STUDY OF INTERPOLATION AND EXTRAPOLATION TECHNIQUES FOR SAR MEASUREMENTS N. Stevens(1), W. Bracke(1) and L. Martens(2) (1)
Department of Information Technology, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium; Tel. +32 9 264 34 27; fax: +32 9 264 35 93; E-mail:
[email protected] (2)
“As (1) above, but E-mail:
[email protected]”
ABSTRACT In this paper, we compare different models for the interpolation and extrapolation of the SAR (Specific Absorption Rate) in biological tissues produced by an electromagnetic source. The configuration under study is a spherical benchmark at 900 MHz. The spatial distribution of the SAR is well-known [1] for this configuration. Based on these data, one can do a virtual measurement on a realistically large grid. These measurement data can be used to evaluate the efficiency of each model. The best models provide an accurate evaluation of the SAR in a large volume with a limited number of measurement points. INTRODUCTION Devices like cellular telephones, walkie-talkies and other wireless handheld radio transmitters are used close to the body and contain the radiating structure in the handheld unit operating with frequencies in the range of 30 MHz to 3 GHz. For these devices, compliance with the SAR (Specific Absorption Rate) limits has to be verified. Several standards and recommendations are defined for the limits of localized SAR. The Council of the European Union proposed a basic restriction of 2 W/kg averaged over 10 g of contiguous tissue. Another commonly used guideline is the FCC limit of 1.6 W/kg averaged SAR over a volume of 1 g in the shape of a cube. In practice, there are physical limitations to the accurate evaluation of the averaged SAR. During the measurement, only a discrete number of samples of the local SAR can be obtained. In order to know the SAR values in-between these measurement points, an accurate interpolation technique is necessary. The sensor of the measurement probe has also finite dimensions, with the immediate consequence that one cannot obtain SAR values very close to the boundary of the biological tissue. It is known that there is a rapid decay of the SAR at wireless device frequencies in biological tissues with the maximum at the surface, so a well-designed extrapolation procedure can significantly improve the evaluation of the maximal averaged SAR. SAR DISTRIBUTION IN ABSORBING MEDIA In order to construct a reasonable model for the SAR distribution in general, it is useful to consider the very simple configuration of a plane wave penetrating a half infinite conducting medium with parameter εr and σ for respectively the relative permittivity and the conductivity of the medium. Suppose that the wave is propagating in the positive zdirection, then we have the following z-dependency for the magnitude of the electrical field in the medium:
E ~ exp(− k 0 ⋅ δ ⋅ z )
(1)
with k0 the propagation constant in free space and ε0 is the permittivity in free space.
δ = − Im ε r − j
σ ωε 0
(2)
The SAR is proportional with the square of the magnitude of the electrical field. So one can rapidly see that the spatial dependency of the SAR will be mainly exponential in nature as in equation (3). (3) SAR ~ exp(− 2 ⋅ k 0 ⋅ δ ⋅ z ) =ˆ exp(− α ⋅ z ) This example contains the main physical properties of more complex configurations such as a mobile phone close to an absorbing irregular volume such as the head. Though it is clear that such a configuration is far more complicated than
the one of a plane wave propagating in a half infinite medium with a flat boundary. Nevertheless, since a complex wave can always be expanded in an (infinite) series of plane waves, the relationship (3) remains useful and will form the base of the different models. THE SPHERICAL BENCHMARK CONFIGURATION The dielectric sphere and cube filled with tissue-simulating liquids and exposed to dipole sources are traditional benchmark problems. They have been proposed by several groups such as the IEEE SCC34-SC2 [1], COST 244, etc. A detailed description of the setup and the obtained SAR distributions can be found in [2]. On Fig. 1, one can see the specific configuration we studied for the evaluation of different models. The operating frequency is 900 MHz and the distance of the dipole center to the bottom of the sphere is 1 cm. The radius of the sphere is 106.5 mm. The parameters of the conducting liquid are 41.5 and 0.86 S/m for the relative permittivity respectively conductivity. On Fig 2, one can see the typical distribution of the SAR in the sphere. The SAR is normalized to the peak SAR, which is 26.2 W/kg when the antenna radiates a power of 1 W continuous wave.
106.5 mm x 3.6 mm y
x Conducting liquid
h=10 mm z 149 mm
Fig. 1: Configuration under study.
Fig. 2: SAR distribution
EVALUATION PROCEDURE AND DEFINITION OF ERROR On Fig. 3, the cubical area is shown where the different models are evaluated. The measurement points are located on a fixed grid of 4 mm within a 8 cm3 cube. The interpolation and extrapolation points are on a 1 mm grid. The offset due to the probe tip is taken 2 mm, which is a realistic value. The total number of measurement points is 125 (=53), while the number of interpolation and extrapolation points is 8821 (=213-440, with 440 the number of points belonging to the cube but not to the sphere). In order to define the error, we applied the relationship as defined in [3]. It is constructed so that errors in areas where the SAR is higher have a more important contribution to the total error. The reason of this choice is a consequence of the fact that higher SAR values form the main contribution to the volume averaged SAR. 4 mm
z 2 mm 2 cm
4 mm
2 cm : Locations to evaluate the model. : Measurement points.
