Hey guys, I have a telecomms exam next and I am bit confused on one of the formulae we have. The formula is: C = Co / √εr which is used to calculate the speed of the wave in a coaxial cable for example. Where C is the speed in the desired medium, Co is the speed of light in a vacuum and εr is the relative permittivity in a medium

Whats bothering me is shouldnt there be a relative permeability term in there as well? Like we have been shown in another subject that the speed of light in a vacuum is equal to 1/ √(εoεrμoμr) [i think]

 

Hey guys, I have a telecomms exam next and I am bit confused on one of the formulae we have. The formula is: C = Co / √εr which is used to calculate the speed of the wave in a coaxial cable for example. Where C is the speed in the desired medium, Co is the speed of light in a vacuum and εr is the relative permittivity in a medium

Whats bothering me is shouldnt there be a relative permeability term in there as well? Like we have been shown in another subject that the speed of light in a vacuum is equal to 1/ √(εoεrμoμr) [i think]

Yes you are correct, strictly there should be a relative permeability term in there as well.

However the thinking runs like this to get to your first equation:

As an electromagnetic wave moves from free space into a medium it is attenuated by that medium. This is in addition to moving more slowly.

We acount for this by attributing the attenuation to the relative permittivity, which now becomes a complex quantity.

εr = ε\(_{1}\) - jε\(_{2}\)

thus √(εoεrμoμr) = √(εo(ε\(_{1}\) - jε\(_{2}\))μoμr)

= (n-j\(\eta\))√(εoμoμr)

where n = √ε\(_{1}\)

and \(\eta\) = ε\(_{2}\)/2√ε\(_{1}\)

Now here is the bit they probably haven't told you

Assume ε\(_{1}\) >> ε\(_{2}\)

and that μr = 1

The electromagentic wave equation is

E = E\(_{0}\)exp{jω(t-z/v)} exp(-ω\(\eta\)z/c)

this reduces to

v = c/√εr as required for the large real term.

 

Whats bothering me is shouldnt there be a relative permeability term in there as well? Like we have been shown in another subject that the speed of light in a vacuum is equal to 1/ √(εoεrμoμr) [i think]

You and studiot are correct. In simpler terms we can say that waves usually do not propagate well in magnetic materials, so as a practical matter, we just forget about the relative permeability which is very close to 1 in most cases of interest to engineers.

 

Ah yes well the absolute main thing was if i was going going to use the corect formula in the exam. Obviously its important for me as an engineer but we are starting fields and waves next year which i am guessing will be dealt with in detail then. (the entire class in my course last year failed this module... shudder shudder shudder)

Thanks very much for that though and you are quite right we have only been mildly introduced to this material so I had no way of knowing where that came from .


Thanks!

 

The luck of the green in the exam!

 

Thanks I am going to need it!

 

We might add that in PRACTICAL transmission lines, such as dielectric fillled coax, the velocity factor can be reduced to as little as .66! Air filled coax, however (HELIAX and such) have velocity factors approaching 1. For all practical purposes, air is about the same permeability as a vacuum.

Eric