[TowerTalk] Walt Maxwell responds to Steve Best
Jim Reid
kh7m@hsa-kauai.net
Thu, 11 May 2000 13:10:19 -1000
A few minutes ago, I received the following from Walt,
W2DU, with his request that it be posted to TowerTalk.
Walt says he cannot post directly, so am glad to
forward this on for him.
Here is a copy of Walt's e-mail to me:
"Jim, I have no way of posting on the 'Reflector', but I'd like
very much to rebut Best's post of May 11.
I copied a portion of his post to Word and interspersed my
comments between his paragraphs. I have attached the
results of my editing in the accompanying file. I have made
my comments in large bold font, but this font may not be
saved in html. If it is not that's ok.
Could you possibly post it in TT with a sufficient explanation
for the readers to get the drift? Sure would appreciate having
it in prior to Dayton so at least some of the audience will
have the opportunity to see my comments.
73, Walt"
As rcvd. from Walt:
"Steve Best says:
Several people have mentioned that both Walt's and my
analysis yield the same steady state power levels so what's
my problem with Walt's work. My issue with Walt's work is
not the steady state conditions of delivered, forward and
reflected power. My issue is with his explanation of how these
steady state power levels are related to the wave reflections
within the transmission line.
In Walt's analysis, he begins knowing the steady state
conditions and then develops his version of wave reflection
mechanics to relate these steady state conditions to the
wave reflections occurring at the output of a matching
device or tuner. He starts with the known answer so no
matter what concepts he develops, he will end up back
at the known answer. If anyone were to look very closely
at the math and concepts that Walt uses in his discussion
of wave reflection mechanics, the inconsistencies in his
methods as they relate to classical transmission line and
circuit theory would be evident."
W2DU's response:
There are no inconsistencies in my in my methods as they
relate to classical transmission line and circuit theory, as
Steve asserts. As we proceed I'll show why it's Steve's own
misunderstanding of the theory that's responsible for the
difference in our results.
Steve:
"Point 1) What is the mechanism responsible for wave
reflections at the end of a transmission line?
When a traveling voltage (using voltage in the discussion
is just a personal preference) arrives at the end of a
transmission line, the level of reflected voltage is a direct
function of the impedance terminating the line, nothing else.
When a forward traveling voltage arrives at the antenna,
the level of reflected voltage is a direct function of the
antenna impedance. When a reflected or rearward
traveling voltage arrives at the input of the transmission
line (i.e. at the output of a transmitter, tuner, or other
matching device) the level of re-reflected voltage is still
a direct function of the impedance terminating the line.
In these cases, the impedance "terminating" the line will
be the output impedance of the transmitter, tuner or
matching device. From this perspective, the output
impedance is simply the impedance seen looking rearward
into these devices."
W2DU:
This is not true. The impedance one measures looking
back into the matching device is entirely different from
the impedance seen by waves reflected from a
mismatched line termination on reaching the matching
device when the forward waves from the source are
present simultaneously. This is one of the many points
Steve fails to understand, even after I've tried to explain
to him with no success.
Steve:
"In order for Walt to tie his reflection analysis and math
together, he developed the concept of total power
re-reflection via the virtual short or open circuit. With
any matching device, Walt states that with the impedance
match, a simultaneous, totally reflecting virtual short or
open circuit is created. In this case, a total re-reflection
of power occurs in-phase with the delivered power such
that the two powers add together to become the
total forward traveling power in the transmission line.
There are a few inconsistencies with these concepts
relative to basic transmission line and circuit theory.
These inconsistencies will be identified in points 2 and 3
below."
W2DU:
As I said earlier, I will show that the inconsistencies Steve
describes below are inconsistencies in his own thinking,
and are not inconsistencies in my method.
Steve:
"Point 2) Do two "in-phase" forward traveling powers add
to become the total forward traveling power in the
transmission line? "
W2DU:
Yes.
Steve:
'In one of the examples in his text, Walt presents a case
where 100 watts of power (steady state) is delivered to
the input of a matched lossless tuner. The tuner is used
to match an antenna with an impedance of 150 ohms. The
steady state forward traveling power in the transmission
line (connecting the tuner and antenna) is 133.33 watts and
the reflected power in the transmission line is 33.33 watts.
I agree with these levels of steady state power. Now, here
is where I start to disagree.
Walt's hypothesis is that the 33.33 watts of reflected power
is totally re-reflected at the tuner in phase with the 100
watts of delivered power and the two add in-phase to
become the 133.33 watts of total forward traveling
power in the transmission line. On the surface, this
seems reasonable but let's look at this concept in more
detail.
Let's assume that Zo is real (50 ohms as in Walt's
example). With Zo real, the traveling power in the
transmission line is equal to the magnitude of the
traveling voltage squared divided by Zo. P = |V|^2 / Zo."
