[TowerTalk] Jim Reid's (KH7M) comments (2)

[Steven Best]

This message has only taken me about 5 tries to post.  Jim has already
responded at least twice.  This is the post that Jim refers to re: his
measurements.  I still need to look at Jim's new posts and get back to him.

Given some of Jim's (KH7M) additional comments (re: the conjugate match) and
those posted by others, I thought it appropriate to discuss exactly why I
disagree with some of the material and analysis presented in Walt's (W2DU)
textbook.  Although most readers are aware of the fact that I disagree with
Walt in some areas, I don't think everyone appreciates why.

This is LONG!

Overall, Walt's book is quite good and presents a wide variety of valuable
and very useful information for both the amateur radio and engineering
communities.  However, there are probably three or four areas where I think
Walt made a few technical mistakes regarding wave reflections and the
conjugate impedance match.

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.

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.

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.

Point 2)  Do two "in-phase" forward traveling powers add to become the total
forward traveling power in the transmission line?

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.  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?

Two "in-phase" powers (P1 and P2) never add to become the algebraic sum of
P1 + P2.  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.  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.

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.

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






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