Maryam Jalalinia, Carey M' Rappaport

1
Routh Stability Test In Modeling 2D FDTD
Propagation
Maryam Jalalinia, Carey M. Rappaport mjalali_at_ece.n
eu.edu, rappapor_at_ece.neu.edu
Abstract
In this work we model wave propagation in
two-dimensional dispersive medium using FDTD
(Finite Difference Time Domain) method. For a
lossy, dispersive medium, it is significant to
model the wave velocity and attenuation over a
wide RF bandwidth with a simple and efficient
model. Using a four-zeros single-pole rational
function of the Z-transform variable (Z
exp(j?t)) to model conductivity with constant
real dielectric constant, we generate a
discretized time domain equation which matches
measured values over more than two decades of
frequency. The simulations represent good
agreement between measured and modeled
propagation constant and attenuation rate. The
modeled propagation and attenuation parameters
are compared to measured data. Stability is
studied using Routh stability test where the
stable region is the intersection of positive
first column coefficients of Routh table.
Finally, the propagation of a 2-D modulated
Gaussian wave in the medium is modeled by FDTD
formulation.
Model Accuracy
Four-Zero Conductivity Rational Function
Stability Analysis
The transformed difference equation, with average
current value between adjacent steps is
For a given time step, the stability region is
the intersection area where all first column
cofficients of Routh table for numerator of
following stability equation are positive using
bilinear transform from z-plane to s-plane
Dispersive FDTD form of Amperes law
Routh Stability Table
The model is implemented by fitting Res to
measured conductivity and Ims/?e0 to measured
real dielectric constant. An initial guess for a1
allows a simultaneous solution for other
parameters. The conductivity and dielectric
constant at three representative frequencies for
the measured data and the model are equated, and
further simple optimization is performed by trial
and error. The choice of Dt defines the model,
and determines the usable frequency range.
State of The Art

• Other dispersive media models
• Frequency domain dispersive complex dielectric
  constant with rational functions of j? Debye and
  Lorentz models 1,2.

Significance
Since the complex permittivity is
, its real part becomes
Accurate 2-D forward model that can give insight
on how to successfully design detection systems.
Stability Example
Technology Transfer
Modeling Example
Following figure shows the stable region for
breast fat tissue with respect to parameters a1
and Dz where all elements of first column of
Routh table are positive.

• This method proposes a more accurate model for
  modeling dispersive materials and can be used in
  problems involving wave propagations in
  dispersive media like
• Body tissues (breast, brain, prostate, ) for
  developing medical imaging systems
• Soil for civil engineering environmental
  problems.

Modeled electric properties and complex
wave-number of breast fat tissue vs. data are
plotted in following figures which show good
agreement.
This work was supported in part by
Gordon-CenSSIS, the Bernard M. Gordon Center for
Subsurface Sensing and Imaging Systems, under the
Engineering Research Centers Program of the
National Science Foundation (Award Number
EEC-9986821).