[TowerTalk] Walt Maxwell responds to Steve Best

[Steven Best]

Hi Tom:

At least we can agree that we disagree.  I think some of this discussion is
starting to turn into a matter of semantics.  In this e-mail, I will
directly address your comments and then give my version of the geometric
series discussion that Walt presented in last night's e-mail.

Tom wrote: Shutting off the generator and measuring the impedance looking
BACK into the source does NOT indicate the impedance a
reflected wave sees.

Tom: I think that you may have partially misinterpreted some of what I said.
I agree with your statement above, that's why its very difficult to
interpret measurements made with the transmitter.  The transmitter has
components whose impedance may not be linear with power, the transmitter may
reduce power output depending upon SWR, the transmitter may have an internal
tuning circuit whose component values change with steady state load.  So
clearly, measuring output impedance with the transmitter off is pointless.
For this reason, it's better to use an external tuner so that you have more
control over these variables and run the analysis between the tuner and the
antenna.

My real point is that the level of re-reflection is based upon the impedance
(whatever it is) that the reflected wave sees when it arrives at the input
end of the transmission line.  This is determined looking rearward into the
transmitter or tuner.

Tom wrote: I also have not found anything that agrees with your theory. When
measurements agree with your theory, and disagree with Walt,
then I will quickly change my opinion.

Tom:  Do you believe that 100 watts of power plus 33.33 watts of power added
in phase add to 133.33 watts?  Check the theory re: the addition of forward
traveling voltage and resulting forward traveling power.

Additionally, all we can measure are the steady state levels of forward and
reflected power.  Using my reflection analysis, based upon the concepts I
have presented, I determine the correct steady state power levels.  This
analysis is based on the fact that 100 watts + 33.33 watts do not add to
133.33 watts but 75 watts plus 8.33 watts do add to 133.33 watts.

Tom, as a final note, I will write a short description of the geometric
series discussion that Walt presented yesterday from my perspective and
relate it to Walt's example.  All I can ask is for you and Walt to think
about it.

As I have stated many times, the forward driving voltage a transmitter
delivers to a transmission line is the voltage it delivers to a Zo load.  I
define this voltage as Vi.  Also, the wave reflections at the antenna are a
function of Za and can be determined using rhoA.  The wave reflections at
the transmitter (or tuner) are based upon the output impedance Zs of the
transmitter or tuner and can be determined using rhoS.  If you correctly go
through the wave reflection process FROM THE INITIAL STATE you will find the
following voltages in the system:  G is gamma = alpha + jbeta.  L is the
transmission line length.

At the Transmitter/Tuner		At the Antenna

1. Start Vi >				2. Vi e^-GL >

4. < rhoA Vi e^-2GL			3. < rhoA Vi e^-GL

5. rhoS rhoA Vi e^-2GL >		6. rhoS rhoA Vi e^-3GL >

8. < rhoS rhoA^2 Vi e^-4GL		7. < rhoS rhoA^2 Vi e^-3GL

9. rhoS^2 rhoA^2 Vi e^-4GL >

This process continues until the steady state conditions are reached.  The
total forward traveling voltage delivered to the transmission line in the
steady state is simply the sum of all of the forward traveling voltage terms
at the input to the transmission line.  Yes, this is a geometric series that
can be represented in the form of 1/1-a.  In this case, a = rhoS rhoA
e^-2GL.  The total steady state forward traveling voltage is given by

Vforward = Vi 1/(1 - rhoS rhoA e^-2GL)

Let's check this with Walt's 150 ohm antenna and tuner example.  The
transmission line is 1 wavelength long and lossless.  The tuner is lossless.

At the output of the tuner, I calculate the voltage Vi delivered to
transmission line to be -21.89 + j57.19 volts (75 watts).  rhoA = 0.5.
Since the tuner is lossless, a conjugate match exists and rhoS looking
rearward into the tuner output is also 150 ohms, therefore rhoS = 0.5.  Now,
since the transmission line is 1 wavelength long and lossless, e^-2GL = 1.
Plugging these numbers into the above equation:

Vforward = Vi (1.3333)

Vforward = -29.19 + j76.25 volts.  Pforward = 133.33 watts.

This is it.  The reflection analysis, transmission line theory and the
steady state conditions all tie together. Quite frankly, Walt simply
misunderstood Johnson's analysis when he developed the concept that a total
re-reflection of power occurs and that the impedance looking rearward into
the tuner is not responsible for power re-reflection.  You must start with
the initial voltage delivered to the transmission line to see all of this.
You can start with the effective net power delivered to the transmission
line to determine the forward power but you miss the wave reflections in the
system since the effective net power is the steady state result of how the
forward and reflected powers develop.

Examining the formula for Vforward above we see that the steady state
forward voltage is a direct function of Vi and rhoS.

At this point, I think Walt should ponder the above analysis.  I probably
will not address this anymore more on towertalk as I want to go through all
of this for the audience at Dayton.  I will give my word to everyone that I
will stick to the facts and NOT mention Walt by name.

73,
Steve, VE9SRB





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