Body Area Networks: Analytical Characterization and Investigations in … · 2011. 12. 15. · Body Area Networks: Analytical Characterization and Investigations in Optimal Antenna

Body Area Networks: Analytical Characterizationand Investigations in Optimal Antenna Design

by

Noman Murtaza

a thesis for conferral of Master of Science in CSE.

Dr. Jon Wallace, Jacobs University Bremen

Dr. Buon Kiong Lau, Lund University Sweden

Date of Submission: 24. July 2009

School of Engineering and Science

Declaration

I hereby confirm that this thesis is an independent work that has not been submittedelsewhere for conferral of a degree.

Noman Murtaza

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Acknowledgements

First of all, I would like to thank my parents and my wife, who have always supported andencouraged me. This work would not have been possible without kind supervision of Prof. JonWallace. He has supported me since the very first days of my stay at Jacobs University Bremen.In particular, I would like to thank him for introducing me to the subject of this thesis andhis careful guidance throughout this work and his patience and time for proof-reading of thisthesis. Many thanks to Dr. Buon Kiong Lau for accepting to review this thesis. I would alsolike to thank Mr. Rashid Mehmood for being a subject for measurements.

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Contents

1 Introduction 11.1 Body Area Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History and Development of BAN . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Optimal BAN - Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 BAN - Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Dominant Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.2 Full-Wave Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.3 3D Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Investigations in Optimal Antennas for BANs . . . . . . . . . . . . . . . . . . . . 6

2 Closed-Form Analytical Solution for Body Area Networks 72.1 Line Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Point Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 ẑ-directed Line source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Incident field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 Scattered field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.3 Total Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 ρ̂-directed Line source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.1 Incident field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.2 Scattered field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.3 Total Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 φ̂-directed Line source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.1 Incident field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.2 Scattered field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5.3 Total Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Instability at ρ = ρ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.7 Validation of Derived Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7.1 Line source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.7.2 Point Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Optimal Antenna Design for Body Area Networks 403.1 Diversity-based Antenna Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.1 Diversity Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1.2 Shadow region at different sensor heights . . . . . . . . . . . . . . . . . . 413.1.3 Excess Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1.4 Optimum Inter-sensor Spacing . . . . . . . . . . . . . . . . . . . . . . . . 443.1.5 Diversity Gain at Optimum Inter-sensor Spacing . . . . . . . . . . . . . . 463.1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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3.2 Antenna Design for Constrained Apertures . . . . . . . . . . . . . . . . . . . . . 473.2.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.3 Case I: Point transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.4 Case II: Transmit Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.5 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 Measurements 554.1 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Indoor Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 Relative Gain of Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Conclusion and Future Work 635.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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List of Figures

1.1 Data Rate Vs Power for BANs (by IEEE 802.15 Task Group 6) . . . . . . . . . . 21.2 Proposed two-step procedure for body area modeling . . . . . . . . . . . . . . . . 51.3 Geometry and coordinate system for our analysis . . . . . . . . . . . . . . . . . . 6

2.1 A parabolic contour integral around singularity kz = k . . . . . . . . . . . . . . . 82.2 Displaced cylindrical harmonic and its displaced coordinate system . . . . . . . . 92.3 Orientation of Receive/Transmit fields (ρ̂) . . . . . . . . . . . . . . . . . . . . . . 152.4 Orientation of Receive/Transmit fields (φ̂) . . . . . . . . . . . . . . . . . . . . . . 242.5 Channel Comparison for ẑ-directed line/point source/sensor with FDTD . . . . . 372.6 Channel Comparison for φ̂-directed line/point sources with FDTD . . . . . . . . 382.7 Channel Comparison for ρ̂-directed line/point sources with FDTD . . . . . . . . 39

3.1 Adding Diversity to BAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 ẑ-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 ρ̂-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 ρ̂-φ̂-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5 φ̂-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6 φ̂-ρ̂-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.7 ẑ-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.8 ρ̂-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.9 ρ̂-φ̂-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.10 φ̂-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.11 φ̂-ρ̂-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.12 Shadow Region Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.13 Excess Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.14 Optimum inter-sensor spacing in degrees . . . . . . . . . . . . . . . . . . . . . . . 453.15 Optimum inter-sensor spacing in wavelengths (λ) . . . . . . . . . . . . . . . . . . 463.16 Comparison of diversity gain for different polarizations . . . . . . . . . . . . . . . 473.17 Optimal current distribution for ẑ-directed source/sensor . . . . . . . . . . . . . 523.18 Optimal current distribution for ρ̂-directed sources . . . . . . . . . . . . . . . . . 533.19 Optimal current distribution for φ̂-directed source . . . . . . . . . . . . . . . . . 54

4.1 λ/6-monopole antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Measurement belt - white lines are measurement points . . . . . . . . . . . . . . 564.3 Measurement setup for indoor measurements . . . . . . . . . . . . . . . . . . . . 574.4 Measurement and model comparison for ρ̂-directed line/point sources . . . . . . . 584.5 Measurement and model comparison for φ̂-directed line/point sources . . . . . . 594.6 Measurement and model comparison for ẑ-directed line/point source/sensor . . . 604.7 Circuit Models for Measurement Scenario . . . . . . . . . . . . . . . . . . . . . . 61

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Chapter 1

Introduction

1.1 Body Area Networks

A body area network (BAN) is a system of devices close to a person’s body that cooperate for thebenefit of the user. Different from any other wireless network, nodes are located on the clothes,body or even implanted inside the body. Depending on the implementation, the nodes consistof sensors and actuators. BAN offers promising applications in medicine, military, security,consumer electronics, multimedia etc., all of which make use of the freedom of movement that abody area network offers. The most obvious of these applications is in medicine where a patienthas several sensors for measuring temperature, blood pressure, heart rate, electrocardiogram(ECG), etc., and with all these devices, the patient has high freedom of movement. Thisimproves the quality of life of a patient and could reduce hospital costs. Apart from medicalapplications, a person using a number of devices like mobile phone, PDA, pocket TV, etc.,might need different devices to communicate. Resource sharing could be one reason, e.g., onedisplay for several devices.

The development of any communication link should start with a model for the channel. Thechannel for a body area network is quite complex because of the complicated propagation mech-anisms near the human body, which may be affected by complex shapes shapes and differentlayers of tissues, bones etc., where each layer has different permittivity and conductivity. Theelectromagnetic (EM) waves can propagate around the body via two paths: (1) penetrationthrough the body, and (2) creeping waves that propagate on the surface of the body. Simu-lations have shown that the loss due to penetration inside the body is very high [1]. Hence,the contribution of the penetrating waves can be neglected, especially for body worn antennas,which is the focus of this thesis. Therefore, for body worn BANs, the human body can be mod-eled as a volume filled with a lossy material having dielectric properties comparable to averagedielectric properties of human tissues.

Another important aspect of BAN is that most of the radiated energy should be confined tothe surface of the body. Therefore an antenna should be designed in such a way that the wavestravel around the body in a guided fashion [2]. The wearable antennas should be designedto favorably propagate trapped surface waves present because of the non-perfect conductingnature of the body. In this way, the skin-air interface is used to guide the signal around thebody. Energy reflecting from nearby structures or furniture is another potential propagationmechanism. However, since BANs should function regardless of the random operating environ-ment, this propagation mechanism should not be relied on. Therefore, this work focuses onbody-centric propagation only.

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Figure 1.1: Data Rate Vs Power for BANs (by IEEE 802.15 Task Group 6)

In this thesis, an analytical model for the BAN channel is derived that includes the three pos-sible orthogonal transmit current distributions: ẑ, ρ̂ and φ̂. Actual BAN measurements on ahuman subject are performed to validate and understand any potential limitations of the an-alytical model. Investigations on optimal antenna design for BANs are performed, includingtwo-antenna spatial diversity and the derivation of optimal current distributions for limitedtransmit/receive apertures. Therefore, this work serves as an important first step in the devel-opment of optimal antennas for BANs.

1.2 History and Development of BAN

BAN technology has emerged as a natural by-product of sensor network technology and biomed-ical engineering. Professor Guang-Zhong Yang was the first person to formally define the phraseBody Sensor Network (BSN) with publication of his book Body Sensor Networks in 2006. BSNtechnology represents the lower bound of power and bandwidth from the BAN application sce-narios. However, BAN technology is quite flexible and there are many potential uses for BANtechnology in addition to BSNs. Some common applications of BAN technology are Body Sen-sor Networks (BSN), sports and fitness monitoring, wireless audio, mobile devices integration,personal video devices, security, etc. Each of these applications has unique requirements interms of bandwidth, latency, power usage, and signal distance.

IEEE 802.15 is the working group for Wireless Personal Area Networks (WPAN). The WPANworking group realized the need for a standard for use with devices inside and around closeproximity to the human body. This group established Task Group 6 to develop the standardsfor BANs. IEEE 802.15 defines a body area network as “a communication standard optimizedfor low power devices and operation on, in or around the human body (but not limited tohumans) to serve a variety of applications including medical, consumer electronics / personalentertainment ...” The BAN task group has drafted a standard that encompasses a large rangeof possible devices, which also gives device developers a target for balancing data rate andpower. Figure 1.1 from IEEE 802.15 describes the ideal position for BANs in the power vs.data rate spectrum. As seen in Figure 1.1, the range of BAN devices can vary greatly in termsof bandwidth and power consumption.

BANs are still in the research phase, and efforts are being made to gain insight to BAN channels

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and the potential technologies that could be used to implement BANs. In [1], channel modelshave been developed for body area networks at 400 MHz, 900 MHz and 2.4 GHz using FDTDsimulations, and it was argued that the propagation around the body is mainly because of thecreeping waves that travel on the surface. Much less energy penetrates into the body, and hencethis mechanism can be neglected for applications involving body worn networks. The work in[2] suggests robust communication strategies for BAN and discusses different simulation andmeasurement scenarios. Measurements indicate that diversity is practical for on-body systemswhen the mode of communication is on-body surface waves. A simple Green’s function basedanalytical model is developed in [3], which models the body as an infinite lossy cylinder, andthis served as the starting point for the work in this thesis. Wave propagation in inhomogeneousmedia is discussed in [7] and dielectric properties of human tissues that are required for BANanalysis are given in [8].

1.3 Optimal BAN - Challenges

The body area network channel is quite different from a standard long range (far-field) channelfor a number of reasons: (1) it is a close-range (near-field) channel, (2) signals undergo extremeshadowing due to very close proximity to the body, and (3) low-power operation is desirable forbattery-operated devices. The challenges that need to be addressed for practical implementationof BAN are

• Interoperability: BAN systems should be compatible with existing standards like blue-tooth, Zigbee, etc., since in certain cases, especially for medical applications, data needto be transferred to the base station using these technologies. Hence the capacity thatBAN provides should be comparable to these standards in order to ensure seamless datatransfer.

• Security: BAN transmission should be secure and accurate. The data generated from theBAN should have secure and limited access for intended users only.

• Privacy: Social acceptance of BANs is key to the growth of this technology. People mightconsider BAN technology as a potential threat to freedom, if the applications go beyondsecure medical usage.

• Power Consumption and Confinement Low power is another requirement for BAN sinceuser would like to conserve energy, especially in the cases when devices are implantedinside the body. An optimal BAN should also ensure that most of the energy emittedfrom the devices should stay on the body and ideally no energy should radiate from thebody in order to avoid interference with other BANs.

