Note on an Generalized Solution of the Three-Conductor Transmission Line Equations

Here we present an abridged investigation on the electromagnetic compatibility aspects of three lossless transmission lines terminated by linear loads introduced by C. Paul. Taking into account the mutual interaction between two lines we do not neglect the influence of the receptor line i.e. we do not apply the weak-coupling approximation. This leads to a more general mathematical model than the model of C. Paul. We formulate a mixed problem for the hyperbolic system describing the three-conductor transmission line. It is proved that the mixed problem is equivalent to an initial value problem for a functional system on the boundary of hyperbolic system's domain. The unknown functions in this system are the lines' voltages and currents. The obtained system of functional equations can be solved by a fixed-point method that enables us to find an approximated but explicit solution. Usually such problems are treated by numerical methods or Laplace transformation method, which are applicable to linear problems only.