On Mar 9, 10:35 am, Cecil Moore wrote:
Keith Dysart wrote:
My issue was that you seemed, in that sentence, to be saying
that the reflected energy was dissipated in the source
resistor. But earlier you had stated that was not your claim.

My earlier claim was that the average power in a reflected
wave is dissipated in the source resistor when the forward
wave is 90 degrees out of phase with the reflected wave at
the source resistor. In that earlier claim, I didn't care
to discuss instantaneous power and thus excluded instantaneous
power from that claim.

For instantaneous values, it will be helpful to change the
example while leaving the conditions at the source resistor
unchanged. Here's the earlier example:

Rs Vg Vl
+----/\/\/-----+----------------------+
| 50 ohm |
| 1/8 WL |
Vs 45 degrees 12.5 ohm
100v RMS 50 ohm line Load
| |
| |
+--------------+----------------------+
gnd

Here's the present example:

Rs Vg Vl
+----/\/\/-----+----------------------+
| 50 ohm |
| 1 WL |
Vs 360 degrees 23.5+j44.1
100v RMS 50 ohm line ohm Load
| |
| |
+--------------+----------------------+
gnd

If I haven't made some stupid mistake, the conditions at
the source resistor are identical in both examples.

No silly mistakes. This number is computed by the spreadsheet at
http://keith.dysart.googlepages.com/...oad,reflection
as 23.529411764706+44.1176470588235j, so you are pretty close.

But
in the second example, it is obvious that energy can
be stored in the transmission line during part of a cycle
(thus avoiding dissipation at that instant in time) and be
delivered back to the source resistor during another part
of the cycle (to be dissipated at a later instant in time).
That is the nature of interference energy and is exactly
equal to the difference between the two powers that you
calculated. You neglected to take into account the ability
of the network reactance to temporarily store energy and
dissipate it later in time.

I don't think I did.

The power delivered by the generator to the line is
Pg(t) = 32 + 68 cos(2wt)
The average power delivered to the line is 32W while
the peak power is 100W towards the load and 36W from
the line to the generator.
The exact same function describes the power delivered
to the line for 12.5 ohm load 45 degree line example,
and the 23.5+44.1j ohm load 360 degree line example.

And if the load was connected directly to the generator,
the same power would be delived directly to the load.

In all cases the power dissipated in the source resistor
is
Prs(t) = 68 + 68 cos(2wt-61.92751306degrees)
This power varies between 0.0W and 136W.

This is true even when the load is connected directly to
the generator without a line to created reflected power.

The power provided by the source is
Ps(t) = 100 + 116.6190379 cos(2wt-30.96375653degrees)
= Prs(t) + Pg(t)
so all of the energy delivered by source is nicely
accounted for. All of the energy dissipated in the
source resistor and delivered to the line originates
in the voltage source.

Looking at the line where it connects to the generator,
we find that the forward power is
Pf.g(t) = 50 + 50cos(2wt)
and the reflected power is
Pr.g(t) = -18 + 18cos(2wt)

Pf.g(t) + Pr.g(t) = 32 + 68cos(2wt)
= Pg(t)
As expected, the sum of the forward and reflected power
at the generator terminals is exactly the power delivered
by the generator to the line.

So you are correct, energy is stored in and returned from
the line on each cycle, but this is the net energy, not
the forward or reflected power since these are both
computed into non-reactive impedances, their respective
voltage and current are always in phase, so their powers
always flow in only one direction.

All of the reflected energy is dissipated in the source
resistor, just not at the time you thought it should be.

So as yet, there is no mechanism to explain storage of
reflected power so it can be dissipated at a different
time in the source resistor.

I stand by my contention that the reflected power is not
being dissipated in the source resistor because the
dissipation does not occur at the correct time.

You could convince me to change my contention by locating
the entity that is storing the energy, and showing some
analysis that explains why the amount of the energy
time shift is a function of the magnitude of the
reflected power.

....Keith