• Robustness BANs should be robust, such that the antenna orientation, distance from thebody, position around the body, and location on the body do not effect the performance,reliability, and efficiency of the network.

The work in this thesis concentrates on the last challenge, or specifically, how the antennasystem should be designed to provide a reliable link from transmit to receive. This aspectis studied by considering two-antenna diversity for BANs as well as optimal antenna currentdistributions for on-body antennas with finite aperture.

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1.4 BAN - Model Development

Accurate characterization of specific body area network scenarios is possible using numeri-cal electromagnetic analysis methods like method of moments (MOM), finite-difference time-domain (FDTD), the finite-element method (FEM), etc., but these methods tend to be compu-tationally expensive. Statistical methods, such as multipath far-field models, could also be usedfor BANs, but due to the near-field propagation mechanisms governing BANs, these methodsmay have limited accuracy. The work in this thesis takes a middle approach by developing sim-ple analytical models that possibly capture the important propagation mechanisms for BANchannels, but at a fraction of the computation required for numerical methods.

Two possible approaches for developing a simple analytical model for BANs are discussed below.

1.4.1 Dominant Mode Analysis

In this method, the fields existing in and near the body are expanded in terms of eigenmodesthat relate the fields to underlying electric or displacement currents. For example, considera circular PEC cylinder with infinite extent in the z direction. The z-directed current andincident field on the cylinder are related by

Ei(x) = −jk0η∫

Sdx′g(x,x′)I(x′), (1.1)

where x is a 2D coordinate on the surface of the cylinder, k0 is the free-space wave number,η is the intrinsic impedance of the surrounding medium, g(x,x′) is the 2D free-space Green’sfunction, I(x′) is current, Ei(x) is incident field, and S is a contour on the surface of thecylinder.

The 2D Green’s function can be expressed as an expansion of cylindrical modes, or

g(x,x′) =j

4H

(2)0 (k|x− x′|) (1.2)

=j

4

∞∑

`=−∞J`(ka)H

(2)` (ka)e

j`(φ−φ′), (1.3)

where a is the radius of the cylinder, and φ and φ′ are the polar angles corresponding to x andx′. It is easily shown that currents of the form ejmφ′ are eigenfunctions of this equation, orgiven a current of

I(φ′) = ejmφ′, (1.4)

the incident field must be

Ei(φ) =πk0η

2Jm(ka)H(2)m (ka)︸︷︷︸

λm

ejmφ, (1.5)

where λm is the eigenvalue associated with this eigenfunction.

The dominant mode can be defined as the mode yielding the highest current on the surface ofthe cylinder for unit incident field. Since for the mth mode,

Ei(φ) = λmI(φ), (1.6)

the mode with the smallest eigenvalue would be dominant in this sense. A goal for optimalantenna design could be to identify which type of antenna best excites this mode.

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fourier transform

lossy cylinder(body)

infinite

inverse

lossy cylinder

point source(antenna)

(body)

line source

Figure 1.2: Proposed two-step procedure for body area modeling

One difficulty in identifying the dominant mode is that even if a large current may be excitedon the surface of the scatterer, this current may not effectively couple to a receive antenna.Although this idea of identifying and exciting dominant modes is a promising concept, morework is needed to understand how to apply the principle correctly to this problem.

1.4.2 Full-Wave Analysis

A more straightforward approach than the dominant mode analysis is to solve for the exactscattered fields arising from a specified current source. This method is more precise, since allmodes are taken into account. However, it may be more difficult to understand the behavior ofthe scatterer in this case due to the complexity of all superimposed modes.

In order to better understand propagation near the body, we can develop an approach directlyfrom Maxwell’s equations. An important goal is to develop expressions that are valid for arbi-trary distance of the antennas from the body surface as well as all possible antenna polarizationsand excitation frequencies.

In this work, the BAN scenario is modeled with a point source (antenna) near an infinitedielectric cylinder (body). A 3D model is developed starting with the solution for an infiniteline source in the vicinity of a lossy cylinder and then calculating the inverse Fourier transform,numerically, to obtain the field due to a point source.

Figure 1.2 depicts the approach to body area modeling in this thesis. The body is modeledas a lossy cylinder and the antenna is assumed to be a point source. A lossy cylinder is areasonable approximation for the human body since it takes into account many propagationphenomena, including diffraction around the curved lossy surface, reflections off the body andpenetration into the body. All these factors play a role in body area propagation, thoughthe relative importance depends on factors like frequency, polarization, radius of curvature,and tissue properties. Thus the chosen geometry allows exploration of many important bodyarea propagation phenomena, while still remaining analytically tractable so that a solutioncan be derived directly from fundamental principles. Also note that since the response due tothe point source is the Green’s function of this system, more complicated antennas could beaccommodated using the same method with Green’s function analysis.

1.4.3 3D Model

In order to derive the electric fields due to a point source, the two step procedure depicted inFigure 1.2 is followed. First, the response due to a 2D line source with e−jkzz variation in the

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z direction is found, which is accomplished by writing fields inside/outside cylinder in terms ofincident and scattered fields and then matching tangential components at the boundary. Theresponse to a point source is then found by taking the Fourier transform of the line sourceresponse. This approach takes advantage of the fact that the point source represented by theDirac function δ(z − z0) and a line source represented by exponential e−jkz(z−z0) are a Fouriertransform pair. Figure 1.3 provides a more detailed diagram of our geometry. An infinitecylinder of radius a is oriented along the z-axis with center at the origin. An infinite linearcurrent source is located at cylindrical co-ordinates (ρ′, φ′). The development of closed-form

(ρ′, φ′)x

y

z

a

φρ

Figure 1.3: Geometry and coordinate system for our analysis

analytical models for ẑ-, ρ̂- and φ̂-directed point sources is explained in Chapter 2. This chapteralso deals with validation of the developed solution using FDTD simulations. Investigations inoptimal antenna design for body area networks are performed in Chapter 3. Measurementresults for different BAN channels are compared with analytical model in Chapter 4.

1.5 Investigations in Optimal Antennas for BANs

Two approaches for optimal transmission in BANs are investigated in Chapter 4. One approachis to employ diversity techniques where signals from a number of antennas at some fixed spacingare combined to provide a reliable, high-gain link. The other approach calculates optimal currentdistributions for transmit and receive antennas with fixed apertures using the covariance matrixof the communications link for BAN.

Optimal antenna design involving the whole communication link (transmit antenna, channeland receive antenna) in the analysis [5] has been suggested for wireless communications channelin [4] such that superdirectivity is avoided [6]. This approach is extended to near-field to findthe optimal current distribution of antennas for BAN in Chapter 4.

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Chapter 2

Closed-Form Analytical Solution forBody Area Networks

This chapter derives the closed-form solution for the simplified BAN environment studied in thisthesis, which is depicted in Figure 1.2. The idealized transmit antenna is a current source withsome arbitrary polarization at some specified distance from an infinite lossy dielectric cylinder.The idealized receive antenna is assumed to sample an arbitrary polarization of the electricfield at an arbitrary position outside of the cylinder. Both 2D (line) and 3D (point) sources willbe considered in later modeling work. The line source is computationally more efficient, whilethe point source is more accurate for modeling small antennas. The idealized “channel” in thiswork is the transfer function from either a line or point current to electric field at a point oralong a line in the space around the body.

2.1 Line Source

To formulate a closed-form analytical model for BANs, we begin by deriving the solution forthe non-homogeneous wave equation linking the electric current and vector potential, or

(∇2 + k2)A = −µJ (2.1)

where k is the free space wave number and A is the vector potential. The current distributionJ for a point source is represented by a Dirac function δ(z − zo) and that of a line source bye−jkz(z−z0). The solution of this wave equation can be obtained as a sum of the homogeneoussolution (∇2 + k2)As = 0 and particular solution. The homogeneous solution corresponds toany field that can exist in an isotropic homogeneous medium with the wave number k in theabsence of any sources in that medium. For our problem, this can be interpreted as the scatteredfield, or the part of the field that is different from the incident field (the source). The particularsolution corresponds to the incident field from the line source propagating in free space withoutthe presence of the lossy cylinder. Hence the problem of finding the fields at different pointsin space reduces to finding the scattered field for a known imposed incident field (source). Thescattered field is the homogeneous solution of (2.1),

(∇2 + k2)As = 0. (2.2)

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�� b = 0

kz = k

singularity in line source

b = d

„1−

“a−k

k

”2«for 0 < a < 2k

a = Re {kz}

b = Im {kz}

Figure 2.1: A parabolic contour integral around singularity kz = k

2.2 Point Source

The electric and magnetic fields due to a line source near a lossy cylinder can be transformedto a point source, which represents a small body worn antenna more accurately, by taking theadvantage of the following Fourier transform pair as explained in Figure 1.2, or

δ(z − z0) = 12π∫ ∞−∞

e−jkzzejkzz0dkz. (2.3)

Since our system is linear, we can write the response to the point source as a superposition ofthe responses due to line sources that make up the point source.

Epoint =12π

∫ ∞−∞

Elineejkzz0dkz. (2.4)

Thus the field due to a point source can be computed as the sum of the fields due to aninfinite number of line sources with currents ejkzz. The integration in (2.4) must be performednumerically using a contour integral in the complex plane to avoid a singularity in Ez whenkz = k, the free space wave number. A parabolic contour integral defined in Figure 2.1 togetherwith Simpson’s rule provides a practical numerical integration technique that rapidly converges(changing d, converges at small value) to a good approximation of the solution.

In the following sections, fields for line sources have been derived. The fields for all point sourcescan be found using the same contour integral explained above.

2.3 ẑ-directed Line source

2.3.1 Incident field

As already mentioned, incident vector potential is the particular solution of (2.1). Consideringan incident line source at (ρ′,φ′),

J = − 1µ

δ(x− x′)δ(y − y′)e−jkzz ẑ. (2.5)

It is clear from the geometry that ẑ-directed component of the electric field radiated by the linesource takes the form of an outgoing cylindrical traveling wave represented mathematically by

8

Rρ

ρ′

x

y

φ

φ′

(x′, y′)

(x, y)

Figure 2.2: Displaced cylindrical harmonic and its displaced coordinate system

Hankel function of second kind [7]. Hence,

Ain =e−jkzz

4jHo

(2)(kρR)ẑ. (2.6)

where k2 = kz2 + kρ2 and R =√

(x− x′)2 + (y − y′)2 is the distance from source to theobservation point. To match the fields at the boundary, we need to write the incident field dueto the source in terms of functions centered at origin and exponential variation in φ. Hencerelative distance between source and observation point (R) should be defined as the distancefrom the origin. This can be expressed well by writing (x, y) and (x′, y′) in terms of cylindricalcoordinates as shown in Figure 2.2 and calculating R as

x = ρ cosφ, x′ = ρ′ cosφ′

y = ρ sinφ, y′ = ρ′ sinφ′

R =√

(ρ cosφ− ρ′ cosφ′)2 + (ρ sinφ− ρ′ sinφ′)2

R =√

ρ2 + ρ′2 − 2ρρ′ cos(φ− φ′). (2.7)From [9],

Cm(w) cos mx = Σ∞k=−∞Cm+k(u)Jk(v) cos kα (|ve±jα| < |u|) (2.8)Cm(w) sin mx = Σ∞k=−∞Cm+k(u)Jk(v) sin kα (|ve±jα| < |u|) (2.9)

where w =√

u2 + v2 − 2uv cosα and Cm is any Bessel function. Combining the two equations,we get

Cm(w)ejmx = Σ∞k=−∞Cm+k(u)Jk(v)ejkα. (2.10)

Using (2.10), we can find the incident field for ρ < ρ′ and ρ > ρ′. Also note that for our solution,(2.6), w = k′ρR.

