A Novel Approach To Constructing The Green’s Function for Layered Media and Its Application to MMIC, RFIC, and EMC Probl

1. A Novel Approach To Constructing The Green’s Function for Layered Media and Its Application to MMIC, RFIC, and EMC Problems Raj Mittra*, Jonathan Bringuier * and Nadar Farahat** *Electromagnetic Communication Laboratory, Penn State **2Polytechnic University of Puerto Rico P.O. box 192017 San Juan, PR 00919

2. Space waves Transmit Antenna Receive Antenna Layered Model For human body Surface waves Canonical Geometry for Conformal Antennas

3. Space waves Transmit Antenna Receive Antenna Surface waves Green’s function approach using FDTD Methodology Outline: 1) Simulate Antenna structure in isolation to obtain terminal parameters and fields on an aperture just above the conformal antenna. This aperture distribution will be treated as equivalent sources J and M in the forgoing anaylsis. We assume that this information can either be obtained or given a priori. 2) Using BOR-FDTD, simulate horizontal electric and magnetic ideal dipole sources resting on the human body layered model to estimate the Green’s function of the media. 3) Since coupling distances can reach a large number of wavelengths, we use a Prony analysis of the fields at distances from the source where only surface waves are sustained and higher order behavior is neglible. 4) We break up the obtained transmitting aperture into a discrete number of ideal dipole sources (both magnetic and electric) and apply superposition to an arbitrary receive aperture size. 5) Using the fields obtained on the receive aperture, we can apply the reaction concept to calculate the coupling. Layered Model For human body

4. z z HED Ex y y x x Simulating Horizontal Electric dipoles (HED) in FDTD Theoretical BOR-FDTD implementation Theory

5. Theory z Normalization required Ex y x

6. Theory z Normalization required Ex y x

7. Normalizing the results of the BOR-FDTD with the dipole moment calculated, the results are the fields generated by an ideal dipole with unit moment in BOR-FDTD. Subsequent simulations (specific to the simulation parameters mentioned above) for different material distributions in the computation domain must be normalized by this factor. Note: There is a near perfect agreement, both in magnitude and phase, between the normalized fields from FDTD and the Theoretical expressions for the ideal dipole with a unit moment.

8. Prony extrapolation Field Magnitude FDTD result Higher order terms negligible. Only propagating surface waves Remain. Sample region Data used in Prony anaylsis Therefore, the extrapolation will be valid for values of Prony method FDTD result Sample region Extrapolation region

9. Prony estimation for Ephi of ideal HED FDTD data Extrapolation region Sample region End of FDTD simulation Note: 5th order Prony estimation was used but only one term dominates which corresponds to the surface wave.

10. R R FDTD grid w/ aperture distribution Transmitting Aperture approximated by a discrete number of ideal dipole sources At large separation distances We can treat the aperture mesh As a planar array. Use Prony results: Ideal dipole source

11. R R FDTD grid w/ aperture distribution Using Array Theory, the resulting expression for the any of the field components is: AF with arbitrary spacing and weigting At large separation distances We can treat the aperture mesh As a planar array. Ideal dipole source

12. transmit receive • FDTD generated mesh • (196 x 114 x 105 matrix) (b) Human torso 3D CT scan data with 1 mm cubic voxel resolution. Human Body composite 3 layer model: skin, fat, muscle.

13. y Receive Aperture grid: dx=dy=7.1429e-005 Number of cells=21x21 Space waves Transmit Antenna Receive Antenna Layered Model For human body Surface waves Simulation of two square patchs resting on the human body torso Transmit Aperture grid: dx=dy=7.1429e-005 Number of cells=21x21 x FDTD Receive grid R y R FDTD grid w/ aperture distribution x Sinusiodal aperture distribution along x

14. Application of method to multilayer model for human body Surface wave Surface wave Region 1: Skin epsr=46.7 sigma=0.69 height=1/6*lambda Observation plane Jx Region 2: Fat epsr=11.6 sigma=0.08 height=1/6*lambda Region 1 Region 2 Region 3 Region 3: Muscle epsr=58.8 sigma=0.84 height=infinite half plane PML

15. Sample Field used for Prony Prony results BOR result (blue) Bn_Ephi=0.239623547159758E+03 alphan_Ephi=0.178558442838438E+02 A_Ephi= -0.572584898018054E+01-j*0.120430818055098E+01 Bn_Erho=0.210600035140825E+03 alphan_Erho=0.458742417351554E+01 A_Erho=0.677026493838244E+03+j*0.567776370035642E+03 Bn_Ez=0.209850321425741E+03 alphan_Ez=0.427832694143299E+01 A_Ez=-0.399837251051164E+04 +j*0.314069187401468E+04 Bn_Hphi=0.210256687643315E+03 alphan_Hphi=0.445572941127469E+01 A_Hphi= 0.101106161268781E+02 +j*0.899268946273489E+01 Bn_Hrho=0.225717493571084E+03 alphan_Hrho=0.832370230201144E+0 A_Hrho=-0.749370952261226E-01 +j*0.132739082925593E+00 Bn_Hz=0.216866491191321E+03 alphan_Hz=0.242139604271817E+02 A_Hz=-0.481638724981019E-01 -j*0.654421958326780E-01 Sample for Prony (red)

16. Space waves Transmit Antenna y x Receive Antenna Layered Model For human body Surface waves Field Magnitudes on the receive aperture Ex Ey source Hy Hx

17. Calculation of coupling between 64 element X-band and 64 element Ku-band arrays separated by 40 wavelengths • Using the Green’s function approach with the convolution of sources on the on the transmitting X-band array we can obtain the fields induced on the defined aperture of Ku-band the receiving array. • Both arrays are separated by 40 free-space wavelengths with a RAM material having relative permittivity of 3.1, conductivity of 0.134 and height of ½ inch (1.27 cm).

18. Space waves Transmit Antenna Receive Antenna RAM Surface waves X-band Array All elements excited Ku-band Array Single element excited 40 lambda 8.16 lambda 11.04 lambda 11.52 lambda 8.64 lambda

19. Ex for X-band spiral array

20. Ex for Ku-band single active element

21. Ex magnitude on Ku-band aperture Hy magnitude on Ku-band aperture Index y-axis Index y-axis Index x-axis Index x-axis Space waves Transmit Aperture Receive Aperture RAM Surface waves

22. Ex phase on Ku-band aperture Hy phase on Ku-band aperture Index y-axis Index y-axis Index x-axis Index x-axis From non-uniform grid

23. Reaction Calculation: Using the fields obtained on the receive aperture and the source information, we can apply the reaction concept to calculate the coupling. For coupling calculations we must have the isolated Port characteristics of the transmitting and receiving antennas. The coupling can be directly computed using the reaction information on the aperture.

24. Conclusion • Direct simulation of antennas mounted on the human body can be done if the computing resources are available. However, such simulations may not be feasible if computational power is limited. • The Green’s function approach using BOR-FDTD has many distinct advantages: 1) The computational domain can be reduced from 3D to 2D by exploiting azimuthal symmetry. 2) Fields can be extrapolated to very large distances using Prony’s method. 3) The total fields on a receiving aperture can be obtained by simple superposition of equivalent sources on the transmitting aperture. 4) Coupling can be directly calculated using the Reaction Theorem.

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