Fig. 3: Measurement and model evaluation locations.
The error is defined as [3]: 8821 8821 ERR = m(rn ) − c(rn ) ⋅ c(rn ) n=1 n =1
∑
∑
−1
(4)
where m(rn ) and c (rn ) are respectively the interpolated/extrapolated and correct value of the SAR at position rn . The summation is done over all the points on the 1 mm grid of Fig. 3. The unknown parameters of the interpolation/extrapolation model are chosen so that ERR is minimized. DIFFERENT MODELS AND THEIR EVALUATION The first model is the spherical model with fixed attenuation, which forms the most simplified model. The attenuation constant is chosen exactly α from equation (3) for all directions: SAR(x, y, z ) = A exp − α
(x − x0 )2 + ( y − y0 )2 + (z − z 0 )2
(5)
The parameters to be chosen are A, x0, y0 and z0. As initial estimation of the parameters, a good choice is A=26.2 W/kg (the peak SAR), x0=0 m, y0=0 m and z0=106.5 mm. The minimal ERR was 0.1233. It is clear that the simple physical attenuation can hardly model the SAR distribution of the configuration. The spherical model with parametric attenuation is similar to the previous model, but the attenuation is chosen as an unknown parameter: SAR(x, y, z ) = A exp − γ
(x − x0 )2 + ( y − y 0 )2 + (z − z 0 )2
(6)
The choice of the initial estimation of the parameters can be done as in the first model. The ERR was equal to 0.0537. The error is clearly much smaller. Though, in order to take the anisotropy introduced by the source into account, the spherical models are not sufficient. Therefore, an ellipsoidal model is defined. The simplest ellipsoidal model is formed by equation (7) 2 2 2 x − x0 y − y0 z − z0 + + SAR(x, y, z ) = A exp − a b c
(7)
Here, the anisotropy of the antenna can be taken into account since a, b and c can differ from each other. As an initial estimation of these parameters, one can choose α −1 . The ERR improves a lot by this model and is only 0.0258. On Fig. 4 and 5, one can see some sample plots of the three models for different x and y. MORE GENERAL SAR DISTRIBUTIONS It is clear that a more general configuration of a mobile device with a more realistic head model will need a more sophisticated model than previous models. The ellipsoidal model can still be used if one introduces a coordinate transformation by use of a rotation transformation. The Euler angles of this transformation can be left as parameters to minimize ERR. For the ellipsoidal model, the model with rotation transformation looks like: 2 2 2 x′ y′ z ′ SAR (x, y, z ) = A exp − + + a b c 0 0 cos(φ ) sin(φ ) 0 x − x 0 x ′ cos(ψ ) sin(ψ ) 0 1 with, y ′ = − sin(ψ ) cos(ψ ) 0 ⋅ 0 cos(θ ) sin(θ ) ⋅ − sin(φ ) cos(φ ) 0 ⋅ y − y 0 z′ 0 0 1 0 − sin(θ ) cos(θ ) 0 0 1 z − z 0
(8)
This model contains 10 parameters. Another model, which also makes use of rotation transformations, can be found in [4]. SAR along the z-axis in the cube for x=y=0 m
30
20 18
Measured values Exact values Spherical model with fixed attenuation Spherical model with parametric attenuation Ellipsoidal model
16
20 SAR [W/kg]
SAR [W/kg]
25
SAR along the z-axis in the cube for x=y=4 mm
22
Measured values Exact values Spherical model with fixed attenuation Spherical model with parametric attenuation Ellipsoidal model
15
14 12 10
10
8 6
5
0.088
0.09
0.092
0.094
0.096 0.098 z [m]
0.1
0.102
Fig. 4: Results at x=y= 0 m.
0.104
0.106
4
0.088
0.09
0.092
0.094
0.096 0.098 z [m]
0.1
0.102
0.104
0.106
Fig. 5: Results at x=y= 4 mm.
CONCLUSIONS Based on physical reasoning, we were able to construct a model for the SAR distribution of a simple benchmark configuration. The model can be extended for less symmetrical configurations by making use of rotation transformations. The method is still to be evaluated for more complicated configurations, such as a wireless device radiating in a head phantom. REFERENCES [1] IEEE SCC 34, WG1, “Spherical phantom experimental protocol, I draft”, 1997. [2] N. Stevens, L. Martens, C. Gabriel, N. Kuster and K. Poković, “Numerical and Experimental Evaluation of the SAR Employing a Spherical Benchmark,” in Abstract collection Bioelectromagnetics Society Annual Meeting, June 10-14, 2001. St Paul, Minnesota, USA, pp. 180-182. [3] M. Brishoual, C. Dale, J. Wiart and J. Citerne, “Methodology to Interpolate and Extrapolate SAR Measurements in a Volume in Dosimetric Experiment,” IEEE Transactions on Electromagnetic Compability, vol. 43, no. 3, August 2001, pp. 382-389. [4] O. Merckel, G. Fleury and J-C Bolomey, “Rapid SAR Measurement via Parametric Modeling,” in Proceedings of the 5th International Congress of the European BioElectromagnetics Association (EBEA), 6-8 September 2001, Helsinki, Finland, pp. 75-77.
