Xem mẫu
- Spheroidal Wave Functions in Electromagnetic Theory
Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong
Copyright 2002 John Wiley & Sons, Inc.
ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic)
7
SAR Distributions in a
Spheroidal Head Model
7.1 INTRODUCTION
The development of cellular telephones and mobile communication systems
has led to a growing awareness of the vital role that wireless systems play in
communication networks and the biological effects of EM fields on users. Since
cellular hand phones are operated in close proximity to human heads while in
use, there has been increasing public concern about the health effects of the
human head exposed to EM energy emitted from mobile handset antennas.
To avoid the harmful effects of EM fields radiated by mobile phones, designers
of hand-phone antennas must be able to develop highly efficient, low-profile
antennas that can be mounted on handheld transceivers and operated in the
proximity of human tissues.
To evaluate potential health effects, the specific absorption rate (SAR) is
normally used to evaluate the rates of EM energy deposition in animal or
human tissues. An important motivation for researchers is to gain a detailed
understanding of the EM field distribution and power absorption distribution
inside the human head [138,139]. There exist a variety of techniques that can
be used to perform the required analysis of SAR deposition in human heads,
each with its own strength and weakness. In the analysis, the model obtained
from medical imaging serves as a realistic model but requires a lot of compu-
tational time as in the methods of finite element [140], finite difference [141-
1451, and moments [146,147]. To increase computational speed, the spherical
human head model is also often adopted as an idealized representation of a
real biological human head in the analysis [ 14%1511, but it. is less accurate.
191
- 192 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
In view of the fact that the geometry of a human head can be better approxi-
mated by a prolate spheroid than a simple sphere and that the computational
time can be saved as compared with those using the finite difference time
domain (FDTD) technique and the finite element method (FEM), a dielectric
prolate spheroidal model serves as a compromise for the full-wave analysis
of EM field distributions. Investigation of the prolate spheroidal head model
presented in this chapter can serve as a complement of the existing analyses
of analytical models of the human head.
7.2 MULTILAYERED PROLATE SPHEROIDAL HEAD MODEL
In this analysis the human head is modeled as a multilayered dielectric prolate
spheroid, as shown in Fig. 7.1. The multilayered spheroidal model is defined
similarly to that of multilayered spherical head models in previous analyses
[148,151,152]. The dielectric spheroidal model consists of six layers: brain,
CSF (cerebrospinal fluid), dura, bone, fat, and skin, respectively. To make
reasonable assumptions, the maximum major and minor semiaxial lengths of
the outer layer (the layer of skin) is assumed to be 10 cm and 7.5 cm, re-
spectively. The thickness of each layer along the direction of the minor axis
is assumed to be the same as that employed in 11511. The region of each
layer of the human head is labeled as 1, 2, . . ., 6, and the wave propagations
within the layer is k1, k2, . . ., and k6, respectively. The outside region of the
head model (where the mobile antenna is located) is labeled as 7, whose wave
propagation constant is k7. To simplify the computation, each spheroidal in-
terface is assumedto have the same interfocal distance d. This is a reasonable
approximation when the first inner region of brain is much thicker than that
of other modeled layers of tissues [33,49], as shown in Table 7.1.
In this book, two frequencies of mobile antennas are studied: the GSM
(Pan European Cellular System-Group Special Mobile, with center frequency
about 900 MHz) and the PCN (Personal Communications Network, with cen-
ter frequency about 1800 MHz). The dielectric constants of each region at the
GSM and PCN frequencies, as well as those used in spherical head models or
FDTD analyses [139,141,145,148,153,154], are provided in Table 7.1.
The antenna is modeled as a A/4 monopole or dipole (as shown in Fig. 7.1)
located at a distance s away from the multilayered spheroidal model. To
simplify the computation, the orientation of the monopole or dipole is chosen
to be on the plane parallel to the z, y-plane. The inclination angle of the
antenna is denoted by p, which is the angle between the linear antenna and
the z-axis. The feed point for the monopole or dipole is located at 7’ = 0
and 4’ = 0. The wire antennas are assumedto have a negligible diameter and
their current distributions are assumedto have a center-fed sinusoidal current
distribution such as that described in [155].
- MULTILAYERED PROLATE SPHEROIDAL HEAD MODEL 193
Fig. 7.1 Multilayered spheroidal head model exposed to a mobile antenna.
