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Semi-infinite cable

In this case, we assume that at x=0 the voltage is clamped to The boundedness of the voltage as means that we must take


It is now clear why is called a space-constant, it determines the voltage attenuation with distance.

Alternatively, we could demand that the current at x=0 be specified as . From (14) we see that

and since we get

The input resistance is the ratio of is thus


The input conductance for the semi-infinite cable is thus

Note: It is often convenient to introduce the dimensionless space variable, and the electrotonic length of the cable, This gets rid of all the divisions by in the exponentials and hyperbolic functions.

Bard Ermentrout
Mon Jan 5 13:18:36 EST 1998