Topband: Re: Coax capacitor losses

[Peter Wardenier]

Summary: 
- coax stubs make quite acceptable capacitors @ 1.8 MHz 
- the remaining -call it theoretical- discussion: are the added  losses just 'low',  
  (see my previous post), or are they completely negligible (as predicted by a 
  popular transmission line simulator program)? 
 
As a follow-up to my first post on this subject (on 22 December 2004), and
as a response to some questions, let's again look at the formula for the input 
impedance Zi following from the basic transmission line equations:
Zi = Zo * (Zl + Zo * tanh(gamma*L)) / (Zo + Zl * tanh(gamma*L))
 
Note: the function tanh(z) is given by tanh(z) = (exp(z)-exp(-z)) / (exp(z)+exp(-z)),
where z may be a complex number.
The formula for Zi is found in almost any textbook on Electromagnetics, and for
instance also in the ARRL handbook (chapter 19 on the 1999 V3.0 CD).
For the math involved see http://www.sm.luth.se/~urban/master/Theory/4.html,
leading to equation (82).
 
Luckily, the equation for Zi can be greatly simplified for an open stub (Zl=infinite)
to: Zi = Zo / tanh (gamma*L)      - try it for yourself in MathCad, or even in Excel-
(Only!) for short, not too lossy cable stubs (gamma*L<<1) we have tanh(z)~z
 
and with Zi = Ri + jXi we find:
Ri ~ (Zo/L) * alpha /(alpha**2+beta**2), and Xi ~ (-Zo/L) * beta /(alpha**2+beta**2) 
and  so: Q = |Xi|/Ri ~ beta/alpha
 
This leaves the -maybe surprising- conclusion that Q is independent of the coax 
stub's length! Once again: this applies to short stubs only; up to some 10m @
1.8MHz is considered 'short' in this respect. 
At a given frequency (i.e. given beta), the Q is only directly influenced by the coax 
cable attenuation (alpha).
 
However, for one reason or another, some transmission line simulators erroneously 
seem to show an ever increasing Q (up to infinitity...) for the stub length approaching zero! 
This unrealistic outcome is present in the handy online version (applet) by W9CF, 
which is based on TL.EXE, please refer to http://fermi.la.asu.edu/w9cf/tran/
 
Concluding: some care is needed using transmission line simulators for this application 
(Zl=infinite, L~0), for it may lie outside the permissible scope of the simulator.
Validity of simulations can easily be checked by taking a quick look at the original 
transmission line equations.
 
73, Peter PA3AUC
 


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