Planar elliptically shaped dipole antenna for UWB Impulse Radio

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Planar elliptically shaped dipole antenna

for UWB Impulse Radio

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The picture on the cover represents co-designed an UWB elliptically shaped dipole antenna with a pulse generator on the same PCB. (See also Chapter 6 of this thesis).

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Planar elliptically shaped dipole antenna

for UWB Impulse Radio

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. J.T.Fokkema voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 14 april 2008 om 15:00 uur

door

Alexander VOROBYOV

Specialist (Master of Sciences) in

Applied Electromagnetic radiophysics and electronics Karazin Kharkiv National University, Oekraïne

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Dit proefschrift is goedgekeurd door de promotoren: Prof.dr.ir. L.P. Ligthart

Prof.dr.sc. O.G. Yarovyi

Samenstelling promotiecommissie: Rector Magnificus voorzitter

Prof. dr. ir L.P. Ligthart Technische Universiteit Delft, promotor

Prof. dr. sc. O.G. Yarovyi Karazin Kharkiv National University, promoter

Prof. ir. P. van Genderen Technische Universiteit Delft

Prof. dr. J. Long Technische Universiteit Delft

Prof. dr. A.G. Tijhuis Technische Universiteit Eindhoven

Prof. dr. J.-Y. Dauvignac University de Nice-Sophia Antipolis

Dr. M. Geissler IMST GmbH

ISBN/EAN: 978-90-76928-14-2

Keywords: UWB IR, ultra-wideband antenna, impulse radio, antenna integration, antenna shield.

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from any author.

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To my parents,

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i

4.5 Miniaturization of a butterfly antenna using antenna bending techniques 77 4.5.1 First scenario, 50% antenna size reduction 78 4.5.2 Second scenario, 40% antenna size reduction 79 4.5.3 Third scenario, 30% antenna size reduction 80 4.5.4 Filled –in bended antenna with a dielectric substrate 81 4.5.5 Conclusion 84 4.6 Antenna flair area minimization based on the surface current distribution85 4.7 Conclusions ... 90

5. Feeding of an electrically small antenna ... 93

5.2 Antenna common mode current ... 95

5.3 Different approaches to a balanced antenna feeding ... 97

5.3.1 Differential antenna feeding ... 97

5.3.2 Antenna feeding via a shielded loop ... 98

5.4 Realization of antenna feeding via loop ... 102

6. Antenna integration with a pulse generator ... 111

6.1 Introduction ... 111 6.2 Antenna feed by a pulse generator on a PCB via a loop feed

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iv

6.3.1 Numerical model ... 119

6.3.2 Electromagnetic design of the antenna ... 122

6.3.3 Antenna feeding ... 122

6.3.4 Experimental verification ... 127

7. Planar dipole operation in realistic environment ... 135

7.2 Near –field wireless channel ... 137

7.2.1 Planar elliptically shaped antenna near-field ... 137

7.2.2 Pulse Transmission Through an Antenna Pair ... 140

7.3 Antenna on human body ... 144

7.3.1 Antenna measurements set-up ... 144

7.3.2 Time domain peak-to-peak analysis ... 146

8. Cavity-backed elliptically shaped dipole antenna... 155

8.2 Numerical model of the cavity-backed antenna ... 157

8.3 Shield optimisation ... 159

8.3.1 Antenna location alongside ‘b‘ 161

8.3.2 Antenna location alongside ‘a‘ 163

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v

8.3.4 Shield height optimization 169

8.3.5 Antenna shield welt influence 170

8.4 Antenna measurements ... 172

9. Conclusions and discussions ... 181

9.2 Utilization and dissemination of results ... 186

9.3 Recommendations ... 187

APPENDIX A ... 189

APPENDIX B ... 193

APPENDIX C ... 197

List of symbols and abbreviation ... 203

Reference ... 201

Summary ... 211

Acknowledgment ... 217

About the Author ... 219

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Chapter 1 1. Introduction

1.1 Background of the research

The recent rapid growth in technology and the successful commercial deployment of wireless communications are significantly affecting our daily lives. The transition from analog to digital cellular communications, the rise of third –and fourth – generation radio systems, and the replacement of wired connections with Wi-Fi and Bluetooth are enabling consumers to access a wide range of information from anywhere and at any time. As the consumer demand for higher capacity, faster service, and more secure wireless connections increases, new enhanced technologies have to find their place in the overcrowded and scarce radio frequency (RF) spectrum. This is because every radio technology allocates a specific part of the spectrum; for example, the signals for TVs, radios, cell phones, and so on are sent at different frequencies to avoid interference to each other. As a result, the constraints on the availability of the RF spectrum become more and more strict with the introduction of new radio services [Nekoogar, 2005].

Expectations are that the number of computers in the equipment and personal objects that surround us will solely increase. These computer systems will remain largely invisible to the user and, thanks to wireless communication facilities, will carry out various useful tasks. Technology that can support this development to a limited extent already exists: Bluetooth. However, there is a requirement for a technology that will enable large-scale, broadband-based applications, for instance home networks that utilize high-quality multimedia communication. Projects are going on focusing on the development of such a technology, on the basis of impulse radio or Ultra Wideband (UWB) radio, [Scholtz R., 1993]. This is a promising innovative technology for short distances, which has enormous potential and complements Bluetooth. Compared to existing solutions for wireless local aria connections (WLAN) and personal networks (PAN), UWB promises to have enormous advantages, particularly for short-distance communications. It is

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anticipated that the equipment will be compact and cheap. UWB is harder to listen into and the transmission speeds and the number of simultaneous users will be high: in a limited space dozens of pieces of equipment can exchange information flows of 100 Mbit/s each. This technology does not modulate the information on a carrier wave like all the existing wireless systems (AM/FM radio, TV, GSM, UMTS, WLAN, Bluetooth), but transmits the data by means of a series of short pulses, each with a length of less than a nanosecond. This means that the bandwidth used is more than a Gigahertz and covers all allocated for wireless communication frequencies with a minimal capacity density. The transmit power is very low: microwatts rather than the milliwatts used in existing technologies. In addition, environments with a lot of reflection do not affect reception. These characteristics make UWB extremely suitable for high- quality, wireless multimedia applications. It therefore has an excellent chance of becoming an important standard for so-called fourth generation wireless and mobile networks.

Ultra-wideband communications is not a new technology; in fact, it was first employed by Guglielmo Marconi in 1901 to transmit Morse code sequences across the Atlantic Ocean using spark gap radio transmitters. However, the benefit of a large bandwidth and the capability of implementing multi-user systems provided by electromagnetic pulses were never considered at that time.