Table 7.2 Electric Properties of the Multilayered Prolate Spheroidal Model of a Hu-
man Head at 900 MHz/1800 MHz
Region Layer Thickness (cm) 0 (S/m) e:r P (Wm3)
1 Brain b-l.10 1.10/1.42 49147 1030
2 CSF b-o.80 2.1312.50 72172 1060
3 Dura bO.70 0.9811.29 45145 1050
4 Bone b0.27 0.1yo.15 818 1850
5 Fat bO.15 0.08/0.26 619 920
6 Skin b 0.60/0.57 35132 1100
- 194 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
7.3 FORMULATION OF THE PROBLEM
7.3.1 Expansions of EM Fields Using Spheroidal Wave Functions
The EM waves excited by the wire antenna can be expressed in terms of the
spheroidal vector wave functions by means of the formulated dyadic Green’s
functions for spheroidal structures introduced in Chapter 3. The electric fields
inside (El N I&) and outside (ET) the multilayered spheroidal model are
expressed as
El(r) = F 2 2 { [AlfM M;$$cl,J) +dlZN N”(‘>
emn ,mn e
omn
n=m m=O
X 2-6”“@),(k7, c’)I&‘) dV’
JJJ V’ Nmn
emn(cj9t)+ ‘fHEnNZO
+ t3$zn Mzt3)
Ornlz
(cf9c)]
X v,
JJJ$$%$;Jh,dv’
5’)&(f) mn
f = 2,3,4,5, and 6, (7.lb)
- FORMULATION OF THE PROBLEM 195
(7.lc)
where the prime symbol denotes the source point location, Nm, is the nor-
malization factor of the angular function, and ci = 4 kid (i = 1,2). 6mo is
the Kronecker delta function, Ia = I(t’) .s (where 6 = 2, s), and I(
- 196 SAR DISTRMJTIONS IN A SPHEROIDAL HEAD MODEL
Thus, the localized SAR is related directly to the internal field, and all the
numerical procedures involve determination of the electric field distribution
within the biological human head. The ANSI/IEEE C95.1-1992 RF Safety
Guidelines proposes a detailed procedure to satisfy the safety guidelines for
uncontrolled environments, which are defined as situations where there exists
exposure of persons who have no control of exposure [1571.
The SAR in each layer of the multilayered prolate spheroid is defined as
aj JE12/2pj, where aj and pj represent the average conductivity and density
of the j th layer, respectively. Values of aj and pj are presented in Table 7.1.
7.4 NUMERICAL COMPUTATION
On the basis of the EM fields calculated inside the multilayered prolate spher-
oidal model, the SAR in each layer of the spheroidal head model can be ob-
tained. To simplify the calculation, the transmitted power of the dipole or
monopole is assumedto be 1 W at both the GSM and PCN frequencies.
The convergence of the matrix equation system for the determination of un-
known scattering and transmission coefficients is discussed in Chapter 3. For
the multilayered spheroidal structure presented in this chapter, the truncation
number is generally chosen to be the maximum number of Integer( lkal + 4).
Four Mathematics packages were developed on the basis of the previously
verified software package to calculate the multilayered dielectric spheroidal
structure.
l The first is Source.nb, which is used for calculation of the integral
about the wire antenna in Eq. (7.1).
l The second is Itemexpansion.nb, which is used for calculation of the
intermediates Ittyln(c) described in Appendix B.
l The third package is Matrixequation.nb, which is applied to deter-
mine the unknown scattering coefficients using the matrix equation sys-
tem. In practical computation, the truncation number is chosen to be
48.
l The last package, Efield.nb, is designed for computation of the electric
field inside the multilayered dielectric spheroid.
For convenience in the investigation of a multilayered spheroidal head
model, the results of spheroidal wave functions and intermediate items used
in the functional expansion for each layer are first calculated using packages
introduced in Chapter 2 and saved as numerical tables. EM fields due to
different antennas and their positions are then obtained using those equation
packages.
- RESULTS AND DlSCUSSION 197
7.5 RESULTS AND DISCUSSION
The various SAR distributions inside the multilayered spheroidal model of the
human head for a quarter-wavelength GSM dipole, PCN dipole, GSM mono-
pole, and PCN monopole have been calculated and the results are presented
in Figs. 7.2 to 7.4, 7.6 to 7.8, 7.10 to 7.12 and 7.14 to 7.16, respectively. The
inner SAR distributions of the multilayered spherical head model for these
various antennas are presented for comparison in Figs. 7.5, 7.9, 7.13, and
7.17, respectively. Here the spherical head is modeled as a six-layer sphere
with the maximum radi us r = +(a+ b), and the thicknesses of the second
through fifth layers are the same-asthose of the prolate spheroidal model. In
all these figur ‘es,the antenna is placed on the right of the multilayered model,
and the SAR values in each figure are normalized to the peak value in the
model.
