Hi Martin,
> I didn't want to crack open my E&M text, so I will accept that you've done
> the math.
Actually, I did the math with intention of proving that the formula is wrong :-)
The formula represents just a limit of accumulated loss
caused by an infinite number of two-way reflections.
Its derivation assumes only that the line is linear,
uniform and "isotropic". Therefore, it is very general.
> On the qualitative/intuitive/hand-waving level,
> it's hard to see where the power is dissipated
> when you're at a voltage maximum & current minimum.
If we consider the case of a constant voltage applied to a short line having
Zo = 50 + j * 0 Ohms and terminated with loads of 0.5 Ohms and 5000 Ohms,
the amount of power dissipated in the line will be, of course, much smaller
with 5000 Ohms load.
But the power delivered to the 5000 Ohms load will also be much smaller,
causing the loss expressed as a percentage to stay the same.
Nevertheless, it is possible to get a smaller loss with 5000 Ohms load
than with 0.5 Ohms load when used with a line having SQRT (L' / C') = 50 Ohms,
along with R' large enough to cause Zo to differ significantly from 50 Ohms,
in spite of L' / C' ratio. But then the actual SWR is different,
and the formula is correct again. This is illustrated with the following
examples.
* * * * *
Let's consider a transmission line with following parameters,
roughly corresponding to RG213 on 14 MHz, assuming that
all the loss comes from the conductor:
R' = 0.295 Ohms/m
L' = 250 nH/m
G' = 0 S
C' = 100 pF/m
We calculate the characteristic impedance
and attenuation constant at 14 MHz:
Zo = 50.001112 - j * 0.3353547 Ohms
a = 0.029499336 Np/m = 0.0256228 dB/m
Let's consider 10 m of the line terminated in
resistive loads of 0.5 Ohms and 5000 Ohms.
With 0.5 Ohms load SWR is 100.0067, giving total loss of 5.9702 dB.
With 5000 Ohms load SWR is 99.99775, giving total loss of 5.9699 dB.
Both SWR and loss are almost the same.
* * * * *
Let's now increase R' 100 times, leaving other line parameters unchanged:
R' = 29.5 Ohms/m
L' = 250 nH/m
G' = 0 S
C' = 100 pF/m
Characteristic impedance and attenuation constant at 14 MHz:
Zo = 57.80534 - j * 29.00789 Ohms
a = 0.2551667 Np/m = 2.21635 dB/m
Zo is quite different from 50 Ohms, although the L/C ratio is unchanged.
With 0.5 Ohms load SWR is 144.7259, giving total loss of 37.81 dB.
With 5000 Ohms load SWR is 86.50011, giving total loss of 35.61 dB.
Both SWR and loss are different, which is to be expected, given that
Zo differs significantly from 50 + j * 0 Ohms.
73,
Sinisa YT1NT, VE3EA
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