Roy:

[snip]
Here you've lost me. a is the forward voltage in the transmission line.
What can be the meaning of its facing a driving point impedance? The
forward wave sees only the characteristic impedance of the line; at all
points the ratio of forward voltage to forward current is simply the Z0
of the line. So I don't believe that it sees 2r anywhere.

This is where I'm stuck. If you can show where along the line the
forward voltage wave "faces" 2r, that is, Vf/If = 2r, I can continue.

Roy Lewallen, W7EL
[snip]

I appologize if I am "going to fast" for the limitations inherent in
NewsGroup postings.

Let me take it a little more slowly here...

To see that there is essentially no difference between my (apparently two)
different
definitions of the incident and reflected waves a and b, consider the
following scenario
of an actual transmission line having surge impedance [characteristic
impedance] Zo.
Let Zo be in general complex. i.e. Zo = sqrt[(R + jwL)/(G + jwC)] where R,
L, G,
and C are the primary parameters of R Ohms, L Henries, G Siemens, and C
Farads
per unit length.

Now at any particular frequency w = 2*p*f you will find that this general
complex
surge impedance Zo evaluates to a complex number, say Zo(jw) = r + jx.
Later
let's let r = 50 Ohms and x = 5 Ohms so that we can work out a numerical
example.

Consider either a semi-infinite length of this Zo line, or even a finite
length of the Zo line
terminated in an impedance equal to Z0. I am sure that you will agree that
both the
semi-infinite Zo line or the finite length Zo line terminated in Zo have the
same
driving point impedance namely Zo.

Now excite this semi-infinte Zo line by an ideal generator of open circuit
voltage
Vi = 2*a behind an impedance equal to the surge impedance Zo. In other
words
this is a Thevenin generator of ideal constant voltage 2a behind a complex
impedance
of Zo. And so... since an ideal voltage source has zero impedance, the
termination
at the source end of this semi-infinite Zo line is Zo, and so the line which
has a
driving point impedance of Zo is terminated in Zo. Thus the port or
reference plane
between the generator and the semi-infinite line is an "image match" point.

For reference, a rough ASCII style schematic of this situation would be as
follows.

Generator = Vi = 2aVolts - generator impedance Z0 - impedance Zo of
semi-infinite line.

Clearly the the generator impedance of Zo and the line's driving point
impedance of Zo
constitute a voltage divider hanging off the generator with open circuit
volts Vi = 2a Volts,
and since Zo = Z0 they divide the generator voltage, which equals 2a volts
exactly by 2, and
so the voltage across the input of the semi-infinte line is exactly a volts.
This a is the incident
voltage wave of the Scattering Formalism and it is exactly the same as the
classical forward
voltage for transmission lines. Note that it is exactly half of the
generator's open circuit volts.
i.e. the voltage across the perfectly terminated [image matched] line is
exactly a Volts or 1/2
of Vi Volts. [Aside: That is why I said in an earlier post that your
forward wave and my
incident wave only differ by a factor of two! They are really no different
it's only a scale
factor. You can compensate as I have by calling the Zo generator voltage
2a, but for
simplicity I often just call it a to eliminate the factors of 2 that occur
all over the place. Sorry
if that is confusing, I'll try not to do that any more. :-).

Now everybody knows and everybody will agree that at the junction of the Zo
line
terminated in Zo [NOT a conjugate match] where the Thevenin generator is
connected
to the semi-infinite Zo line there is an image match AND there will be no
reflected voltage
waves. What is more since the Zo line is semi-infinite, or the alternative
of a finite length
and terminated in Zo at the far end, there will be no reflections at the
"far end" either. i.e. in this
situation there are no reflections and the incident wave "a" just propagates
into the line and
there is no reflected wave b to interfere with it. Note that this does not
mean that maximum
power is transmitted into the line. You and I both agree that when a
generator is image
matched [Cecil likes to call "image match" a Zo match] that in the general
case of complex
Zo maximum power is not transferred but there is NO reflected voltage when
measured
with a reflectometer which uses Zo as it's reference impedance.

