General solution of transmission-line equations

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Equations of a transmission line:

Initial conditions:

General solutions in the Laplace transform 's' domain:

General solutions in the time domain:

where u is the propagation speed and Z0 is the characteristic impedance:

We will derive the general solution of transmission line equations.

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Equations of a transmission line are:

The Laplace transform of the voltage v(x,t) and current i(x,t) are:

and the Laplace transform of its time derivative is:

Therefore, the Laplace transform of the equations are:

Here we assume the initial conditions at t=0 as:

, then we obtain the wave equation.:

Its general solutions are:

Using the propagation speed u and the characteristic impedance Z0 instead of L and C:

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Multiplying exp(-ƒŃs) in the 's' domain is equivalent to shifting in the time domain by ƒŃ.

Therefore the general solutions in the time domain are

where the va/ vb are a forward/ backward moving waves at the propagation speed u.


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