Since the United States Federal Communications Commission (FCC) adopted the first UWB Report and Order in 2000 (www.fcc.gov), the interest in UWB technology has increased substantially in both academic and market places. Interest is stimulated by the expectation that UWB can solve the shortage of available frequency resources. UWB also promises enhanced data throughput with low-power consumption.

Ultra wide band signals are also used in military applications for detection of buried mines and for search for human beings in buildings as these pulses can penetrate ground, walls, etc. Rescue workers can use this technology by rescuing people inside buildings when disasters happened.

UWB communication devices can be also used in wireless distributed services like phone and computer networking throughout a building or home. They can also be utilized by police, fire, and rescue personnel to provide secure and efficient communication devices.

There are many issues involved in designing UWB systems, such as antenna design, interference, propagation and channel effects, and modulation methods. Designing the UWB antenna can be one of the most challenging of these issues. UWB antennas must cover an extremely wide band, 3.1GHz to 10.6GHz for the indoor and

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3

Chapter 1

handheld UWB applications have electrically small size, and hold a reasonable impedance matching over the operating band for high efficiency. In addition, they are required to have a non-dispersive characteristic in time and frequency, providing a narrow, pulse duration to enhance a high data throughput.

1.2 Major types of UWB antennas

Historically, there have been at least three different classes of UWB antennas. These classes are based on applications. First is the ―DC –to –daylight‖ class. These antennas are designed to have maximum bandwidth. Typical applications are ground penetrating radars, field measurements or electromagnetic compatibility, impulse radars, and shelter communication systems. The design goal of these antennas is to grab as much spectrum as possible. Second is the ―multi-narrowband‖ class. These antennas are designed as scanner or signal –intelligence antenna for receiving or detecting relatively narrowband signals through certain frequencies. The design goal of multi-narrow band antennas is similarly to grab as much spectrum as possible but to only use small sub –bands at any given time. Third is what might be called ―modern‖ UWB antennas. These are antennas designed for use in conjunction with the approximately 3:1 bandwidth, as 3.1-10.6 GHz UWB systems authorized by the FCC. The bandwidth requirements for a modern UWB antenna are narrower than for ―DC –to –daylight‖ antennas. These antennas have certain implication that distinguish them from the other more traditional classes of UWB antennas. First, instead of trying to grab maximal bandwidth, these modern UWB antennas must operate within a certain spectral mask. In this context, excessive bandwidth degrades system response and is counterproductive. Second, unlike multi-narrowband antenna, a modern UWB antenna potentially uses much, if not all, of its bandwidth at the same time. Thus, a modern UWB antenna must be well behaved and consistent across the antenna‘s operational band. Its properties include radiation pattern, gain, antenna matching, and requirement for low or no dispersion. A wide variety of antennas meets the demands of modern UWB system.

From the other hand the antennas for UWB can be conditionally divided into the following group [Schantz H.G., 2005]:

Frequency independent antennas: Antennas whose mechanical dimensions are short

compared to the operating wavelength are usually characterized by low radiation resistance and large reactance[Rumsey V., 1965]. This combination results in a high Q and consequently a narrow bandwidth. The current distribution on a short conductor is ideally sinusoidal with zero current at the free end, but because the

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conductor is so short electrically, typically less than 30° of a sine wave, the current distribution will be approximately linear. By end loading to give a constant current distribution, the radiation resistance is increased four times, thus greatly improving the efficiency but not noticeably altering the pattern. Because the effective source of the radiated fields varies with frequency, these antennas tend to be dispersive. Examples of frequency-independent antennas include spiral, log periodic, and conical spiral antennas [Taylor J.D., 1995].

Small element antenna: These antennas tend to be small, omni -directional antennas

well suited for commercial applications. Examples of small-element antennas include Lodge‘s biconical (Schelkunoff S.A., 1952) and bow-tie antennas [Brown G.H., 1952], diamond dipole [Fulerton L., 2001], ellipsoidal antennas [Lee K.S.H., 1974], and Thomas‘s circular dipole.

Horn antennas: A horn antenna is an electromagnetic ―funnel‖ concentrating energy

in a particular direction. Horn antennas tend to have high gain and relatively narrow beams. Horn antennas also tend to be large and bulkier than small-element antennas. These antennas are well suited for point-to-point links or other applications where a narrow field of view is desired. As an example we can mention the TEM horn antenna [Evans S., 1983].

Reflector antennas: A reflector antenna also concentrates energy in a particular

direction. Like horn antennas, reflector antennas tend to have high gain and are relatively large. Reflector antennas tend to be structurally simpler than horn antennas and are easier to be modified and adjusted by manipulating the antenna feed.

The Taylor table [Taylor J.D., 1995] (Appendix A) summarizes major parameters and properties of major types of UWB, such as: dipole, bicone, TEM horn, ridged horn, spiral and LPDA antennas.

1.3 The state of the art in UWB antennas for Impulse Radio

In today‘s marketplace for emerging communication technologies, the focal point of attention is ultrawideband radio as it not only promises enhanced data throughput with low-power consumption, but also provides high immunity against electromagnetic interference (EMI) and robustness to fading. It is expected that future short-range indoor ultra-wideband (UWB) telecommunication systems will operate in the frequency band from 3.1-10.6GHz according to the Federal Communications Commission (FCC) mask. One form of UWB technology is impulse radio, in which information is transmitted by very short EM pulses [Roy S.,

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5

Chapter 1

2004]. An impulse generator and special (so-called transient) antennas [Harmuth H.F., 1981] are thereby employed in order to radiate these very short pulses.

Antennas for impulse radio ultra-wideband (IR UWB) are designed to fulfil rigid specifications. Typical requirements are that they must radiate ultra-short pulses, that is, with a pulse width shorter than 1ns, without considerable ringing. An acceptable level of the antenna ringing is of about -40dB (with respect to the main pulse amplitude) after twice the duration of the pulse [Yarovoy A.G. , 2004], thus having an intrinsic operational bandwidth in excess of several gigahertz (i.e., ultimately from 3.1 to 10.6GHz). In practice all antennas have non –linear phase characteristic, but this non –linearity should be limited in such a way that maximum variation of the group delay within operational frequency band should be less 1ns (pulse duration). Moreover, UWB antennas must be well matched (i.e. achieve a VSWR<2) to the generator/transmitter or the receiver and have radiation efficiencies close to 100%. For many applications the antenna should have omni-directional radiation patterns. But there are many UWB applications in which antenna should be integrated in to a device which will prevent antenna radiation in one of hemispheres or the antenna can be located on a body. Thus the antenna should radiate just in one of hemispheres. Finally, an UWB antenna for mass market applications should be inexpensive in production without this affecting its performance.

