W9DMK (Robert Lay) wrote:
On Sat, 02 Jul 2005 08:21:07 -0500, Cecil Moore
wrote:


The first example was much too easy. How about this one?

---50 ohm feedline---+---300 ohm feedline---
Pfwd1=100w-- Pfwd2 not given--
--Pref1=0w --Pref2 not given

Given a Z0-match at point '+':
Solve for Vfwd1, Ifwd1, Vref1, Iref1, Pfwd2, Vfwd2, Ifwd2,
Pref2, Vref2, Iref2, including magnitudes and phase angles
for all voltages and currents. Source is unknown. Load is
unknown. Lengths of feedlines are unknown.


Without disclosing the answers or the exact procedure for solving the
"brain teaser", I would like to draw attention to some of the implicit
relationships that "ought" to help.
1) It is assumed that both feelines have purely resistive
characteristic impedances (imaginary component, Xo, is zero).
2) Regardless of the length of the 300 ohm line and its termination
impedance, the standing wave pattern and the voltages and currents,
both incident and reflected as a function of distance x along that
line are determined completely by the requirement/condition that there
is a Z0 match at point "+".
3) There are an infinite number of lengths of the 300 ohm line and a
corresponding infinite number of termination impedances for that line
that will produce a Z0 match at point "+". However, because of (2),
above, some of those combinations are well known combinations with
well understood results (e.g., odd multiple of quarter wavelength or
an integer number of half wavelengths).
4) Due to conditions (1) and (2) above, the phase relations between
all of the voltages and currents immediately adjacent to either side
of point "+" are trivial (i.e., any two quantities chosen will be
either exactly in phase or exactly 180 degrees out of phase with one
another).

Due to (3) and (4) above, it would seem that an arbitrary choice of
either a quarter wave line with an 1800 ohm termination or a half wave
line with a 50 ohm termination would provide a convenient example with
which to begin an analysis. However, that is not necessary and only
provides a crutch to get off dead center.

If all of the above elements are kept in mind, then it becomes a
matter of solving a simple algebraic relationship involving 4
equations with 4 unknowns (the incident and reflected voltages and
currents at the right hand side of point "+").

The actual numerical answer to such a problem is irrelevant. The
points to be learned from all this are really the implicit
relationships (2), (3) and (4) above. Without an understanding of
those points, it is virtually impossible to even know where to start.
I think that is the real point that Cecil is trying to make.

Bob, W9DMK, Dahlgren, VA
Replace "nobody" with my callsign for e-mail
http://www.qsl.net/w9dmk
http://zaffora/f2o.org/W9DMK/W9dmk.html


Cecil already defined the voltage and current at
the match point when he gave the characteristic
impedances of the two lines and the rate of
energy transfer through them. Knowing the voltage
and current, anyone can calculate
Pfwd2 and Prev2 using Pfwd2 = |(V+IZ0)/2sqrt(Z0)|^2 and
Prev = |(V-IZ0)/2sqrt(Z0)|^2, where Z0 is the characteristic
impedance of the second transmission line.
Cecil's ability to add powers together, which he did in
this instance, isn't anything unique, and doesn't
really teach anything about the general case.
In fact, for a quarter wave transformer, you can
do the following trick: compute the value of the
power as it just comes through the impedance discontinuity
for the first time and call it Pa. Call Rho^2 at the
load P. Then the power delivered to the load will be
Pa( 1 + P + P^2 + P^3 + P^4 ....) which looks the
same as if the power reflection coefficient looking
back toward the generator was 1 and the power at the
load was the result of the addition of an infinite
number of reflections. Such an interpretation, though,
can be shown to be absolutely wrong. Can anyone see why?
73,
Tom Donaly, KA6RUH

Tom Donaly wrote:
Cecil's ability to add powers together, which he did in
this instance, isn't anything unique, and doesn't
really teach anything about the general case.

I'm glad you agree, Tom. Other experts on this newsgroup
will argue with you as they have with me for four years
ever since Dr. Best posted his infamous Z0-match equation:

Ptot = 75w + 8.33w = 133.33w

to which I objected back then, only to have most of
the rest of the posters agree with Dr. Best. I was
dumbfounded to see so many otherwise knowledgable
engineers agree to a violation of the principle of
conservation of energy. I was told not to worry about
conservation of energy - that it takes care of itself.

In fact, for a quarter wave transformer, you can
do the following trick: compute the value of the
power as it just comes through the impedance discontinuity
for the first time and call it Pa. Call Rho^2 at the
load P. Then the power delivered to the load will be
Pa( 1 + P + P^2 + P^3 + P^4 ....) which looks the
same as if the power reflection coefficient looking
back toward the generator was 1 and the power at the
load was the result of the addition of an infinite
number of reflections. Such an interpretation, though,
can be shown to be absolutely wrong. Can anyone see why?

