Mathematical basis for Measurements on Scaled Models--The derivation of relations between quantities in the full-size and scaled system is outlined here. Readers interested only in the results should skip to the next subsection.
Each point in Ã is obtained by reducing the coor-dinates of a point in A by the scale factor S. Therefore, the coordinate values are related by
where is the complex permeability and is the complex relative permittivity (see Sections 3.2.6, 3.3.3, and 4.1). The effective conductivity, , is related to e" by
Fields in the scaled model, on the other hand, must satisfy
Equations 7.21 and 7.22 will be the same as 7.16 and 7.17, respectively, if
If material properties and scaling factors are selected so that Equations 7.25 and 7.26 are true, then measured quantities in the scaled model can be related to those in the full-size system, because Equations 7.19 and 7.20, which describe fields in the scaled system, are equivalent to Equations 7.16 and 7.17, respectively, which describe fields in the full-size system.
Second, it is most convenient to have both the full-size and the scaled models surrounded dby air. Then Equation 7.26 must be valid when both and represent air; that is From Equation 7.26 for this condition,
Equations 7.27 and 7.28 together require
which is equivalent to making the intensity of the source fields in the two systems equal. This can be seen from Equations 7.13 and 7.14, which are valid for all pairs of coordinate points (x, y, z, t) and . Let the corresponding points be far enough away from the object so that scattered fields are negligibly small and only source fields are present. Then, corresponds to the source fields in the two systems having equal intensities. This assumes that the sources in the two systems are correspondingly similar. Since scattered-field intensities are proportional to source field intensities, the interpretation that "setting is equal to making the source intensities equal" is valid at all points but easier to understand at points where the scattered fields are negligible.
The relationship with the effective conductivity, , is usually used. From Equations 7.26, 7.30, and 7.31,
Relating and to and (see Equation 7.18),
Using Equation 7.23 gives
and using Equation 7.30 gives
From Equations 7.32, 7.13, and 7.34, the general relationship for SAR is
When both models are in air and the intensities of the sources are equal (so Equations 7.30 and 7.31 apply) and for the usual case when , Equation
7.37 reduces to
From Equation 7.38, the scaled-model SAR is seen to be higher than that in the full-size model by scale factor S. This is often a significant advantage because it generally means that making measurements in a scaled model requires less generator power. This is particularly important when temperature measurements are made because it means that less power is required to get a measurable temperature rise in a scaled model than in a full-size model.
Another quantity that sometimes is of interest is the Poynting vector. The scaling relationship is easily obtained from Equations 7.13 and 7.14:
Adjusting the conductivity of the model material is often important in scaling techniques, as illustrated in Table 7.15. This can usually be done by varying the amount of NaCl in the mixture. Fortunately the amount of NaCl can be varied enough to adjust without affecting drastically. Figure 7 .5 shows conductivity as a function of percentage of NaCl for various percentages of the gelling agent TX-150 (see Section 7.2.5). Doubling the percentage of TX-150 has a relatively small effect on the conductivity, which is largely controlled by the percentage of NaCl. Figures 7.6 and 7.7 show the conductivity values as a function of percentage of NaCl. These graphs can be used to simulate muscle tissue in saline form for a wide range of frequencies and scale factors.
Table 7.16 shows nine compositions that can be used to simulate muscle material over a wide range of parameters. For example, to get a conductivity of = 4.6 S/m, Figure 7.5 shows that the NaCl concentration should be about 3.5% of the total mixture. From Table 7.16, we see that mixture VIII could be adjusted to accommodate the 0.4% difference needed in NaCl concentration. The ' for the mixtures in Table 7.16 are all about that of water. In many cases the " for biological tissue is the dominant factor in determining the SAR, especially at frequencies from 10 to 20 MHz, and the value of ' is not critical to the measurements.
Table 7.17 contains a summary of published work in experimental dosimetry, including references.
Go to Chapter 8.
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