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FRACTALCOMS Exploring the limits of Fractal Electrodynamics for the future telecommunication technologies

IST-2001-33055

Deliverable reference: D8

Contractual Date of Delivery to the EC: July 31, 2003

Author(s): R. Gómez Martín, A. Rubio Bretones, M. Fernández Pantoja, F. García Ruiz , R. Godoy Rubio, S. González García.

Participant(s): UGR

Workpackage and task: WP3, T3.4

Security: Public

Nature: Deliverable

Version and date: Final, 30-6-2003

Total number of pages: 23

Keyword list: Time domain, method of moments, Koch monopoles, visualization, spectral analysis

Task 3.4 Final Report

Abstract:

This deliverable describes the work done in enhancing the simulation algorithms for the analysis of pre-fractal antennas in the time domain and the results obtained for examples such as the Koch, Hilbert, Sierpinsky and binary tree monopoles and the Koch loop antennas, etc., modelled by thin wires and strips. Because the antennas are assumed to be perfectly electric conductors (PEC) the analysis is carried out using the Method of Moments in the Time Domain (MoMTD).

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RELATED WP AND TASKS (FROM THE PROJECT DESCRIPTION)

WP3: Software simulation tool. Task 3.4: Simulation of pre-fractal structures in the time domain.

1 INTRODUCTION

Our goal has been to simulate and analyse, in the time domain, the behaviour and characteristics of pre-fractal antennas. Because the antennas was assumed to be perfectly electric conductors (PEC) the analysis has been carried out using the method of moments in the time domain (MoMTD) [1], [2]. This task was divided, throughout the project, into two main parts:

a) Extensions of our previous code DOTIG5, [3] [4], to study different PEC pre-fractal antenna geometries formed either by thin wires or by continuous surfaces

b) Simulation and analysis of different pre-fractal geometries taking advantage of

the possibilities of simulations in the time domain of different pre-fractal antennas. This allowed us to isolate interactions using time range (cause and effect can be distinguished providing an easier physical interpretation of the results), to simulate transient phenomena, visualization of the time history of the physical magnitudes and possibility of obtaining, with a single run, broadband information.

2 EXTENSIONS OF THE DOTIG5 CODE

2.1 Extension of DOTIG5 for thin wire structures DOTIG5 is a code based on the time solution of the electric field integral equation in the time domain (TD-EFIE) using the MoMTD. For the case of a perfect electric conducting thin wire, the EFIE simplifies to [2]-[4]

where s and s’ are tangent vectors to the wire axis of contour C(s’) at positions s (r) = s and s(r’)=s’, I(s’,t’) and q(s’,t’) are the unknown current and linear charge distribution respectively at source point s’ at retarded time t’ = t – R/c; E(s,t) is the field applied to the observation point, and R = |r-r’|. The charge q(s’,t’) can be expressed in terms of I(s’,t’) by means of the continuity equation. The scheme for the MoMTD for thin wires is given in figure 1.

∫

⋅−

∂∂⋅

+∂∂⋅

=⋅)'(

320

')','(ˆ

)','('

ˆ)','('ˆˆ

41),(ˆ

sC

i dstsqR

RstsIscR

RstsItcR

sstsEsπε

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Figure 1: MoMTD scheme (thin wire) In order to model pre-fractal antennas formed by thin wires it has been necessary to extend our initial code :

1. To be able to model the thin wire structures using non-uniform segmentation.

2. To save computational resources we have extended the code in order to take advantage of the symmetry of monopole antennas in front of a perfect conducting ground.

3. To include the possibility of modeling junctions between wires modelled with a non-uniform segmentation. Figure 2a shows, as an example of prefractal antennas including junctions, several iterations of a binary tree antenna. Figure 2b shows the way of treating junctions by means of overlapping segments. The basic idea is to force the current and charge per unit length to be continuous at the junctions

20

2 3

ˆ ˆ1 'ˆ ( , ) [ ( ', ')4 '

ˆ ˆ( ', ') ( ', ') ]

T h in -w ire T D -E F IE e q u a tio

''

n

i

C

s ss E s t s tc R t

s R s Rs t

I

I q s t d sc R t R

π ε⋅ ∂

⋅ = +∂

⋅ ∂ ⋅+ −

∂

∫

BBaassiiss FFuunnccttiioonnss:: 22nndd oorrddeerr LLaaggrraannggiiaann iinntteerrppoollaattiioonn iinn ssppaaccee aanndd iinn ttiimmee WWeeiigghhttiinngg FFuunnccttiioonnss::DDeellttaa ooff DDiirraacc

