coupling

1 Dimensional Coupling and Propagation

From Ohm's law

$\displaystyle V_i(x+dx) - V_i(x)$	$\textstyle =$	$\displaystyle -I_i(x) r_idx,$	(1)
$\displaystyle V_e(x +dx) - V_e(x)$	$\textstyle =$	$\displaystyle -I_e(x) r_e dx,$	(2)

In the limit as $dx \rightarrow 0$ ,

$\displaystyle I_i$	$\textstyle =$	$\displaystyle -{1 \over r_i} {dV_i \over dx},$	(3)
$\displaystyle I_e$	$\textstyle =$	$\displaystyle -{1 \over r_e} {dV_e \over dx}.$	(4)

Next, from Kirchhoff's laws

$\begin{displaymath} I_i(x) - I_i(x+dx) = I_t dx = I_e(x+dx) - I_e(x) \end{displaymath}$

(5)

In the limit as $dx \rightarrow 0$ , this becomes

$\begin{displaymath} I_t = - {\partial I_i \over \partial x} = {\partial I_e \over \partial x}. \end{displaymath}$

(6)

Using that

, we find that

$\begin{displaymath} I_t = {\partial \over \partial x} \left({1 \over r_i + r_e} {\partial V \over \partial x}\right), \end{displaymath}$

(7)

and, thus,

$\begin{displaymath} I_t = p( C_m {\partial V \over \partial t} + I_{\rm ion})=... ...eft({1 \over r_i + r_e} {\partial V \over \partial x}\right). \end{displaymath}$

(8)

This equation is usually referred to as the cable equation.