
Radiofrequency Radiation Dosimetry Handbook
(Fourth Edition)

Chapter 5. Theoretical Dosimetry
5.1. METHODS OF CALCULATION
In principle, the internal fields in any object
irradiated by electromagnetic fields can be calculated by
solving Maxwell's equations. In practice, this is very
difficult and can be done only for a few special cases.
Because of the mathematical complexities involved in
calculating SARs, a combination of techniques has been used
to obtain SARs as a function of frequency for various models
(Durney, 1980). Each technique gives information over a
limited range of parameters. The combined information gives a
reasonably good description of SAR versus frequency over a
wide range of frequencies and for a number of useful models.
Figure 5.1 summarizes the combination of techniques used in
the various frequency ranges to obtain the average SAR versus
frequency for a model of an average man.
With spheroidal models we used a method called the
longwavelength approximation up to frequencies of about 30
MHz; the extendedboundarycondition method (EBCM) up to
approximately resonance (80 MHz); and the iterative
extendedboundarycondition method (IEBCM), an extension of
the EBCM, up to 400 MHz. With cylindrical models we used the
classical solution of Maxwell's equations to obtain useful
average SAR data for E polarization from about 500 to 7,000
MHz, and for H polarization from about 100 to 7,000 MHz;
above approximately 7,000 MHz, we used an approximation based
on geometrical optics. Up to about 400 MHz we used the
momentmethod solution of a Green's function integral equation
for the electric field; for K polarization we used the
surfaceintegralequation (SIE) technique with a model
consisting of a truncated cylinder capped on each end by
hemispheres. An empirical relation developed for E
polarization gives a good approximation for the average SAR
over the entire frequency spectrum up to 10 GHz. For K
polarization we used estimated values based on experimental
results for the range between 400 and 7,000 MHz because
calculations are not yet possible in this frequency range.
Each of these techniques will be briefly described.
Figure 5.1.
Illustration of different techniques, with their frequency limits, used for calculating SAR data for
models of an average man.
LongWavelength ApproximationIn the frequency range
where the length of the irradiated object is approximately
twotenths or less of a freespace wavelength, we have
approximated the SAR calculations, based on the first order
term of a power series expansion in k of the electric and
magnetic fields, where k is the freespace propagation
constant (Durney et al., 1975). This is called a
perturbation method because the resulting fields are only a
small change from the static fields. Equations for SAR have
been derived for homogeneous spheroidal and ellipsoidal
models of humans and animals (Johnson et al., 1975; Massoudi
et al. , 1977a, 1977b, 1977c) . Detailed relations (given in
the referenced articles) have been used here to calculate the
SAR in the lowfrequency range.
ExtendedBoundaryCondition MethodThe EBCM is a matrix
formulation based on an integral equation and expansion of
the EM fields in spherical harmonics. This method was
developed by Waterman (1971) and has been used to calculate
the SAR in prolate spheroidal models of humans and animals
(Barber, 1977a, 1977b). The EBCM is exact within the limits
of numerical computation capabilities; but for prolate
spheroidal models of humans, numerical problems limit the
method to frequencies below about 80 MHz. In SAR calculations
for these models, the longwavelength approximation and the
EBCM give identical results up to about 30 MHz, where the
longwavelength approximation begins to be inaccurate.
IterativeExtendedBoundaryCondition MethodThe EBCM
has been extended (Lakhtakia et al., 1983b) to a technique
(the IEBCM) that is capable of SAR calculations up to at
least 400 MHz in prolate spheroidal models of man. The IEBCM
differs from the EBCM in two main respects. By using more
than one spherical harmonic expansion, the IEBCM allows
better convergence for elongated bodies and at higher
frequencies; and it uses iteration, beginning with an
approximate solution, to converge to the solution. These two
features have significantly extended the calculation range of
the IEBCM over that of the EBCM.
The Cylindrical ApproximationIn the frequency range
where the wavelength is very short compared to the length of
the spheroid, the SAR calculated for an appropriately long section of an infinitely long
cylinder is a good approximation to the SAR of spheroids. The
lowest frequency at which the approximation is useful depends
both on the length of the spheroid and on the ratio of the
major axis to the minor axis. For mansized spheroids, the
lower frequency limit occurs for E polarization when the
wavelength is about four tenths the length of the spheroid.
(Massoudi et al., 1979a).
MomentMethod SolutionA momentmethod solution of a
Green'sfunction integral equation for the Efield has been
used to calculate the internal Efield in block models,
socalled because the mathematical cells of which the model
is composed are cubes (Chen and Guru, 1977a, 1977b, 1977c;
Hagmann et al., 1979a, 1979b). Wholebody average SARs
calculated by this method are very close to those calculated
for spheroidal models. Although the block modelwith
simulated arms, legs, and headhas the advantage of
resembling the human body better than a spheroid, the
calculations of the spatial distribution of the internal
