The solution of voltage along an RC transmission line as a function of distance and time
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Consider the case of an infinite RC delay line:

If we isolate just a small part of the line, we can define currents and voltages:

So,

If we take each element to represent a length, x, and rewrite voltages as a function of x, and rewrite each R and C as a capacitance per unit length, times a the length  x, the equations can be rewritten as a partial differential equation

In the last line  I wrote for simplicity (later).

Let us assume (without proof) that we can write v(x,t) as a product of two functions, one an equation of position alone (X(x)), and the other an equation of time alone (T(t)), so v(x,t)=X(x)T(t).  This is a standard technique that works for a wide variety of partial differential equations.

Now on the left hand side we have something that is a function of position alone, and the right hand side is a function of time alone.  For these equations to hold for any value of x and/or t, obviously they must both be constant.  So we can choose an arbitrary constant, chosen to be -u2 for simplicity (later).

,  or

You have seen these equations before and they have solutions of the form

Continue on to the response of the line to a step input.

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