Case I : ρ < ρ′

v = kρρ note that |u| > |v|u = kρρ′

α = φ− φ′w =

√u2 + v2 − 2uv cosα

=√

kρ2(ρ2 + ρ′2 − 2ρρ′ cos(φ− φ′))

= kρR (2.11)

9

Putting this value of w in (2.10) and taking Cm(w) = Ho(2)(kρR),

Ho(2)(kρR) = Σ∞n=−∞Hn

(2)(kρρ′)Jn(k′ρρ)ejn(φ−φ′) (2.12)

Case II : ρ > ρ′ Just exchange ρ and ρ′

Ho(2)(kρR) = Σ∞n=−∞Hn

(2)(kρρ)Jn(kρρ′)ejn(φ−φ′) (2.13)

2.3.2 Scattered field

We can use the separation of variables method to find the particular solution of the waveequation, represented by As in cylindrical coordinates. Separating an arbitrary component ofAs as

As(ρ, φ, z) = R(ρ)Φ(φ)Z(z) (2.14)

where ∇ in cylindrical coordinates is given by:

∇2As = 1ρ

∂

∂ρ(ρ

∂

∂ρAs) +

1ρ2

∂2

∂φ2As +

∂2

∂z2As

=1ρ

∂

∂ρ(ρR′(ρ))Φ(φ)Z(z) +

1ρ2

Φ′′(φ)R(ρ)Z(z) +

R(ρ)Φ(φ)Z′′(z) (2.15)

putting 2.15 and 2.15 in (2.2) we obtain

1ρ

∂

∂ρ

(ρR′(ρ)

)Φ(φ)Z(z) +

1ρ2

Φ′′(φ)R(ρ)Z(z) +

R(ρ)Φ(φ)Z′′(z) + k2R(ρ)Φ(φ)Z(z) = 0(R′′(ρ) +

R′(ρ)ρ

)Φ(φ)Z(z) +

1ρ2

Φ′′(φ)R(ρ)Z(z) +

R(ρ)Φ(φ)Z′′(z) + k2R(ρ)Φ(φ)Z(z) = 0 (2.16)

dividing both sides by R(ρ)Φ(φ)Z(z),

1R(ρ)

(R′′(ρ) +

R′(ρ)ρ

)+

1ρ2

Φ′′(φ)Φ(φ)︸︷︷︸

−kρ2

+Z′′(z)Z(z)︸︷︷︸−kz2

+k2 = 0 (2.17)

where k2ρ and k2z are constants since they add up to k

2 to get 0. Solving the above two identitiesseparately,

Z′′ (z)Z (z)

+ kz2 = 0

Z′′(z) + kz2Z(z) = 0

the solution of this equation leads to

Z(z) = C1e−jkzz + C2ejkzz. (2.18)

10

solving the second identity,

1R(ρ)

(R′′(ρ) +R′(ρ)

ρ) +

1ρ2

Φ′′(φ)Φ(φ)

+ kρ2 = 0

ρ2

R(ρ)

(R′′(ρ) +

R′(ρ)ρ

)+

Φ′′(φ)Φ(φ)︸︷︷︸−kφ2

+ρ2kρ2 = 0

gives solution for Φ

Φ′′(φ)Φ(φ)

+ kφ2 = 0 (2.19)

⇒ Φ(φ) = C3e−jkφz + C4ejkφz. (2.20)1 And

ρ2

R(ρ)

(R′′(ρ) +

R′(ρ)ρ

)+ ρ2kρ2 −m2 = 0

ρ2(R′′(ρ) + ρR′(ρ)

)+ (ρ2kρ2 −m2)R(ρ) = 0

comparing (2.21) with Bessel’s equation, we get the solution for R as

R(ρ) = C5Jm(kρρ) + C6Ym(kρρ) (2.21)

or

R(ρ) = C7H(1)m (kρρ) + C8H(2)m (kρρ) (2.22)

or any linearly independent combination of H(1), H(2), J or Y .To get the final solution for As, we put (2.21), (2.20) and (2.18) in (2.14) and obtain

As(ρ, φ, z) = Σ∞m=−∞[AmJm(kρρ) + BmYm(kρρ)]ejmφ[Cme−jkzz + Dmejkzz], (2.23)

where k2 = kρ2 + kz2.

2.3.3 Total Field

To enforce boundary conditions, we need E and H. From Maxwell’s equations,

E = −jω[A +

1k2∇ (∇ ·A)

]

=ω

jk2[∇ (∇ ·A) + k2A] . (2.24)

Also

∇ ·A = 1ρ

∂ (ρAρ)∂ρ

+1ρ

∂Aφ∂φ

+∂Az∂z

. (2.25)

Since we have J in z-direction only, we would only have Az and our line source has variationonly in e−jkzz, so

∇ ·A = −jkzAz. (2.26)1Φ must be periodic in φ with Φ(φ) = Φ(φ + 2π), hence kφ is an integer; say kφ = m. Also if m can be

positive and negative, we only need one exponential term

11

Similarly, considering z component for ∇,

∇ (∇ ·A) = ∂ (∇ ·A)∂z

= kz2Az. (2.27)

Putting (2.27) in (2.24), we obtain

Ez =ω

jk2[k2 − kz2

]Az

=ωkρ

2

jk2Az. (2.28)

(2.29)

Also, from Maxwell’s equations

H =1µ∇×A. (2.30)

where

∇×A = ρ̂(

1ρ

∂Az∂φ

− ∂Aφ∂z

)+ φ̂

(∂Aρ∂z

− ∂Az∂ρ

)

+ẑ

ρ

(∂ (ρAφ)

∂ρ− ∂Aρ

∂φ

)(2.31)

Since we only have Az

H = ρ̂1ρ

∂Az∂φ

− φ̂∂Az.

∂ρ (2.32)

Fields can be divided into three regions; (1) inside cylinder, (2) between cylinder and Sourcepoint and (3) outside Source point. The vector potential for these three regions is different asdiscussed below.

Inside cylinder: ρ ≤ aAz = Σ∞m=−∞AmJm(kρ1ρ)e

jmφe−jkzz (2.33)

Note that we can remove Ym(kρρ) from the scattered field potential since it approaches infinityat the origin (ρ = 0). Also, we retain only e−jkzz from (2.23) because this is the only solutionpossible when the source has e−jkzz variation.

Between cylinder and source point: a ≤ ρ ≤ ρ′

Az = Ain + As (2.34)

where

Ain = Σ∞m=−∞Jm(kρρ)Hm(2)(kρρ′)e−jkzzejm(φ−φ

′) (2.35)

As = Σ∞m=−∞[BmHm

(2)(kρ2ρ)]e−jkzzejmφ (2.36)

where kρ22 = k22− kz2 and k2 is the wave number outside cylinder and we can ignore the term

Hm(1) in (2.22) since the scattered field only goes outwards. As before, only e−jkzz variation

needs to be retained.

Az = Σ∞m=−∞[Jm(kρ2ρ)Hm

(2)(kρ2ρ′)e−jmφ

′+ BmHm(2)(kρ2ρ)

]e−jkzzejmφ.

(2.37)

12

Outside source point: ρ ≥ ρ′ Scattered field has the same form. Only incident field changesthe form as per (2.13).

Az = Σ∞m=−∞[Jm(kρ2ρ

′)Hm(2)(kρ2ρ)e−jmφ′ + BmHm(2)(kρ2ρ)

]e−jkzzejmφ.

(2.38)

Fields in all these three regions has the same functional form and therefore can be written as

Az = Σ∞m=−∞Rm(kρnρ)e−jkzzejmφ. (2.39)

where

Rm(kρn) = AmJm(kρ1ρ) ρ ≤ aJm(kρ2ρ)Hm

(2)(kρ2ρ′)e−jmφ′ + BmHm(2)(kρ2ρ) a ≤ ρ ≤ ρ′

Jm(kρ2ρ′)Hm(2)(kρ2ρ)e−jmφ

′+ BmHm(2)(kρ2ρ) ρ ≥ ρ′ (2.40)

In terms of physical field,

Ez =ωkρ

2

jk2Σ∞m=−∞Rm(kρnρ)e

−jkzzejmφ, (2.41)

Hφ = −kρnΣ∞m=−∞R′m(kρnρ)e−jkzzejmφ, (2.42)Hρ =

jm

ρΣ∞m=−∞Rm(kρnρ)e

−jkzzejmφ. (2.43)

2.3.4 Boundary Conditions

The boundary conditions can now be applied to determine the constants Am and Bm. Theseconditions state that the tangential components of electric and magnetic fields are continuousat the surface of the dielectric cylinder (ρ = a). Expressed mathematically as

Ez2 (ρ = a) = Ez1 (ρ = a) , (2.44)Hφ2(ρ = a) = Hφ1(ρ = a) (2.45)

where Ez2, Hφ2 and Ez1, Hφ1 represent the z component of the fields just outside and justinside the cylinder respectively. Combining (2.41) and (2.44),

Ez:

ωkρ12

jk12 Σ

∞m=−∞AmJm(kρ1a) =

ωkρ22

jk22 Σ

∞m=−∞Jm(kρ2a)Hm

(2)(kρ2ρ′)e−jmφ

′

+BmHm(2)(kρ2a)(2.46)

⇒ Am = k12

kρ12

kρ22

k22

︸︷︷︸x

Jm(kρ2a)Hm(2)(kρ2ρ

′)e−jmφ′ + BmHm(2)(kρ2a)Jm(kρ1a)

(2.47)

13

Hφ: Combining (2.43) and (2.45),

−kρ1Σ∞m=−∞AmJ ′m(kρ1a) = −kρ2Σ∞m=−∞J ′m(kρ2a)Hm(2)(kρ2ρ′)e−jmφ′

+BmHm(2)′(kρ2a)

⇒ Am = kρ2kρ1︸︷︷︸y

J ′m(kρ2a)Hm(2)(kρ2ρ

′)e−jmφ′ + BmHm(2)′(kρ2a)

J ′m(kρ1a). (2.48)

Let φ′ = 0 for simplicity, since we can always rotate solution, and equating (2.47) and (2.48)we obtain

Bm

[x

Hm(2)(kρ2a)

Jm(kρ1a)− yHm

(2)′(kρ2a)J ′m(kρ1a)

]= −xJm(kρ2a)Hm

(2)(kρ2ρ′)

Jm(kρ1a)+

yJ ′m(kρ2a)Hm

(2)(kρ2ρ′)

J ′m(kρ1a)

Bm

[xHm

(2)(kρ2a)J′m(kρ1a)− yHm(2)

′(kρ2a)Jm(kρ1a)

]

= −xJm(kρ2a)Hm(2)(kρ2ρ′)J ′m(kρ1a) + yJ ′m(kρ2a)Hm(2)(kρ2ρ′)Jm(kρ1a) (2.49)