This scenario at the Zo generator driving the Zo line is the same situation
found when you cut
into an infinite length of Zo line and insert a reflectometer with reference
impedance Zo,
the reflectometer will read out the reflected voltage b as:

b = rho*a = (Zo-Zo)/(Zo+ Zo) = 0 Zero, nada, nil... no reflections.

Now lets do a simple numerical example.

Consider that at some frequency the surge impedance evaluated to say Zo = r
+ jx = 50 + j5
Ohms and that we set the generator open circuit voltage to be Vi = 2a = 2
volts. Thus the "incident
voltage wave is a = 1 Volt.

The maximum power available from this Zo generator will occur when the load,
call it Z, on the
generator is a conjugate match to the generator impedance Zo. Thus for
maximum power
transfer Z must be the complex conjugate of Zo, i.e. we must let Z = 50 -
j5, to extract maximum
power not Z = 50 +j5.

Under these conjugate matched conditions, the total impedance faced by the
generator is the sum of
its' internal impedance Zo and the external Z, i.e. the total impedance is

Zo + conj(Zo) = 50 + j5 + 50 - j5 = 100 Ohms

And so the generator impresses it's 2 Volts of open circuit volts across the
resulting totalof 100 Ohms
supplying a current of I = 2/100 = 20mA to the real part of the load of 50
Ohms for a maximum
power of:

Pmax = I*I*R = 0.02*0.02*50 = 0.0004*50 = 0.02 = 20 mWatts.

Under this conjugate matched condition, where the maximum power of 20mW is
transferred, if we
use the classical definition of rho = (Z - Zo)/(Z + Zo) with Z = 50 - j5 and
Zo = 50 + j5
we get:

rho = b/a = ((50 - j5) - (50 + j5))/((50 - j5) + (50 + j5)) = (-j5 -
j5)/(100) = -j10/100 = -j/10

Since the incident wave is a = 1 Volt, then:

b = rho* a = -j/10 Volts.

With a conjugate match, and using the classical definition of rho, as
measured by a reflectometer using
Zo as it's reference impedance, there will be a reflected voltage of
magnitude 0.1 Volts, or one tenth of a volt at a phase lag of ninety
degrees.

It is interesting to see what value of reflected voltage would be indicated
by a Bird Model 43 in
this circumstance of a perfect conjugate match.

A Bird Model 43 uses an internal reference impedance of R = 50 Ohms and
it implements the "classical" definition of rho. i.e. The Bird Model 43
calculates the reflected
voltage b as b(Bird):

b(BIRD) = rho * a = (Z - 50)/(Z + 50) * 1 = (50 - j5 - 50)/(50 - j5 + 50)
= -j5/(100 - j5)

Actually the Bird cannot indicate phase angles, rather it just computes the
approximate magnitude
of the reflected voltage which in this case would be

|b(Bird)| = 5/sqrt(100*100 + 5*5) = 5/sqrt(1025) = 0.156 Volts.

Compare this to the magnitude of the "true" reflected voltage which is 0.1
Volts as computed by a true reflectometer which used Zo as its' internal
reference impedance.

And so... contrary to popular opinion, the Bird does not indicate zero
reflected power when it is inserted into a perfect conjugate matched
[maximum power transfer] situation when the Zo of the system is not a pure
50 Ohms!

Now if we choose to use a different definition of rho, say the one proposed
by Slick why then we will get different results for the reflected voltage,
in fact with his somewhat erroneous formula for rho he will read rho as zero
in the conjugate matched situation. However. even though one can use any
definition as long as one consistently uses it in all theoretical
developments and measurement, as in one of my other posts to this thread, I
do not believe Slick's or anybody elses re-definition of rho to be approved
by "Mother Nature".

Mother Nature uses rho = (Z - Zo)/(Z + Z0) simply because that is the
reflection coefficient that exists at any point along any infinitely long
transmission line of constant surge impedance Zo where looking to the right
and to the left at any point in the line one sees the same driving point
impedance Zo in both directions and there are no discontinuities in Zo to
cause reflectons, and so the reflection coefficient must be zero and (Zo -
Zo)/(Zo + Zo) is the only formulation of a reflection factor that supports
that condition.

I hope this "treatise" helps you to understand my thoughts on this.

--
Peter K1PO
Indialantic By-the-Sea, FL.

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