The peak SAR values in the multilayered spheroidal head model vary with
the inclination angle of the antenna at both the GSM and PCN frequencies.
The SAR value increaseswhen the inclination angle /? increases, as illustrated
in Figs. 7.2 to 7.4, 7.6 to 7.8, 7.10 to 7.12, and 7.14 to 7.16 (at inclination
angles of p = O”, 30’) and 60° for different antennas), respectively. In all
cases,the SAR values decreaserapidly inside the multilayered spheroidal head
model 7 from the right of the head model to the left of the head model. It is
also found that the peak SAR values for monopole antennas are higher than
their dipole counterparts at the same frequency (e.g., compare Fig. 7.2 with
Fig. 7.10, or Fig. 7.6 with Fig. 7.14). This conclusion agrees with those from
FDTD calculations [144,158).
From the rear view of the SAR distribution, it is found that the SAR values
inside the spheroidal head model decrease faster when the inclination angle
p of the dipole becomes smaller [e.g., Figs. 7.2(a), 7.3(a), and 7.4(a)]; while
for GSM and PCN monopoles, there is no such obvious phenomenon for inner
EM fields and the SAR values show an asymmetric distribution for the upper
and lower parts of the spheroid, as shown in Figs. 7.10 to 7.12 and 7.14 to
7.16.
From the figures presented, it can be seenthat the SAR distribution inside
the multilayered spheroidal head model for a GSM dipole or monopole differs
from that for its PCN counterpart, especially for the SAR distribution from
the rear view (e.g., the comparison of Fig. 7.2 with Fig. 7.6, or Fig. 7.10 with
Fig. 7.14). For dipole antennas, the peak SAR value at the GSM frequency
(shown in Figs. 7.2 to 7.4) occurs at the CSF layer and is smaller than its
counterpart at the PCN frequency (shown in Figs. 7.6 to 7.8), where the peak
value occurs at the surface of the model. For monopole antennas, the peak
SAR value also increases with the operating frequency of the antenna, and
the peak value occurs at the surface. For GSM dipoles, although the EM
fields inside the head model decrease from the right of the model to the left,
the SAR value inside the head model shows a peak value at the right part of
the CSF layer, of which the electric conductivity is much higher than those
- 198 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
1 1 1 L I 1 L a t
lO.O-
8.0-
CO-
4.0-
2.0-
o.o-
-2.o-
-4.o-
-6.O-
-8.O-
-1 o.o-
I I 1 1 1 1 I I I 1
-10.0 -6.0 -2.0 2.0 6.0
(a) C/I= 0 and x (rear view)
10.0
8.0
6.0
-4.o-
-CO-
-8.O-
-1 o.o-
I 1 I I 1 1 ’ I I 1
-10.0 -6.0 -2.0 2.0 6.0 1
(b) 4 = ~12 and 3x12 (side view)
Fig. 7.2 SAR distributions (in dB) inside the multilayered prolate spheroidal head
model (GSM dipole), normalized to 1.86 W/kg. s = 1.5 cm, p = O”, and the unit of
the coordinate is centimeters.
- RESULTS AND OKUSSION 199
lO.O-
8.0-
6.0-
4.0-
2.0-
o.o-
-2.o-
-4.o-
-6.O-
-8.O-
-lO.O-
-10.0 -6.0 -2.0 2.0 6.0 10.0
(a) 4 = 0 and 7~ (rear view)
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
I I 1 I 1 1 1 I I
-1 I.0 -6.0 -2.0 2.0 6.0 l(
(b) 4 = ~~12 and 3~ /2 (side view)
Fig. 7.3 SAR distributions (in dB) inside the multilayered prolate spheroidal head
model (GSM dipole), normalized to 2.11 W/kg. s = 1.5 cm, p = 30’) and the unit of
the coordinate is centimeters.
- 200 SAR DISTUlBUTlONS IN A SPHEROlDAL HEAD MODEL
10.0.
8.0,
6.0,
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
-6.0 -2.0 2.0 6.0 11
(a) 4 = 0 and T (rear view)
1 1 I 1 I 1 ’ ’ L
1o.o-
8.0-
6.0-
4.0-
2.0-
o.o-
-2.o-
-4.o-
-6.O-
-8.O-
-1 o.o-
t I 1 1 1 I 1 I I I
-10.0 -6.0 -2.0 2.0 6.0 I
(b) 4 = r/2 and 3x12 (side view)
Fig. 7.4 SAR distributions (in dB) inside the multilayered prolate spheroidal head
model (GSM dipole), normalized to 2.60 W/kg. s = 1.5 cm, p = 60”, and the unit of
the coordinate is centimeters.