Bm =−xJm(kρ2a)Hm(2)(kρ2ρ′)J ′m(kρ1a) + yJ ′m(kρ2a)Hm(2)(kρ2ρ′)Jm(kρ1a)

xHm(2)(kρ2a)J ′m(kρ1a)− yHm(2)

′(kρ2a)Jm(kρ1a)(2.50)

2.4 ρ̂-directed Line source

Consider the case when the transmitter is placed vertical to the body surface as shown in Figure2.3. In this case, the field around the body would consist of ρ and φ components pertaining todirection of unit vector ρ̂′ specifying the orientation of transmitted signal. ρ̂′ can be written asa sum of ρ̂ and φ̂ for the ρ̂′-directed source as under:

ρ̂′ = −ρ̂ cos(φ− φ′) + φ̂ sin(φ− φ′) (2.51)


Assuming a current distribution given by

J = − 1µ

δ(x− x′)δ(y − y′)e−jkzzρ̂′, (2.52)

the incident vector potential is then given by

Ain =e−jkzz

4jHo

(2)(kρ2R)(−ρ̂ cos(φ− φ′) + φ̂ sin(φ− φ′)) (2.53)

where

R =√

ρ2 + ρ′2 − 2ρρ′ cos(φ− φ′). (2.54)

Similar to the ẑ case, when the source is not at origin (0,0,0), the Hankel function can be writtenas a summation. For

14

Âinc = ρ̂′

2D cross-section of body

R

(ρ′, φ′)

location of sensor

location of source

ρ̂′

ρ̂

φ̂

Figure 2.3: Orientation of Receive/Transmit fields (ρ̂)

Case I : ρ < ρ′

Ho(2)(kρR) = Σ∞m=−∞Hm

(2)(kρρ′)Jm(kρρ)ejm(φ−φ′) (2.55)

and for



(2)(kρρ)Jm(kρρ′)ejm(φ−φ′) (2.56)

Incident Field inside source ρ < ρ′

We would like to expand cos(φ − φ′) and sin(φ − φ′) in such a way that it facilitates theapplication of boundary conditions. Expanding in terms of exponentials

cos(φ− φ′) = [ej(φ−φ′) + e−j(φ−φ′)]

2, (2.57)

sin(φ− φ′) = [ej(φ−φ′) − e−j(φ−φ′)]

2j. (2.58)

Using these values for cos(φ− φ′) and sin(φ− φ′) we can write the ρ and φ components of thevector potential as

Aρin = −e−jkzz

8j[ej(φ−φ

′) + e−j(φ−φ′)]Σ∞m=−∞Hm

(2)(kρρ)Jm(kρρ′)ejm(φ−φ′),

(2.59)

15

Aρin = −e−jkzz

8jΣ∞m=−∞[Hm−1

(2)(kρρ)Jm−1(kρρ′) + Hm+1(2)(kρρ)Jm+1(kρρ′)]

ejm(φ−φ′). (2.60)

Aφin =e−jkzz

8j2[ej(φ−φ

′) − e−j(φ−φ′)]Σ∞m=−∞Hm(2)(kρρ)Jm(kρρ′)ejm(φ−φ′)

=e−jkzz

8Σ∞m=−∞[−Hm−1(2)(kρρ)Jm−1(kρρ′)︸︷︷︸

Xm−1

+Hm+1(2)(kρρ)Jm+1(k′ρρ′)︸︷︷︸

Xm+1

]

ejm(φ−φ′). (2.61)

Ain =e−jkzz

8Σ∞m=−∞

[− ρ̂

j(Xm+1 + Xm−1) + φ̂(Xm+1 −Xm−1)

]

ejm(φ−φ′). (2.62)

Now, in order to get Ein, we need

∇ ·A = 1ρ

[ρ[

∂Aρ∂ρ

+ Aρ

]+

1ρ

∂Aφ∂φ

+∂Az∂z

,

=∂Aρ∂ρ

+Aρρ

+1ρ

∂Aφ∂φ

+∂Az∂z

=18Σ∞m=−∞(

−1j

(X ′m+1 + X′m−1)−

1jρ

(Xm+1 + Xm−1) +

jm

ρ(Xm+1 −Xm−1))ejm(φ−φ′)e−jkzz

=18j

Σ∞m=−∞

[−(X ′m+1 + X ′m−1)−

1ρ

(Xm+1(m + 1) + Xm−1(m− 1))]

︸︷︷︸Tm

ejm(φ−φ′)e−jkzz. (2.63)

Then

∇ (∇ ·A) = ∂ (∇ ·A)∂ρ

ρ̂ +1ρ

∂ (∇ ·A)∂φ

φ̂ +∂ (∇ ·A)

∂zẑ (2.64)

=e−jkzz

8jΣ∞m=−∞[ρ̂(−(X”m+1 + X”m−1)−

1ρ(X ′m+1(m + 1)

+X ′m−1(m− 1)) +1ρ2

(Xm+1(m + 1) + Xm−1(m− 1)))

+φ̂jm

ρTm − ẑjkzTm]ejm(φ−φ′). (2.65)

Also

k2Ain =e−jkzz

8k2Σ∞m=−∞[−

ρ̂

j(Xm+1 + Xm−1) + φ̂(Xm+1 −Xm−1)]ejm(φ−φ′). (2.66)

16

Using the above identities, we can write incident electric field as

Ein =w

jk2[∇ (∇ ·A) + k2A] , (2.67)

=w

jk2e−jkzz

8jΣ∞m=−∞[ρ̂(−(X ′′m+1 + X ′′m−1)

−1ρ(X ′m+1(m + 1) + X

′m−1(m− 1))

+1ρ2

(Xm+1(m + 1− k2ρ2) + Xm−1(m− 1− k2ρ2)))

+φ̂[jm

ρTm + jk2(Xm+1 −Xm−1)]

−ẑjkzTm]ejm(φ−φ′). (2.68)

Magnetic field is given by

Hin =1µ∇×Ain, (2.69)

where

∇×A = ρ̂(

1ρ

∂Az∂φ

− ∂Aφ∂z

)+ φ̂

(∂Aρ∂z

− ∂Az∂ρ

)+

ẑ

ρ

(∂ (ρAφ)

∂ρ− ∂Aρ

∂φ

)

=e−jkzz

8Σ∞m=−∞

[−jkzρ̂

j(Xm+1 −Xm−1) + φ̂kz(Xm−1 + Xm+1)

]

+ẑ

ρ(ρ(X ′m+1 −X ′m−1) + (Xm+1 −Xm−1)

+m(Xm−1 + Xm+1))ejm(φ−φ′). (2.70)

Therefore

Hin =e−jkzz

8µ2Σ∞m=−∞[−kzρ̂(Xm+1 −Xm−1) + φ̂kz(Xm−1 + Xm+1)

+ẑ

ρ(ρ(X ′m+1 −X ′m−1) + (Xm+1(m + 1)

+Xm−1(m− 1)))]ejm(φ−φ′). (2.71)

Incident Field outside source : ρ > ρ′

For the region outside the source point ρ > ρ′, we only need to change Hm(2)(kρρ) Jm(kρρ′) toJm(kρρ)Hm(2)(kρρ′). This would change the values of Xm and Tm. We can write the ρ and φcomponents of the vector potential outside the source point as

Aρin = −e−jkzz

8j[ej(φ−φ

′) + e−j(φ−φ′)]Σ∞m=−∞Hm

(2)(k′ρρ)Jm(k′ρρ′)ejm(φ−φ

′),

= −e−jkzz

8jΣ∞m=−∞[Jm−1(k

′ρρ)Hm−1

(2)(k′ρρ′) + Jm+1(k′ρρ)Hm+1

(2)(k′ρρ′)]

ejm(φ−φ′). (2.72)

17

Aφin =e−jkzz

8j2[ej(φ−φ

′) − e−j(φ−φ′)]Σ∞m=−∞Hm(k′ρρ)Hm(2)(k′ρρ′)ejm(φ−φ′),

=e−jkzz

8Σ∞m=−∞[−Jm−1(k′ρρ)Hm−1(2)(k′ρρ′)︸︷︷︸

Ym−1

+Jm+1(k′ρρ)Hm+1(2)(k′ρρ

′)︸︷︷︸Ym+1

]

ejm(φ−φ′). (2.73)

Ain =e−jkzz

8Σ∞m=−∞[−

ρ̂

j(Ym+1 + Ym−1) + φ̂(Ym+1 − Ym−1)]ejm(φ−φ′). (2.74)


∇ ·A = 1ρ

[ρ[

∂Aρ∂ρ

+ Aρ

]+

1ρ

∂Aφ∂φ

+∂Az∂z

,

=∂Aρ∂ρ

+Aρρ

+1ρ

∂Aφ∂φ

+∂Az∂z

=e−jkzz

8Σ∞m=−∞[

−1j

(Y ′m+1 + Y′m−1)−

1jρ

(Ym+1 + Ym−1)

+jm

ρ(Ym+1 − Ym−1)]ejm(φ−φ′)

=e−jkzz

8jΣ∞m=−∞ [−(Y ′m+1 + Y ′m−1)−

1ρ(Ym+1(m + 1) + Ym−1(m− 1))]

︸︷︷︸Sm

ejm(φ−φ′).

∇ (∇ ·A) = ∂ (∇ ·A)∂ρ

ρ̂ +1ρ

∂ (∇ ·A)∂φ

φ̂ +∂ (∇ ·A)

∂zẑ (2.75)

=e−jkzz

8jΣ∞m=−∞[ρ̂(−(Y ′′m+1 + Y ′′m−1)−

1ρ(Y ′m+1(m + 1) + Y

′m−1(m− 1))

+1ρ2

(Ym+1(m + 1) + Ym−1(m− 1)))

+φ̂jm

ρSm − ẑjkzTm]ejm(φ−φ′). (2.76)

Also

k2Ain =e−jkzz

8k2Σ∞m=−∞[−

ρ̂

j(Ym+1 + Ym−1) + φ̂(Ym+1 − Ym−1)]ejm(φ−φ′). (2.77)

Making use of the above identities, we can write incident electric field as

Ein =w

jk2[∇ (∇ ·A) + k2A] , (2.78)

=w

jk2e−jkzz

8jΣ∞m=−∞[ρ̂(−(Y ”m+1 + Y ”m−1)−

1ρ(Y ′m+1(m + 1) + Y

′m−1(m− 1))

+1ρ2

(Ym+1(m + 1− k2ρ2) + Ym−1(m− 1− k2ρ2))) + φ̂(jmρ

Sm

+jk2(Ym+1 − Ym−1))− ẑjkzSm]ejm(φ−φ′). (2.79)

Magnetic field can be calculated using identity

Hin =1µ∇×Ain, (2.80)

18

where

∇×A = ρ̂(

1ρ

∂Az∂φ

− ∂Aφ∂z

)+ φ̂

(∂Aρ∂z

− ∂Az∂ρ

)

+ẑ

ρ

(∂ (ρAφ)

∂ρ− ∂Aρ

∂φ

)

=e−jkzz

8Σ∞m=−∞[−jkzρ̂

j(Ym+1 − Ym−1) + φ̂kz(Ym−1 + Ym+1)

+ẑ

ρ(ρ(Y ′m+1 − Y ′m−1) + (Ym+1 − Ym−1) + m(Ym−1 + Ym+1))]

ejm(φ−φ′). (2.81)

Hence

Hin =e−jkzz

8µΣ∞m=−∞[−kzρ̂(Ym+1 − Ym−1) + φ̂kz(Ym−1 + Ym+1)

+ẑ

ρ(ρ(Y ′m+1 − Y ′m−1) + (Ym+1(m + 1) + Ym−1(m− 1)))]

ejm(φ−φ′). (2.82)


As specified already, the scattered field is the homogeneous solution of (2.1),

(∇2 + k2)As = 0.Instead of using separation of variables method to find the solution to the above equation, wecan make use of the e−jkzz variation. With this kind of a variation, transverse electric andmagnetic field can be written in terms of Ez and Hz as

ET =−j

k2 − kz2(wµ∇T × ẑHz + kz∇TEz) (2.83)

HT =−j

k2 − kz2(−w²∇T × ẑEz + kz∇THz) (2.84)

where

k2 − kz2 = kρ2,

wµ = w√

µ²

√µ

²= wη,

w² = w√

µ²

√²

µ

=w

η.

