Finding the Velocity Factor of Coaxial Cable
The importance of knowing velocity factor and how to calculate it lies in the fact that radio frequency energy (RF) travels slower in wire/cable than it does in free space (air).
On occasion we find a length or roll of coax that’s in fairly good condition but with little to no information about its characteristics. Such as, is it 50 W, is it 75 W, what’s the velocity factor (speed of propagation). Why would we want to know this anyway especially the speed of propagation. Well, the purest wants to feed his/her antennas with an even number of half wavelengths so that the instantaneous terminal impedance of the antenna is presented at the other end of the transmission line. The antenna designer want to be able to shift the phase by exactly 180 electrical degrees when using a “T” match feed system. The building of a coaxial collinear antenna requires this knowledge. Another instance is when a quarter wavelength of line is used to phase the dipoles by 90° of a Turnstile antenna. A half wavelength of coaxial cable is determined by;
Wavelength (l) in coax / 2 (feet) = (492 X Vp) / F(MHz)
Generally we find the characteristics of coaxial line on the cable reel on which it is rolled or we can look it up on the Internet or from tables in our reference library. Now and then some strange cable appears at hamfests or surplus emporiums that is the remnant from some military or government project and the characteristics are relatively unknown. Impedance is somewhat easily discernable as the ratio of center conductor diameter to outer conductor inter-diameter is rather large. In air a 75 W line would have a ratio of ~3.5:1 and 50 W line has a ratio of 2.3:1. Remember, these ratios are when the dielectric is air. These ratios are calculated as follows;
Zo = 138 Log D/d
D = the inner diameter of the outer conductor,
d = the outer diameter of the inner conductor.
Coaxial cable will have dielectrics ranging from Solid Polyethylene to Teflon foam. The range of the constants of the dielectric material will be as low as “1.0” of dry air to “8” of Polyvinylchloride (PVC). The dielectric constants of these materials will modify the above equation in that the constant “138” will be divided by the square root of the dielectric constant which is designated by using the Greek letter Epsilon ( e ). Therefore other than an air dielectric the constant (138) in the above equation is modified thusly; 138/Öe.
Now, on to the procedure of finding the velocity factor of an unknown coaxial cable. I’ll use a roll of stuff I have hanging here in the ‘lab’. On the outer covering it is printed “Consolidated RG-213/U”. I could probably look this up on the Internet and find that this is 50 W cable but, I may or may not find the Vp listed. Also, in practice, I have found with one brand of cable the manufacture was in error. With my sample cable I measure the diameter of the center conductor with a caliper or micrometer and find it to be 0.090”. To find the inner diameter of the outer conductor I simply measure the diameter of the dielectric, which I find to be 0.285. To find the ratio I divide the diameter of the dielectric by the center conductor diameter and find the ratio of the two to be 3.1667:1, not quite the air dielectric ratio of 75 W cable but RG-213 is a common 50 W cable so I basically know the characteristic impedance of this cable to be ~50 +/-5 W. With these ratios plugged into my derivated impedance equation, I wish to find the Epsilon (e);
e = (138/50 Log 3.1667)^2
e = 2.3669
By inspection and a little experience, having calculated the Epsilon to be ~2.4, and an educated guess would be that the dielectric is Solid Polyethylene.
The equation for dielectric constant (Vp) once the Epsilon has been calculated/determined is;
Vp = 1/Öe
Vp = 0.65, or 65%
of free space propagation
Armed with this information, let’s say I need a half wavelength balun for a 2 meter antenna using the subject coax.
L (ft) = (492 X 0.65) / 144.2
L = 2.2178
If you prefer the length in inches;
L (in) = (5616 X 0.65 / 144.2
L = 25.3148
(note) Actually the constant 5600
would be close enough for
anything we do in ham radio.
Good luck, I hope this information will be of use to the group in both understanding and construction.
Dave W6OAL –
Olde Antenna Laboratory 41541 Dublin Drive Parker, CO 80138
updated 8/11/2011 w6oal/n0poh