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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 dV/dx=0, (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 V_{0}. Then the general solution
to the steadystate equation is:
where B_{L} is an arbitrary constant. This general solution is
equivalent to asserting that the boundary condition at X=0 is V_{0}
and that at X=L
The free parameter, B_{L} is the ratio of the input conductance for
the cable, G_{L} to that of the semiinfinite cable, That
is,
For example, if we want the
sealed end condition at X=L we take B_{L}=0 so that
If we want the open end conditions, we take so that
If we choose B_{L}=1 then
V(X) = V_{0} e^{X}
which is precisely the solution to the semiinfinite 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

Next: Solving boundary value problems
Up: Steady state and boundary
Previous: Semiinfinite cable
G. Bard Ermentrout
1/10/1998