(2.85)

Using these identities to modify (2.83) and (2.84)

ET =−jkρ

2 (kη∇T × ẑHz + kz∇TEz) (2.86)

HT =j

kρ2

(k

η∇T × ẑEz + kz∇THz

)(2.87)

19

For a ρ̂-directed source we will only have Hz component and not Ez component, we can writethe fields using (2.86) and (2.87) in terms of Hz where Hz can be written as a sum of basisfunctions.

Scattered Field inside cylinder ρ < a

Let

Hzscat = wΣ∞m=−∞AmJm(kρ1ρ)e

jm(φ−φ′)e−jkzz (2.88)

Also

∇T = ∂∂ρ

ρ̂ +1ρ1

∂

∂φφ̂ (2.89)

Putting (2.88) and (2.89) in (2.86) and making use of right hand rule, we obtain

ET =−jkρ1

2 k1η1∇T × ẑHz

=−jkρ1

2 k1η1

(1ρ1

∂Hz∂φ

ρ̂− ∂Hz∂ρ

φ̂

)

⇒ Eρscat =−jkρ1

2 k1η1 ·jm

ρ1wΣ∞m=−∞AmJm(kρ1ρ)e

jm(φ−φ′)e−jkzz

Eρscat =wk1η1

kρ12ρ

Σ∞m=−∞mAmJm(kρ1ρ)ejm(φ−φ′)e−jkzz (2.90)

⇒ Eφscat =j

kρ12 k1η1 · wkρ1Σ∞m=−∞AmJ ′m(kρ1ρ)ejm(φ−φ

′)e−jkzz

Eφscat =jwk1η1

kρ1Σ∞m=−∞AmJ

′m(kρ1ρ)e


Scattered Field outside cylinder ρ > a

Outside the cylinder, field is a wave propagating outwards, which can be modeled with a second-order Hankel function. Therefore

Hzscat = wΣ∞m=−∞BmHm

(2)(kρ2ρ)ejm(φ−φ′)e−jkzz. (2.92)

Also

∇T = ∂∂ρ

ρ̂ +1ρ2

∂

∂φφ̂ (2.93)

20


ET =−jkρ2


=−jkρ2

2 k2η2

(1ρ2

∂Hz∂φ

ρ̂− ∂Hz∂ρ

φ̂

)


2 k1η1 ·jm

ρwΣ∞m=−∞BmHm

(2)(kρ2ρ)ejm(φ−φ′)e−jkzz

Eρscat =wk2η2

kρ22ρ

Σ∞m=−∞mBmHm(2)(kρ2ρ)e


⇒ Eφscat =j

kρ22 k2η2 · wkρ2Σ∞m=−∞BmH ′m


Eφscat =jwk2η2

kρ2Σ∞m=−∞BmH

′m

(2)(kρ1ρ)ejm(φ−φ′)e−jkzz (2.95)

2.4.3 Total Field

Inside Cylinder : ρ < a

Eρ =wmk1η1

kρ1ρ12 Σ

∞m=−∞AmJm(kρ1ρ)e


Eφ =jwk1η1


′m(kρ1ρ)e


Hz = wΣ∞m=−∞AmJm(kρ1ρ)ejm(φ−φ′)e−jkzz (2.98)

Between cylinder and source point : a < ρ < ρ′

Eρ =wmk2η2

kρ2ρ22 Σ

∞m=−∞BmHm

(2)(kρ2ρ)ejm(φ−φ′)e−jkzz +

w

jk22

e−jkzz

8jΣ∞m=−∞[−(X ′′m+1 + X ′′m−1)−

1ρ(X ′m+1(m + 1)

+X ′m−1(m− 1)) +1ρ2

(Xm+1(m + 1− k2ρ2)

+Xm−1(m− 1− k2ρ2))]ejm(φ−φ′) (2.99)

Eφ =jwk2η2


′m


w

jk22

e−jkzz

8jΣ∞m=−∞

(jm

ρTm + jk2(Xm+1 −Xm−1)

)ejm(φ−φ

′) (2.100)

Hz = wΣ∞m=−∞BmHm(2)(kρ2ρ)e

jm(φ−φ′)e−jkzz +

e−jkzz

8µ2Σ∞m=−∞

1ρ2

(ρ2(X ′m+1 −X ′m−1) + (Xm+1(m + 1) + Xm−1(m− 1))

)

ejm(φ−φ′) (2.101)

21

Outside source point : ρ > ρ′

Eρ =wmk2η2

kρ2ρ22 Σ

∞m=−∞BmHm(2)(kρ2ρ)e


w

jk22

e−jkzz

8jΣ∞m=−∞−(Y ′′m+1 + Y ′′m−1 −

1ρ(Y ′m+1(m + 1) + Y

′m−1(m− 1))

+1ρ2

(Ym+1(m + 1− k2ρ2) + Ym−1(m− 1− k2ρ2))ejm(φ−φ′) (2.102)

Eφ =jwk2η2

kρ2Σ∞m=−∞BmH ′m


w

jk22

e−jkzz

8jΣ∞m=−∞

(jm

ρSm + jk2(Ym+1 − Ym−1)

)ejm(φ−φ

′) (2.103)



e−jkzz

8µ2Σ∞m=−∞

1ρ2

(ρ2(Y ′m+1 − Y ′m−1) + (Ym+1(m + 1) + Ym−1(m− 1))

)

ejm(φ−φ′) (2.104)


Equate mode m at boundary for tangential components

Hz:

wAmJm(kρ1a) = wBmHm(2)(kρ2a) +

18µ2ρ2

ρ2(X ′m+1(a) −X ′m−1(a))

+ (Xm+1(a)(m + 1) + Xm−1(a)(m− 1))

AmJm(kρ1a) = BmHm(2)(kρ2a) +

18 µ2w︸︷︷︸

k2η2

ρ2(ρ2(X ′m+1

(a) −X ′m−1(a)) + (Xm+1(a)(m + 1) + Xm−1(a)(m− 1))

︸︷︷︸Cm

Am =BmHm

(2)(kρ2a) + CmJm(kρ1ρ)

(2.105)

where Xm(a) represents the value of Xm for ρ = a.

22

Eφ:

jwk1η1kρ1

AmJ′m(kρ1a) =

jwk2η2kρ2

BmH′m

(2)(kρ2a) +

w

jk22

18j

(jm

aTm

(a) + jk2(Xm+1(a) −Xm−1(a)))

k1η1kρ1︸︷︷︸b1

AmJ′m(kρ1a) =

k2η2kρ2︸︷︷︸b2

BmH′m

(2)(kρ2a) +

−18k22

(ma

Tm(a) + k2(Xm+1(a) −Xm−1(a))

)

︸︷︷︸Dm

b1AmJ′m(kρ1a) = b2BmH

′m

(2)(kρ2a) + Dm

⇒ Am = b2BmH′m

(2)(kρ2a) + Dmb1J ′m(kρ1a)

(2.106)

Comparing (2.105) and (2.106), we obtain

BmHm(2)(kρ2a) + CmJm(kρ1ρ)

=b2BmH

′m

(2)(kρ2a) + Dmb1J ′m(kρ1a)

BmHm(2)(kρ2a) + Cmb1J

′m(kρ1a) =

b2BmH′m

(2)(kρ2a) + DmJm(kρ1a)

Bm

(Hm

(2)(kρ2a)b1J′m(kρ1a)− b2Jm(kρ1a)H ′m(2)(kρ2a)

)

= DmJm(kρ1a)− Cmb1J ′m(kρ1a)Bm =

DmJm(kρ1a)− Cmb1J ′m(kρ1a)Hm

(2)(kρ2a)b1J ′m(kρ1a)− b2Jm(kρ1a)H ′m(2)(kρ2a)(2.107)

2.5 φ̂-directed Line source

Consider the case when the transmitter is placed vertical to the body surface as shown in Figure2.4. In this case, the field around the body would consist of ρ and φ components. The unitvector φ̂′ specifying the orientation of transmitted signal can be written as a sum of ρ̂ and φ̂for the φ̂-directed source.

φ̂′ = φ̂ cos(φ− φ′) + ρ̂ sin(φ− φ′) (2.108)


Assuming a current distribution given by

J = − 1µ

δ(x− x′)δ(y − y′)e−jkzzφ̂′, (2.109)

the incident vector potential can be written as

Ain =e−jkzz

4jHo

(2)(kρ2R)(φ̂ cos(φ− φ′) + ρ̂ sin(φ− φ′)) (2.110)

23

Âinc = φ̂′

2D cross-section of body

R

location of sensor

(ρ′, φ′)

location of source

φ̂′ρ̂

φ̂

Figure 2.4: Orientation of Receive/Transmit fields (φ̂)

where

R =√

ρ2 + ρ′2 − 2ρρ′ cos(φ− φ′). (2.111)

As for the ρ̂-directed case, when the source is not at origin (0,0,0), the Hankel function can bewritten as a summation

Case I : ρ < ρ′


(2)(kρρ′)Jm(kρρ)ejm(φ−φ′) (2.112)



(2)(kρρ)Jm(kρρ′)ejm(φ−φ′) (2.113)

Incident field inside source ρ < ρ′

We would like to expand cos(φ − φ′) and sin(φ − φ′) in such a way that it facilitates theapplication of boundary conditions, hence expanding them in terms of exponentials

cos(φ− φ′) = [ej(φ−φ′) + e−j(φ−φ′)]

2, (2.114)

sin(φ− φ′) = [ej(φ−φ′) − e−j(φ−φ′)]

2j. (2.115)

24

Using these values for cos(φ− φ′) and sin(φ− φ′) we can write the ρ and φ components of thevector potential as

Aρin =e−jkzz

8j2[ej(φ−φ

′) − e−j(φ−φ′)]Σ∞m=−∞Hm(2)(kρρ)Jm(kρρ′)ejm(φ−φ′),

=e−jkzz

8Σ∞m=−∞[−Hm−1(2)(kρρ)Jm−1(kρρ′)︸︷︷︸

Xm−1

+Hm+1(2)(kρρ)Jm+1(kρρ′)︸︷︷︸Xm+1

]

ejm(φ−φ′). (2.116)

Aφin =e−jkzz

8j[ej(φ−φ

′) + e−j(φ−φ′)]Σ∞m=−∞Hm


= −e−jkzz

8jΣ∞m=−∞[Hm−1

(2)(kρρ)Jm−1(kρρ′) + Hm+1(2)(kρρ)Jm+1(kρρ′)]

ejm(φ−φ′). (2.117)