W2DU:
50 ohms is NOT used in Walt's example. This is the
first of many errors in Steve's method, which I pointed
out to him weeks and weeks ago via email correspondence,
but he would not agree.
Steve's error here is in using the characteristic impedance
Zo to apply the voltage. Here Zo is irrelevant, because
when a 50-ohm line has reflections there is no place on
the line where the line impedance is 50 + j0 ohms. There
are two locations on the line where the line impedances
are purely resistive, 150 + j0 and 16.667 + j0 ohms. There
are also two locations on the line where the resistive
component of the line impedance is 50 ohms, but they
are complex line impedances, 50 + j57.7 and 50 - j57.7ohms.
Steve does not use the line impedance in his calculations
-he uses Zo, the CHARACTERISTIC impedance of the
line, which is incorrect. The result of this error? His
calculations using voltage with Zo results in a totally
erroneous value of forward power, 248.8 watts instead
of the correct 133.33 watts.
Steve:
"100 watts of forward traveling power must result from a
traveling voltage having a magnitude of 70.711 volts.
33.33 watts of forward traveling power must result from
a traveling voltage having a magnitude of 40.825 volts.
In all transmission lines, the traveling voltages add together
to develop the total traveling voltage and hence the total
traveling power. If the powers are in phase, the voltages
must be in phase as well. If Walt's hypothesis is
correct, I should be able to calculate the 133.33 watts
of forward power from the in-phase addition of these
voltages. For sake of discussion, pick any absolute phase
for the voltages so long as they are in-phase, say both
voltages have 0 degrees phase. Let's calculate the total
forward power using these voltages. The total forward
traveling voltage in the transmission line would be
111.536 volts and the resulting total forward traveling
power is calculated to be 248.8 watts NOT 133.33 watts.
So, we now see an inconsistency when calculating power
using these voltages. The real issue is the following:
Can in-phase powers traveling in a transmission line
be "added" to determine the total traveling power?"
W2DU:
All values of voltage and power in the paragraph above
are incorrect. Steve's calculation of 248.8 watts of
forward power is incorrect, because of using Zo as the
line impedance, which it is not, instead of the line
impedance determined by the position of the standing
waves on the line. I will now explain why 133.33 watts
is correct, using for a reference the work of Princeton
University's Walter Johnson's "Transmission Lines
and Networks." On page 99 of this well known reference
Johnson shows that the summation of an infinite number
of reflections follows the infinite geometric series of the
form '1 + a + a'squared + a'cubed.' that converges to
1/(1 - a) for values of 'a' less than 1. This expression is
appropriate for determining the forward power in a lossless
line. In our case where the mismatch is 3:1 the voltage
reflection coefficient rho is 0.5 and the power reflection
coefficient rho squared is 0.25. Now substituting rho
squared in the expression 1/(1 - a), we get 1/(1 - 0.25)
= 1/0.75 = 1.3333. When multiplying this factor by 100
watts we get 133.33 watts of forward power.
With the line mismatch of 3:1, and power reflection
coefficient of 25 percent and a power transmission
coefficient of 75 percent, we find that 75 percent of
133.33 watts is 100 watts absorbed, and 25 percent
of 133.33 watts is 33.333 watts reflected. When the
33.333 watts of reflected power reaches the matching
device it is totally reelected, adding to the 100 watts
from the source to achieve 133.33 watts of total forward
power.
Unfortunately Steve fails to understand the nature and
concept of the reflection mechanics that achieves the
total re-reflection of the reflected waves. There isn't
sufficient space here to explain that concept, so I suggest
you refer to this subject as I explain it in detail in Reflections.
However, please be aware that Steve has trashed my
explanation appearing in Reflections by his totally
erroneous article in the Fall 1999 issue of CommQuart,
in which he asserted that my concept of reflection
mechanics in impedance matching is incorrect.
Steve:
"Two "in-phase" powers (P1 and P2) never add to become
the algebraic sum of P1 + P2. "
W2DU:
The above statement is totally untrue. When the
corresponding voltages and currents of the forward and
reflected powers are in phase the two powers add
algebraically. In the paragraph below Steve talks about
powers where the voltage and currents are out of phase.
This is irrelevant, because when the antenna tuner is
correctly adjusted to deliver maximum output power the
voltage and current phases are exactly equal at zero degrees.
Steve:
"The general rule for addition of power (really voltage) is the
following (Zo is real): Assume that V1 has 0 degrees
relative phase. If V2 has a phase between 0 and 90
degrees then Ptotal will be greater than P1 + P2. If V2
has a phase of 90 degrees then Ptotal = P1 + P2. If V2 has a
phase between 90 and 180 degrees then Ptotal is less
than P1 + P2. The extreme cases in power addition occur
when V1 and V2 are in phase where maximum power develops
and where V1 and V2 are 180 degrees out of phase
where 0 power develops.
