Calculating velocity coefficient of arbitrary transmission lines?

[kastein]

note: I really hope I'm not rehashing a thread that already happened or posting this in the wrong forum... I did some searching, and couldn't really find anything related, but if this is in the wrong place or old news, feel free to yell at me.

semi-irrelevant background:
I'm currently attempting to build what is known as a "j-pole" antenna for two-meter ham radio usage. The j-pole, for those not familiar with it, is a half-wave dipole antenna end-fed by a quarter-wave twin-wire matching stub with the same characteristic impedance, with a small inductive tuning stub attached to the far side of the quarter-wave matching stub to bring the impedance mismatch between the two down. It is fed by either a coaxial cable with a small coax choke (coil of coax) in it just before the feed point to limit RF currents in the shield, or by a balanced-unbalanced ("balun") transformer properly designed for the coax cable and transmission line in use.

background links:
http://snow.prohosting.com/%7Ew0rcy/Jpole/jpole.html This does a really good job of explaining how and why the design works.
http://www.hamtechnet.com/jpole/ve3xkv/ This page also does a good job, but is not quite as heavy on the math and explanations, and details why the balun transformer is needed.
http://en.wikipedia.org/wiki/Ladder_line Has two equations for finding the characteristic impedance of the twin-wire line to be used

My problem - I need to calculate the velocity coefficient of the transmission line used for the quarter wave stub, the inductive tuning stub, and the half-wave dipole. I've found equations I can use to calculate the characteristic impedance of the line, though they do not give the same results for the same spacing and diameter conductors (wikipedia link). The velocity coefficient I need so that I can start with a good guess at what the quarter-wave and half-wave electrical lengths will be, and thus greatly reduce tweaking/tuning time. I know that:
v = c/n where v = EM propagation velocity in medium, c = EM propagation velocity in free space, n = refractive index
and also that:
v = 1/sqrt(mu*epsilon) where v = EM propagation velocity in medium or transmission line and mu and epsilon are the permittivity and permeability of the transmission line or medium.

How do I get mu and epsilon for a given transmission line though? I know the conductivity of the metal used (copper), the conductor diameter, the conductor spacing, and the dielectric constant of the material between the two conductors (air, 1.00054).

Also, how were the two equations for characteristic impedance on that wikipedia page derived, and why do they give different but similar answers?

(this isn't a homework or test problem, I would like to understand why the magical formulas I've found are true, and any hints, pointers, or answers are more than welcome)

 

[f95toli]

There is no simple, generic way of calculating properties of transmission lines. You can find approximate formulas for most of the popular types (coaxial, coplanar, microstrip etc)but in general you need to use numercial methods.
Hence, the way to do this is to first figure out what type of transmission lines you have and then look up the relevant formulas. Note that many of the formulas you can find in the litterature is little more than interpolation polynoms for numerical solutions (and in some cases experimental data) meaning they were not actually derived using EM theory.

 

[Danger]

Welcome to PF, Kastein.
I know absolutely nothing about this subject. The only reason for this post is to let you know that the Mentors will move this to a different sub-forum if appropriate. My thought is that it might end up in Electrical Engineering. If you look for it where you posted it and see an arrow with 'moved' beside it, just click the link and you'll be in the new spot. Science is a very complicated field of endeavour, and it can be difficult to know what category something belongs in. The Mentors are used to that, and do a very fine job of redirecting things. You could have posted this in Biology and it would have been moved to where it belongs. A good approach is to scan through the site to see what the various sub-forums are and choose whatever you think is most appropriate. There's a lot of overlap.