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