K7ITM wrote:
On Feb 18, 7:13 am, Cecil Moore wrote:
While working on an energy-based presentation of W7EL's
data from the following web page, I came across an
instance where my energy analysis differed from W7EL's
results under the "Food for Thought: Forward and Reverse
Power" section. Assuming Roy was correct, I attempted to
find my error and failed to do so. . .

Why did you bother with Hecht? It's simple enough to go back to the
second equation above the line you disagree with, the one at the
bottom of page 7 in that pdf, and see it does not agree there either.
It's obviously a typo and should be corrected. I'll drop Roy a line
about it, in case he doesn't see this.

Thanks very much to both Cecil, for finding the error, and Tom, for
passing it along.

Tom is correct, that the information in the table should follow directly
from the equations at the bottom of the preceding page. The table entry
was in error, but not the equations or underlying principles. For Rl = 0
+ j0 the equation at the bottom of page 7

Pa(R0) = |Ilrms|^2 * R0 = (Vrms^2 * R0) / [(R0 + Rl)^2 + Xl^2]

gives the correct result of 400 watts, not 0 as shown in the table. The
table has been corrected, and the comments following it have been
modified to reflect the corrected value. Here's the corrected table and
text:

****************************

Zl fPa rPa Pa(tx) Pa(src) Pa(R0) Pa(Rl) frac R0 frac Rl
50 + j0 100 0 100 200 100 100 0.50 0.50
100 + j0 100 11.1 88.9 133 44.4 88.9 0.33 0.67
25 + j0 100 11.1 88.9 267 178 88.9 0.67 0.33
37 +/-j28(*)100 11.4 88.6 209 120 88.6 0.58 0.42
0 +/-j50 100 100 0 200 200 0 1.00 0
0 +/-j100 100 100 0 80.0 80.0 0 1.00 0
0 + j0 100 100 0 400 400 0 1.00 0
infinite 100 100 0 0 0 0 - -

(*) For any Zl that causes exactly a 2:1 SWR, rPa will equal 11.1 and
Pa(Rl) = 88.9. The values shown for 37 +/-j28 are slightly different
because this impedance doesn’t result in quite exactly a 2:1 SWR.

For the second, third, and fourth entries, the SWR is 2:1. The forward
and reverse powers are the same for all three, and the source impedance
(50 ohms) is the same for all the above cases. So here we have three
cases where the reverse powers are the same, and the impedance match
looking back toward the source is the same (1:1), yet the dissipation in
the source resistor Pa(R0) is very different. The obvious conclusion is
that THE POWER DISSIPATED IN THE SOURCE RESISTANCE ISN’T DETERMINED
DIRECTLY BY THE SOURCE MATCH, THE SWR, OR THE REVERSE POWER. Otherwise
it would be the same in all three cases, since all these quantities are
the same for all three.

For the last four entries, the SWR is infinite, and the reverse power is
a full 100 watts. The source is perfectly matched to the line for all
table entries. Yet the source resistor dissipation varies from 0 to 400
watts depending on the load impedance – despite no difference in source
match, or forward or reverse power for the four entries.

The last two entries are particularly interesting. When the line is open
circuited at the far end (last table entry), there is no power at all
dissipated in the source resistor. So none of the reverse power is
dissipated in the source resistor. Yet when the line is short circuited
at the far end (next to last table entry), the source resistor
dissipates twice the sum of the forward and reverse powers.

From the last entry alone we can conclude that THE REVERSE POWER IS NOT
DISSIPATED IN OR ABSORBED BY THE SOURCE RESISTANCE. And the table
clearly shows that the source resistor dissipation bears no relationship
to the amount of reverse power.

**************************

The corrected essay has been uploaded to replace the previous one at
http://eznec.com/misc/Food_for_thought.pdf. Please note the uppercase
"F" -- it has to be entered exactly as shown.

Again, thanks very much for the corrections. It's my sincere intention
to present material that's accurate, and I appreciate the help in
finding and correcting errors I've made.

Roy Lewallen, W7EL