Before the decade 1990‘s, all proposed UWB antennas were based on general volumetric and partly on planar structures, such as Schelkunoff‘s spheroidal antenna (1941), Lodge‘s & Carter‘s biconical antenna (1898, 1939), Lindenblad‘s coaxial horn element (1941), Brillouin‘s omni-directional and directional coaxial horn antenna (1948), King‘s conical horn antenna (1942), Katzin‘s rectangular horn antenna (1946). A number of small UWB antennas have been proposed. They are bow-tie antenna by Brown and Woodward (1950), Stohr‘s ellipsoidal monopole and dipole antenna (1968), Harmuth‘s large current radiator (1985), etc. From 1992, several microstrip, slot and planar monopole antennas with simple structure, such as circular, elliptical or trapezoidal shapes have been proposed [Seok H. Choi, 2004], [Kim Y., 2004], [Ying C., 2004], [Liang J., 2004], [Lui W.J., 2005].

1.3.1 Small element UWB antennas

In the Section 1.2 we discussed different UWB antenna classes. Our focus here is ―modern‖ UWB antennas or as named by Schantz, small element antennas. We are particularly interested in an electrically small size planar UWB antenna (as an example antenna dimensions are from 25×50mm2 to 70×60mm2, from application point of view). Herein we would like to show the state of the art in small element UWB antennas in comparison with above mentioned requirements for the IR UWB antennas.

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Nowadays research of UWB small antennas is focused on microstrip stacked patches, planar dipoles (bow-tie, almond, shaped dipoles), open aperture and planar monopole antennas with different matching techniques to improve the bandwidth ratio without degradation of its radiation pattern properties [Peyrot-Solis M.A., 2005], [Dauvignac J.Y., 1998], [Tzyh-Ghuang Ma, 2005], [Seong-Youp Suh, 2004], [Zhi Ning Chen, 2005], [Saou-Wen Su, 2004], [Peyrot-Solis M. A., 2005]. All of these antennas have advantages and disadvantages with respect to their use in IR UWB.

The first among the above –mentioned antenna classes is the microstrip antenna class [Deschamps G.A., 1953]. In the past, one serious limitation of microstrip antennas was the narrow bandwidth characteristic, being 15 to 50% that of commonly used antenna elements such as dipoles, and slots [Zurcher J.F., 1995], [Sainati R.A., 1996]. To increase the impedance matching bandwidth ratio it was necessary to increase the size, height, volume or adopt proper feeding techniques [Garg R., 2001]. An efficient method to increase operational bandwidth was adding additional patches. A microstrip antenna using a multiresonance behavior (additional radiative (parasitic patch) elements) [Targonski S.D., 1998] is capable to cover a frequency band from 3.1GHz to 10.6GHz. The size of the patch should be around

40×50mm2

for operating in 3.1-10.6GHz bandwidth (see Figure 1.1). Microstrip antenna gain is around 0dBi, and with an additional patch it can be up to 5dBi [Chih-Yu Huang, 2003], [Row J.S., 2005], [Yoharaaj D.Azmir, 2006].

a b c

Figure 1. 1 Geometry of the broadband CP coplanarsquare-patch antenna

[Wojciech J. Krzysztofik, 2006] (a), Geometry of a microstrip patch on a finite

grounded dielectric slab(b) and Layout Configuration of differentially fed aperture coupled microstrip patch antenna [K. M. Chan, 2005] (c).

The main disadvantages of microstrip antennas are: the antenna radiates in only one of hemisphere (we can‘t use them as omni-directional) and it is not able to transmit a short pulse without late –time ringing due to multiple resonances behavior.

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7

Chapter 1

antennas. Since the first aperture coupled antenna [Pozar D.M., 1985] was proposed, a large number of variations in geometry have been suggested. These are rectangular shape, circular shape, etc. One of the useful features of the aperture coupled antennas is that they can provide substantially improved operating bandwidths. Aperture coupled elements have been demonstrated with a bandwidth up to 10-15% using a single layer [Zurcher J.F., 1988], [Crog F., 1990], [Targonski S., 1993], and up to 30-50% using a stacked patch configuration [Crog F., 1991], [Edimo M., 1994], [Targonski S.D., 1996]. This improvement in bandwidth is primarily a result of the additional degrees of freedom offered by the stub length and coupling aperture size. The tuning stub length can be adjusted to compensate for the inductive shift in impedance that generally occurs when thick antenna substrates are used, and the slot can bring close to resonance to achieve a double tuning effect. The use of a stacked patch configuration also introduces a double tuning effect.

a b

Figure 1. 2 Quasi-omni directional slot antenna (a) and Frequency notched UWB microstrip slot antenna with fractal tuning stub.

One of the latest open aperture antennas is the frequency notched UWB microstrip antenna with fractal tuning stub (see Figure 1.2) [Lui W.J., 2005]. The notched function is obtained with the fractal tuning stub. The matching impedance bandwidth is from 2.66 to 10.76 GHz with a notched band from 4.95 to 5.85 GHz. The radiation pattern for this antenna is omni-directional. The antenna gain is close to 5dBi level in maximum.

The drawbacks of this antenna type are: the antenna is very sensitive to any changes in antenna slot dimension (very hard to tune the antenna); the antenna radiates in one of hemispheres.

The third UWB antenna class is planar monopole over a ground plane. Their main features are simple geometry and construction [Agrawall N.P., 1997]. The shape of the monopole might be different. One of the most interesting antenna, is the planar

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inverted cone antenna (PICA) (Figure 1.3 a) with radiation patterns similar to monopole disk antennas, but it possesses smaller dimensions. Its design provides more than 10:1 matching impedance bandwidth [Seong-Youp Suh, 2004]. A second version with two circular holes (Figure 1.3 b) extends the high end of the operating frequency range, improving the omni-directional radiation pattern bandwidth [Seong-Youp Suh, 2004]. A square planar monopole UWB antenna, which provides a good impedance matching in the frequency band from 1.979 to 12.738 GHz with a quasi-omni-directional radiation pattern, was reported in [Saou-Wen Su, 2004].

a b

Figure 1. 3 Geometries of PICA (a) , PICA antenna (without holes) and two-circular-hole PICA antenna (b) [Seong-Youp Suh, 2004].