Ain =e−jkzz

8Σ∞m=−∞

[− ρ̂

j(Xm+1 + Xm−1) + φ̂(Xm+1 −Xm−1)

]

ejm(φ−φ′). (2.118)


∇ ·A = 1ρ

[ρ[

∂Aρ∂ρ

+ Aρ

]+

1ρ

∂Aφ∂φ

+∂Az∂z

,

=∂Aρ∂ρ

+Aρρ

+1ρ

∂Aφ∂φ

+∂Az∂z

=18ρ

Σ∞m=−∞(ρ(X′m+1 −X ′m−1) + Xm+1 −Xm−1

+jm

j(Xm+1 + Xm−1))ejm(φ−φ

′)e−jkzz

=18Σ∞m=−∞

[X ′m+1 −X ′m−1) +

1ρ

(Xm+1(m + 1) + Xm−1(m− 1))]

︸︷︷︸Um

ejm(φ−φ′)e−jkzz. (2.119)

∇ (∇ ·A) = ∂ (∇ ·A)∂ρ

ρ̂ +1ρ

∂ (∇ ·A)∂φ

φ̂ +∂ (∇ ·A)

∂zẑ

=e−jkzz

8Σ∞m=−∞[ρ̂(X

′′m+1 −X ′′m−1) +

1ρ(X ′m+1(m + 1)

+X ′m−1(m− 1))−1ρ2

(Xm+1(m + 1) + Xm−1(m− 1)))

+φ̂jm

ρUm − ẑjkzUm]ejm(φ−φ′). (2.120)

Also

k2Ain =e−jkzz

8k2Σ∞m=−∞[ρ̂(Xm+1 −Xm−1) +

φ̂

j(Xm+1 + Xm−1)]ejm(φ−φ

′). (2.121)

25


Ein =w

jk2[∇ (∇ ·A) + k2A] , (2.122)

=w

jk2e−jkzz

8Σ∞m=−∞[ρ̂(X”m+1 −X”m−1)

+1ρ(X ′m+1(m + 1) + X

′m−1(m− 1))

+1ρ2

(Xm+1(m + 1− k2ρ2)−Xm−1(m− 1− k2ρ2)))

+φ̂[jm

ρUm − jk22(Xm+1 + Xm−1)]− ẑjkzUm]

ejm(φ−φ′). (2.123)


Hin =1µ2∇×Ain, (2.124)

where

∇×A = ρ̂(

1ρ

∂Az∂φ

− ∂Aφ∂z

)+ φ̂

(∂Aρ∂z

− ∂Az∂ρ

)+

ẑ

ρ

(∂ (ρAφ)

∂ρ− ∂Aρ

∂φ

)

⇒ Hin = e−jkzz

8µ2Σ∞m=−∞[kzρ̂(Xm+1 + Xm−1)− φ̂jkz(Xm+1 −Xm−1)

+ẑ

ρ

(ρ(X ′m+1 + X

′m−1) + (Xm+1(m + 1)−Xm−1(m− 1))

)]

ejm(φ−φ′). (2.125)

Incident Field outside source : ρ > ρ′

Given that the point source is not at the origin, we have different summations which in turncorrespond to two regions. For the region outside the source point ρ > ρ′, we only need tochange Hm(2)(kρρ)Jm(kρρ′) to Jm(kρρ)Hm(2)(kρρ′). This would change the values of Xm andUm. We can write the ρ and φ components of the vector potential as

Aρin =e−jkzz

8j2[ej(φ−φ

′) − e−j(φ−φ′)]Σ∞m=−∞Hm(kρρ)Hm(2)(kρρ′)ejm(φ−φ′),

=e−jkzz

8Σ∞m=−∞[−Jm−1(kρρ)Hm−1(2)(kρρ′)︸︷︷︸

Ym−1

+Jm+1(kρρ)Hm+1(2)(kρρ′)︸︷︷︸Ym+1

]ejm(φ−φ′). (2.126)

Aφin =e−jkzz

8j[ej(φ−φ

′) + e−j(φ−φ′)]Σ∞m=−∞Hm


=e−jkzz

8jΣ∞m=−∞[Jm−1(kρρ)Hm−1

(2)(kρρ′) +

Jm+1(kρρ)Hm+1(2)(kρρ′)]ejm(φ−φ′). (2.127)

Ain =e−jkzz

8Σ∞m=−∞[ρ̂(Ym+1 − Ym−1) +

φ̂

j(Ym+1 + Ym−1)]

ejm(φ−φ′). (2.128)

26


∇ ·A = 1ρ

[ρ[

∂Aρ∂ρ

+ Aρ

]+

1ρ

∂Aφ∂φ

+∂Az∂z

,

=∂Aρ∂ρ

+Aρρ

+1ρ

∂Aφ∂φ

+∂Az∂z

=e−jkzz

8Σ∞m=−∞[(Y

′m+1 − Y ′m−1) +

1ρ(Ym+1 − Ym−1)

+jm

jρ(Ym+1 + Ym−1)]ejm(φ−φ

′)

=e−jkzz

8Σ∞m=−∞

[Y ′m+1 − Y ′m−1 +

1ρ(Ym+1(m + 1) + Ym−1(m− 1))

]

︸︷︷︸Vm

ejm(φ−φ′).

Then

∇ (∇ ·A) = ∂ (∇ ·A)∂ρ

ρ̂ +1ρ

∂ (∇ ·A)∂φ

φ̂ +∂ (∇ ·A)

∂zẑ (2.129)

=e−jkzz

8Σ∞m=−∞[ρ̂(Y

′′m+1 − Y ′′m−1 +

1ρ(Y ′m+1(m + 1) + Y

′m−1(m− 1))

− 1ρ2

(Ym+1(m + 1) + Ym−1(m− 1)))

+φ̂jm

ρVm − ẑjkzVm]ejm(φ−φ′). (2.130)

(2.131)

Also

k2Ain =e−jkzz

8k2

2Σ∞m=−∞[φ̂

j(Ym+1 + Ym−1) + ρ̂(Ym+1 − Ym−1)]ejm(φ−φ′). (2.132)


Ein =w

jk2[∇ (∇ ·A) + k2A] , (2.133)

=w

jk2e−jkzz

8Σ∞m=−∞[ρ̂(Y ”m+1 − Y ”m−1 +

1ρ(Y ′m+1(m + 1) + Y

′m−1(m− 1))

− 1ρ2

(Ym+1(m + 1− k2ρ2)− Ym−1(m− 1 + k22ρ2))) + φ̂(jmρ

Vm

−jk2(Ym+1 + Ym−1))− ẑjkzVm]ejm(φ−φ′). (2.134)


Hin =1µ∇×Ain, (2.135)

27

where

∇×A = ρ̂(

1ρ

∂Az∂φ

− ∂Aφ∂z

)+ φ̂

(∂Aρ∂z

− ∂Az∂ρ

)

+ẑ

ρ

(∂ (ρAφ)

∂ρ− ∂Aρ

∂φ

)

⇒ Hin = e−jkzz

8µ2Σ∞m=−∞[kzρ̂(Ym+1 + Ym−1)− φ̂jkz(Ym+1 − Ym−1)

+ẑ

jρ(ρ(Y ′m+1 + Y

′m−1) + (Ym+1(m + 1)− Ym−1(m− 1)))]

ejm(φ−φ′). (2.136)


As we know, the scattered field is the homogeneous solution of (2.1),

(∇2 + k2)As = 0.

For a φ̂-directed source we will only have Hz component and not Ez component, we can writethe fields using (2.86) and (2.87) in terms of Hz where Hz can be written as a sum of basisfunctions. Since the scattered field has no influence of the source, it is the same as calculatedfor ρ̂-directed source.

Scattered Field inside cylinder ρ < a

Let

Hzscat = wΣ∞m=−∞AmJm(kρ1ρ)e


Also

∇T = ∂∂ρ

ρ̂ +1ρ1

∂

∂φφ̂ (2.138)


ET =−jkρ1


=−jkρ1

2 k1η1

(1ρ1

∂Hz∂φ

ρ̂− ∂Hz∂ρ

φ̂

)


2 k1η1 ·jm

ρwΣ∞m=−∞AmJm(kρ1ρ)e

jm(φ−φ′)e−jkzz

Eρscat =wk1η1

kρ12ρ

Σ∞m=−∞mAmJm(kρ1ρ)ejm(φ−φ′)e−jkzz (2.139)

⇒ Eφscat = jkρ1

2 k1η1 · wkρ1Σ∞m=−∞AmJ ′m(kρ1ρ)ejm(φ−φ′)e−jkzz

Eφscat =jwk1η1


′m(kρ1ρ)e


28

Scattered Field outside cylinder ρ > a

Outside the cylinder, field is a wave propagating outwards, which can be modeled with a second-order Hankel function. Therefore

Hzscat = wΣ∞m=−∞BmHm

(2)(kρ2ρ)ejm(φ−φ′)e−jkzz. (2.141)

Also

∇T = ∂∂ρ

ρ̂ +1ρ2

∂

∂φφ̂ (2.142)


ET =−jkρ2


=−jkρ2

2 k2η2

(1ρ2

∂Hz∂φ

ρ̂− ∂Hz∂ρ

φ̂

)


2 k1η1 ·jm

ρwΣ∞m=−∞BmHm


Eρscat =wk2η2

kρ22ρ

Σ∞m=−∞mBmHm(2)(kρ2ρ)e


⇒ Eφscat =j

kρ22 k2η2 · wkρ2Σ∞m=−∞BmH ′m


Eφscat =jwk2η2


′m

(2)(kρ1ρ)ejm(φ−φ′)e−jkzz (2.144)

2.5.3 Total Field

Inside Cylinder : ρ < a

Eρ =wmk1η1

kρ1ρ12 Σ

∞m=−∞AmJm(kρ1ρ)e


Eφ =jwk1η1


′m(kρ1ρ)e


Hz = wΣ∞m=−∞AmJm(kρ1ρ)ejm(φ−φ′)e−jkzz (2.147)

29

Between cylinder and source point : a < ρ < ρ′

Eρ =wmk2η2

kρ2ρ22 Σ

∞m=−∞BmHm


w

jk22

e−jkzz

8jΣ∞m=−∞[−(X”m+1 + X”m−1)−

1ρ(X ′m+1(m + 1) +

X ′m−1(m− 1)) +1ρ2

(Xm+1(m + 1− k2ρ2) +

Xm−1(m− 1− k2ρ2))]ejm(φ−φ′) (2.148)Eφ =

jwk2η2kρ2

Σ∞m=−∞BmH′m


w

jk22

e−jkzz

8jΣ∞m=−∞

(jm

ρTm + jk2(Xm+1 −Xm−1)

)ejm(φ−φ

′) (2.149)



e−jkzz

8µ2Σ∞m=−∞

1ρ2

(ρ2(X ′m+1 −X ′m−1) + (Xm+1(m + 1) + Xm−1(m− 1))

)

ejm(φ−φ′) (2.150)

Outside source point : ρ > ρ′

Eρ =wmk2η2

kρ2ρ22 Σ

∞m=−∞[BmHm


w

jk22

e−jkzz

8jΣ∞m=−∞[(−(Y ′′m+1 + Y ′′m−1)−

1ρ(Y ′m+1(m + 1) + Y

′m−1(m− 1))

+1ρ2

(Ym+1(m + 1− k2ρ2) + Ym−1(m− 1− k2ρ2))]ejm(φ−φ′) (2.151)

Eφ =jwk2η2


′m


w

jk22

e−jkzz

8jΣ∞m=−∞

(jm

ρSm + jk2(Ym+1 − Ym−1)

)ejm(φ−φ

′) (2.152)



e−jkzz

8µ2Σ∞m=−∞

1ρ2

(ρ2(Y ′m+1 − Y ′m−1) + (Ym+1(m + 1) + Ym−1(m− 1))

)

ejm(φ−φ′) (2.153)


Equate mode m at boundary for tangential components

30

Hz:

wAmJm(kρ1a) = wBmHm(2)(kρ2a)

+1

8µ2ρ2[ρ2(X ′m+1

(a) + X ′m−1(a))

+(Xm+1(a)(m + 1)−Xm−1(a)(m− 1))]AmJm(kρ1a) = BmHm

(2)(kρ2a)

+1

8 µ2w︸︷︷︸k2η2

ρ2[ρ2(X ′m+1

(a) + X ′m−1(a))

+ (Xm+1(a)(m + 1)−Xm−1(a)(m− 1))]︸︷︷︸Fm

Am =BmHm

(2)(kρ2a) + FmJm(kρ1ρ)

(2.154)

where Xm(a) represents the value of Xm for ρ = a.

