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

TransmitAntenna

Receive Antenna

Space waves

Surface waves

z

Layered ModelFor human body

Canonical Geometry for Conformal Antennas

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.

TransmitAntenna

Receive Antenna

Space waves

Surface waves

z

Layered ModelFor human body

Simulating Horizontal Electric dipoles (HED) in FDTD

HED

momentdipoleIl _

z

y

x

z

y

x

0

0

0

sincos

sin

cos

EEEE

EE

EE

sourcesourcesourcex

source

source

Ex

Theoretical BOR-FDTD implementation

2

32

2

32

1

4sin

1

0

0

0

1

4sin

1

1

4cos

2

2

jIleH

H

H

E

jIle

jE

jIle

jE

rfor

j

z

z

j

j

Theory

2

32

2

32

1

4sin

1

0

0

0

1

4sin

1

1

4cos

2

2

jIleH

H

H

E

jIle

jE

jIle

jE

rfor

j

z

z

j

j

Theory

z

y

x

0

0

0

sincos

sin

cos

EEEE

EE

EE

sourcesourcesourcex

source

source

Ex

Normalizationrequired

2

32

2

32

1

4sin

1

0

0

0

1

4sin

1

1

4cos

2

2

jIleH

H

H

E

jIle

jE

jIle

jE

rfor

j

z

z

j

j

Theory

z

y

x

0

0

0

sincos

sin

cos

EEEE

EE

EE

sourcesourcesourcex

source

source

Ex

Normalizationrequired

Normalizing the results of the BOR-FDTD with the dipole moment calculated, the results are thefields 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.

Prony extrapolation

FieldMagnitude

)(f

)(g)()( 0 gf

0

Higher order terms negligible.Only propagating surface wavesRemain.

FDTD result

Sample regionData used in Prony anaylsis

)(ˆ

ˆ

0ˆ)(

ˆ)(

n

n

kN

nn

kN

nn

eAf

eAg

Prony method0 Therefore, the extrapolation will be valid for values of

)(f

0

Sample regionExtrapolation region

FDTD result

Prony estimation for Ephi of ideal HED

Note: 5th order Prony estimation was used but only one term dominates which corresponds to the surface wave.

Sample region

Extrapolation region

FDTDdata

End of FDTD simulation

RR

FDTD gridw/ aperturedistribution

At large separation distancesWe can treat the aperture meshAs a planar array.Use Prony results:

patternelementeC

eAC

Let

jk

jAAA

eAE

n

n

n

j

nn

nn

nnn

irn

k

nn

_)(ˆ

ˆ)(ˆ

ˆ

ˆ

ˆ)(

)(

)(

)(ˆ

0

0

0

Transmitting Aperture approximated by a discrete number of ideal dipole sources

Ideal dipole source

Using Array Theory, the resulting expression for the any of thefield components is:

AF with arbitrary spacing and weigting

cos)(sin)()( ˆ)(ˆsin

cos),( 0 mDj

n

L

l

L

m

lDjml

jn

xn

y xynn eeSeCF

RR

FDTD gridw/ aperturedistribution

At large separation distancesWe can treat the aperture meshAs a planar array.

Ideal dipole source

transmit

receive

(a) 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.

Simulation of two square patchs resting on the human body torso

),( 00 yx

R

R

FDTD gridw/ aperturedistribution

Transmit Aperture grid:

dx=dy=7.1429e-005

Number of cells=21x21FDTD Receive grid

y

x

22

y

x

Sinusiodal aperture distribution along x

21

)9.0,9.0(),( 00 mmyx Receive Aperture grid:

dx=dy=7.1429e-005

Number of cells=21x2122

21

TransmitAntenna

Receive Antenna

Space waves

Surface waves

z

Layered ModelFor human body

Application of method to multilayer model for human body

z

Region 3

Region 2

Region 1

Observation plane

Region 1: Skin epsr=46.7 sigma=0.69 height=1/6*lambda

Region 3: Muscle epsr=58.8 sigma=0.84 height=infinite half plane

Region 2: Fat epsr=11.6 sigma=0.08 height=1/6*lambda

PML

Surface wave Surface wave

Jx

Sample Field used for Prony

BOR result (blue)

Sample for Prony (red)