From above one can conclude that planar monopoles over ground plane are good UWB antennas. The antenna can be well matched to the feeding line over a large frequency band (2-20 GHz). The antenna gain can be 4- 6dBi. The size of monopole can be small (approximately 30×30mm2

), but at the same time such monopole requires a ground plane with dimensions of around 150×150mm2. It makes monopole a 3-D dimensional antenna which is not well suitable for IR UWB applications.

The asymmetrical monopole antenna is the fourth class of UWB antennas [Pan C.Y., 2004], [Liang J., 2005], [Low Z.N., 2005], [Brzezina G., 2006], [Ma T.G., 2005]. This antenna type is a planar version of the above mentioned monopole antenna over ground plane. Large perpendicular ground plane in this antenna type is realized as an antenna planar ground plain (located on the other side of the substrate relatively to the radiative element). Planar asymmetrical monopoles perform well in commercial applications [Schantz H. S., 2002]. They allow matching with return loss on the order of -15 dB or better. Despite their planar form factor, they also exhibit near omni –directional dipole like patterns over better than 3:1 span frequency band. Planar asymmetrical monopole antennas are small. This antenna also offers a radiation efficiency in excess of 60% in band [Schantz H., 2004]. The antenna gain is around 5dBi. The asymmetrical planar monopole antenna can be implemented on a printed circuit board substrate making it inexpensive and readily manufacturable.

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9

Chapter 1

There are a lot of different shapes of radiative monopole elements. The most common ones are rectangular and elliptical shapes. Rectangular shape monopole

with dimension 14.5×15mm2

can operate from 3.2 GHz to 12GHz (at -10dB reflection coefficient level) with an antenna gain around 5dBi [Seok H. Choi, 2004]. This antenna has a rectangular patch with two steps, single slot on the patch, and a partial ground plane (Figure 1.4). This design provides a quasi-omni-directional radiation pattern.

a b

Figure 1.4 Rectangular patch UWB antenna: (a) front view (b) back view [Seok H.

Choi, 2004].

A planar monopole with an elliptical shape and microstrip feed line is shown in Figure 1.5a [Ying C., 2004]. The impedance matching bandwidth is from 3 to 10.6 GHz with a quasi-omni-directional radiation pattern at low frequencies. A similar design was also proposed in [Liang J., 2004], but in this case, it is a printed circular disc monopole UWB antenna. This design has the ground plane and the radiator structure in the same plane, suitable for integration with printed circuit boards (Figure 1.5b). The impedance matching bandwidth is from 2.78 to 9.78 GHz with an omni-directional radiation pattern.

a b

Figure 1. 5 LTCC UWB slot antenna: top layer and bottom layer [Ying C., 2004] (a), printed circular disc UWB monopole antenna [Liang J., 2004] (b).

A planar monopole antenna provides an alternative way to attach an antenna to enclosure [Liang J., 2005], [Brzezina G., 2006]. In spite of all good features, planar

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monopole antennas have a problem with the common mode currents. This is due to the small ground plane (comparable with SMA connector) and the close location. A common mode current affects the antenna efficiency.

The last class of small UWB antennas is planar dipole antennas. It consists mainly of bowtie antennas and elliptically shaped dipole antennas [Lin Yu-De, 1997], [Filipovic D.F., 1999], [Guiping Zheng, 2004], [Dubrovka F.F., 2006]. Planar elliptical dipoles perform well as UWB antennas. They allow matching with return loss level in the order of –15 dB over a frequency band with a fractional bandwidth 10:1 or better. Despite their planar form factor, they also exhibit near omni-directional dipole-like patterns over a larger than 3 to 1 span in frequency. Planar elliptical dipoles elements are as small as 0.14λ at their lowest frequency of operation. Antenna gain is around 0-3dBi in the azimuth plane, and depends on antenna flair axial ratio. Due to the large flare angle in the vicinity of the feeding point, the antenna has a low input impedance and can be well matched to a 100Ω feeding line. These antennas also offer radiation efficiencies in excess of 90% in the band. Planar elliptical dipoles can be implemented on printed circuit board substrates making them inexpensive and readily manufacturable.

Based on published results the elliptical [Schantz H. S., 2002] dipole seems to be a promising antenna in IR UWB applications. The antenna can be optimized to the size even less than 25×50mm2 for the desired frequency band. Antenna can be modified to radiate in one hemisphere. Thus in this thesis the elliptical dipole is selected as an antenna for further improvement. Antenna will be optimized to operate in frequency band from 3.1GHz to 10.6GHz. The antenna will provide good impedance matching within the operating frequency band (i.e. achieve a VSWR<2) to the generator/transmitter or the receiver and will have a radiation efficiency close to 100%. Antenna gain will be around 3-5dB. An acceptable level of the antenna ringing is of about -40dB (with respect to the main pulse amplitude) after twice the duration of the pulse. The maximum variation of the group delay within operational frequency band should be less then 1ns (pulse duration).

One drawback of this antenna is the balanced feeding. However it is an extremely cumbersome task to design a compact, low-loss transient balun for feeding the antenna from an unbalanced line and matching the output impedance of the feed line to the input impedance of the antenna. Different baluns for UWB antennas will be discussed further in section 1.3.2.

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Chapter 1 1.3.2 UWB antenna baluns

For numerous applications it is desirable to have a miniaturized flat (2D) antenna. Miniaturization of the antenna results not only in degradation of antenna efficiency, but also in increased impact of common mode currents. It is well-known that when the antenna dimensions become comparable to those of the ground plane (if any) and the feeding line, the common mode current becomes comparable in magnitude to the differential mode current, drastically changing the antenna properties. For narrow band antennas different baluns are used to suppress common mode currents and prevent their propagation over feeding lines. For UWB antennas, however, the design of such baluns becomes a very challenging problem.

Ultra Wideband (UWB) antennas can require baluns that are able to operate over a bandwidth in excess of one decade, and Baluns claiming to fall into this UWB class are the Marchand [Marchand N., 1944], Tapered line/Split coax [Duncan J., 1960], log-periodic [Basraoui M., 1998], loop balun [Manteghi M., 2004], Double-Y [Schiek B., 1976].

Refers to how much current flows in the differential mode compared to the common mode. Quantitatively this is defined as the Balun Ratio and is given by:

, arbitrarily set BR < 5 % for discussion.

The Marhand Balun (Figure 1.6) has enjoyed popularity due to its high performance where the point source operation guarantees symmetry and the 4th order band pass model allows for S11 design of below -10dB for bandwidths of a decade or more

[Cloete J.H., 1979]. Figure 1.6a shows a coaxial example where the detail of the point source gap has been highlighted and expanded to clarify how it works.