Eφ:

jwk1η1kρ1

AmJ′m(kρ1a) =

jwk2η2kρ2

BmH′m

(2)(kρ2a)

+w

jk228

(jm

aTm

(a) − jk22(Xm+1(a) + Xm−1(a)))

k1η1kρ1︸︷︷︸b1

AmJ′m(kρ1a) =

k2η2kρ2︸︷︷︸b2

BmH′m

(2)(kρ2a)

+−18k22

(ma

Tm(a) − k2(Xm+1(a) + Xm−1(a))

)

︸︷︷︸Gm

b1AmJ′m(kρ1a) = b2BmH

′m

(2)(kρ2a) + Gm

⇒ Am = b2BmH′m

(2)(kρ2a) + Gmb1J ′m(kρ1a)

(2.155)

Comparing (2.154) and (2.155), we obtain

BmHm(2)(kρ2a) + Fm

Jm(kρ1ρ)=

b2BmH′m

(2)(kρ2a) + Gmb1J ′m(kρ1a)

BmHm(2)(kρ2a) + Fmb1J

′m(kρ1a) =

b2BmH′m

(2)(kρ2a) + GmJm(kρ1a)

Bm

(Hm

(2)(kρ2a)b1J′m(kρ1a)− b2Jm(kρ1a)H ′m(2)(kρ2a)

)

= GmJm(kρ1a)− Fmb1J ′m(kρ1a)Bm =

GmJm(kρ1a)− Fmb1J ′m(kρ1a)Hm

(2)(kρ2a)b1J ′m(kρ1a)− b2Jm(kρ1a)H ′m(2)(kρ2a)(2.156)

31

2.6 Instability at ρ = ρ′

The series form of the incident field has a singularity at ρ = ρ′, making computation of thefields unstable. ρ = ρ′ is the most probable case of our problem since the sensor and actuatorare present around the body, most probably, at the same distance from the surface of body (i.e.ρ = ρ′). This instability was dealt with by avoiding the basis function expansion of the incidentfield when computing total observed field.

For the discussion so far, we used ρ and φ coordinates to find incident field. Since cylindricalcoordinates, ρ and φ, change direction with movement around the body, they were adjusted inthe analytical solutions. As compared to cylindrical coordinates (ρ and φ), Cartesian coordi-nates (x, y, z) do not change directions as we move around the body. These coordinates wereused to avoid singularity at ρ = ρ′. Note that we still need the basis function expansion of theincident field in order to match modes at boundary to get the constants (Am, Bm).

Writing Ain in Cartesian coordinates

Ain =14j

Ho(2)(kρ2R)[axx̂ + ayŷ + az ẑ]e

−jkzz, (2.157)

where ax, ay and az are the vectors representing the direction of source current in terms ofCartesian coordinates. This Ain can be used to derive incident electric and magnetic fieldswhich would be stable for ρ = ρ′ but approach infinity when both φ = φ′ and ρ = ρ′, indicatingcoincidence of sensor and source, which would not happen in practice.

To compute physical fields due to the line source, we write

∇ ·Ain = ∂Ax∂x

+∂Ay∂y

+∂Az∂z

,

=e−jkzz

4j[(axx

R+

ayy

R

)Ho

(2)′(kρ2R)kρ2

+az

(Ho

(2)′(kρ2R)kρ2z

R− jkzHo(2)(kρ2R)

)],

=e−jkzz

4j[Ho

(2)′(kρ2R)kρ2R

(axx + ayy + azz)

−jkzazHo(2)(kρ2R)]. (2.158)

∇ (∇ ·Ain) = ∂ (∇ ·Ain)∂x

x̂ +∂ (∇ ·Ain)

∂yŷ +

∂ (∇ ·Ain)∂z

ẑ. (2.159)

Solving the three terms one by one

(∇ ·Ain)x =e−jkzz

4j[axkρ2 x

Ho(2)′(kρ2R)

R︸︷︷︸Bx

+(ayy + azz) kρ2Ho

(2)′(kρ2R)R︸︷︷︸Cx

−jkzazHo(2)(kρ2R)], (2.160)

differentiating Bx and Cx separately,

32

Bx =kρ2x

2

R2Ho

(2)′′(kρ2R) + Ho(2)′(kρ2R)

(−x2R3

+1R

),

Bx =kρ2x

2

R2Ho

(2)′′(kρ2R) + Ho(2)′(kρ2R)

(R2 − x2

R3

).

Cx =1R

Ho(2)′′(kρ2R)

kρ2x

R+ Ho(2)

′(kρ2R)

−12R3

2x,

Cx =kρ2x

R2Ho

(2)′′(kρ2R)−Ho(2)′(kρ2R)

x

R3. (2.161)

Therefore

∇(∇ ·Ain)x = e−jkzz

4j[ax

kρ22x2

R2Ho

(2)′′(kρ2R) + axkρ2Ho(2)′(kρ2R)

R2 − x2R3

+(ayy + azz)kρ2(kρ2x

R2Ho

(2)′′(kρ2R)−Ho(2)′(kρ2R)

x

R3)

−jkzazHo(2)′(kρ2R)kρ2x

R]

=e−jkzz

4j[ax

kρ2x2

R2(kρ2Ho

(2)′′(kρ2R)−Ho

(2)′(kρ2R)R

)︸︷︷︸

Q

+axkρ2Ho

(2)′(kρ2R)R

(ayy + azz)kρ2x

R2·

(kρ2Ho(2)′′(kρ2R)−

Ho(2)′(kρ2R)

R)

︸︷︷︸Q


R]

∇(∇ ·Ain)x = e−jkzz

4j[ax

kρ2x2

R2Q + axkρ2

Ho(2)′(kρ2R)

R(ayy + azz)

kρ2x

R2Q


R]. (2.162)

A similar expression can be derived for ∇ (∇ ·Ain)y,

∇(∇ ·Ain)y = e−jkzz

4j[ay

kρ2y2

R2Q

+aykρ2Ho

(2)′(kρ2R)R

(axx + azz)kρ2y

R2Q

−jkzazHo(2)′(kρ2R)kρ2y

R]. (2.163)

33

Now, ∇ (∇ ·Ain)z can be found by following the same steps

∇ (∇ ·Ain)z =14j

[(axx + ayy)∂

∂z

(kρ2

Ho(2)′(kρ2R)

Re−jkzz

)

︸︷︷︸Bz

+∂

∂z

(azze

−jkzzkρ2Ho

(2)′(kρ2R)R

)

︸︷︷︸Cz

− jkzaz ∂∂z

(Ho

(2)(kρ2R)e−jkzz

)

︸︷︷︸Dz

]. (2.164)

Differentiating Bz, Cz and Dz separately,

Bz =∂

∂z

(kρ2

Ho(2)′(kρ2R)

Re−jkzz

),

Bz = kρ2 [Ho

(2)′(kρ2R)R

(−jkze−jkzz) + e−jkzz(

Ho(2)′′(kρ2R)

R

kρ2z

R

)

+Ho(2)′(kρ2R)R

−zR3

],

Bz = kρ2 [−jkzHo(2)′(kρ2R)

Re−jkzz

+e−jkzzz

R2(kρ2Ho

(2)′′(kρ2R)−Ho

(2)′(kρ2R)R

R)︸︷︷︸

Q

],

Bz = kρ2

[−jkzHo(2)′(kρ2R)

R+

z

R2Q

]e−jkzz,

(2.165)

Cz = kρ2az∂

∂z

(ze−jkzz

RHo

(2)′(kρ2R))

Cz = kρ2az

[ze−jkzz

RHo

(2)′′(kρ2R)kρ2z

R+ Ho(2)

′(kρ2R)

∂

∂z

ze−jkzz

R

],

Cz = kρ2az[kρ2z

2

R2e−jkzzHo(2)

′′(kρ2R)

+Ho(2)′(kρ2R)(

1R

(z(−jkz)e−jkzz + e−jkzz

)

+ze−jkzz(−1/2)R−32z)],

Cz = kρ2az[z2

R2e−jkzz

(kρ2Ho

(2)′′(kρ2R)−Ho

(2)′(kρ2R)R

)

︸︷︷︸Q

+Ho

(2)′(kρ2R)R

(1− jkzz) e−jkzz],

Dz = −jkzaz[e−jkzzHo(2)

′(kρ2R)

kρ2z

R− jkzHo(2)(kρ2R)e−jkzz

]

(2.166)

34

Hence,

∇ (∇ ·Ain)z =14j

[kρ2 (axx + ayy) [−jkzHo(2)′(kρ2R)

R+

z

R2Q] + kρ2az[

z2

R2Q

+Ho

(2)′(kρ2R)R

(1− jkzz)]− jkzaz[Ho(2)′(kρ2R)kρ2z

R

−jkzHo(2)(kρ2R)]]e−jkzz. (2.167)

Since we now have ∇ (∇ ·Ain)x, ∇ (∇ ·Ain)y, ∇ (∇ ·Ain)z and Ainx,Ainy,Ainz, we can findEincx ,Eincy ,Eincz using

Einc =w

jk2{∇ (∇ ·Ain) + k2Ain

}.

In order to find magnetic field, we need ∇×A in Cartesian coordinates

∇×A = x̂(

∂Az∂y

− ∂Ay∂z

)+ ŷ

(∂Ax∂z

− ∂Az∂x

)

+ẑ

ρ

(∂Ay∂x

− ∂Ax∂y

)

=⇒ (∇×A)x =14j

[∂

∂y

(azHo

(2)(kρ2R)e−jkzz

)− ∂

∂z

(ayHo

(2)(kρ2R)e−jkzz

)]

=e−jkzz

4j

[Ho

(2)′(kρ2R)kρ2R

(azy − ayz) + jkzayHo(2)(kρ2R)]

.

=⇒ (∇×A)y =14j

[∂

∂z

(axHo

(2)(kρ2R)e−jkzz

)− ∂

∂x

(azHo

(2)(kρ2R)e−jkzz

)]

=e−jkzz

4j

[Ho

(2)′(kρ2R)kρ2R

(axz − azx)− jkzaxHo(2)(kρ2R)]

.