DDiissccrreettiizzaattiioonn iinn ttiimmee aanndd ssppaaccee

Linear system ( ) ( ) ( )+ =s ij j jE t E t ZI t

MMaarrcchhiinngg oonn iinn ttiimmeessoolluuttiioonn

Well suited for PEC thin wires

Current ( )⇒ jI t

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a) Binary tree dipole b) Original thin-wire structure and overlapping segments model. Figure 2

2.2 Extension of DOTIG5 for continuous surface structures Pre-fractal antennas can also be built using printed strip technology. In this case the strip can be modeled using triangular patches to solve the EFIE . For an arbitrary PEC surface the TD-EFIE is given by [3]-[5]:

.

1 2

3 N-1

N

1 2

3 N-1

N

. . . . . .

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' '

0

( ', ') 1 1ˆ ˆ ˆ( , ) ( ', )4 4

tisS S

J r ttn E r t n ds n ds J r d

R R τµ τ τπ πε

∂ ∂× = × − × ∇ ∇

∫ ∫ ∫ i

where n̂ is a unitary vector normal to the surface at position point r . The scheme for the MoMTD for continuous surfaces is given in figure 3. To save computing resources, DOTIG5 for continuous surface structures has been extended to use non uniform triangular patches and parametric techniques based on the Prony and Pencil algorithms used to extrapolate the transient response of the antennas [5]. Moreover, we have also extended the DOTIG5 code for surface structures in order to take advantage of the symmetry of monopole antennas in front of a perfect conducting ground. Figure 4 shows the basis of this extension: two adjacent triangular patches and their mirror image where the triangles have been changed so that the right mirror-image current is obtained in the lower part of the structure. For a given structure modelled with N triangular patches and located in front of a PEC ground, applying symmetry considerations greatly reduces the computational resources needed for its simulation. In particular the memory is reduced in approximately 50% and the computational time in more than 50%.

Figure 3: MoMTD scheme (continuous surfaces)

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Figure 4. Triangular patches and their mirror images

2.3 Validation

• Thin-wire structures The extensions of DOTIG5 have been validated for pre-fractal Koch monopole antennas, figure 5, of several fractal interactions (K0-K3). Figure 6 shows the input impedance and the reflection coefficient for these antennas. It can be observed that there is a good agreement between the results obtained using the NEC code in the frequency domain and the results obtained, via the fast Fourier transform (FFT) algorithm, with the DOTIG5 code. Figure 7 shows the Q factor of the Koch monopoles. Again it can be seen that there is a good agreement between the results obtained with NEC and DOTIG 5. Figure 8 shows the input impedance for K03 calculated using NEC, DOTIG 5 and the commercial software WIPL. In order to validate DOTIG5 for the triangular model of the strip Koch antennas shown in figure 9, figure 10 compares the results obtained for the thin wire and strip models of the K0 and K1 monopoles. For thin-wire structures with junctions, the extensions of DOTIG5 have been validated using representative examples. Figure 11 shows for the order 4 pre-fractal thin-wire binary tree antenna of Figure 2, the input impedance (resistance and reactance) calculated using the extended version of DOTIG5 and NEC. It can be observed that there is a excellent agreement between both results.

• Surface structures Regarding the surface version of DOTIG5, this code have been validated solving the same structure with and without using the extension that takes advantage of the symmetry for a monopole in front of a PEC ground and the results were identical.

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Figure 5: Koch monopole (K0-K3)

Figure 6: input impedance and reflection coefficients for K0-K3 monopoles

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Figure 6: Q factor of the K0-K3 monopoles calculated by NEC and DOTIG5

Figure 8: Input impedance for the K3 monopole using NEC, DOTIG1 and WIPL codes

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Figure 9: Strip Koch monopoles (K1-K3) modeled by triangular patches.

a)

b) Figure 10: Results obtained for the K0 (a) and K1 (b) thin wire and strip monopoles using the DOTIG5 code

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Figure 11. Input impedance( resistance and reactance) of the order 4 pre-fractal thin-wire binary tree antenna.