fields have been unreliable (Massoudi et al., 1984). One
problem with this technique is that the Efield in each
mathematical cell is approximated by a constant (called a
pulse function), and this approximate field cannot satisfy
the boundary conditions between cells well enough. Another
problem is that the discontinuities at the sharp corners of
the cells make the calculated fields at the corners between
cells of different permittivities vary rapidly with position,
which causes problems in numerical calculations.
SurfaceIntegralEquation TechniqueThe SIE method,
based on a formulation of the EMfield equations in terms of
integrals over induced currents on the surface of an object
(Wu, 1979; Harrington and Mautz, 1972), has been used to
calculate average SARs, principally for K polarization and
mostly for models consisting of a truncated cylinder capped
on each end by hemispheres. Average SARs for this model are
close to those for a spheroid, depending on how the
dimensions of the cylinderhemispheres model are chosen
relative to the spheroid.
Empirical Relations for FreeSpace IrradiationTechniques for calculating SARs (especially over a wide frequency range) are complex and expensive, so a simplified method for calculating approximate
average SAR over a broad range of frequencies could be very
useful, even if it gave results within 10% or 15% of those
calculated by more sophisticated methods. Kucia (1972) and
Gandhi and Hagmann (1977a) made some approximate calculations
based on antenna theory. Gandhi and Hagmann found from
experimental data that the resonant frequency for E
polarization occurs when the length of the object is equal to
approximately 0.4 where is the freespace wavelength.
They also noticed that the SAR decreases approximately as 1/f
(f is frequency) in the postresonance region. Using a
combination of antenna theory, circuit theory, and curve
fitting, we have developed empirical relations for
calculating the average SAR over the whole frequency range of
interest for a prolate spheroidal model of any human or
animal (Durney et al., 1979). We have also developed
semiempirical methods for calculating the average SAR of an
irradiated object near or on a ground plane or connected to a
ground plane by a resistive connection. These relations are
described below.
Based on available calculated and experimental data, we
formulated the following expression of average SAR for an
incidentpower density of 1 mW/cm² and E polarization
for a spheroid with semimajor axis, a, and semiminor axis, b, in
meters:
(Equation 5.1)
where (as given by Equations 5.5  5.9) A_{1}, A_{2}, A_{3}, and A_{4}
are functions of a and b, and A_{5} is a function of . Unit
step function u (f  f _{ol} ) is defined by
and u (f  f _{o2} ) is similarly defined. Also, f_{o} < f_{ol}
< f_{o2}. The resonant frequency, f_{o} , is given by the
following empirical relation:
(Equation 5.2)
We obtained Equation 5.2 by constructing a function of a
and b with adjustable parameters and using a
leastsquareserror procedure to fit the function to
calculated values of f_{o}. The function was constructed from
the observation that resonance is a combination of the length
being near a half wavelength and the circumference being near
a wavelength. Values calculated from Equation 5.2 are within
5% of all resonant frequency values calculated by more
accurate methods.
The empirically derived quantities f_{ol} and f _{o2} are defined by
(Equation 5.3)
(Equation 5.4)
By requiring Equation 5.1 to provide a best leastsquares fit to all the data available, we obtained the following expressions:
A_{1 }= 0.994  10.690 a + 0.172 a/b + 0.739 a^{1} + 5.660 a/b^{2} (Equation 5.5)
A_{2} = 0.914 + 41.400 a + 399.170 a/b  1.190 a^{1} 2.141 a/b^{2} ((Equation 5.6)
A_{3} = 4.822 a  0.084 a/b  8.733 a^{2} + 0.0016 (a/b)^{2} + 5.369 a^{3} (Equation 5.7)
A_{4 }= 0.335 a + 0.075 a/b  0.804 a^{2}  0.0075 (a/b)^{2} + 0.640 a^{3} (Equation 5.8)
A_{5} =  / _{20} ^{ 1/4 } (Equation 5.9)
where _{20} is the complex permittivity at 20 GHz.
A_{5} is a function of , the complex permittivity of muscle, and is used to describe the SAR in the geometrical optics region.
Equation 5.1 is a powerful relation because it allows
using a hand calculator to get good approximate values of SAR
for any prolate spheroidal model between rat size and man
size, whereas the SAR data in this handbook and its previous
editions require sophisticated and expensive calculation
methods and are plotted only for specific cases. Numerical
results from Equation 5.1 are shown in Figure 5.2 as data
points on the E polarization curve.
This empirical formula (Equation 5.1) is included,
however, to complement but not substitute for the SAR data
given in the handbook. In its present form, Equation 5.1 is
useful for calculating the SAR for models of intermediate
sizes between humans and rats. Although the coefficients A_{1},
A_{2}, . . ., A_{5} were derived by fitting available SAR data for 18
models, the accuracy is rather limited in the transition
regions at f = f_{ol} and f = f_{o2}, where step functions begin to
be effective. Because of the abrupt nature of the step
function, SAR values in close proximity to f_{ol} and f_{o2} are
usually inaccurate. Also, since the frequencydependent
permittivity is not explicitly included in Equation 5.1,
SARvalue fluctuations caused by the variation of with
frequency are not always accurately represented.
William D. Hurt and Luis Lozano (USAFSAM) modified
Equation 5.1 to eliminate the step functions. Their equation
is
(Equation 5.10)
where


