There are a variety of possible end conditions we can apply to the
cable. Among them are the (a) * sealed end* where no current can
pass and so , (b) * short circuit* or * open end* where
the voltage is clamped to 0, (c) * leaky ends* which is a mixture
of the two, some current escapes but not an infinite amount. Let's
revert to the dimensionless equations and
Assume that the voltage at **X=0** is . Then the general solution
to the steady-state equation is:

where is an arbitrary constant. This general solution is
equivalent to asserting that the boundary condition at **X=0** is
and that at **X=L**

The free parameter, is the ratio of the input conductance for the cable, to that of the semi-infinite cable, That is,

For example, if we want the
sealed end condition at **X=L** we take so that

If we want the open end conditions, we take so that

If we choose then

which is precisely the solution to the semi-infinite cable. In figure 3 we plot the steady state voltages for a variety of different cables and at different electronic lengths. These could be solved analytically, but the plots were in fact generated by using XPPAUT.

**Figure 3:** Steady state voltages for a variety of electrotonic lengths
and for different end conditions

Mon Jan 5 13:18:36 EST 1998