Of interest, due to its non-perfect symmetry, is the planar (or printed) Marchand balun. It provides a transition from unbalanced transmission line media such as microstrip, coplanar waveguide (CPW) and coplanar waveguide with finite ground plane (CPWFGP), to a balanced medium such as the coplanar strip (CPS) or slot line.

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a b

Figure 1. 6 Coaxial example of a Marchand Balun with expanded detail of the point source gap (a) and balun ratio of the planar Marchand balun [Palmer K.D.,

2004].

As an anti-phase feed balun, the log-periodic balun structure (Figure 1.7) provides two output signals with equal amplitude and 180 degree phase difference. The design presented here consists of six λ/2 resonator sections connected by λ/4 sections and produces a balance bandwidth of 40% [Sturdivant R., 1993]. Further bandwidth improvement has been reported when the lengths of the λ/2 sections are varied (gradually decreasing) [Basraoui M., 1998].

a b

Figure 1. 7 Model of the log-periodic Balun (microstrip ground plane not shown) (a) and balun ratio of the log-periodic balun (b) [Palmer K.D., 2004].

The double-Y balun also provides a transition from an unbalanced to a balanced medium (see Figure 1.8). In theory, double-Y baluns are all passive networks, but the bandwidth is limited by the media. At low frequencies the circular slot open circuit is only a good open circuit in a 2-3 octave range but can be improved with CPS [Jokanović B., 1994]. The balun approaches the upper edge of its bandwidth where the electrical length of the junction radius becomes 450. Also, the distance from the short circuit to open circuit should be made smaller than a quarter of a wavelength, otherwise the structure would start radiating and the circuit losses would increase [Schiek B., 1976]. For the structure to operate as a balun it is necessary to design the short circuited and open circuited lines to have the same input reactance magnitude (i.e. correct characteristic impedance and electrical

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13

Chapter 1

length). This condition is difficult to satisfy simultaneously for all the lines in both media, making the design of the balun rather complicated.

Figure 1. 8 Example of a Double-Y Balun implemented in CPWFGP-CPS

[Jaikrishna B.V., 2003].

Another type of balun is the loop. As a method for feeding UWB impulse radiating antennas, in particular the TEM horn antenna (Figure 1.9), the loop balun has been introduced [Palmer K.D., 2004] [Manteghi M., 2004].

a b

Figure 1. 9 Simulation model of the TEM horn antenna with a double-loop balun (a) and balun ratio of the loop balun feeding a TEM horn antenna [Palmer K.D., 2004].

The coaxial feeding cable is attached to the antenna's body all the way from the feed point to an area of low current density. It is then looped back via a path along which the tangential electric field is much smaller than at the feed point. This greatly reduces the current on the outside of the coaxial cable, and as a result, the current balance on the two conductors of the antenna is not disturbed. A dummy loop can be added to the other side of the antenna to ensure symmetry.

In literature, emphasis has been placed on tapered line/split coax balun to achieve a desired impedance match between the unbalanced transmission line and the antenna. In [Duncan J., 1960] an impedance transformer was developed by peeling away the outer conductor of the coaxial cable in a gradual fashion until an equal dimension twin line remained. This was done in such a way that the S11 had a Chebycheff

response in the pass band. Similar designs have gradually reduced the size of a microstrip ground plane to achieve geometrical symmetry. The reasoning behind the

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theory of balanced operation is that the taper will force all the microstrip ground plane currents to flow onto the antenna arm. This is not valid, as nothing prevents the current from curling round the edges and flowing back on the bottom of the ground plane and down the outside of the feed cable. Even if the geometrically symmetrical transmission line structure is extended a long way, separating the antenna and the taper by some distance, the balance performance is not improved. As conclusion of this section we would like to stress that UWB balun design is a quite challenging problem. Almost all above mentioned baluns cannot be printed on the same PCB with the antenna, except of the Double-Y Balun, because of there 3-D structure. Only the loop balun is capable to operate in frequency band from 3.1GHz to 10.6 GHz.

1.4 Challenges taken in this thesis and the research approach

Antenna engineers understand that the antenna design and antenna optimization should be incorporated into a device (or system). For numerous applications it is desirable to have a miniaturized flat (2D) antenna which is incorporated with a device and supports operation of the whole wireless system. In this thesis small UWB antennas for mobile communications will be designed using the just-mentioned system-oriented approach.

This thesis addresses several antenna problems commonly encountered in impulse UWB radio applications. The problems can be summarized as following:

1.

The technological demands require that the antenna should be mounted on a dielectric substrate. The infinite antenna dielectric substrate model is commonly used during the antenna optimization. Most of the engineers during the antenna design and antenna optimization do not take in to account that in reality the antenna dielectric substrate is finite with definite dimensions. Influence of the finite antenna substrate should be taken into account during antenna (antenna into device) optimization. Hence, the problem is how to define the optimal size of the dielectric substrate with a certain characteristics. We will solve this problem theoretically. The computational model will be built based on the volumetric mixed-potential integral equation and takes into account the finite size of the substrate. Influence of the substrate size on the antenna input reflection coefficient, the antenna gain and the radiation patterns will be investigated. This problem is addressed in Chapter 4.

2.

Miniaturization of the antenna results not only in degradation of antenna efficiency, but also in increased impact of common mode currents. It is

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15

Chapter 1

well-known that when the antenna dimensions becomes comparable to those of the ground plane (if any) and the feeding line, the common mode current becomes comparable in magnitude to the differential mode current and drastically changes antenna properties. For narrow band antennas different baluns are used to suppress common mode currents and prevent their propagation along feeding lines. For UWB antennas, however, the design of such baluns becomes a very challenging problem [Roberts, 1957], [Huifang Gu, 1999]. Thus, the problem is how the influence of common mode current can be decreased with a proper design of feeding circuit. In order to increase the common mode impedance of the antenna several designs of the feeding circuit will be numerically investigated. One of these designs shows a satisfactory performance both in terms of increase of the common mode impedance and improvement of the antenna matching to the generator will be experimentally verified. This problem is addressed in Chapter 5.

3.

