[i] Any Waves
-------------
The Google search tool returns ~6/49 references,
in the rec.radio.amateur.antenna newsgroup,
for "Kurokawa", the author of the paper
"Power Waves and Scattering Matrix",
IEEE Transactions on Microwave Theory and Techniques,
vol. MTT-13, March 1965, pp 194-202.

This search result indicates a subject of some interest.

Let us take Kurokawa's conclusion for granted:

"...the power waves are the result
of just one of an infinite number
of possible linear transformations
of voltage and current...".

Following this,
it seems that,
at least mathematically,
we can choose
any four complex constants,

p, q, r, s

and define

a, b

by the linear transformation

a = p.V + q.I
b = r.V + s.I

But as far as we consider both of

V, I

to be waves,

since they are solutions of the
-notably the same-
"One-Dimensional Wave Equation":

X'' - (g^2).X = 0

where X stands for either V or I,
it is trivial to show that each of the just defined

a, b

verify exactly the same equation.

Therefore,
it seems legitimate to call

a, b

"Any Waves", too.

Thus, as we see, we can define,
from a unique couple of (V, I) Voltage and Current waves
"an infinite number" couples of (a, b) "Any Waves",
since, the four mentioned constants can take obviously
an infinite number of values.

[II] Mathematical Restrictions
-----------------------------
If we intend to use the Any Waves mathematically,
e.g. perhaps to facilitate a manipulation of formulas
which requires in some step
the inverse expression of V, I from the a, b,
then we have to impose
a restriction for the existence
of the inverse transformation,
that is

(1): p.s =/= q.r

After that
the

V, I

can be result from the

a, b

as

V = A.a + B.b
I = C.a + D.b

where the new four complex constants

A, B, C, D

are given in terms of the old constants

p, q, r, s

by the relations

A = [ s/(p.s - q.r)]
B = [-q/(p.s - q.r)]
C = [-r/(p.s - q.r)]
D = [ p/(p.s - q.r)]

since the denominator is non-zero.

The condition (1)
although restricts somehow the otherwise totally independent cases,
"The Any Waves" remain of "an infinite number".

[III] Dimension Balance Restrictions
------------------------------------
Now if we would like to certify physically,
the validity of the introduced transformations,
then we have to establish
the "(Dimensional) Unit Balance" of these equations.

To do this,
let us note by

{y}

the Unit(s) of any Physical Quantity y.

Then, from the above equations,
it is almost obvious that
the following relations must be valid:

(2.1): {q} = {Ohms}.{p}
(2.2): {s} = {Ohms}.{r}
(2.3): {a} = {Volt}.{p}
(2.4): {b} = {Volt}.{r}

The added conditions (2),
although reduce the number of
the independently varied Physical Quantities (Units) to two,
they do not affect in essence "The Any Waves"
which are still of "an infinite number".

[IV] Restrictions referenced to Physical Principles
---------------------------------------------------
Next let us impose some specific
physical considerations or principles
on "The Any Waves",
e.g. that of the conservation of energy.

As an example of this
we can consider
as special cases of "The Any Waves"
those mentioned in the referenced paper,
that is the
"Power Waves" and "Traveling Waves".

Indeed;
these waves fulfill the linear transformations,
with all constants specifically expressed
in terms of some impedances,
measured as follows

{p} = {r} = 1/Sqrt(Ohms} = {Ohms}^(-1/2),
{q} = {s} = Sqrt{Ohms} = {Ohms}^(+1/2)

and from them,
the "Physical Units" of both the "Waves" are the same:

(3): {a} = {b} = Sqrt{Watt}.

Once again the additional restriction (3)
although forces the a and b to be of the same physical entity,
one which somehow is connected to the power,
still keeps "The Any Waves" to "an infinite number".

[V] Measurement Restrictions
----------------------------
Further we can assume, for the sake of completeness,
that maybe there is a -necessarily finite- number
of appropriate materials and/or devices,
other than the well known reflectometer,
capable to set apart the Two Any Wave Components
on the basis of some Physical Properties.

If this is the case then it is reasonable to try to build
a linear transformation
in terms of a couple of wave quantities appropriate to
express the mentioned physical properties.

But neither this case can reduce the possibly of
"The Any Waves" "of an infinite number".