Bn_Ephi=0.239623547159758E+03alphan_Ephi=0.178558442838438E+02A_Ephi= -0.572584898018054E+01-j*0.120430818055098E+01

Bn_Erho=0.210600035140825E+03alphan_Erho=0.458742417351554E+01A_Erho=0.677026493838244E+03+j*0.567776370035642E+03

Bn_Ez=0.209850321425741E+03alphan_Ez=0.427832694143299E+01A_Ez=-0.399837251051164E+04 +j*0.314069187401468E+04

Bn_Hphi=0.210256687643315E+03alphan_Hphi=0.445572941127469E+01A_Hphi= 0.101106161268781E+02 +j*0.899268946273489E+01

Bn_Hrho=0.225717493571084E+03alphan_Hrho=0.832370230201144E+0A_Hrho=-0.749370952261226E-01 +j*0.132739082925593E+00

Bn_Hz=0.216866491191321E+03alphan_Hz=0.242139604271817E+02A_Hz=-0.481638724981019E-01 -j*0.654421958326780E-01

nnn

irn

k

nn

jk

jAAA

eAE n

ˆ

ˆ

ˆ)( )(ˆ0

Prony results

source Ex Ey

Hx Hy

yx

Field Magnitudes on the receive aperture

TransmitAntenna

Receive Antenna

Space waves

Surface waves

z

Layered ModelFor human body

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).

TransmitAntenna

Receive Antenna

Space waves

Surface waves

z

RAM

X-bandArray

All elements excited

Ku-band Array

Single element excited

40 lambda

8.64 lambda

8.16 lambda

11.52 lambda

11.04 lambda

Ex for X-band spiral array

Ex for Ku-band single active element

Ex magnitude on Ku-band aperture Hy magnitude on Ku-band aperture

Index x-axisIn

dex

y-

axis

Index x-axis

Ind

ex y

-ax

is

TransmitAperture

Receive Aperture

Space waves

Surface waves

z

RAM

Hy phase on Ku-band apertureEx phase on Ku-band aperture

From non-uniform grid

Index x-axis

Ind

ex y

-ax

is Ind

ex y

-ax

isIndex x-axis

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.

Reaction Calculation:

S

S

sdMHJE

sdMHJE

)(1,2

)(2,1

1212

2121

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.

Reference A. Alomainy, Y. Hao, X. Hu, C.G. Parini, P.S. Hall “UWB on-body radio propagation

and system modelling for wireless body-centric networks.”IEE Proc.-Commun, Vol 153 No. 1, February 2006

M.R. Kamarudan, Y.I. Nechayev, P.S. Hall “Performance of antennas in the on-bodyenviornment.” Second International Symposium, Year:19-20 Oct. 1998, Page(s):116-122

Jaehoon Kim, Yahya Rahmat-Samii “Implanted antennas inside a human body: simulationsdesign, and characterization.” IEEE MTT, Vol.52, No. 8, August 2004

Federal Communications Commission (FCC) Homepage : www.fcc.gov

Wideband slot antennas for wireless communications, Yeo, J.; Lee, Y.; Mittra, R.; Microwaves, Antennas and Propagation, IEE Proceedings-Volume 151,  Issue 4,  15 Aug. 2004 Page(s):351 - 355

Design of a wideband planar volcano-smoke slot antenna (PVSA) for wireless communications, Junho Yeo; Yoonjae Lee; Mittra, R.; Antennas and Propagation Society International Symposium, 2003. IEEE Volume 2,  22-27 June 2003 Page(s):655 - 658 vol.2

Study of CPW-fed circular disc monopole antenna for ultra wideband applications, Liang, J.; Guo, L.; Chiau, C.C.; Chen, X.; Parini, C.G.; Microwaves, Antennas and Propagation, IEE Proceedings- 9 Dec. 2005 Page(s):520 – 526

Printed circular disc monopole antenna for ultra-wideband applications, Liang, J.; Chiau, C.C.; Chen, X.; Parini, C.G.; Electronics Letters, Volume 40,  Issue 20,  30 Sept. 2004 Page(s):1246 – 1247

Modeling Of Interaction Between Body-mounted Antennas, Raj Mittra; Jonathan Bringuier; Kyungho Yoo; Joe Wiart; submitted to EuCAP’06