Roger wrote:
Roy Lewallen wrote:

Can you use your theory to show the voltage at all times at the input
or, if you choose, just inside the input of the line, as I've done
using conventional theory?
Yes. Here is the difference. I am using a short cut by looking at the
way waves travel and superimpose. My assumption is that the input
impedance matches the impedance of the line.

That's a different analysis than I did. I did two, one in which the
mismatch was infinite (perfect voltage source at the input) and one in
which the voltage reflection coefficient at the source was 0.5. The case
where the input is matched is simpler, because there is no re-reflection
from the source.

Can your method be used to analyze a circuit which is mismatched at the
source?

Thus 1v generates a 1v
sine wave.

Sorry, I don't understand that sentence.

For a simple example, such as any multiple of 1/2 wave, this
causes current proportional to applied voltage and impedance of the
line. For example, 50v applied to a 50 ohm line gives a 1 amp current.

It does for the time for one round trip. Then the current drops to zero
if the line is matched at the source.

We know a line 1/2 wavelength long will absorb energy for the length of
time it takes for a wave to travel the length of line and return, so
power must be applied during the entire time.

Yes.

Obviously, 2 half waves
will be applied over that time period so the amount of energy on the
line is equal to the amount of energy contained in the power applied
over time by two half waves, so total power applied is 2 * initial power.

You've lost me here. What's "total power"? Can you express it as an
equation?

Equally obvious, by leaving the source connected after the reflected
wave has returned, the question of how the source reacts to the external
application of power must be answered.

Well, the bit about "total power" wasn't obvious, so I'm not surprised
that what follows isn't either. You're apparently assuming that there
are waves of traveling power, which has led to demonstrably
self-contradictory results before, but let's see where it leads this time.

This is a second question,
unrelated to traveling waves on a transmission line. My answer to this
question is to disconnect the source and replace the source with a
connection identical to the condition at the far end of the transmission
line.

Which I presume is the open circuited example.

You can see that this leaves the transmission line isolated with
power circulating on the line (is is POWER, time is involved).

The power at any point can easily be calculated, and it shows that
energy is moving back and forth on the line. There are no waves of power
moving about.

Your solution (as presented in Analysis 2) and the Power Analysis was to
keep the source supplied at all times in an attempt to create continuous
real world conditions, as contrasted to my method of a quick and
correctly timed disconnect. Both experiments are real world simulations
and can be performed.

Ok. My method correctly predicts what will happen at any instant in
either case. Let's see how yours does.

Your analysis is identical to what happens at a real world transmitter.
A very good, automatically tuned transmitter!

I won't attempt to argue that, because of the great deal of controversy
about what the characteristics of a "real world transmitter" are. I make
no representation for my analysis other than what I stated: it uses an
ideal voltage source, which is rigorously defined and with
characteristics which are completely known, in conjunction with a
perfect resistance.

Your choice of an ideal
voltage source amounts to setting conditions that automatically adjust
for impedance changes as fast as the change happens.

No, there is no change in impedance. The perfect source has a zero
impedance at all times. The resistance has an impedance equal to its
resistance at all times. The voltage of the voltage source stays
constant (that is, it puts out a sine wave of constant amplitude and
phase) at all times.

The transmission
line sees an outgoing impedance change going from 150 ohms to infinity
in your example.

No, it sees a constant 150 ohms at all times.

The source sees an outgoing impedance change from 200
ohms to infinity during the same analysis period.

If you consider the analysis period to extend to steady state, then yes,
that's correct. You might recall that the power into the line increased
for one of the reflection periods because of the improved impedance
match between the source and line (impedances of 150 and 250 ohms
respectively) compared to the impedance match at other periods.

It is interesting to compare the final voltage ratios ( directly
proportional to energy levels and directly proportional to power levels)
on the transmission line to the SWR ratio.

The energy stored = 1/2 * C * v^2 + 1/2 * L * i^2, where v and i are the
voltage and current at a very short section of line, and C and L are the
capacitance and inductance, respectively, per that short section. As you
can see, energy is not directly proportional to V by any means. There
are places and times where energy is stored entirely in the magnetic
field, and v is zero.

You used a SWR ratio of 3:1
and I used 1:1.

No, the SWR on the line was infinite in my example. It's determined
entirely by the load impedance and the line's Z0. If you're assuming an
open ended line, the SWR is infinite in your example also.

You found that the total energy stored on the line was
4 times the initial energy applied,

No, the "initial energy" applied was zero. The analysis began with the
assumption that the line was initially completely discharged. It sounds
like you're confusing power, which had an initial value, and the energy,
which accumulated over time.

and I predicted that the total
energy stored was 2 times the initial.

Where and when did you predict that? What initial energy did you apply
to the line? Can you do an analysis with the line initially discharged
as I did?

The link here is that the power
storage factor(SF) (I know you dislike the term) is found from

SF = SWR + 1.

Your storage factor was SF = 3 + 1 = 4. My storage factor was
SF = 1 + 1 = 2.

First of all, the SWR in my example was infinite, so by your
calculation, the "power storage factor" was infinite.

If we can accept that the storage factor is always SF = SWR + 1 for all
situations, then we have a very convient way to predict the voltage
increase that results form impedance mismatch at a discontinuity. I am
not yet ready to accept this generalization, but we should look at it
more carefully.

I still don't know what you mean by "power storage factor" or why you
need to coin a new and vague term. So I guess I can accept that it's
whatever you want it to be. But it has no relationship to power or
energy that I can see.

Ending comment. Both your and my analysis are correct. Congratulations
to both us!

I've presented an analysis which gives, quantitatively, the exact
voltage at any point on the line, at any instant, from turn-on until
steady state. I've also shown the power being applied at any instant and
the total energy on the line at any time.

You've presented an analysis whose only result is a poorly defined
"power storage factor", derived from invalid premises.

Yes, congratulations to us both.

By the way, I didn't see from your "analysis" how many watts are stored
in the line. If power is stored as you say, you surely must be able to
determine how much.

Roy Lewallen, W7EL