormalisation, well-known method in
theoretical physics, to the calculation of the
radiation characteristics of the fractal
Sierpinski’s Carpet antenna. An estimation of
the computing time and memory resources
showed (for determination of the fixed point)
the considerable reduction we obtain on these
parameters. As far as we know, today, there is
not any other method allowing the
characterization of this antenna, which is - by
definition extended - until the infinite.
However, the various satisfying results are
tributary of dimensions of the studied
structures which should be small in front of the
wavelength. This restriction on the geometrical
fields does not decrease of anything the
[9]
[10]
[11]
[12]
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