- RESULTS AND DISCUSSION 201
8.0-
6.0-
4.0-
2.0-
o.o-
-2.o-
-4.o-
-6.O-
-8.O-
-1 o-o- I I 1 t t I 8 I I I
-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0
(a) 4 = 0 and T (rear view)
8.0-
6.0-
4.0-
2.0-
o.o-
-2.o-
-4. o-
-6. O-
-8. O-
1 I I r I 1 I I 1 I
-lO.O!
-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0
(b) 4 = 42 and 3~12 (side view)
Fig. 7.5 SAR distributions (in dB) inside the multilayered spherical head model (GSM
dipole), normalized to 2.23 W/kg. s = 1.5 cm, p = 30”) and the unit of the coordinate
is centimeters.
- 202 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
1 1 1 I 1 1 1 1 1
-1 .O -6.0 -2.0 2.0 6.0 l(
(a) 4 = 0 and x (rear view)
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
1 1 I I 1 1 1 I 1
-1 .O -6.0 -2.0 2.0 6.0 10.0
(b) 4 = 7~12 and 3x12 (side view)
Fig. 7.6 SAR distributions (in dB) inside the multilayered prolate spheroidal head
model (PCN dipole), normalized to 2.37 W/kg. s = 1.5 cm, p = O”, and the unit of
the coordinate is centimeters.
- RESULTS AND DISCUSSION 203
1 o.o-
8.0-
6.0-
4.0-
2.0-
O.O-
-2.o-
-4.o-
-6.O-
-8.O-
-lO.O- , 1
-10.0 -6.0 -2.0 2.0 6.0 10.0
(a) 4 = 0 and T (rear view)
lO.O-
8.0-
6.0-
4.0-
-4.o-
-6.O-
-8.O-
-lO.O-
t I 1 I 1 I I 1 I 1
t
-10.0 -6.0 -2.0 2.0 6.0 10.0
(b) 4 = ~12 and 3x12 (side view)
Fig. 7.7 SAR distributions (in dB) inside the multilayered prolate spheroidal head
model (PCN dipole), normalized to 2.83 W/kg. s = 1.5 cm, 0 = 30”, and the unit of
the coordinate is centimeters.
- 204 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
1 I I 1 I I I I 1
-1 .O -6.0 -2.0 2.0 6.0 10.0
(a) $ = 0 and T (rear view)
10.0
8.0
6.0
4.0
2.0
o.o-
-2.o-
-4.o-
-6.O-
-8.O-
-10.0 -6.0 -2.0 2.0 6.0 10.0
(b) 4 = x/2 and 3x12 (side view)
Fig. 7.8 SAR distributions (in dB) inside the multilayered prolate spheroidal head
model (PCN dipole), normalized to 3.41 W/kg. s = 1.5 cm, p = 60”, and the unit of
the coordinate is centimeters.
- RESULTS AND DISCUSSION 205
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-lO.OI 1 1 1 I I I I 1 1 I
-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0
(a) + = 0 and T (rear view)
8.0-
6.0-
4.0-
2.0-
o.o-
-2.o-
-4.o-
-6.O-
-8.O-
-lO.O! 1 I 1 1 I I r I 1
-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 li
(b) C#J 7~12 and 3x12 (side view)
=
Fig. 7.9 SAR distributions (in dB) inside the multilayered spherical head model (PCN
dipole), normalized to 3.02 W/kg. s = 1.5 cm, p = 30”) and the unit of the coordinate
is centimeters.
- 206 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
1 1 , , , , I L ,
10.0
8.0
6.0
4.0 I
2.0-
o.o- 0
-2.o-
-4.o-
-6.O-
-8.O-
-lO.O-
I I I I I , , , , 1
-10.0 -6.0 -2.0 2.0 6.0 10.0
(a) 4 = 0 and T (rear view)
8.0-
6.0-
-6.O-
-8.O-
-10.0 -6.0 -2.0 2.0 . 610 ’
(b) 4 = n/2 and 3x12 (side view)
Fig. 7.10 SAR distributions (in dB) inside the multilayered prolate spheroidal head
model (GSM monopole), normalized to 4.94 W/kg. s = 1.5 cm, p = O”, and the unit
of the coordinate is centimeters.
