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 th
Trang 17
Spheroidal Head Model
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
inside the human head [138,139] There exist a variety of techniques that can
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-
real biological human head in the analysis [ 14%1511, but it is less accurate
191
Spheroidal Wave Functions in Electromagnetic Theory
Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong
ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic)
Trang 2192 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
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 assumed to 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 assumed to have a negligible diameter and their current distributions are assumed to have a center-fed sinusoidal current distribution such as that described in [155]
Trang 3MULTILAYERED PROLATE SPHEROIDAL HEAD MODEL 193
Trang 4194 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
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
Trang 5FORMULATION 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(<‘) is the
from the EM boundary conditions
The unknown scattering and transmission coefficients can be solved by sub- stituting Eqs (7.la), (7.lb), and (7.1~) into the following EM boundary conditions at the spheroidal interfaces of the multilayered head model:
$X Ej = ixEj+l,
(7.2a) (7.2b)
the seven regions are assumed to be the same (i.e., pj = ~0) The functional expansion method described in Chapter 3 should be employed to expand Eqs (7.2a) and (7.2b) in matrix form and the coupled unknowns in the equations above are then solved for uniquely
The major issue concerning the dosimetric assessment of the absorption of
EM energy by biological subjects is how much the EM power is absorbed and where it is deposited This is usually quantified by the specific absorption rate @AR) [156] The SAR quantifies the power absorbed per unit mass of tissue and is a fundamental parameter used when discussing the health risk
of EM power absorption in the human head or body The SAR is defined as
dEl 2
where E is the peak value of the sinusoidal electric field, g the material
Trang 6196 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
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
of the j th layer, respectively Values of aj and pj are presented in Table 7.1
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 assumed to 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
about the wire antenna in Eq (7.1)
intermediates Ittyln (c) described in Appendix B
mine the unknown scattering coefficients using the matrix equation sys- tem In practical computation, the truncation number is chosen to be
48
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
Trang 7RESULTS AND DlSCUSSION 197
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-as those 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 increases when 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,
model
the SAR values decrease rapidly inside the multilayered spheroidal head
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 seen that 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
Trang 8198 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
lO.O-
8.0- CO- 4.0- 2.0- o.o- -2.o-
-4.o- -CO- -8.O- -1 o.o-
(a) C/I = 0 and x (rear view)
(b) 4 = ~12 and 3x12 (side view)
Trang 9RESULTS AND OKUSSION 199
lO.O-
8.0- 6.0- 4.0- 2.0- o.o- -2.o-
(b) 4 = ~~12 and 3~ /2 (side view)
Trang 10200 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
1
1 o.o-
8.0- 6.0- 4.0- 2.0- o.o- -2.o- -4.o- -6.O- -8.O- -1 o.o-
Trang 11RESULTS AND DISCUSSION 201
is centimeters
Trang 12202 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
10.0
8.0 6.0 4.0 2.0 0.0 -2.0
(b) 4 = 7~12 and 3x12 (side view)
Trang 13RESULTS AND DISCUSSION 203
(b) 4 = ~12 and 3x12 (side view)
Trang 14204 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
10.0
8.0 6.0 4.0 2.0 0.0 -2.0
(b) 4 = x/2 and 3x12 (side view)
Trang 15RESULTS AND DISCUSSION 205
10.0
8.0 6.0 4.0 2.0 0.0 -2.0
is centimeters
Trang 16206 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
10.0
8.0 6.0 4.0
I 2.0- o.o- -2.o-
-6.O-
-8.O-
Trang 17
RESULTS AND DISCUSSION 207
Trang 18208 SAR DISTRIBUTIONS IN A SPHEROIDAL HEAD MODEL
10.0
8.0 6.0 4.0 2.0 0.0 -2.0
(b) 4 = 7r/2 and 3x/2 (side view)
Trang 19EFFECTS 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
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 cases decay 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
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
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