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3 SIMULATIONS AND VISUALIZATION.

The resulting code has been applied to carry out simulations and numerical experiments to facilitate understanding of the behaviour of pre-fractal antennas. To this end we have studied the time history of the physical magnitudes such as the current induced in several fractals antennas and the fields created by these currents. As a first example of visualization, figure 12 shows the time evolution of the current in a K2 monopole with a Gaussian voltage pulse excitation. It is observed that, due to the initial acceleration of the charges at the feed point and at the bends and corners in the wire, almost all the energy is radiated at the beginning and the amplitude of the current pulse decreases very quickly [6]. Figure 13 (a) shows the fields created at a certain time step by a K1 monopole excited by a Gaussian pulse. And figures 13 ( a and b) shows the time history of the current along the K1 monopole and its mirror image. Several interesting effects can be noted:

a) The current pulse widens as it propagates along the antenna. This effect is associated to the loss of energy of the pulse mainly due to the radiation of its high frequency components.

There are induced currents on some parts of the structure at instants previous to those at which the original pulse of current reaches these points from the feed point. This is due to the fact that the field radiated at the feed point and discontinuities (bends and corners) of the geometry which interact with those parts of the structure follows a shorter path than the one taken by the conduction current (shortcut effect). This effect can also be appreciated in figures 14 and 15. Finally figure 16 shows an example of the spectral analysis of the current along the wire carried out in order to study the effect on the spectrum of the current of the geometrical details of a Koch structure. An important part of the simulations and numerical experiments has been focused on electrically small pre-fractal antennas. As a first example, Figure 17a shows the space-time diagram for a Koch monopole of order 2 excited by a wide Gaussian signal whose maximum spectral component is such that the monopole is a electrically small antenna even for that frequency. It can be observed that current suddenly appears at certain zones of the structure at times before the pulse of current propagating along the structure does. These shortcuts, as happened in electrically large pre-fractal antennas, play a significant role in the behavior of electrically short pre-fractal antennas. The shortcut effects take place mainly at certain segments, which are indicated with arrows in Figure 17b, with a specific orientation tangential to the wave front originated at the feed point. In consequence, it is clear that the orientation of the segments in the pre-fractal antennas plays also an important role that need further investigation After performing different numerical experiments it has been found that, for low frequencies, the shortcut is almost exclusively due to the field created at the feed points and that this effect increases, for wire antennas, with the electric size of the radius of the wire or, and for strip antennas, with the width of the trip. For example Figure 18 shows the resonance frequency corresponding to two Koch monopole strip antennas of

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order 3 and the same length but different widths. It can be observed that the resonance frequency increases with the width, the reason is that the effect of the shortcuts is stronger for the wider strip because there exists a greater coupling between the radiated field and the strip. The same results have been found for Koch monopoles of higher order.

Figure 12: Time evolution of the current for a K2 monopole excited by a Gaussian pulse

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(a)

(b) (c) Figure 13: (a) Field created at a certain time step by a K1 monopole excited by a Gaussian voltage. This field is the sum of the one created at the feed point and those created at the bends and corners of the structure [6], and it reaches the upper corner of the structure before the current propagating along the wire does. The effect of the interaction of this field on the current induced along the whole wire can be observed in (b) and in more detail in (c) which is a zoom of the part of (b) enclosed by a rectangle. It can be observed that approximately at the time interval 70 and the space interval between 45 and 65 (indicated with an arrow in the figure) the current suddenly appears at this zone of the structure at a time before the current propagating along the structure does. The effect of these shortcuts is thought to play a significant role in the behaviour of pre-fractal antennas that needs further investigation.

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(a) (b) Figure 14: These two pictures show, for a K2 monopole, the same effects as figure 11 for a K1 monopole. Because the number of corners and bends is bigger than in K1, the effect of the shortcut taken by the field and as a consequence the sudden increase of the current at certain parts of the structure occurs more often. Some of these increases can be seen in (b) which is a zoom of the part of (a) enclosed by a rectangle.

