Permit me please to conclude
my point of view on the Complex Z0 subject
with three
_claims_:

1.
The well known relation

|rho| = 1 + Sqrt[2]

has been sharpened
to be exactly

|rho| = Sqrt[(1+Sin|to|)/(1-Sin|to|)]

where 'to' is the 'angle' of rho.
This is a
_claim_
that no one else has verified it.

2.
Also,
the well known relation

|Xo|/Ro = 1

is sharpened too, to be exactly

|Xo|/Ro = min{ a/b, b/a }.

This is as well a
_claim_
no one else has verified it yet.

3.
The well known non-violation of the
Principle of Conservation of Energy
at any point of a line,
has been found to be surely valid
only at the terminal load.
There is no proof yet of its validation
at any other point.
Once again this is a
_claim_
no one else has verified it.

The impressive worm with which
these facts
have been generally welcomed
by the members of the current Newsgroup
-devoted to the Amateur Truth I presume-
it seems that is a characteristic
of the contemporary Amateur Ethics.

In any case,
I would delightedly look forward
for any contribution
to these three matters.

But until then

-Farewell my Darling, Complex Z0!

Sincerely,

pez
SV7BAX

The a, b are the propagation constants
and of course,
'to' is not the 'angle' of the
Reflection Coefficient rho
but that of
Z0 = Ro +j Xo.

It has rather to do with
the emotional charge of the moments...

Sincerely,

pez
SV7BAX

In articles and ,
"pez" wrote:

1.
The well known relation

|rho| = 1 + Sqrt[2]

has been sharpened
to be exactly

|rho| = Sqrt[(1+Sin|to|)/(1-Sin|to|)]

where 'to' is the 'angle' of rho.


'to' is not the 'angle' of the
Reflection Coefficient rho
but that of
Z0 = Ro +j Xo.

It has rather to do with
the emotional charge of the moments...

Sincerely,

pez
SV7BAX

In article
the argument I gave showed that if you look at an ellipse with foci at
Z_0 and at - Z_0 then among the points Z_L on any such ellipse that are
in the fourth and first quadrants, the one for which the magnitude of
the reflection coefficient (non-conjugate definition, of course) is
maximized is the one on the imaginary axis near - Z_0 and far from Z_0,
i.e. having opposite reactance to that of Z_0. If you then use geometry,
or algebra, or even calculus if you feel compelled to use it, to see
which such point on the imaginary axis is best (i.e., which of the
ellipses of various eccentricities has the best "best" point) you will
find that the maximum occurs when the magnitude of Z_L is the same as
that of Z_0. In symbols, you will want to have
Z_L = +/- j sqrt ((R_0)^2 + (X_0)^2), where Z_0 = R_0 + j X_0 and the
sign to be chosen (+/-) is the opposite to that of X_0. When you use
this value of Z_L you will indeed get a reflection coefficient
magnitude of sqrt [(1 + sin |t_0|)/(1 - sin |t_o|)], thereby
confirming your formula
|rho| = sqrt [(1 + sin |t_o|)/(1 - sin |t_o|)].

This is a
_claim_
that no one else has verified it.

Now some one has. Details left for the reader -- you can think of what I
wrote above as a "Heathkit proof" ;-)! I used to enjoy giving "Heathkit
proofs" for my students to build; as they became more proficient, the
construction manuals offered became sketchier and sketchier. Eventually
the students got to the point where they could design and build their own
rigs (er, proofs) and the days of "Heathkit proofs" were over for them.

2.
Also,
the well known relation

|Xo|/Ro = 1

is sharpened too, to be exactly

|Xo|/Ro = min{ a/b, b/a }.

This is as well a
_claim_
no one else has verified it yet.

I'll see if I can sketch a nice demonstration of this second result,
preferably one that does *not* involve calculus ;-)!

David, ex-W8EZE, retired math professor, and not quite finished with either
of these things

--
David or Jo Anne Ryeburn

To send e-mail, remove the letter "z" from this address.

Dear Keith,

Your conclusions are surely
(specifically) valid
_if_
we recall their assumptions,
which require a transmission line
with purely _Real_ Z0.

But,
lets state the problem
following your terms.

If we have a piece of a proper Complex Z0 line
- that is one with Xo =/= 0 -
terminated by a passive Impedance
then
we still have to prove that
its Input Impedance Formula,
-which, don't forget, is a result of
a _mathematical_ theory-
results in a non negative real part,
equivalent with a passive lumped Impedance,
even if
it has already the _physical_ dimensions
of the Resistance.

In other words
it is neither enough that
it has already
the _physical_ dimensions of the Resistance
nor
it is a valid one
in the special Real Z0 case.

Sincerely,

pez
SV7BAX

wrote in message ...
| pez wrote:
| 3.
| The well known non-violation of the
| Principle of Conservation of Energy
| at any point of a line,
| has been found to be surely valid
| only at the terminal load.
| There is no proof yet of its validation
| at any other point.
| Once again this is a
| _claim_
| no one else has verified it.
|
| Is the following a proof?
|
| At any point on the line, the line can be cut and the load
| side replaced with a lumped impedance which presents the same
| impedance as was presented by the line with its load.
|
| The conditions on the source side of the cut do not change
| when this is done. Since this new lumped impedance is a
| terminal impedance, Conservation of Energy is not violated.
|
| Therefore Conservation of Energy will not be violated when
| this lumped impedance is removed and the line is restored
| to its former state.
|
| ...Keith

Good day Pez,

Thank you. I now understand the challenge.

....Keith

pez wrote:

Dear Keith,

Your conclusions are surely
(specifically) valid
_if_
we recall their assumptions,
which require a transmission line
with purely _Real_ Z0.

But,
lets state the problem
following your terms.

If we have a piece of a proper Complex Z0 line
- that is one with Xo =/= 0 -
terminated by a passive Impedance
then
we still have to prove that
its Input Impedance Formula,
-which, don't forget, is a result of
a _mathematical_ theory-
results in a non negative real part,
equivalent with a passive lumped Impedance,
even if
it has already the _physical_ dimensions
of the Resistance.

In other words
it is neither enough that
it has already
the _physical_ dimensions of the Resistance
nor
it is a valid one
in the special Real Z0 case.

Sincerely,

pez
SV7BAX

wrote in message ...
| pez wrote:
| 3.
| The well known non-violation of the
| Principle of Conservation of Energy
| at any point of a line,
| has been found to be surely valid
| only at the terminal load.
| There is no proof yet of its validation
| at any other point.
| Once again this is a
| _claim_
| no one else has verified it.
|
| Is the following a proof?
|
| At any point on the line, the line can be cut and the load
| side replaced with a lumped impedance which presents the same
| impedance as was presented by the line with its load.
|
| The conditions on the source side of the cut do not change
| when this is done. Since this new lumped impedance is a
| terminal impedance, Conservation of Energy is not violated.
|
| Therefore Conservation of Energy will not be violated when
| this lumped impedance is removed and the line is restored
| to its former state.
|
| ...Keith