General solution of transmission-line equations
Equations of a transmission line:
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.
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:
Multiplying exp(-Ńs) in the 's' domain is equivalent to shifting in the time domain by Ń.
Therefore the general solutions in the time domain are
where the va/ vb are a forward/ backward moving waves at the propagation speed u.