The impedance presented by a length of transmission line is a constant resistance value "R" when the characteristic impedance value of that transmission line is "R" and a load resistance "R" is connected, but if the load value is not matched in that way, then the presented impedance varies as a function of the cable's length.

Let's call the transmission line a cable and let's assume a load "R" that does NOT equal the cable's characteristic impedance.

If we choose a particular frequency, look at the cable as a cascade of inductance and capacitance values per differential bits of unit length and do a repeated calculation, over and over and over again, of the presented impedance, we can plot that impedance graphically and find out at what cable length the presented impedance again equals the load resistance. That cable length is one-half wavelength at the frequency of the observation.

Double that length to find the full-wave length and then compare that to the free-space wavelength at the chosen frequency. The ratio of the two is the velocity factor of the cable.

In the example below, R = 30 for a 50 ohm characteristic impedance. (You can make the load an R + jX too. Then the trace is still a circle, but it starts and ends at that R + jX point.)

If anybody would like to play with this, please e-mail me at ambertec@ieee.org and I will send you some code in GWBASIC that does all this.

It's fun!

Heard anything about the Smith Chart ?

This was already figured out on the '30s.

Speaking of the velocity factor, prove how it is related to the line's dielectric relative permittivity.

Posted by: Jacques Fortin | November 14, 2010 at 10:27 PM

Of course! This is obviously the basis of the Smith Chart. This calculation is just a way of looking at the underlying fundamentals.

The key to this is that a transmission line has some specified characterisitc impedance and some particular capacitance per unit length from which the inductance per unit length is found.

Then taking unit lengths in very, very small bits yields a recursive calculation of presented impedance. If you keep track of the length itself, when the presented impedance again equals the load impedance, that length yields the velocity factor.

Posted by: John D. | November 15, 2010 at 05:58 AM

Electrical Transmission Lines are designed to hold up in adverse weather conditions, the transmission lines primarily use ACSR conductors.

Posted by: electrical transmission lines | December 19, 2012 at 05:57 AM