(a) (b) (c)

Figure 15: this figure shows, for a strip monopole, the current distribution along the antenna at three different time steps, (a) just before the antenna was excited. (b) at a later time step where it can be appreciated how in the zone indicated by the arrow, there is an induced current before the conduction current arrives there following its path along the strip. this current has been induced by the field radiated at other points of the structure

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(a) Figure 16: In order to study the effect on the current of the geometrical details of a Koch structure, several numerical experiments have been carried out. As an example this figure shows the geometries (top left and top right) of the structures used to estimate this effect. for the basic generator. The spectrum of the current measured at point B in both geometries is compared in the figure. It can be observed that the current pulse loses mainly high frequency components after propagating through the bends which results in a wider pulse in the time domain. This effect can be also observed in figures 10-12. The peaks at certain frequencies in the spectrum may be related to resonant frequencies corresponding to different parts in the structure and to the shortcut effects previously described. ……………………….

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a)

b) Figure 17. a) Space-time diagram for a Koch monopole of order 2 excited by a wide Gaussian signal. b) The arrows indicate the segments were the shortcut effects take place.

Figure 18. Resonance frequency for two Koch monopole strip antennas of order 3 and the same length but different width.

4 ANALYSIS OF OTHER THIN-WIRE PREFRACTAL ANTENNAS.

Other pre-fractal geometries studied are the binary tree monopole, the Hilbert antenna and the Koch loop antenna. For example, Figure 19 shows the comparison of these two antennas and the Koch monopole in terms of the resonance frequency and total length. Figures 19a and 19b give, for these antennas, the resonance frequency and the input

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resistance at the resonance frequency as a function of their total length. Figure 20 presents the results obtained for the Koch loop where it can be observed that in this pre-fractal geometry, the resonance frequency presents a similar behavior than that of the other pre-fractal antennas previously analyzed, that is, it decreases as the order of the iteration increases but it stagnates as the iteration increases.

Figure 19. Comparison of the binary tree monopole and the Hilbert antenna and the Koch monopole in terms of the resonance frequency and total length.

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a)

b) c) Figure 20. Resonance frequency for the Koch loop antenna.

5 PERFORMANCE OF FRACTAL ANTENNAS IN FRONT OF AN INHOMOGENEOUS BODY.

In practice the fractal antennas will work near of arbitrarily inhomogeneous bodies and, consequently, it is useful to study how the parameters of such antennas are affected in such situation. However, the total geometry formed by the inhomogeneous body and the antenna is so complex that there is not a single numerical method appropriate to simulate the whole problem. One alternative is to use for each part of the structure the numerical method that best fits to it. The numerical tool that results of this combination is called hybrid method. In this project we have used a hybrid method combining the method of moments in the time domain (MoMTD) and the Alternating Direction Implicit Finite Difference Time Domain (ADI-FDTD) method [7]-[9] to study the performance of fractal antennas in front of an inhomogeneous body. The ADI-FDTD method is based on an implicit-in-space formulation of the FDTD which offers

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unconditional numerical stability with little extra computational effort. The ADI-FDTD method removes the stability limit for the time increment, making it possible to choose the time increment independently of the space increment [7]-[9]. Figure 21 represents an arbitrarily oriented thin-wire antenna located close to an inhomogeneous scatterer limited with a curved boundary. The antenna is embedded in a homogeneous lossless dielectric medium. The hybridization is based upon the use of the surface equivalence principle. The computational domain D (the area inside surface S1) is sub-divided into different sub-problems. The first sub-problem, which will be handled by the MoMTD, is composed of just the thin-wire antenna extracted from the previous configuration and located in an unbounded, homogeneous environment. The surface S2 is the Huygens’ box that links the MoMTD solution with the rest of the problem which solved using the ADI-FDTD method as explained in [7]-[9].

5.1 Results Figures 22 and 23 show the results obtained when two pre-fractal antennas (Koch and Hilbert monopoles of order two) are in front of a human head that has been simplified to three concentric spheres with different constitutive parameters. It seems that the presence of the head do not disturb appreciably the behaviour of the pre-fractal antennas.

