Radiofrequency Radiation
Dosimetry Handbook

(Fourth Edition)

Chapter 5. Theoretical Dosimetry

5.1. METHODS OF CALCULATION

5.1.1. Planewave Dosimetry

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 long-wavelength approximation up to frequencies of about 30 MHz; the extended-boundary-condition method (EBCM) up to approximately resonance (80 MHz); and the iterative extended-boundary-condition 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 moment-method solution of a Green's function integral equation for the electric field; for K polarization we used the surface-integral-equation (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.

Long-Wavelength Approximation--In the frequency range where the length of the irradiated object is approximately two-tenths or less of a free-space 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 free-space 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 low-frequency range.
Extended-Boundary-Condition Method--The 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 long-wavelength approximation and the EBCM give identical results up to about 30 MHz, where the long-wavelength approximation begins to be inaccurate.
Iterative-Extended-Boundary-Condition Method--The 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 Approximation--In 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 man-sized 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).
Moment-Method Solution--A moment-method solution of a Green's-function integral equation for the E-field has been used to calculate the internal E-field in block models, so-called because the mathematical cells of which the model is composed are cubes (Chen and Guru, 1977a, 1977b, 1977c; Hagmann et al., 1979a, 1979b). Whole-body average SARs calculated by this method are very close to those calculated for spheroidal models. Although the block model--with simulated arms, legs, and head--has 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 E-field 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.
Surface-Integral-Equation Technique--The SIE method, based on a formulation of the EM-field 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 cylinder-hemispheres model are chosen relative to the spheroid.
Empirical Relations for Free-Space Irradiation--Techniques 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 free-space 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 incident-power 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) A1, A2, A3, and A4 are functions of a and b, and A5 is a function of . Unit step function u (f - f ol ) is defined by

and u (f - f o2 ) is similarly defined. Also, fo < fol < fo2. The resonant frequency, fo , 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 least-squares-error procedure to fit the function to calculated values of fo. 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 fol and f o2 are defined by

(Equation 5.3)

(Equation 5.4)

By requiring Equation 5.1 to provide a best least-squares fit to all the data available, we obtained the following expressions:

A1 = -0.994 - 10.690 a + 0.172 a/b + 0.739 a-1 + 5.660 a/b2 (Equation 5.5)

A2 = -0.914 + 41.400 a + 399.170 a/b - 1.190 a-1 -2.141 a/b2 ((Equation 5.6)

A3 = 4.822 a - 0.084 a/b - 8.733 a2 + 0.0016 (a/b)2 + 5.369 a3 (Equation 5.7)

A4 = 0.335 a + 0.075 a/b - 0.804 a2 - 0.0075 (a/b)2 + 0.640 a3 (Equation 5.8)

A5 = | / 20 | -1/4 (Equation 5.9)

where 20 is the complex permittivity at 20 GHz.
A5 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 A1, A2, . . ., A5 were derived by fitting available SAR data for 18 models, the accuracy is rather limited in the transition regions at f = fol and f = fo2, where step functions begin to be effective. Because of the abrupt nature of the step function, SAR values in close proximity to fol and fo2 are usually inaccurate. Also, since the frequency-dependent permittivity is not explicitly included in Equation 5.1, SAR-value 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, A1, A2, A4, A5, and fo are as defined previously.

Figure 5.2.
Average SAR calculated by the empirical formula compared with the curve obtained by other calculations for a 70-kg 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 long-wavelength 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 Al, A 3 and A5 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 Plane--Knowing 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 half-spheroid 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 5.19--and keeping in mind that input voltage aEo is applied across the input impedance and the radiation impedance of a monopole rather than a dipole antenna--the parameters R, L, and C of Equation 5.19 can be expressed in terms of Al, A2 and f o. Therefore, we first compute the parameters Al, A2, and fo so that the power calculated from Equation 5.1 will fit (with least-squares 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 half-spheroid in direct contact with a perfectly conducting ground plane. Introducing a small separation distance between the half-spheroid and the ground plane, in the form of an air gap or a resistive gap representing shoes, will correspond to adding the following Rg and Xg parameters in series with the previously derived resonance circuit:

(Equation 5.20)

where R1and C1 are the parallel combination describing the gap impedance. Rg and Xg 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 Cg 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 Rg 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 free-space value.

Penetration as a Function of Frequency--The concept of skin depth discussed in Section 3.3.4 shows that for the special case of a planewave incident on a lossy dielectric half-space, 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 half-space having a permittivity equal to two-thirds 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 equal-volume elements V. Then Pn, 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 Pn's from highest toward least:

(Equation 5.21)

The number N is selected so that

(Equation 5.22)

where PM is the total power absorbed in the object, given by

(Equation 5.23)

Then the volume fraction VF 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, VF 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 half-space 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, VF, as a function of frequency for a cylindrical model of an average man. VF is the fraction of the volume in which 90% of the power is absorbed (see Equation 5.24).

Figure 5.6
The volume fraction, VF, as a function of frequency for two spheres of muscle material. VF is the fraction of the volume in which 90% of the power is absorbed (see Equation 5.24).


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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 78235-5301