Complex Z0 Clearness
[Complex Z0 mini-Compendium for Example Builders - Part III]

With the help of
Mrs. yin,SV7DMC
it was mechanically proved
that:

The transformation
between any set of
Propagation Characteristics
(a, b, Ro, Xo),
and its corresponding set of
Distributed Circuit Coefficients
(R', wL', G', wC')
as well as of its inverse,
for any proper complex Z0,
namely one with X=/=0,
is

one-to-one

everywhere
on its domain and range of values,
since its Jacobian equals to

4.|g|^2/|Zo|^2

that is to say
an obviously non-zero number.


This result
definitively clears any reservation
on the building of examples
within the frame of the
Uniform Transmission Line Theory,
as long as
the propounded conditions
are checked for their validation.

Sincerely,

pez
SV7BAX
&
yin
SV7DMC

Hello,

Can you explain that in a simple sentence as what you have copied from a
book doesn't really make much sense!
Do you actually know what it means.

"pez" wrote in message
...
Complex Z0 Clearness
[Complex Z0 mini-Compendium for Example Builders - Part III]

With the help of
Mrs. yin,SV7DMC
it was mechanically proved
that:

The transformation
between any set of
Propagation Characteristics
(a, b, Ro, Xo),
and its corresponding set of
Distributed Circuit Coefficients
(R', wL', G', wC')
as well as of its inverse,
for any proper complex Z0,
namely one with X=/=0,
is

one-to-one

everywhere
on its domain and range of values,
since its Jacobian equals to

4.|g|^2/|Zo|^2

that is to say
an obviously non-zero number.


This result
definitively clears any reservation
on the building of examples
within the frame of the
Uniform Transmission Line Theory,
as long as
the propounded conditions
are checked for their validation.

Sincerely,

pez
SV7BAX
&
yin
SV7DMC

| Geoff wrote:
|
| Hello,
|
| Can you explain that in a simple sentence
| as what you have copied from a
| book doesn't really make much sense!
| Do you actually know what it means.
|

Dear Geoff,

I am not so sure about what I have to explain
to prove that
I did not made just a copy of a book sentence...

Okay,
I will try,
I will do my best,
but I am afraid,
I am not in position
to do it in a simple sentence.
I need something more than this...

Let us suppose that
-for some reason, see below at (*)-
we begin by choosing, a set of four
"Propagation Characteristics"-"PCs" values
that is,
a couple of non negative values for a and b, not both zeroes,
a positive value for Ro
and a negative/zero/positive value for Xo.

This choice belongs necessarily
to one of the eleven cases referenced in the opening message
at the thread
"Complex Z0 Consistency".

From the above categorization and
according to which of additional condition
-if there is any-
is fulfilled,
we know definitely,
the kind of values of
the four "Distributed Circuit Parameters"-"DCCs":

These values are either zero or positive.

After that
we can apply a set of book formulas
-yes, here we need some copies from a _book_-
to calculate the DCCs.

It is now the time when
it comes the "Clearness" property
to assure us
that there is no reason to worry
about the calculated DCCs.
Apart from the fact that
these are the proper ones
we are sure,
without any further doubt,
that in addition this is
_the_only_one_
set of parameters
which obey the complete set of our requirements.

What is the next step, it depends.

I can think, off the cuff, two possibilities (*):

I.
If our task was to design and construct
a line presenting the
desired PCs
we have just completed only the easy very first step,
that of the design.
To go to the construction step we have to look for
the proper geometry and materials for a line,
capable to produce the calculated DCCs.
This is a truly difficult practical problem, indeed.

II.
If our desire was to built a working example
to enlighten or demonstrate
some aspects of the theory
then we are now fully equipped with the right
eight in total parameters plus frequency
to go on
without to worry that
any inconsistency surprise skulks in every next corner.

And conversely.

All the story is to be told once again,
but now beginning from DCCs to end with PCs.
Plus we have to deal with an analysis problem this time.

As I told you at the beginning,
-lets take into account the language barrier-
I am not so sure
if I attained to convince you,
this is the only way
by which we can gain a foothold
to concentrate exclusively,
devoid of cares,
in our central aim,
which usually is
to construct a line or build an example.

But I believe it is now
a complete and a right one.


Sincerely,

pez
SV7BAX

| Geoff wrote:
|
| Hello,
|
| Can you explain that in a simple sentence
| as what you have copied from a
| book doesn't really make much sense!
| Do you actually know what it means.
|

Dear Geoff,

I am not so sure about what I have to explain
to prove that
I did not made just a copy of a book sentence...

Okay,
I will try,
I will do my best,
but I am afraid,
I am not in position
to do it in a simple sentence.
I need something more than this...

Let us suppose that
-for some reason, see below at (*)-
we begin by choosing, a set of four
"Propagation Characteristics"-"PCs" values
that is,
a couple of non negative values for a and b, not both zeroes,
a positive value for Ro
and a negative/zero/positive value for Xo.

This choice belongs necessarily
to one of the eleven cases referenced in the opening message
at the thread
"Complex Z0 Consistency".

From the above categorization and
according to which of additional condition
-if there is any-
is fulfilled,
we know definitely,
the kind of values of
the four "Distributed Circuit Parameters"-"DCCs":

These values are either zero or positive.

After that
we can apply a set of book formulas
-yes, here we need some copies from a _book_-
to calculate the DCCs.

It is now the time when
it comes the "Clearness" property
to assure us
that there is no reason to worry
about the calculated DCCs.
Apart from the fact that
these are the proper ones
we are sure,
without any further doubt,
that in addition this is
_the_only_one_
set of parameters
which obey the complete set of our requirements.

What is the next step, it depends.

I can think, off the cuff, two possibilities (*):

I.
If our task was to design and construct
a line presenting the
desired PCs
we have just completed only the easy very first step,
that of the design.
To go to the construction step we have to look for
the proper geometry and materials for a line,
capable to produce the calculated DCCs.
This is a truly difficult practical problem, indeed.

II.
If our desire was to built a working example
to enlighten or demonstrate
some aspects of the theory
then we are now fully equipped with the right
eight in total parameters plus frequency
to go on
without to worry that
any inconsistency surprise skulks in every next corner.

And conversely.

All the story is to be told once again,
but now beginning from DCCs to end with PCs.
Plus we have to deal with an analysis problem this time.

As I told you at the beginning,
-lets take into account the language barrier-
I am not so sure
if I attained to convince you,
this is the only way
by which we can gain a foothold
to concentrate exclusively,
devoid of cares,
in our central aim,
which usually is
to construct a line or build an example.

But I believe it is now
a complete and a right one.


Sincerely,

pez
SV7BAX