(Equation 5.11) 


(Equation 5.12) 
and a, b, A_{1}, A_{2}, A_{4}, A_{5}, and f_{o} are as defined previously.
Figure 5.2.
Average SAR calculated by the empirical
formula compared with the curve obtained by other
calculations for a 70kg man in E polarization. For the
prolate spheroidal model, a = 0.875 m and b = 0.138 m; for
the cylindrical model, the radius of the cylinder is 0.1128 m
and the length is 1.75 m.
Figure 5.2 (continued).
Equation 5.10 has the advantage of being continuous
because it contains no step functions. It is identical to
Equation 5.1 for low, resonance, and high frequencies but
differs somewhat in the immediate postresonance frequency
range, where it gave values within 30% of those calculated
for specific models (data in Chapter 6) except for the
average endomorphic man, for which it gave results that were
40% below the handbook values.
William D. Hurt (USAFSAM) developed another empirical
relation that incorporates in one continuous expression both
the longwavelength approximation for prolate spheroids on
the low end of the frequency spectrum and the geometrical
optics approximation on the high end. This empirical equation
is
(Equation 5.13)
where A_{l}, A _{3} and A_{5} are defined in Equations 5.5, 5.7,
and 5.9, and

(Equation 5.14) 

(Equation 5.15) 

(Equation 5.16) 

(Equation 5.17) 

