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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
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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)
7
-
��������������������������������������������������������������������������������
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)
2.4.1 Incident field
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
Case II : ρ > ρ′ Just exchange ρ and ρ′
Ho(2)(kρR) = Σ∞m=−∞Hm
(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)
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
=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)
2.4.2 Scattered field
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
õ
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
jm(φ−φ′)e−jkzz (2.91)
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
-
Putting (2.92) and (2.93) in (2.86) and making use of right hand
rule, we obtain
ET =−jkρ2
2 k2η2∇T × ẑHz
=−jkρ2
2 k2η2
(1ρ2
∂Hz∂φ
ρ̂− ∂Hz∂ρ
φ̂
)
⇒ Eρscat =−jkρ2
2 k1η1 ·jm
ρwΣ∞m=−∞BmHm
(2)(kρ2ρ)ejm(φ−φ′)e−jkzz
Eρscat =wk2η2
kρ22ρ
Σ∞m=−∞mBmHm(2)(kρ2ρ)e
jm(φ−φ′)e−jkzz (2.94)
⇒ Eφscat =j
kρ22 k2η2 · wkρ2Σ∞m=−∞BmH ′m
(2)(kρ1ρ)ejm(φ−φ′)e−jkzz
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
jm(φ−φ′)e−jkzz (2.96)
Eφ =jwk1η1
kρ1Σ∞m=−∞AmJ
′m(kρ1ρ)e
jm(φ−φ′)e−jkzz (2.97)
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
kρ2Σ∞m=−∞BmH
′m
(2)(kρ1ρ)ejm(φ−φ′)e−jkzz +
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
jm(φ−φ′)e−jkzz +
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
(2)(kρ1ρ)ejm(φ−φ′)e−jkzz +
w
jk22
e−jkzz
8jΣ∞m=−∞
(jm
ρSm + jk2(Ym+1 − Ym−1)
)ejm(φ−φ
′) (2.103)
Hz = wΣ∞m=−∞BmHm(2)(kρ2ρ)e
jm(φ−φ′)e−jkzz +
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)
2.4.4 Boundary Conditions
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)
2.5.1 Incident field
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′
Ho(2)(kρR) = Σ∞m=−∞Hm
(2)(kρρ′)Jm(kρρ)ejm(φ−φ′) (2.112)
Case II : ρ > ρ′ Just exchange ρ and ρ′
Ho(2)(kρR) = Σ∞m=−∞Hm
(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
(2)(kρρ)Jm(kρρ′)ejm(φ−φ′),
= −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)
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
=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
-
Using the above identities, we can write incident electric field
as
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)
Magnetic field is given by
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
(2)(kρρ)Jm(kρρ′)ejm(φ−φ′),
=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
-
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
=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)
Using the above identities, we can write incident electric field
as
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)
Magnetic field is given by
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)
2.5.2 Scattered field
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
jm(φ−φ′)e−jkzz (2.137)
Also
∇T = ∂∂ρ
ρ̂ +1ρ1
∂
∂φφ̂ (2.138)
Putting (2.137) and (2.138) 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
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
kρ1Σ∞m=−∞AmJ
′m(kρ1ρ)e
jm(φ−φ′)e−jkzz (2.140)
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)
Putting (2.141) and (2.142) in (2.86) and making use of right
hand rule, we obtain
ET =−jkρ2
2 k2η2∇T × ẑHz
=−jkρ2
2 k2η2
(1ρ2
∂Hz∂φ
ρ̂− ∂Hz∂ρ
φ̂
)
⇒ Eρscat =−jkρ2
2 k1η1 ·jm
ρwΣ∞m=−∞BmHm
(2)(kρ2ρ)ejm(φ−φ′)e−jkzz
Eρscat =wk2η2
kρ22ρ
Σ∞m=−∞mBmHm(2)(kρ2ρ)e
jm(φ−φ′)e−jkzz (2.143)
⇒ Eφscat =j
kρ22 k2η2 · wkρ2Σ∞m=−∞BmH ′m
(2)(kρ1ρ)ejm(φ−φ′)e−jkzz
Eφscat =jwk2η2
kρ2Σ∞m=−∞BmH
′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
jm(φ−φ′)e−jkzz (2.145)
Eφ =jwk1η1
kρ1Σ∞m=−∞AmJ
′m(kρ1ρ)e
jm(φ−φ′)e−jkzz (2.146)
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
(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′) (2.148)Eφ =
jwk2η2kρ2
Σ∞m=−∞BmH′m
(2)(kρ1ρ)ejm(φ−φ′)e−jkzz +
w
jk22
e−jkzz
8jΣ∞m=−∞
(jm
ρTm + jk2(Xm+1 −Xm−1)
)ejm(φ−φ
′) (2.149)
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′) (2.150)
Outside source point : ρ > ρ′
Eρ =wmk2η2
kρ2ρ22 Σ
∞m=−∞[BmHm
(2)(kρ2ρ)ejm(φ−φ′)e−jkzz +
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
kρ2Σ∞m=−∞BmH
′m
(2)(kρ1ρ)ejm(φ−φ′)e−jkzz +
w
jk22
e−jkzz
8jΣ∞m=−∞
(jm
ρSm + jk2(Ym+1 − Ym−1)
)ejm(φ−φ
′) (2.152)
Hz = wΣ∞m=−∞BmHm(2)(kρ2ρ)e
jm(φ−φ′)e−jkzz +
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)
2.5.4 Boundary Conditions
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
−jkzazHo(2)′(kρ2R)kρ2x
R]
∇(∇ ·Ain)x = e−jkzz
4j[ax
kρ2x2
R2Q + axkρ2
Ho(2)′(kρ2R)
R(ayy + azz)
kρ2x
R2Q
−jkzazHo(2)′(kρ2R)kρ2x
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