As it was mentioned earlier, one of the UWB antenna design challenges is antenna feeding. The antenna must be well-matched to a generator or to receive circuit and have a radiation efficiency close to 100%. A co-design ―antenna plus generator‖ can be a good solution. The positive issues of the antenna integration with a generator on chip are the following: first of all the chip is located on the same PCB with an antenna which leads to a reduced antenna transmitting (receiving) system size. Second, there is no need for any transmitting line (or balun) between antenna and generator. It therefore avoids the common mode currents and makes a challenging design of the UWB balun needless. Third, the antenna integration with a chip gives an extra flexibility in terms of antenna matching to the generator, because any (complex) antenna impedance can be selected and the antenna and RF circuit designers are not limited to the commonly used 50Ohm interface. Hence, the problem is how to co –design the antenna with a pulse generator on the same PCB, and find optimal position of a RF device (i.e., pulse generator) with respect to the antenna. To this end we will perform a co-design of the system ”antenna plus generator”. The co-co-design is a three-step approach: firstly, the interface between the RF device and the antenna is identified. Secondly, an electromagnetic design encompassing the generator, antenna and the PCB characteristics is performed, resulting in the antenna geometry, the position of the RF device with respect to the antenna flairs and the input impedance of the antenna. And finally, the output impedance of the generator is designed accordingly. The output circuits of the generator

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(i.e., voltage buffers) match the generator directly to the antenna input impedance. This problem is addressed in Chapter 6.

4.

The antenna plays a primary role in communication systems. Antenna design becomes more challenging for wearable devices where the influence of the body on the antenna characteristics, such as input impedance, current distribution, gain and radiation patterns, needs to be taken into account. UWB communication systems for a personal wireless area network can‘t be successfully implemented without achieving an understanding of the influence of the body on the antenna characteristics. The problem is how to quantify a human body influence on the UWB antenna performance for different antenna location on a human body. We are going to perform UWB propagation channel measurements for scenarios when transmit antenna is located on a human body. Measurements will be done for different antenna locations: on a head, on a torso, on a belt. For reference purposes we measured signal propagation when the transmit antenna will be in free space. This problem is addressed in Chapter 7.

5.

For a wide variety of applications, omni-directional radiation patterns are highly desirable. But there are many UWB applications in which antenna should be integrated in to a device or the antenna is located on a body which will block antenna radiation into a hemisphere. In both examples an antenna with radiation in hemisphere is required. Such radiation can be achieved by antenna shielding. The shield should decrease antenna back radiation, but at the same time should not influence the antenna performance. This problem is addressed in Chapter 8.

The major approach in solving the abovementioned problems is to analyze the problem based on an accurate electromagnetic model, determine physical relations between the performance of an antenna and its shape, size and environment, optimize antenna performance using the above-mentioned physical relations and verify the solution experimentally.

1.5 Novelties of approaches and main results

The research reported in this thesis has utilized several novel approaches, which have never been used by other parties. Additionally, the research has led to a number of scientific results. In the following we list the major novel approaches and main novel results achieved in this thesis.

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17

Chapter 1

Novel approaches

I investigated possibility to use so-called shielded loop to feed small antennas [Vorobyov A. V., 2005], [Yarovoy A.G., 2005]. The designed feeding circuit drastically reduces the common mode current and removes spurious radiation from the feeding cable. As a result a generic problem for all small-size UWB antennas has been completely solved.

I demonstrated principal possibility of integration the RF chips with dipole balanced antenna on the same printed circuit board [Vorobyov A. V., 2006]. We implemented the above suggested approach to demonstrate principal possibility to embed RF chips directly in an UWB antenna.

I developed a shield for UWB antenna without use of absorber materials and thus without decrease of antenna efficiency. In order to avoid resonances without shield, a staircase design has been implemented.

Main novel results

In order to increase the common mode impedance of the antenna I proposed a novel design for a small planar antennas feeding line which is based on so-called shielded loop. We developed and measured two concept models for the loop feed circuit: with a single input and two differential inputs. Optimization of the loop size resulted in the loop size of 11mm. For loop feed line we used 50 Ohm coaxial cable with 2.1mm diameter. The length of the coaxial wire from SMA connector to the loop itself was around 30cm. From the measured antenna radiation patterns we concluded that the loop feed line works very well as symmetrical device. And finally, using detailed theoretical model, I checked surface currents on the feeding lines. It was shown that the surface current level for loop antenna is much lower in comparison with just coaxial cable. The difference is about in two times for a different loop surface part. This is a novel result as in commercially available antennas (e.g., Pulse On antenna) the common mode current is suppressed by absorbers placed on the feeding cable. This solution is addressed in Chapter 5.

I implemented integration of a pulse generator into an UWB antenna to demonstrate a principal possibility to embed RF chips directly in an UWB antenna. To reduce manufacturing costs and simplify technological

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implementation of RF chips we aimed at experimental realization of the co-design in the frequency band from 3 GHz till 6GHz. The developed demonstrator transmits a 300ps pulse with approximately 223ps ringing covering bandwidth from approximately 2.8GHz till 6.7GHz at -3dB level. This is a wider than the originally aimed frequency range. As nobody before performed integration of a miniaturized UWB antenna with an impulse generator on chip we claim to develop and demonstrate the first ever active miniaturized UWB antenna. This problem is addressed in Chapter 6.

I designed a shielded UWB antenna and optimized the antenna back shield. In comparison with a known design, the elevation profile of the shield has been changed from E-plane to H-plane; the relative dimensions of the shield and its profile have been changed. Thus the developed back shield can be considered as a new design. Measurements of antenna with a back-shield show that the back shield reduces the antenna back radiation by approximately 20-30dB (this value depends from the frequency), and increases the antenna gain approximately two times (6dB) with respect to the unshielded antenna. The operational bandwidth (from 4.5 GHz till 10.2GHz at -10dB level) remains as wide as for the unshielded antenna however it is shifted now towards higher frequencies. Additional measurements with a metal object behind the antenna shield show that the antenna radiation is almost not affected by this object. Thus the antenna can be easily integrated into different devices or can be placed on a human body. The antenna shield optimization and shielded antenna measurements are presented in Chapter 8.

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19

Chapter 1 1.6 Embedding in European and national research agendas

The thesis has been done in the framework of the AIRLINK (Ad –hock Impulse Radio: Local Instantaneous Networks) project which has started on the 1st of August 2002 and involves TU Delft (ISC, DIMES, IRCTR) and TNO-FEL.

The AIRLINK project investigates the possibilities of UWB for wireless communication over short distances, at high speed and with low power usage. In additional, AIRLINK investigates the possibilities of UWB for combining data communication with localization and the design of an ad –hoc communication network. This research has been performed within one of the work packages of the AIRLINK project.

