Consider a voltage wave travelling from the supply source end towards the far end, and the progressive charging of the line capacitances will account for the associated current wave. Assume that in a very small time δt the conditions of a current I and a voltage E are established along a length δx of the line (fig 2). The emf E is balanced by the back emf generated by the magnetic flux which is produced by the current in this length of the line. The inductance of the length δx is L δx, (L is inductance of line per unit length) so that the flux built up is I Lδx and the back emf is the rate of build up viz. I L (δx/δt)

So we have E = I L (δx/δt) = I L v (1)

where v is the velocity of propagation of wave.

The current I carries a charge I δt in the time δt, and this charge remains on the line to charge it upto the potential of E. Since the capacitance of length δx of the line is C δx (C is the capacitance of the line per unit length), its charge is E C δx, so we have

I δt = E C δx

I = E C (δx/δt) = E C v (2)

The switching of an emf E on to the line results therefore in a wave of current I and velocity v where E and I are given by equations (1) and (2). Dividing equation (1) by equation (2), we have

The expression is a ratio of voltage and current which has the dimensions of impedance and is therefore here designated as surge impedance of the line. It is also called the natural impedance because this impedance has nothing to do with the load impedance, but depends only on the line constants. The value of this impedance is 400 Ω to 600 Ω for an overhead line and 40 Ω to 60 Ω for a cable.

Multiplying equations (1) and (2), we have

Substituting the values of L and C for overhead lines in above expression, we have

Since the product of L and C is the same for all overhead lines, it follows that the velocity of propagation is also the same. This velocity is the same as the velocity of light, but as we have assumed a resistance less line in the above analysis, the velocity in practice will be from 5 to 10 percent less than this. Normally a velocity of approximately 285 m/μs is assumed. The velocity of propagation over the cables will be smaller than that over the overhead lines because in case of overhead lines ε _r = 1 while for cables ε _r > 1, being the dielectric constant. The velocity of wave propagation in case of cables can be given as

where ε _r varies from 2.5 to 4 in case of cables

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