Figure 21

S2:Huygen’s surface MoMTD+Scattered field ADI-FDTD zone

Scattere

ADI-FDTD total field zone

S3

S1

S1 (PML)

S1

HHyybbrriidd ((AADDII--FFDDTTDD--MMooMM))TTDD

S1 (PML)

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Figure 22

FFrraaccttaall aanntteennnnaa--hhuummaann hheeaadd

100 200 300 400 500 600 700

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10-3

Without headWith head

0 0.5 1 1.5 2

x 109

0

1000

2000

3000

4000

5000

6000W ithout headW ith head

0 0.5 1 1.5 2

x 109

-3000

-2000

-1000

0

1000

2000

3000

Without HeadWith Head

Time

Frequency Frequency

Current (A)BBrraaiinn

BBoonnee

SSkkiinn--FFaatt

6cm

20cm

8, 0.11ε σ= =

34.5, 0.6ε σ= =

55, 1.23ε σ= =

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Figure 23 REFERENCES [1] R. Gómez Martín, A. Salinas, A. Rubio Bretones. “Time-Domain Integral Equations Methods for Transient Analysis” IEEE-Antennas and Propagation Magazine”. (Vol.. 34), pp 15-23, 1992. [2] A. Rubio Bretones “DOTIG1, un Programa para el Cálculo en el Dominio del Tiempo de la Interacción de Ondas Electromagnéticas con Estructuras de Hilo” Tesis Doctoral Universidad de Granada. 1.988. [3] Mario Fernández Pantoja “Técnicas numéricas híbridas en el dominio del tiempo”. Tesis Doctoral Universidad de Granada. 2001. [4] A. Rubio Bretones, R. Gómez Martín and A. Salinas.”DOTIG1, a Time-Domain Numerical Code for the Study of the Interaction of Electromagnetic Pulses with Thin-Wire Structures”.The International Journal for Computation and Mathematics in Electrical and in Electrical and Electronic Engineering (COMPEL). (Vol. 8), pp 39-69, 1989. [5] J. Fornieles Callejón, A. Rubio Bretones, R. Gómez Martín. “On the application of parametric models to the transient analysis of resonant and multiband antennas. IEEE Antennas and Propagation., Vol.. 46, No. 3, pp 312-317, March 1998.

0 1 2 3 4 5 6

x 10-8

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

x 10-3

Time (seconds)

Cur

rent

(am

pere

s)

Without headWith head

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

x 108

0

2

4

6

8

10

12x 104

Frequency (hertzs)

Inpu

t Im

peda

nce

- Rea

l (oh

ms)

Without headWith head

2 2.1 2.2 2.3 2.4 2.5 2.6

x 108

-6

-4

-2

0

2

4

6

x 104

Frequency (hertzs)

Inpu

t Im

peda

nce

- Im

ag (o

hms)

Without headWith head

FFrraaccttaall aanntteennnnaa-- hhuummaann hheeaadd

BBrraaiinn

BBoonnee

SSkkiinn--FFaatt

6c

20c

8, 0.11ε σ= =

34.5, 0.6ε σ= =

55, 1.23ε σ= =

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[6] R. Gómez Martín, A. Rubio Bretones, and S. González García. "Some Thoughts about transient Radiation by Straight Thin Wires". IEEE Antennas and Propagat. Magazine , Vol.. 41, n. 3, pp. 24-33, June 1999 [7]. S. González García, A. Rubio Bretones, M. A. Hernández-López and R. Gómez Martín, “A new hybrid method combinig the ADI-FDTD and the MoMTD techniques” Invited paper in Electromagnetics, vol. 23, pp. 103-118, 2003 [8] R. Gómez Martín, A. Rubio Bretones, A. Monorchio, M. Fernández Pantoja, S. González "Time-domain hybrid methods" Chapter in the book: Time Domain Techniques in Computational Electromagnetics, Edited by D. Poljak, WIT Press / Computational Mechanics, 2003. [9] S. González García, R. Godoy Rubio, A. Rubio Bretones, R. Gómez Martín, "Extension of the ADI-FDTD method to Debye media", IEEE Trans. Antennas and Propagation (To appear), 2003

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DISCLAIMER

The work associated with this report has been carried out in accordance with the highest technical standards and the FRACTALCOMS partners have endeavoured to achieve the degree of accuracy and reliability appropriate to the work in question. However since the partners have no control over the use to which the information contained within the report is to be put by any other party, any other such party shall be deemed to have satisfied itself as to the suitability and reliability of the information in relation to any particular use, purpose or application. Under no circumstances will any of the partners, their servants, employees or agents accept any liability whatsoever arising out of any error or inaccuracy contained in this report (or any further consolidation, summary, publication or dissemination of the information contained within this report) and/or the connected work and disclaim all liability for any loss, damage, expenses, claims or infringement of third party rights.