On Feb 18, 7:58 pm, Cecil Moore wrote:
Roy Lewallen wrote:
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.

Such is the nature of *total destructive interference* as
described by Hecht in "Optics". All of the reflected energy
is redistributed back toward the load.

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.

Such is the nature of *total constructive interference* as
described by Hecht in "Optics". All of the reflected energy
plus some more supplied by the source is dissipated in the
source resistor.

From the last entry alone we can conclude that THE REVERSE POWER IS NOT
DISSIPATED IN OR ABSORBED BY THE SOURCE RESISTANCE.

Of course not from only the last entry when total destructive
interference is occurring. 100% of the reflected energy is
redistributed back toward the load.

OTOH, when total constructive interference is occurring, not
only is 100% of the reflected energy dissipated in the source
resistor but the source has to supply twice as much energy as
the forward power plus the reflected power combined.

Perhaps the following energy analysis will shed some light on
the misconceptions. "Shedding some light" seems appropriate
since these concepts are from the field of optical physics.

This posting will provide an energy analysis approach to the
same previous W7EL data specifically avoiding any reference
to voltage and current.

The example that Roy provided in "Food for Thought: Forward
and Reflected Power" is:

Rs
+----/\/\/-----+----------------------+
| 50 ohm |
| |
Vs 1/2 wavelength ZLoad
141.4v 50 ohm line |
| |
+--------------+----------------------+

http://eznec.com/misc/Food_for_thought.pdf

We will create a new chart, step by step, that doesn't use
voltages or currents. Note that the first two columns are
copied from W7EL's chart. The Gamma reflection coefficient
is calculated at the load and |Rho|^2 is the power reflection
coefficient. The reflected power is the forward power multiplied
by |Rho|^2. 'GA' is the reflection coefficient Gamma Angle.

Zl fPa Rho GA,deg |Rho|^2 rPa
1. 50 + j0 100 0.0 0 0.0 0
2. 100 + j0 100 0.3333 0 0.1111 11.1
3. 25 + j0 100 0.3333 180 0.1111 11.1
4. 37 +/-j28 100 0.3378 97.1 0.1141 11.4
5. 0 +/-j50 100 1.0 -90 1.0 100
6. 0 +/-j100 100 1.0 -53.2 1.0 100
7. 0 + j0 100 1.0 -180 1.0 100
8. infinite 100 1.0 0 1.0 100

So far, everything agrees with W7EL's chart. We will now use
the following power equation not only to predict the dissipation
in the source resistor but also to explain the redistribution of
energy associated with interference. The power equation is:

Pa(R0) = fPa + rPa + 2*(fPa*rPa)cos(180-GA)

Could you expand on why the expression on the right is equal to
the average power dissipated in R0(Rs)? How was the expression
derived?

As well, what would be the equivalent expression for the following
example?

+-------+-------------+----------------------+
| | |
^ | Rs |
Is +-/\/\/-+ 1/2 wavelength ZLoad
2.828A 50 ohm | 50 ohm line |
| | |
+---------------+-----+----------------------+

The forward power is the same, the source impedance is the same,
but the conditions which cause maximum dissipation in the source
resistor are completely different.

Why is it not the same expression as previous since the conditions
on the line are the same?

What is the expression that describes the power dissipated in the
source resistor?

How is the expression derived?

Where 'GA' is the reflection coefficient Gamma angle and the last
term, 2*SQRT(fPa*rPa)cos(180-GA), is known as the *INTERFERENCE TERM*.

fPa rPa (180-GA) Pa(R0) interference term
1. 100 0 180 100 0
2. 100 11.1 180 44.4 -66.7
3. 100 11.1 0 177.8 +66.7
4. 100 11.4 82.9 119.8 + 8.35
5. 100 100 270 200 0
6. 100 100 233.2 80.2 -119.8
7. 100 100 360 400 +200
8. 100 100 180 0 -200

Except for the error that W7EL made in the Pa(R0) for example
number 7, these values of Pa(R0) agree with W7EL's posted values.
Therefore, the power-interference equation works. Not only does
it work, but it tells us the magnitude of interference between
the forward wave and the reflected wave when they interact at
the source resistor. Line by line:

1. There is zero interference because there are no reflections.

2. There is 66.7 watts of destructive interference present.

3. There is 66.7 watts of constructive interference present.

4. There is 8.35 watts of constructive interference present.

5. There is zero interference because the forward wave and
reflected waves are 90 degrees apart.

6. There is 119.8 watts of destructive interference present.

7. There is 200 watts of constructive interference present.

8. There is 200 watts of destructive interference present.
All of the reflected energy is redistributed back toward the load.

Wonder no more where the power goes. Constructive interference
requires extra energy from the source. Destructive interference
redistributes some (or all) of the reflected energy back toward
the load. Under zero interference conditions, all of the reflected
power (if it is not zero) is dissipated in the source resistor.
--
73, Cecil http://www.w5dxp.com