Destructive interference between the external reflection
at the match point and the internal reflection from the
load supplies additional constructive interference
energy to the forward wave in the quarter wave transformer.
You didn't include that constructive interference energy
above. Hint: That virtual power reflection coefficient looking
rearward into the match point doesn't reach 1 until steady-
state is reached (wrong premise above). The virtual power
reflection coefficient looking forward into the match point
also doesn't reach 0 until steady-state is reached. Those
two virtual power reflection coefficients actually start out
the same value and proceed in opposite directions during
the transient buildup to steady-state.
--
73, Cecil http://www.qsl.net/w5dxp


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Cecil Moore wrote:
Tom Donaly wrote:

Cecil's ability to add powers together, which he did in
this instance, isn't anything unique, and doesn't
really teach anything about the general case.


I'm glad you agree, Tom. Other experts on this newsgroup
will argue with you as they have with me for four years
ever since Dr. Best posted his infamous Z0-match equation:

Ptot = 75w + 8.33w = 133.33w

to which I objected back then, only to have most of
the rest of the posters agree with Dr. Best. I was
dumbfounded to see so many otherwise knowledgable
engineers agree to a violation of the principle of
conservation of energy. I was told not to worry about
conservation of energy - that it takes care of itself.

In fact, for a quarter wave transformer, you can
do the following trick: compute the value of the
power as it just comes through the impedance discontinuity
for the first time and call it Pa. Call Rho^2 at the
load P. Then the power delivered to the load will be
Pa( 1 + P + P^2 + P^3 + P^4 ....) which looks the
same as if the power reflection coefficient looking
back toward the generator was 1 and the power at the
load was the result of the addition of an infinite
number of reflections. Such an interpretation, though,
can be shown to be absolutely wrong. Can anyone see why?


Destructive interference between the external reflection
at the match point and the internal reflection from the
load supplies additional constructive interference
energy to the forward wave in the quarter wave transformer.
You didn't include that constructive interference energy
above. Hint: That virtual power reflection coefficient looking
rearward into the match point doesn't reach 1 until steady-
state is reached (wrong premise above). The virtual power
reflection coefficient looking forward into the match point
also doesn't reach 0 until steady-state is reached. Those
two virtual power reflection coefficients actually start out
the same value and proceed in opposite directions during
the transient buildup to steady-state.

Hi Cecil,
you come up with the right answer, but is your
interpretation correct? Can you do the same thing in a
general sense? If there is no Z0 match between the two
transmission lines, does your method still work? The
little conundrum I posed is an example of a procedure
that will actually give the right answer, but the
interpretation I gave of how it works is wrong. Can you
be sure your method doesn't have the same flaw?
73,
Tom Donaly, KA6RUH
(P.S. The method of using V and I and the junction of
the two xmission lines to find the forward and reverse
powers on a transmission line doesn't prove the powers exist.
It works just as easily with a pair of resistors and is
more an algebraic stunt that works than anything else. It
does agree with you, however.)

Tom Donaly wrote:
you come up with the right answer, but is your
interpretation correct? Can you do the same thing in a
general sense? If there is no Z0 match between the two
transmission lines, does your method still work?

As a stand alone analysis, it yields two possible solutions
but the purpose of this discussion is not to come up with
a new stand alone method of analysis. The purpose is, given
a standard analysis, to add TRACKING OF THE ENERGY COMPONENTS
through an impedance discontinuity, something many people
believe to be impossible.

It wasn't designed to work as a stand alone analysis but
it does for Z0-matched systems, the most usual ham
configuration. However, an additional piece of information
is required in the general case to be able to tell which
voltage is leading and which is lagging.

(P.S. The method of using V and I and the junction of
the two xmission lines to find the forward and reverse
powers on a transmission line doesn't prove the powers exist.

Do you think the powers defined in HP App Note 95-1 exist?
Remember my one second long transmission line example where
the number of stored joules exactly equaled the number of joules
required by the forward wave and the reflected wave? If the
energy is not in those waves, where is it? Nobody has
provided any explaination of how standing waves can exist
without forward and reflected waves. Under "standing wave",
The IEEE Dictionary says: "A pure standing wave results
from the interference of two oppositely directed traveling
waves of the same frequency and amplitude." i.e. standing
waves are the result (effect), two oppositely directed
traveling waves are the *cause*. Most of my references
agree. The forward and reflected wave energy components
must exist as causes before standing waves can materialize
as an effect.
--
73, Cecil http://www.qsl.net/w5dxp


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