At the end of the AIRKINK project we demonstrated a working prototype of the UWB antenna. This prototype of the antenna is fed in two different ways: by coaxial feeding line, based on the shielded loop and by complete integration antenna with a pulse generator. By that, we completely fulfil our obligations in AIRLINK project. Optimization of an elliptically shaped dipole and development of the active UWB antenna by integration of a pulse generator into the antenna were highly valued as major achievements within the WP2.2 (small UWB antennas) of the ACE2 (Antenna Center of Excellence) –a EU –sponsored Network of Excellence. The research results have been presented in the following reports [Small UWB antennas WP2.2 Report, 2006].

Furthermore, some research within this project has been done in the framework of

EUROPCOM (Emergency Ultrawideband Radio for Positioning and

COMmunication) project –EU –sponsored STREP project. Study on the environmental impact of the performance of an UWB antenna and its radiation patterns were used in this project to determine an optimal position of an UWB antenna on a firefighter suit [Investigation of the impact of a human body on UWB antenna radiation Report, 2006].

The developed shielded elliptically-shaped dipole antenna will be downscaled to be used in RADIOTECT (Ultra Wideband Radio application for localisation of hidden people and detection of unauthorized objects) project –EU –sponsored CRAFT project.

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Chapter 2 2. Tools for theoretical

analysis

2.1 Introduction

The rapid growth in the communication and high-speed electronics industries is demanding more of electromagnetic simulation software. Electronic design engineers now use sophisticated electromagnetic simulation software (EMS) to accurately analyze and optimize designs prior to constructing a physical prototype [M.S. Mirotznik, 1997]. Electromagnetic simulations are widely used to reduce antenna development time.

It is impossible to classify all electromagnetic devices and problems that a researcher may wish to solve. Still, three broad classes of applications can be identified for which EM software tools are readily available.

One class consists of low-frequency devices such as electric motors, transformers, and wires. Here a variety of two- dimensional and three-dimensional finite-element packages are used to predict electric and magnetic fields. Some of the first EM simulation programs were intended for low-frequency devices.

Broadband electronic circuits, such as computers, are another class of devices for which EM software tools are available. Here an engineer can use these programs to predict signal integrity, cross-talk, or EMI.

Devices in the last category are those that require high-frequency electromagnetic analysis and include antennas, radar systems, and microwave components. In this case, electromagnetic simulators serve to predict S-parameters, radar cross sections, and radiation patterns.

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If EM simulators are to predict electromagnetic behavior as accurately as possible, they should be able to model all structural and material properties of a device. Most modern EM simulators do so by incorporating a graphically based preprocessor into the simulation package. Through the preprocessor, users can define the geometry--using modeling tools similar to those found in common computer-aided design (CAD) software, and can assign boundary conditions, material properties, and energy sources.

Once the device has been geometrically modeled, a so-called solution space must be created and exactly related to that model: a mesh of small computational elements needed for the analysis. Finite-element solvers, for example, require a mesh of non-overlapping 2-D or 3-D elements, like triangles or tetrahedrons, respectively. Many finite-difference solvers, on the other hand, use a grid of rectangular cells, which results in a staircase-like approximation to curved surfaces.

Constructing the mesh is a critical step on the way to a solution. A well-designed mesh produces results that are both accurate and computationally efficient. A perfect mesh, of course, does not exist: trade-offs are always made among accuracy, computation time, and memory requirements.

Once the geometry is at hand, the next step is the analysis. Recent theoretical advances in computational electromagnetics have resulted in new algorithms for EM analysis. These include finite-element method (FEM), moment method (MoM), transmission-line matrix method (TLM), difference method (FDM), finite-element and finite-difference time-domain (FET and FDTD), as well as combinations of these methods.

Each method offers advantages and disadvantages depending on the class of problems. On the one hand the finite-element method well performs at modeling complex structures with curved boundaries, but may less attractive for problems that are electrically large (many wavelengths in size) because so much memory is required. On the other hand, methods such as finite-difference time-domain can tackle larger problems, but may have difficulty conforming to curved surfaces. The aim of this Chapter to give a short overview of numerical methods used in this thesis for analysis and optimization of antenna. These methods are implemented in a commercially available software simulator FEKO, which was mainly used in theoretical analysis. Strong and weak points of selected methods are discussed and a brief comparison with alternative methods is given.

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23 Tools for theoretical analysis

Table 2.1 (Comparison of possible numerical techniques)

Field method

Source method

Base

Electromagnetic field Current and charges

Equations

Differential equation Integral equation

Discretization

3 –Dimensional 2 –Dimensional

Infinity of space

(open problem)

Special ABCs must be introduced

Exact treatment

Methods

Finite differential methods

(FD)

Finite element methods (FEM)

Method of moment (MoM)

Available codes

XFDTD, Empire, MAFIA,

Maxwell Eminence

FEKO, Concept, Aseris, FISC, IE3D

This chapter is organized as follows. Basic explanations of the Methods of Moment (MoM), surface and volumetric integral equations are presented in section 2.2. Alternative numerical methods such as FDTD and FEM are briefly outlined in section 2.3. Finally, this chapter is closed with some conclusions in Section 2.4.

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2.2 FEKO simulation environment

The analysis of the antennas presented in this thesis has been carried out via numerical simulation FEKO (FEKO, 2006).

The program FEKO is a commercial simulator, which is based on the mixed – potential form of the IE formulation, which is solved by the MoM using Rao – Wilton –Glisson (RWG) basic functions. Electromagnetic fields are obtained by first calculating the electric surface currents on conducting surfaces and equivalent electric and magnetic surface currents on the surface of dielectric bodies. The currents are calculated using a linear combination of basic functions, where the coefficients are obtained by solving a system of linear equations. Once the current distribution is known, further parameters such as the near field, the far field, scattering, directivity or the input impedance of antennas can be obtained.

For the modeling of dielectric/magnetic bodies, the MoM, as implemented in FEKO, offers a number of different techniques such as the surface equivalence principle, the volume equivalence principle, special Green‘s functions for planar multilayer media, and approximations for coatings and thin dielectric sheets.

2.2.1 The method of the moments

The method of moment (MoM) was one of the first numerical methods to achieve widespread acceptance in electronic engineering for the analysis antennas and scatterers [Harrington R.F., 1968]. The MoM is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in boundary integral form).

In the method of moments, the radiating (scattering) structure is replaced by equivalent currents. These are normally surface currents. Volumetric currents can be used for inhomogeneous dielectric bodies. This surface current is discretised in to a wire segment and (or) surface patches. A matrix equation than derived, representing the effect of every segment (patch). This interaction is computed using the Green function for the problem. The relevant boundary condition is then applied to all the interactions, yields the (approximate) current on each segment (patch). The resulting matrix which must be factored (or used in an iterative solution scheme) is fully populated, with complex valued entries. Typical matrix dimensions range from some hundreds for small antenna problems to several thousand –the upper limit is imposed by computational limitations, either limited memory or excessive run –time.