=⇒ (∇×A)z =14j

[∂

∂x

(ayHo

(2)(kρ2R)e−jkzz

)− ∂

∂y

(axHo

(2)(kρ2R)e−jkzz

)]

=e−jkzz

4j

[Ho

(2)′(kρ2R)kρ2R

(ayx− axy)]

.

(2.168)

Hincx ,Hincy ,Hincz can be found in Cartesian coordinates using the relation

H =1µ∇×A. (2.169)

Once the fields are obtained in Cartesian co-ordinates, they can be easily converted to cylindricalcoordinates and vice versa using co-ordinate conversion formulas. These incident fields can beadded to the scattered fields calculated above for all the three cases (ẑ, ρ̂, φ̂) and hence thechannels can be found for all the three sources and used for characterization of BAN.

2.7 Validation of Derived Solution

A code implementing the analytical model, as explained above, was written in Matlab. Tocheck that the derived expressions are correct, the analytical code was compared with numericalsimulations performed with an FDTD solver written by Prof. Wallace.

35

2.7.1 Line source

In the FDTD simulation of the line source, a single-frequency simulation was performed witha non-conductive dielectric cylinder of radius 1.0λ0, where λ0 is the free-space wavelength, andrelative permittivity of ²r = 2. The cylinder was placed at the middle of a 10λ0× 10λ0 grid (40cells per wavelength) which was terminated on all sides with a PML of 10 cells. The simulationswere run for 400 sinusoidal periods with 500 steps per period to ensure high accuracy. Thesinusoidal line source was placed at φ′ = 0 and ρ′ = 1.5λ0.

This analytical model can be compared with the FDTD simulation by letting z = z0 = 0 andkz = 0. Fields are compared by sweeping the observation angle φ for a fixed observation radiusρ = 1.6λ0. The comparison results for different orientations of receive/transmit sensor/sourceare shown in Figures (2.5), (2.6) and (2.7). Some receive/transmit sensor/source combinationswere found to be 0. This includes ẑ-ρ̂, ẑ-φ̂, φ̂-ẑ, ρ̂-ẑ source and sensor respectively. The resultsfrom the FDTD code showed a very good match with the derived channels for all the differentpolarizations of transmit and receive sensors.

2.7.2 Point Source

FDTD simulations of a point source were also performed, but due to the need for a true 3Dsimulation, the resolution had to be reduced. A cylinder with a = 1.0λ0 was simulated, butonly on a 5λ0× 5λ0 grid with 20 cells per wavelength in x and y and 10 cells per wavelength inz. The simulation was run for 400 sinusoidal periods with 100 steps per period.

The channel due to a point source was calculated using integral (2.4) where the integration rangewas 0 − 2k. The simulation results were also very close for the point source and it was foundthat the point source channel was also close to the line source channel. It was also discoveredthat ẑ-directed source radiated most maximum power and φ̂-directed source radiated minimumpower.

36

0 50 100 150 200 250 300 350 400−30

−25

−20

−15

−10

−5

0

5

10

15

20Comparison of line source, point source and FDTD simulations (z − directed source)

φ

No

rma

lise

d |E

z|

point

line

FDTD−point

FDTD−line

Figure 2.5: Channel Comparison for ẑ-directed line/point source/sensor with FDTD

37

50 100 150 200 250 300 350

−120

−100

−80

−60

−40

−20

0

20

40

Comparison of line source, point source and FDTD simulations (φ − directed source)

φ

No

rma

lise

d |E

ρ|

point

line

FDTD−point

FDTD−line

(a) ρ̂-directed line/point sensor

0 50 100 150 200 250 300 350 400−40

−30

−20

−10

0

10

20

30

Comparison of line source, point source and FDTD simulations (φ − directed source)

φ

No

rma

lise

d |E

φ|

point

line

FDTD−point

FDTD−line

(b) φ̂-directed line/point sensor

Figure 2.6: Channel Comparison for φ̂-directed line/point sources with FDTD

38

0 50 100 150 200 250 300 350 400−40

−30

−20

−10

0

10

20

30

40

Comparison of line source, point source and FDTD simulations (ρ − directed source)

φ

No

rma

lise

d |E

ρ|

point

line

FDTD−point

FDTD−line

(a) ρ̂-directed line/point sensor

50 100 150 200 250 300 350 400

−120

−100

−80

−60

−40

−20

0

20

40

Comparison of line source, point source and FDTD simulations (ρ − directed source)

φ

No

rma

lise

d |E

φ|

point

line

FDTD−point

FDTD−line

(b) φ̂-directed line/point sensor

Figure 2.7: Channel Comparison for ρ̂-directed line/point sources with FDTD

39

Chapter 3

Optimal Antenna Design for BodyArea Networks

Two approaches for optimal transmission in BANs are investigated in this thesis. One approachis to employ diversity techniques where signals from a number of antennas at some fixed spacingare combined to provide a reliable, high-gain link, and the main problem is to determine theoptimal spacing of the multiple sensors to achieve the best diversity. The other approach thatwas investigated is similar to the work in [4], where arbitrary transmit and receive antennas areassumed to have fixed apertures, and given the covariance matrix of the BAN channels, optimalcurrent distributions for transmit and receive antennas can be obtained. These two approachesand their respective results are discussed in the following sections.

3.1 Diversity-based Antenna Design

This approach suggests that an antenna can consist of a number of sensors separated by opti-mal spacing and the signals present at the antenna terminals can be combined using diversitycombining techniques. The derived analytical models were used to study this diversity-basedantenna design approach. Optimal inter-sensor spacing was found and diversity gain was calcu-lated at this optimal spacing. A general observation of this study is that the gain and optimalspacing are dependent on the height of the antenna from the body surface.

Figure 3.1 shows the configuration used for simulations. Transmit and receive sensors areplaced at a certain height h from the body, where the receivers are separated by spacing d.The variation in certain parameters with respect to height as well as different polarizations ofsensors was studied.

A diversity-based antenna system was studied at a simulation frequency of 2.45 GHz, radiusof the cylinder a = 12.7 cm (corresponds to the approximate radius of the human subjectfor the measurements in Chapter 4), relative permittivity ²r = 52.7, relative permeabilityµr = 1 and conductivity σ = 1.74 S/m. These values correspond to the dielectric properties ofhuman muscle tissue at 2.45 GHz [8]. Maximal ratio combining (MRC) was used as a diversitycombining technique for the analysis, whose post-processing gain is given by the simple formula

C = ΣiwiHi= Σi|Hi|2 for wi = H∗i . (3.1)

where Hi represents the channel to the ith antenna, and wi are the optimal weights that are to

40

h

ρ′Tx

d

Rx2

Rx1

Figure 3.1: Adding Diversity to BAN

be applied to achieve maximum receive power at ith antenna.

Our main interest for diversity-based antenna design is the “shadow region” behind the bodywhere communication is likely to be most challenging. We will consider the ability of a diversityreceiver to overcome the fading occurring in this region as a function of sensor spacing (d) andheight of the sensors relative to the surface of the body (h).

3.1.1 Diversity Channels

The channels used to study the diversity-based antenna design approach are shown in Figure(3.2)-(3.6) for d = 6.8 mm, which was the minimum height used in this study.

Figure (3.7)-(3.11) show the results for d = 1.4 cm, which was the maximum height consideredin this analysis. It can be seen from the above mentioned figures that the channel is periodicwith equidistant peaks and nulls close to the body surface. The channel away from the bodysurface consists of fewer peaks and nulls and hence less diversity gain is expected in this area.

The diversity gain should be 3dB when the fading profiles at the two sensors are identical(overlapping), which arises from simple array gain, and this is considered to be the worst-casefor diversity. The optimal case, on the other hand, is when the fading profiles of the two sensorsare offset, such that when one sensor is in a null, the other sensor is at a peak, ensuring thatat least one sensor will always give a strong link. Since the shadow region is periodic nearthe body surface, a number of sensors can be placed at regular angular intervals, to increaseredundancy/diversity in this area.

3.1.2 Shadow region at different sensor heights

As expected, the channel gain is strong when the receive antenna is in the front region (sameregion as the transmit antenna) and a shadow region is observed as we move the receivertoward the back where the channel is obstructed by the body. It is clear from the analyticalcomputations that the width of the shadow region decreases as the transmit and receive antennaare moved away from the cylinder, which is also intuitive. Therefore, the most challengingscenario is when the transmit and receive sensors are very close to the surface of the body.

41

100 150 200 250−50

−40

−30

−20

−10

0

10

φ

Norm

aliz

ed E

z(d

B)

Figure 3.2: ẑ-channel

100 150 200 250−30

−25

−20

−15

−10

−5

0

5

10

φ

Norm

aliz

ed E

ρ(d

B)

Figure 3.3: ρ̂-channel

100 150 200 250−35

−30

−25

−20

−15

−10

−5

0

5

10

φ

Norm

aliz

ed E

φ(d

B)

Figure 3.4: ρ̂-φ̂-channel

100 150 200 250−70

−60

−50

−40

−30

−20

−10

0

10

φ

Norm

aliz

ed E

φ(d

B)

Figure 3.5: φ̂-channel

100 150 200 250−35

−30

−25

−20

−15

−10

−5

0

5

10

φ

Norm

aliz

ed E

ρ(d

B)

Figure 3.6: φ̂-ρ̂-channel

42

100 150 200 250−25

−20

−15

−10

−5

0

5

10

15

φ

Norm

aliz

ed E

z(d

B)

Figure 3.7: ẑ-channel

100 150 200 250−20

−15

−10

−5

0

5

10

φ

Norm

aliz

ed E

ρ(d

B)

Figure 3.8: ρ̂-channel

100 150 200 250−30

−25

−20

−15

−10

−5

0

5

10

φ

Norm

aliz

ed E

φ(d

B)

Figure 3.9: ρ̂-φ̂-channel

100 150 200 250−15

−10

−5

0

5

10

φ

Norm

aliz

ed E

φ(d

B)

Figure 3.10: φ̂-channel

100 150 200 250−30

−25

−20

−15

−10

−5

0

5

10

φ

Norm

aliz

ed E

ρ(d

B)

Figure 3.11: φ̂-ρ̂-channel

43

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

10

20

30

40

50

60

70

80

90

100Shadow width as a function of sensor height

Sensor height, h(m)

Sh

ad

ow

wid

th(d

eg

)

Zz

Rr

Rp

Pr

Pp

Figure 3.12: Shadow Region Width

The size of the shadow region was quantified by applying a smoothing window to the fieldintensity as a function of angle, sufficient to remove the oscillations behind the body. Theshadow region is then defined as the range of angles that are less than 2 dB above the point ofminimum power (usually directly behind the body from the source).

As seen from Figure (3.12), by moving a distance approximately equal to the cylinder radius, theshadow width has already decreased to a relatively small value for all of the transmit and receiveantenna orientations. The legends in (3.12) and the subsequent figures indicate the orientationof transmitter (capital alphabet) and receiver respectively, for example, Rp represents ρ̂-directedtransmitter and φ̂-directed receiver.

3.1.3 Excess Loss

Since the channel is in less deep fade as we move away from the surface, the excess loss (lossin excess to the free space path loss) decre

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