Now that the various typo mistakes have been corrected, and putting
aside for the moment the name calling and ad hominem arguments, could
it be that _both_ sides in this discussion are correct? Camp 'A' says
that the reflection coefficient is computed the classical way, without
using Zo conjugate, and offers various mathematical proofs and
discussions of infinitely long lines. Camp 'B' says the reflection
coefficient is computed with Zo* (Zo conjugate) in the numerator, and
offers explanations dealing with the conservation of energy and
maximum transfer of power.

Both sides may be correct since they are talking about _two different_
meanings for the term "reflection coefficient." One has to do with
voltage (or current) traveling waves and the other has to do with
power. Quoting two references:

###

From Chipman, "Theory and Problems of Transmission Lines," 1968:

Section 7.1, Reflection coefficient for voltage waves:
[Discussion and math, then] ...
rho = (ZL - Zo) / (ZL + Zo)

Section 7.6, Complex characteristic impedance
[Various mathematical manipulations, then] ... "the maximum possible
value for |rho| is found to be 1 + sqrt(2) or about 2.41. ... [T]he
principal of conservation of energy is not violated even when the
magnitude of the [voltage wave] reflection coefficient exceeds unity."

[then more math, then] ... "The conclusion is somewhat surprising,
though inescapable, that a transmission line can be terminated with a
[voltage wave] reflection coefficient whose magnitude is as great as
2.41 without there being any implication that the power level of the
reflected wave is greater than that of the incident wave."

[then a discussion of a source with internal impedance Zo connected to
a line with characteristic impedance also Zo that is terminated with a
load of impedance ZL, then] "... more power will be delivered to a
terminal load impedance Zo* [conjugate of Zo] that produces a
reflected [voltage] wave on the line than to a terminal load impedance
Zo that produces no reflected [voltage] wave."

So Chipman states quite clearly that zero reflected voltage wave
magnitude does _not_ mean maximum power transfer. On the contrary,
maximum power is transferred only when there is a non-zero voltage
wave reflection (assuming a complex Zo line). Counter arguments along
the lines of "Well that doesn't seem right to me so therefore Chipman
must be wrong" don't carry much weight given Chipman's credentials.

###

From Kurokawa, "Power Waves and the Scattering Matrix," IEEE
Transactions on Microwave Theory and Techniques, March 1965:

Section 2, explanation of and mathematical definition of the concept
of "power waves," explicitly noted by the author to be distinct from
the more commonly discussed voltage and current traveling waves.

Section 3, definition of the reflection coefficient [for power waves]:
s = (ZL - Zo*) / (ZL + Zo)
with a footnote "[Only w]hen Zo is real and positive this is the
voltage wave reflection coefficient." Kurokawa takes pains to make it
clear that his "s" power wave reflection coefficient is not the same
as the (usually rho or Gamma) voltage wave reflection coefficient.

Section 9, comparison with [voltage and current] traveling waves:
"... since the [voltage or current] traveling wave reflection
coefficient is given by (ZL - Zo) / (ZL + Zo) [note no conjugate] and
the maximum power transfer takes place when ZL=Zo*, ... it is only
when there is a certain reflection in terms of [voltage or current]
traveling waves that the maximum power is transferred from the line to
the load."

So Kurokawa agrees with Chipman concerning the condition for maximum
power transfer. Kurokawa also defines two different reflection
coefficients, both in the same paper.

[In some of the above quotes I have altered the subscript letter
assigned to Z, merely for consistency between the two references.]

###

So, it seems to me, everybody can agree as long as it is understood
that there are different meanings for the term "reflection
coefficient." One meaning, and its mathematical definition, applies
to voltage or current waves. The other, with a slightly different
mathematical definition, applies to the power transfer from a line to
a load. They are one and the same only when the reactive portion of
Zo (Xo) is ignored. It may or may not be acceptable to do so,
depending on the attenuation of the line and the frequency. Lossy
lines and lower frequencies yield more negative values for the Xo
component of Zo.

You can use Reg's COAXPAIR or my TLDetails program to do the math and
show concrete examples. Try something like 100 feet of RG-174 at 0.1
MHz, terminated with loads equivalent to Zo and then Zo conjugate, and
compare the rho (or SWR) figures versus the power delivered to the
load for each case. When the termination equals Zo conjugate, note
that the total dB loss is actually _less_ than the matched line loss.
As counter intuitive as this may sound, Chipman offers an explanation
on page 139. (And as others are sure to point out, this makes
absolutely no difference in practical applications and is of academic
interest only.)

Copy of the Kurokawa paper, in pdf format, available on request via
private email. I've obtained copies of Chipman, on two separate
occasions, from Powell's in Portland.

Dan, AC6LA
www.qsl.net/ac6la/