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25 Tools for theoretical analysis

Figure 2.1Computational model of the aircraft in FEKO.

In FEKO constant, linear and triangular functions (so –called basic functions) are used for approximations of currents. The boundary conditions on a metal surface in

MoM are verified approximately, namely: only in several points within the limits of

each elementary segment (strictly these conditions should be carried out in all points). As a result of the system of the linear algebraic equations (SLAE) with respect to the magnitude of basis function is derived. The currents are calculated using a linear combination of basis functions, where the coefficients are obtained by solving a system of linear equations. Once the current distribution is known, further parameters such as the near field, the far field, scattering, directivity or the input impedance of antennas can be obtained. In FEKO the elementary surface segment has the triangular form that allows to describe well any surface shape including the curved surfaces.

There are two principal approaches to build a computational model of a dielectric body. One approach is to use the surface integral equations while another one uses volumetric integral equations.

2.2.2 Surface integral equation

Integral equations for either the magnetic or electric fields scattered by a metal surface can be derived for problems with currents flowing on surfaces. The derivation is quite complex, and only the results will be presented here. One integral equation couples the incident electric field to the induced surface current, and is known as Electric Field Integral Equation (EFIE):

( )

( ) ( , )

( )

( , )

,

inc S S S S

n E

r

n

jk J r G r r

J r

G r r

dS

r r

S

jk



 



 

(2-1)

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The ' operator implies differentiation in the source coordinates. n is the unit vector on the surface S.

G r r

( , )

 

is the scalar free-space Green function given by

R

e

r

G

jkR

4 )

,

(



(2-2) r r R , (2-3)

(2-1) is valid for both closed and open surfaces. In the latter case, JS



is the average

of surface currents on both sides of such surfaces.

The other integral equation couples the incident magnetic field to the induced surface current, and is known as the Magnetic Field Integral Equation (MFIE):

S

r

S

d

r

G

r

J

n

r

H

n

r

J

S S imc S



,

)

,

(

)

(

)

(

)

(

2

1

(2-4)

This is valid only for closed surfaces.

In both equations, the presence of singularities raises delicate issues and requires careful treatment. The simple expedient of slightly offsetting field and source points as was done with the one-dimensional wire problem is no longer possible.

Mathematically, the EFIE is a Fredholm integral equation of the first kind (the unknown is present only in kernel). The MFIE is a Fredholm integral equation of the second kind - the unknown is present both inside and outside the kernel. Fredholm type two integral equations are generally more well-posed; this explains much work using the MFIE. Since in a well-posed problem is one whose solution is not strongly depending on a specific problem. However, the requirement for a closed surface S is frequently a problem in applied electromagnetics, with the result that the EFIE is usually preferred in practical codes. Finally, linear combinations of the EFIE and

MFIE have also been used; not surprisingly, this method is known as the Combined

Field Integral Equation (CFIE). The CFIE will not be discussed here.

When dealing with surfaces using the MoM, two aspects need attention. The first is that we need to split the geometry up into small elements. The simplest approach, and the first one explored historically in codes such as NEC2, uses square (or rectangular) patches. However, for general two-dimensional geometries, triangular elements are better for approximating the geometry, and this is the approach which most modern codes (including FEKO) use. The second aspect is that the physical

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27 Tools for theoretical analysis

parameter being approximated is now two-dimensional. The basis function must also incorporate this.

Nowadays, a very widely used basis function is the triangular patch, introduced by Rao, Wilton and Glisson in 1982 [S.M. Rao, 1982]. The basis function is often known simply as the RWG element. Subsequent work led to the realization that this basis function is very closely related to the edge-based elements often used in contemporary finite element analysis.

The basis function includes some new features which have not yet been encountered in this sectoion. Most importantly, the basis function is a vector, which means that the individual scalar components (e.g.

J

_x,

J

_y and

J

_z), can only be recovered with some manipulation. The essential idea is to enforce current continuity over the edge of a patch. The interpolation function used to achieve this is the following:

2 ( )

2

0

n n n n n n n n n

l

r in T

A

l

f r

r in T

A

otherwise



_

(2-5)

(Figure 2.2) defines the vectors



_n and



_n . Note that the basis function is defined over two adjoining triangles

T

_n and

T

_n which share a common edge.

A

_n is the area of triangle

T

_n and similarly

A

_n the aria of

T

_n .

l

n is the length of the shared

edge. The vector



_n is the vector position within triangle

T

_n , with the left-hand node of

T

_n as origin; similarly,



_n is the (opposite-directed) vector position, with the right-hand node of

T

_n as origin.

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Edge 1 Ed_ge 3 E d g e 2 n T Tn n



n



Figure 2.2 The two connected triangles

T

_n and

T

_n , sharing a common edge, which support a RWG basis function.

By introduction a coordinate system known as simplex coordinates the study of interpolation functions on triangles much simpler, this approach used in Finite Element analysis; the vector can be written rather simply in that coordinate system. The terms

l

_n

2 A

_n and

l

_n

2 A

_n are normalizing constants.

Next, we note some characteristics of RWG basis function. The basis function has no component normal to the upper or lower sides of either of the triangles, only the central (shared) edge. Without more detailed theoretical analysis, the following statement is made without proof: the current crossing this shared edge is linearly interpolated in the tangential direction (i.e. along the edge) and interpolated as a constant normal to (i.e. across) the edge. This latter value is usually the "degree of freedom" (the unknown value of current) which is associated with this basis function; the current associated with this edge is thus approximated as

)

(

)

(

r

I

f

r

J



_n



_n



_n



.

What are the currents flowing across the two other edges? To approximate these, one defines additional basis functions on each of the other two connected triangles; thus on anyone triangle, there are three such basis functions, with three associated degrees of freedom (unknowns), which are the normal components of current on each edge. Within the element, the total current then approximated by the sum of these three basis functions.

With the edges numbered as in Figure 2.2, the total current on triangle

T

_n is given by: 1 1 2 2 3 3 ( ) ( ) ( ) ( ) n n J r  I f r  I f r  I f r  r in T (2-6)

We note again that the basis functions carry the vector information and the unknowns for (

I

₁

, I

₂ etc.) are just scalars.

Planar elliptically shaped dipole antenna for UWB Impulse Radio