Consideration of Modal Propagation Velocities

In a real transmission line, the line impedance becomes frequency dependent due to the skin effects of the conductor and the earth as explained in Section 1.5. The propagation velocity is also frequency dependent, and furthermore the modal velocity differs from each other. T e velocity difference causes a very significant effect on the voltage and current waveshapes along the line. The effect is included analytically in a calculation of the voltage and the current in the following manner  

Transforming the earlier traveling waves in the phase domain into a modal domain. 

The earlier modal traveling waves e-io and propagate to the receiving end by the propagation velocities Cq and Cj, respectively. Each traveling time is given by 

Transforming the earlier traveling wave back to the phase domain. 

The receiving-end voltage (Vj) is obtained using the aforementioned traveling waves in the same manner as that in Section 1.6.4.2  

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