\chapter{Circuit Model}

Once a set of capacitance values for the particular geometry are
calculated, a circuit model is utilized to calculate the signal
strengths at the receiver.

\section{Circuit model}

\begin{figure}
\begin{center}
\mbox{\epsfysize=2in \epsfbox{circuit.eps}}
\end{center}
\caption{Circuit model of PAN}
\label{pan:circuit}
\end{figure}


This fully interconnected network of nodes (as shown in Figure
\ref{pan:circuit}) is solved using node equations.
At each node (except the transmitter pair) the total current must be
zero.  This yields the system of equations in Equation
\ref{eqn:matrix}.  All 6 node equations are unnecassary, so one is
arbitrarily dropped (and hence it becomes the reference potential).

\par

In this case we are ignoring any resistive or inductive components of
the network, but these can be reasonably assumed to not substantially
impact the overall node-to-node impedences.

\section{Calculation of Signal Strength}

To solve this network, the following system of equations are solved
simultaneously.  $Z_{ab}$ is the impedence between the pair of nodes $a$ and
$b$.

\begin{equation}
\alpha_i = \sum_{y=0,y \ne i}^{6} -\frac{1}{Z_{iy}}\\
\end{equation}

\begin{equation}\label{eqn:matrix}
\left(
\begin{array}{cccccc}
\alpha_1& \frac{1}{Z_{12}}& \frac{1}{Z_{13}}& \frac{1}{Z_{14}}& \frac{1}{Z_{15}}\\
\frac{1}{Z_{12}}& \alpha_2& \frac{1}{Z_{23}}& \frac{1}{Z_{24}}& \frac{1}{Z_{25}}\\
\frac{1}{Z_{13}}& \frac{1}{Z_{23}}& \alpha_3& \frac{1}{Z_{34}}& \frac{1}{Z_{35}}\\
\frac{1}{Z_{14}}& \frac{1}{Z_{24}}& \frac{1}{Z_{34}}& \alpha_4& \frac{1}{Z_{45}}\\
\frac{1}{Z_{15}}& \frac{1}{Z_{25}}& \frac{1}{Z_{35}}& \frac{1}{Z_{45}}& \alpha_5\\
\end{array}
\right)
\left(
\begin{array}{c}
\phi_1\\
\phi_2\\
\phi_3\\
\phi_4\\
\phi_5\\
\end{array}
\right)
=
\left(
\begin{array}{c}
0\\
0\\
-I\\
+I\\
0\\
\end{array}
\right)
\end{equation}

Here, $I$ is an arbitrary transmitter current.  Node 6 is set
arbitrarily to 0, and the other nodes will be relative to it.

\par

The signal strength will be proportional to the square of the voltage
difference between the receiving electrodes.  The actual value
calculated will be arbitrary, as the currents in the trasmitter were
chosen arbitrarily in Equation \ref{eqn:matrix}.  Values for the
voltage difference at the transmitter will have also been calculated,
and due to the system being a linear network, the output voltage for
any arbitrary input voltage can be calculated.
