Summary:
- coax stubs make acceptable capacitors @ 1.8 MHz
- example 10m RG58A Ri=0.69 ohm, Q~110; loss<0.1 dB when used as series
cap for a slightly long quarter-wave vertical)
- Q scales with (~1/stub length)
- theory and experiments are in agreement(again)
Thanks to Kevin, W9CF the 'mystery' has been fully solved
(refer to my 30 Dec 2004 posting).
Here's Kevin's full answer that I got by e-mail:
> The transmission line equations are fine. The problem with Peter's
> analysis is the incorrect assumption that the characteristic impedance
> is purely resistive. Making this approximation assumes that the loss
> is equally in series resistance of the conductors and in the parallel
> resistance of the dielectric. This approximation has little effect
> if the line length is a multiple of a quarter wave length (or if it
> is so long that it is many wavelengths) or nearly matched. However,
> this resistive Z0 approximation is completely inaccurate for short
> runs of coax at high SWR, which is exactly the case when using an open
> circuited line as a capacitor. Both my applet and TLA use a complex
> Z0 and therefore correctly give the increase of Q when the line gets
> short. This is because, as Tom mentioned, there is less affect from
the
> series resistance for short lengths, and the series resistance of the
> conductors is the dominant loss mechanism.
The effect has recently been illustrated on this forum by the example
given by David, WX7G.
Finally, some noteworthy remarks made by various commentators:
- resistive losses in coaxial cables dominate; dielectric loss can
safely
be ignored in the calculations (esp. at these low frequencies)
- Zo is not your 50 ohms, but may turn into something complex (as stated
above
by Kevin); examples: RG58 @ 1.8 MHz Zo=50-j1.7 ohm, RG213 Zo=50-j0.79
ohm
ref. http://fermi.la.asu.edu/w9cf/tran/
- ignoring the additional imaginary component in Zo leads to
overestimating the
value of Ri for short stubs
- for the mathematically inclined reader: if g=0 (no dielectric losses),
Zo = sqrt[(r+jwl)/jwc], or after some manipulation:
Zo = Ro(1-j*alpha/beta) with
- alpha [Neper/m] = attenuation per unit length [dB/m] / 8.686
- beta [radians/m] = 2*pi / (free space wavelength [m] * coax
velocity factor)
- Ro [ohm] = sqt(l/c)
73, Peter PA3AUC
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