[VI] Physical Hypotheses Restrictions
-------------------------------------
Finally, we can imagine that
we ingeniously predict some extraordinary Any Waves,
with the proof of their existence
to become a subject of a life-long heavy research activity.

For instance;
if for some reason we define the constants
by using some "extreme" expressions of the Impedance,
such as the following

{p} = (Ohms}^k
{q} = {Ohms}^n

with k, n equal to any other non-zero rational number,
e.g.
.... -100, -1, -1/3, 1, 1/3, 100 ...

then "The Any Waves" have now
the extraordinary Physical Meaning
suggested by their corresponding physical units expressions

{a} = {Ohms^(k+(1/2)}.{W^(1/2)}
{b} = {Ohms^(n+(1/2)}.{W^(1/2)}

Fine;
but "The Any Waves" stay "of an infinite number".

[VII] Practical Applications
----------------------------
Let us drop from the clouds...

In the practical application of the Transmission Line,
"The Subtle Detail" which discriminates
the two instances of "The Any Waves",
"The Power Waves" and "The Traveling Waves",
with two different physical meanings for the same physical phenomenon
and the concrete result of "The Third Term" disappearance
in the "Beloved" Complex Characteristic Impedance case,
increases the confusion instead of comprehension.

Accordingly and as far as we are not in place
to reduce the number of "The Any Waves",
regardless of their more or less well established physical meaning,
in something less than infinity,
any further attempt to discuss about "what actually happens"
in the totality of the related practical applications
becomes redundant and worthless.

It seems that we have to content ourselves
with what it is already known...

Sincerely,

pez
SV7BAX
TheDAG

Pez wrote -
It seems that we have to content ourselves
with what it is already known...

==========================

I am content.
----
Reg, G4FGQ

pez wrote:
Accordingly and as far as we are not in place
to reduce the number of "The Any Waves",
regardless of their more or less well established physical meaning,
in something less than infinity,
any further attempt to discuss about "what actually happens"
in the totality of the related practical applications
becomes redundant and worthless.

It seems that we have to content ourselves
with what it is already known...

Assuming an s-parameter analysis is one of the things already known ...

the equations b1 = s11*a1 + s12*a2 and b2 = s21*a1 + s22*a2

should be familiar. HP's AN 95-1 defines some powers. Assume an
impedance discontinuity in a transmission line, e.g. where 50 ohm
coax interfaces with 300 ohm twinlead aka G5RV style:

|a1|^2 is the power incident on the input of the network
|a2|^2 is the power incident on the output of the network
|b1|^2 is the power reflected from the input port
|b2|^2 is the power reflected from the output port

If we take the equation, b1 = s11*a1 + s12*a2 and square it
we get |b1|^2 = |s11*a1|^2 + |s12*a2|^2 + 2*s11*a1*s12*a2

This takes the form of Dr. Best's equation Ptot = P1 + P2 + 2[sqrt(P1)*sqrt(P2)]
in his QEX article which is also the form of Hecht's irradiance equation in _Optics_
which is: Itot = I1 + I2 + 2*sqrt(I1*I2)

In optics, 2*sqrt*I1*I2) is known as the "interference term". It's
pretty obvious that 2*s11*a1*s12*a2 is that same interference term
in an s-parameter analysis of RF waves. It follows that what happens
at a Z0-match point in a transmission line involves destructive and
constructive interference.

What?
--
73, Cecil http://www.qsl.net/w5dxp



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Reg Edwards wrote:
It seems that we have to content ourselves
with what it is already known...

I am content.

Dang Reg, I'm disappointed in you. I'm not content.
I can't wait to learn something new every day.
--
73, Cecil http://www.qsl.net/w5dxp



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Dear Mr. Cecil Moore,

I am terribly sorry but
I am not so sure that
I am in position to follow your argument,
except perhaps that
the equation

| ...
| |b1|^2 = |s11*a1|^2 + |s12*a2|^2 + 2*s11*a1*s12*a2
| ...

needs a modification to
the third term on its right hand side,
in which,
instead of

2*s11*a1*s12*a2

we have to set

2*Re{[s11*a1]*Conjg[s12*a2]}

Therefore,
I can only guess that
this subtle distinction is the source of the trouble
because this is maybe due to
the sure existence of two,
after Kurokawa,
different physical meanings
for the same physical phenomenon.

And finally,
unfortunately enough,
it seems that there are
maybe more than a finite number
of possible such physical meanings
for the same physical phenomenon...