(Equation 5.18) 
Also, is the conductivity of the spheroid in siemens per meter.
The SARs calculated from Equation 5.13 were within about
25% of the values for specific models as given in Chapter 6
except for the small rat, for which the values differed by
about 40% at 5 GHz.
Semiempirical Relations for Irradiation Near a Ground PlaneKnowing how shoes and soles affect the SAR in man on a ground plane is desirable. Such an effect, however, is very difficult to estimate even by
using complicated numerical techniques (Hagmann and Gandhi, 1979). In this section we present a simple semiempirical formula for calculating the SAR of a halfspheroid placed over, but at a distance from, an infinitely large ground plane.
To derive this formula, we first put Equation 5.1 in the form of the power absorbed in a series RLC circuit. Hence
(Equation 5.19)
where
Comparing Equations 5.1 (up to resonance) and 5.19and
keeping in mind that input voltage aE_{o} is applied across the
input impedance and the radiation impedance of a monopole
rather than a dipole antennathe parameters R, L, and C of
Equation 5.19 can be expressed in terms of A_{l}, A_{2} and f _{o}.
Therefore, we first compute the parameters A_{l}, A_{2}, and f_{o} so
that the power calculated from Equation 5.1 will fit (with
leastsquares error) the numerical results of the SAR in a
man model on a ground plane (Hagmann and Gandhi, 1979). The
corresponding R, L, and C parameters will hence be valid for
a halfspheroid in direct contact with a perfectly conducting
ground plane. Introducing a small separation distance between
the halfspheroid and the ground plane, in the form of an air
gap or a resistive gap representing shoes, will correspond to
adding the following R_{g }and X_{g }parameters in series with
the previously derived resonance circuit:
(Equation 5.20)
where R_{1}and C_{1} are the parallel combination describing the gap impedance. R_{g} and X_{g} are frequency dependent and will result in changes in both the SAR values and the resonance frequency at which maximum absorption occurs.
At frequencies higher than 1 MHz and at separation
distances more than 1 cm from a relatively dry earth, R _{g} can
be shown to be negligible with respect to the resistance of
the equivalent circuit in Equation 5.19. For this case the
effect of only X _{g} is shown in Figure 5.3. As the gap
distance increases, the SAR curve continues to shift to the
right toward the limiting case of a man in free space. The
SAR curve reaches this limiting case for a separation
distance of about 7.5 cm. Since the introduction of C_{g} will
not account for any changes in the SAR value at resonance,
the peak value in the figure remains the same.
For wet earth and particularly for spheroids at small separation distances, the effect of gap resistance R_{g} should be taken into account (Spiegel, 1977). Figure 5.4 illustrates such an effect where small reductions in the SAR values are generally observed.
Although the presence of the ground plane shifts the
resonant frequency in each case, it does not significantly
affect the maximum value of the average SAR. At a given
frequency well below resonance (e.g., 10 MHz), however, the
presence of the ground plane increases the average SAR by an
order of magnitude over the freespace value.
Penetration as a Function of FrequencyThe concept of
skin depth discussed in Section 3.3.4 shows that for the
special case of a planewave incident on a lossy dielectric
halfspace, the penetration of the planewave becomes
shallower and shallower as the frequency increases. For
example, from Equation 3.46 the skin depth in a dielectric
halfspace having a permittivity equal to twothirds that of
muscle tissue is only 0.41 cm at 10 GHz. Although the concept
of skin depth in a dielectric half space can give a
qualitative indication of how penetration changes with
frequency in nonplanar objects, it must be used with
caution.
Figure 5.3.
Calculated effect of a capacitive gap, between man model and ground plane, on average SAR.
Figure 5.4.
Calculated effect of grounding resistance on
SAR of man model placed at a distance from ground plane.
To provide more quantitative information about
penetration in nonplanar objects, we have made some
calculations that show how power absorption is distributed
over the volume of the object as a function of frequency for
spheres, cylinders, and spheroids irradiated by planewaves.
These calculations are based on the following procedure.
First, the object is divided into M small equalvolume
elements V. Then P_{n}, the power absorbed in each
V, is calculated and ranked in order from greatest
power absorbed to least. Next, the total power absorbed in N
of the Vs is calculated by summing the ranked P_{n}'s
from highest toward least:
(Equation 5.21)
The number N is selected so that
(Equation 5.22)
where P_{M} is the total power absorbed in the object, given by
(Equation 5.23)
Then the volume fraction V_{F} is defined as that fraction of the volume in which 90% of the power is absorbed:
(Equation 5.24)
As the curves in Figures 5.5 and 5.6 show, V_{F} is nearly unity at low frequencies but decreases to a very small number at high frequencies.
Although like calculations are not practical for a shape
closer to the human body, the similarity of the curves for
the objects shown indicates that similar results would be
expected for the human body. Curves for spheres, cylinders,
and planar halfspace all show that the penetration decreases
rapidly with frequency, and at the higher frequencies almost
all of the power is absorbed in a small percentage of the
volume near the surface.
Figure 5.5
The volume fraction, V_{F}, as a function of frequency for a cylindrical model of an average man. V_{F} is the fraction of the volume in which 90% of the power is absorbed (see Equation 5.24).
Figure 5.6
The volume fraction, V_{F}, as a function of frequency for two spheres of muscle material. V_{F} is the fraction of the volume in which 90% of the power is absorbed (see Equation 5.24).
Go to Chapter 5.1.2
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Last modified: June 14, 1997
© October 1986, USAF School of Aerospace Medicine, Aerospace Medical Division (AFSC), Brooks Air Force Base, TX 782355301