Richard Harrison wrote:

Keith wrote:
"At the quarter wave points where voltage and current are always zero,
there is no energy flowing. Period."

Yes there is energy flowing, and it is flowing in both directions if it
is flowing in one direction. Otherwise there would be no standing wave.

This view that there is energy flowing in both directions at the same
time
leads to some strange conclusions.

The directional wattmeter uses the following expressions to compute its
displayed value:
Vf = (V + IR)/2
Vr = (V - IR)/2
where V and I are the instantaneous voltage and current at the same
point
on the line and then computes power from:
Pf = average(Vf**2/R)
Pr = average(Vr**2/R)
Appropriate scaling (which we'll ignore for simplicity) is needed
depending on the characteristics of the voltage and current sensors.

Let's apply these expressions to some simple examples.

Connect a length of 50 Ohm transmission line to a 9 Volt battery and
wait
for the transient to die:
Vf = (9 + 0)/2 - 4.5 V
Vr = (9 - 0)/2 - 4.5 V
Pf = 0.405 Watt
Pr = 0.405 Watt
Disconnect the battery and the capacitance of the line remains charged
to
9 Volts. Are you really quite comfortable with the notion that this
line,
charged to 9 Volts, with 0 current everywhere has 0.405 Watts flowing
forward canceling at all points the 0.405 Watts flowing in reverse?

Others have stated that lamp cord has an impedance of about 100 Ohms.
Assume this to be correct.
With a table lamp plugged in but the lamp turned off:
Vf = (120 + 0)/2 - 60 V
Vr = (120 - 0)/2 - 60 V
Pf = 36 Watt
Pr = 36 Watt
Again, are you comfortable with this conclusion?

How about phone lines (phone on hook) at 600 Ohms and 48 Volts:
Vf = 24 V
Vr = 24 V
Pf = 0.96 W
Pr = 0.96 W
In a 400 pair cable there is 384 Watts flowing in the forward and
reverse direction. From a hundred thousand line central office we
get 960 Kilowatts forward and reverse, when the current is zero,
everywhere. Comfort level?

With the forward and reverse power view, to completely understand
the behaviour of the circuit, we need to know Z0 or we can not
compute these forward and reverse powers which seem to be
fundamental.

Consider a flashlight with the lamp off: the wire twists and turns
and has a very non-constant Z0. While computationally tedious, it
is possible to determine Pf and Pr at every point along the wire.
Isn't it necessary to do this to obtain a complete understanding
of what is happening in the circuit, if Pf and Pr are flowing?
If not, why not? The necessity for doing this is the logical
conclusion of the Pf and Pr approach.

Presented with a wire charged to high static voltage, what is Pf and
Pr? Presented with a sphere charge to a high static voltage, what
is the Pf and Pr?
Presented with a simple length of wire, how much power is flowing?

Similarly for a capacitor. The plates of a capacitor must have power
reflecting in all directions even when it is just charged to a
constant voltage (according to the Pf and Pr theory).

Given these bizarre results it would seem wise to limit the application
of the watts indicated by a directional 'watt'meter to those things
which are proper and not assume that it necessarily means that there
is real power flowing.

....Keith