- RESULTS AND DISCUSSION 207
L
lO.O-
8.0-
6.0-
4.0-
2.0-
o.o-
-2-o-
-4.0-
-6.O-
-8.O-
-lO.O-
(a) $ = 0 and 7~ (rear view)
8.0
1
-2.o-
-4.o-
-6.O-
-8.O-
1 I 1 ’ ’ 1 ’ ” ’
-10.0 -6.0 -2.0 2.0 6.0 11
09 4 = 7r/2 and 3n/2 (side view)
Fig. 7.11 SAR distributions (in dB) inside the multilayered prolate spheroidal head
model (GSM monopole), normalized to 6.18 W/kg. s = 1.5 cm, /? = 30”, and the unit
of the coordinate is centimeters.
- 208 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
I I 1 1 I 1 I I 1 1
-1 .O -6.0 -2.0 2.0 6.0 10.0
(a) 4 = 0 and T (rear view)
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
1 1 I 1 I I I I I
-1 .O -6.0 -2.0 2.0 6.0 10.0
(b) 4 = 7r/2 and 3x/2 (side view)
Fig. 7.12 SAR distributions (in dB) inside the multilayered prolate spheroidal head
model (GSM monopole), normalized to 7.51 W/kg. s = 1.5 cm, ,0 = 60”, and the unit
of the coordinate is centimeters.
- EFFECTS ON WIRE ANTENNAS 209
of other layers in the spheroidal head model. This coincides with the result
of FDTD [142], in which the inner peak SAR value occurs at the edge of the
brain layer, where the electric conductivity is assumed to be highest. The
relationship of the SAR value with the antenna frequency obtained from the
analytical results from the multilayered prolate model is also coincident with
the results from FDTD [143,144] and the spherical model [151].
For the spheroidal model at the GSM and PCN frequencies, the peak value
of SAR is smaller than its spherical counterpart, as shown in Figs. 7.5, 7.9,
7.13, and 7.17, respectively. There are obvious differences in the inner EM
field distributions between the prolate spheroidal head model and the spherical
head model (e.g., the comparison of Fig. 7.11 with Fig. 7.13). Also, there
is apparently no difference in SAR values in the spherical head model for
different inclination angles of the antenna. In view of the fact that the human
head should be better approximated by a prolate spheroid than a simple
sphere, the full-wave analysis of the EM field distribution inside the human
head using the prolate spheroidal model is more relevant to the actual case
than that obtained using the simple spherical model.
From Fig. 7.18 it is clear that the SAR values in the head model also vary
with the location of the antenna. The peak SAR values in all the casesdecay
rapidly when the distance s between the mobile antenna and the head model is
increased. This fundamental result has already been verified by many authors
using various techniques.
Table 7.2 shows a comparison of the SARs for various head models at
frequencies of 1800 MHz, 900 MHz, and 915 MHz. The close agreement of
the results for SAR of the six-layer sphere at both 1800 MHz and 900 MHz
confirms the validity of the spheroidal function approximation. It is clear
that the result for the six-layer spheroid (SAR of 4.94 W/kg) is closer to the
result for the true anatomical model (SAR of 3.9 W/kg). The simple box or
spherical models of the human head always give overestimated SAR values,
as illustrated by Okoniewski and Stuchly [139]. The inner SAR distributions
(rear view) of the multilayered prolate spheroidal model have been compared
with the FDTD results which are available [139] and found to be similar. This
is fundamentally true because the shape of an anatomical head model is more
like that of a prolate spheroid than that of a simple box or sphere.
7.6 EFFECTS ON WIRE ANTENNAS DUE TO THE PRESENCE OF
THE MULTILAYERED SPHEROID
In the previous study, the mobile antennas were assumed to be very thin
wires with current distributions in sinusoidal form. This approximation is fair
enough for the free-space assumption and the infinite small diameter of the
wire antenna (usually, diameters D of D < 0.05X) [155]. For the diameter D >
0.05& the sinusoidal current distribution of the wire antenna is representative
- 210 SAR DlSTRIt3UTlONS IN A SPHEROIDAL HEAD MODEL
8.0-
6.0-
4.0-
2.0-
o.o-
-2.o-
-4.o-
-6.O-
-8.O-
40.0-l , I I 1 1 I 1 1 1 I
-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0
(a) 4 = 0 and T (rear view)
80
6.0-
4.0
20
-6.
-8.
-10.0-l I I I 1 1 I 1 I 1 !
-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0
(b) + = 7r/2 and 342 (side view)
Fig. 7.13 SAR distributions (in dB) inside the multilayered spherical head model
(GSM monopole), normalized to 6.69 W/kg. s = 1.5 cm, /? = 30”, and the unit of the
coordinate is centimeters.
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