Sincerely yours,

pez
SV7BAX
TheDAG

pez wrote:
Dear Mr. Cecil Moore,

I am terribly sorry but I am not so sure that
I am in position to follow your argument,
except perhaps that the equation

| ...
| |b1|^2 = |s11*a1|^2 + |s12*a2|^2 + 2*s11*a1*s12*a2
| ...

needs a modification to
the third term on its right hand side,
in which, instead of

2*s11*a1*s12*a2

we have to set

2*Re{[s11*a1]*Conjg[s12*a2]}

That is taken care of by the cosine of the angle between
a1 and a2. Note no magnitude bars around that term in my
equation. a1*a2 is phasor multiplication, a1*a2*cos(theta).

Therefore,
I can only guess that
this subtle distinction is the source of the trouble
because this is maybe due to the sure existence of two,
after Kurokawa, different physical meanings
for the same physical phenomenon.

And finally, unfortunately enough,
it seems that there are maybe more than a finite number
of possible such physical meanings for the same physical phenomenon...

What I was trying to point out is the similarities between
Dr. Best's QEX article term, 2*sqrt(P1)*sqrt(P2)*cos(theta),
In _Optics_, Hecht's interference term 2*sqrt(I1*I2)*cos(theta),
and the above 2*s11*s12*a1*a2(cos theta) term. Seems to me, they
are all interference terms. If (0 deg = theta 90 deg) then the
interference is constructive. If (90 deg theta = 180 deg) then
the interference is destructive.
--
73, Cecil, W5DXP

Cecil Moore wrote:
What I was trying to point out is the similarities between
Dr. Best's QEX article term, 2*sqrt(P1)*sqrt(P2)*cos(theta),
In _Optics_, Hecht's interference term 2*sqrt(I1*I2)*cos(theta),
and the above 2*s11*s12*a1*a2(cos theta) term. Seems to me, they
are all interference terms. If (0 deg = theta 90 deg) then the
interference is constructive. If (90 deg theta = 180 deg) then
the interference is destructive.
--
73, Cecil, W5DXP

(a + b) * (a + b) = a^2 + 2ab + b^2

Wow, I made my own "interference term"!

73 de jk

Jim Kelley wrote:

Cecil Moore wrote:
What I was trying to point out is the similarities between
Dr. Best's QEX article term, 2*sqrt(P1)*sqrt(P2)*cos(theta),
In _Optics_, Hecht's interference term 2*sqrt(I1*I2)*cos(theta),
and the above 2*s11*s12*a1*a2(cos theta) term. Seems to me, they
are all interference terms. If (0 deg = theta 90 deg) then the
interference is constructive. If (90 deg theta = 180 deg) then
the interference is destructive.

(a + b) * (a + b) = a^2 + 2ab + b^2

Wow, I made my own "interference term"!

Yep, assuming those are phasor voltages normalized to the
square root of Z0, you sure did. That's one of the advantages
of an s-parameter analysis. With the voltages normalized to
the square root of Z0, the square of any voltage equals
V^2/Z0 = power.

a^2 would be P1, b^2 would be P2, and 2ab would equal to
2*sqrt(P1)*sqrt(P2)*cos(theta) [phase angle between a and b]
--
73, Cecil, W5DXP

Dear Mr. Cecil Moore,

I think now I understand the notation.
If

| ...
| a1*a2 is phasor multiplication, a1*a2*cos(theta)
| ...

then s11*a1*s12*a2
is s11*a1*s12*a2*cos(phi)

which has the same meaning as the

Re{[s11*a1]*Conjg[s12*a2]}

and the only thing which remains
is the specification of the phi range of values.

I think also,
I grasped your point of view
for the existence of a deeper, common, base between
these different physical phenomena,
with which anyone hardly disagrees.

Finally,
I would ask you to tell me please,
the reference details to
Dr. Best's QEX article and Hecht's _Optics_.

Thanking you in advance,

pez
SV7BAX
TheDAG

"Cecil Moore" wrote in message
...
Jim Kelley wrote:
(a + b) * (a + b) = a^2 + 2ab + b^2

Wow, I made my own "interference term"!

Yep, assuming those are phasor voltages normalized to the
square root of Z0, you sure did.

a^2 would be P1

I call the a^2 term Cecil. :-)

73 jk