Walt's direct addition of "in-phase" forward and re-reflected
power is the one obvious aspect of his wave reflection
analysis where his concepts and math fall apart. His
relationships between voltage and power are not
consistent with transmission line or circuit theory. "
W2DU:
Sorry, it's Steve's concept and math that fall apart, not mine,
for the reason explained above, where he gets 248.8 watts
of forward power. There is no way there can be 248.8 watts
of forward power. See my explanation above showing that
he used the wrong value of Z to apply the voltage in
attempting to calculate the forward power. It's Steve's
relationships between voltage and power that are not
consistent with transmission line or circuit theory.
Steve:
"I have no problem with the fact that the forward and
re-reflected powers are in-phase. Walt's two
errors are that he assumes that the 100 watts of steady
state delivered power is the initial forward driving power
delivered to the transmission line input and that that the
33.33 watts of reflected power is totally re-reflected.
The 100 watts of delivered power is the end result of the
wave reflection process and is simply the difference
between forward and reflected power.
Looking at a detailed wave reflection analysis starting
with the system's initial conditions, we would find that
the initial forward driving power delivered to the transmission
line is 75 watts and that the re-reflected power is actually
8.33 watts. Any questions as to how I got these numbers
can be answered by my articles or in a separate e-mail.
When 75 watts and 8.33 watts add in phase, the resulting
total power is 133.33 watts."
W2DU:
Some of the statements in Steve's Winter 1999 Comm
Quart article are erroneous, and practically all of his
statements in the Fall 1999 are erroneous, because
he failed to understand some of the most basic concepts
of transmission line theory. It is in this article that he
totally and inconceivably trashed the concepts of reflection
mechanics I discussed in Reflections. And reviewing the
last sentence in his paragraph above, can you really believe
that 75 watts plus 8.33 watts yields 133.33 watts. Where
did he learn that kind of arithmetic?
Steve:
"Point 3) Effective net power delivered to a transmission
line is always equal to Pforward minus Preflected.
When an impedance match occurs, Walt indicates that a
simultaneous totally reflecting virtual short or open circuit
is created. In this case, the total forward power is then
equal to the delivered power plus the totally re-reflected
reflected power. So, from Walt's perspective, Pforward =
Pdelivered + Preflected."
W2DU:
Nearly all of the following of Steve's statements are untrue
and misleading, and prove that Steve has a large
misunderstanding of transmission line theory to overcome.
Steve:
"Another inconsistency here is the fact that the same
relationship between delivered, forward and reflected
powers still hold even when an impedance match does not
exist. If you were to adjust the tuner components such that
an impedance match did not exist, you would find that the
net power delivered to the transmission line is still equal
to Pforward - Preflected. Mathematically, this can be rewritten
as Pforward = Pdelivered + Preflected.
>From that perspective, should we still assume that a total power
re-reflection occurs at the mismatched tuner? Does this mean
that a virtual short circuit exists under all general conditions?
The answer is no because it does not exist even with the
impedance match.
In any transmission line system, the net power "delivered" to the
transmission line is always equal to the forward power minus
the reflected power. The important concept to realize is that
the effective net power delivered to the transmission line is
the end result of the wave reflection process, it is not the
beginning of the wave reflection process.
The relationship Pdelivered = Pforward - Preflected works
because the powers are traveling in opposite directions.
This equation can be manipulated to determine any one power
value when the other two are known. It is incorrect however
to conceptually rewrite the equation as Pdelivered = Pforward +
Pre-reflected because we know that powers traveling in the
same direction do not add this way. To understand this, the
math of changing the equation around has to be ignored and
the underlying concepts have to be considered.
The correct determination of Pforward results from the correct
determination of Vforward which is equal to the forward driving
voltage + the re-reflected voltage. To determine these values
correctly one must start with the initial voltage delivered to
the transmission line and perform a complete wave reflection
analysis. A steady state power analysis of a transmission
line system will also allow for the correct determination of
forward and reflected power levels, however, it is incorrect
to relate these power levels to the wave reflections within
the system in the manner that Walt does.
Point 4) The Conjugate Match. I don't recall ever disagreeing
with the conjugate match theorem only Walt's statements that
a conjugate match always occurs at the output of a tuner,
when the tuner is impedance matched at its input. This
statement is only true when the tuner components are lossless.
With lossy tuner components, a conjugate match will not
occur at the tuner output when the tuner is impedance
matched at its input. The lossy tuner components can be
adjusted to achieve a conjugate match at the tuner output
but an impedance match will not exist at the tuner input.
Under this conjugate match situation, it can be demonstrated
that more power can be delivered to a load at the tuner output
when an impedance mismatch exists at the tuner input.
This is consistent with the conjugate match theorem. If
anyone really wants proof I can send along a couple of simple
circuit examples to illustrate these points.
73,
Steve, VE9SRB"
Walt Maxwell, W2DU
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