Hi Jeff,
OK. Here is the procedure to find the additional line loss
caused by operating a transmission line into a load which does
not perfectly match the line impedance. I have spread it out
within some explanatory text. So it appears much longer and more
complicated than it really is. Nothing more than a scientific
calculator is required.
First, determine what the loss would be if the load _was_
properly matched to the line. If you have the ability and
instrumentation, measure the matched loss. Or look up the
published value for the cable and scale it to the length of line
you plan to use. This value will be the matched loss (LOSS) in
dB. Convert this value into nepers (a) as follows:
a = (LOSS/-8.6859) units are nepers
Next, determine the level of mismatch (SWR at load end of cable).
Convert this value into "reflection coefficient" (p) as follows:
p = ((SWR-1)/(SWR+1)) dimensionless quantity
Next, calculate the total loss in the transmission line (Lt) as
follows:
Lt = 10*log[ ((1-p^2)*e^(-2a)) / (1-(p^2*e^(-4a))) ]
where e = base of natural logarithms, units for Lt are dB
Finally, subtract the matched loss from the total loss to get the
additional cable loss (d) due to operating the line in a
mismatched condition.
d = Lt - LOSS units are dB
Remember that losses expressed as power ratios in dB are negative
numbers. You can use the following two examples to verify that
you have done the procedure correctly.
1. 200 ft of RG213, 8:1 SWR, 20 MHz
This set of parameters should produce the following results.
Matched loss (LOSS) = 2.0 dB
(from Fig 23 on page 16-14 of '91 ARRL handbook)
Total Loss (Lt) = -4.837 dB
Additional loss due to mismatch (d) = -2.837 dB
2. 200 ft. of 450 ohm open wire, 20:1 SWR, 20 MHz
This set of parameters should produce the following results.
Matched loss (LOSS) = 0.04 dB
(Extrapolated from Fig 23 on page 16-14 of '91 ARRL
handbook. Probably a bit on the high side.)
Total Loss (Lt) = -0.737 dB
Additional loss due to mismatch (d) = -0.657 dB
3. 200 ft of RG213, 3:1 SWR, 20 MHz
This set of parameters should produce the following results.
Matched loss (LOSS) = 2.0 dB
(from Fig 23 on page 16-14 of '91 ARRL handbook)
Total Loss (Lt) = -2.794 dB
Additional loss due to mismatch (d) = -0.794 dB
Now you can calculate your antenna system losses to however many
decimal places you wish!
73, Eric N7CL
>From: "Jeffrey R Davis" <jrdavis@crnotes.collins.rockwell.com>
To: <towertalk@contesting.com>
>Date: Fri, 19 Mar 1999 08:38:42 -0600
>
>Even if no one else is interested in the equations, I would be!
>
>Thanks,
>
>Jeff N0DY
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