General solution of coupled transmissionline equations
Equations of coupled transmission lines:
The voltage and current in even and odd mode:
Initial conditions:
General solutions in Laplace transform 's' domain:
General solutions in the time domain:
where u_{±} are the propagation speed and Z_{±} are the characteristic impedance:

We will derive the general solution of the coupled transmissionline equations.
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Equations of coupled transmission lines are:
Using the even and odd mode instead of each line variables
, the equations will be diagonalized:
Eliminating the i_{±}, we obtain the wave equation of v_{±}:
As in the case of a single transmission line (→General solution of transmissionline equations), assuming the initial condition
, the general solutions in Laplace's transform 's' domain are:
and the solutions in the time domain are:
where u_{±} are the propagation speeds and Z_{±} are the characteristic impedances in even and odd mode:
The expressions of u_{+} and u_{−} look different, but they must be equal in the case of TEM wave. We will confirm it later by an example.(→Differential impedance of two wires over a ground plane)