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\title{PAN Electric Field Modelling\\
Project for {\em Nature of Mathematical Modelling}}
\author{Matthew Gray}
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The goal of this semester project will be to create a three
dimensional electric field model of a PAN system.  This model will
essentially be solved by the solution of LaPlace's equations using
successive over-relaxation (SOR).  There are some differences,
described here.

\par

In this version of the model, the human body will be modeled as an
arbitrarily shaped equipotential containing net zero charge.  This
second condition produces a different sort of boundary condition which
will be described below.  The PAN device will be modeled as two plates
of charge, with values to give the desired potential difference across
the electrodes.  The room will be modeled as flat walls a large
distance (compared to the scale of the body) away.  The walls will
have a fixed potential of zero.

\par

In order to satisfy all of these conditions, an arbitrary potential
for the human body will be selected.  Given, this new boundary
condition, the field will be solved for all space by SOR.  The surface
integral of the field over the body will then be calculated and
compared to zero.  In all likelyhood, after the initial guess for the
potential for the body, this value will not be zero.  Based on the
calculated charge, the guess for the potential for the body will be
adjusted and SOR will be run again.  This will be repeated until the
charge contained in the body is considered to be sufficiently close to
zero.

\par

Once this is solved for the specific transmitter geometry, an
arbitrary receiver geometry can be selected and field strengths, and
hence receiver voltages can be calculated.

\par
{\bf Issues:} I'm not 100\% sure if I'm likely to run into problems
switching back and forth between using field strengths (needed to
calculate the integral over the body), potentials (for ground and the
body), and charges (for the transmitter plates). 


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