Complex Z0 - Power : A Proof
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It seems that
a proof has been carried out
for the validation
of the so-called
Principle of Conservation of Energy
"P.C.E"
in any point of a Transmission Line
with a Complex characteristic impedance,
which is terminated at any passive load.

But,
I have to make an appeal
for some patience...

[1]
---
Let
the Characteristic Impedance

Zo = Ro + j Xo

and the Gamma

g = a + j b

[2]
---
At the Terminal Load

Zt = Rt + j Xt

a "Reflection Coefficient"
is defined by

pt = (Zt - Zo)/(Zt + Zo)
pt = |pt|.Exp(j tt)

On the load
the Average Power
"Wt"
is expressed as[*]

Wt = k.T

where k 0 is a constant
and

T = 1 - |pt|^2 + j (Xo/Ro)(pt - pt*)

[3]
---
With direct substitution of
Zt and Zo in the last one
and after
_enough_but_straightforward_
algebraic manipulation,
which it is impossible to be reproduced here,
we have for the
Average Power at the Terminal Load

[Wt = 0] = [T = 0] = [Rt = 0]

This is an important partial result
which proves that
P.C.E. is valid
on any passive terminal load
of a Transmission Line with complex Zo.

[4]
---
Now,
at any other point on the Transmission Line
at a distance d from the Terminal Load,
a "Reflection Coefficient" is introduced from

pd = pt.Exp(-2.g.d)[*]

and the following are defined

Zd = (1 + pd)/( 1 - pd)
pd = (Zd - Zo)/(Zd + Zo)
pd = |pd|.Exp(j td)

and the relation

|pd| = |pt|.Exp(-2.a.d)

holds between the two
Reflection Coefficients

[5]
---
As in above [2],
the Average Power at any point at distance d is

Wd = k.D

where

D = 1 - |pd|^2 + j (Xo/Ro)(pd - pd*) =
D = 1 - Exp(-4.a.d).|pt|^2 - 2.(Xo/Ro).Exp(-2.a.d).|pt|.Sin(td)

Obviously,
P.C.E. holds,
if and only if

D = 0

or equivalently

2.(Xo/Ro).Exp(-2.a.d).|pt|.Sin(td) = 1 - Exp(-4.a.d).|pt|^2

[6]
---
But, we have

0 Exp(-4.a.d) = 1

and

0 = |pt|^2

therefore

Exp(-4.a.d)|pt|^2 = |pt|^2 =
-|pt|^2 = -Exp(-4.a.d)|pt|^2

and finally

1 - |pt|^2 = 1 - Exp(-4.a.d).|pt|^2

Hence,
for P.C.E validation,
it is sufficient to prove

2(Xo/Ro).Exp(-2.a.d).|pt|.Sin(td) = 1 - |pt|^2

But, in addition, we have
for the maximum value of the left part

0 Exp(-2.a.d) = 1
Sin(td) = 1
(Xo/Ro) = |Xo|/Ro

and therefore the left part
has the relation,
to its maximum value

2.(Xo/Ro).Exp(-2.a.d).|pt|.Sin(td) = 2.(|Xo|/Ro).|pt|

[7]
---
Consequently,
it is sufficient to prove

2.(|Xo|/Ro).|pt| = 1 - |pt|^2

or

|pt|^2 + 2.(|X0|/R0).|pt| -1 = 0

(Note:
(Probably,
(this is possible
(with a direct substitution,
(as in [3]

This is a 2nd degree polynomial in |pt|
with roots:

|pt|- = -|Xo|/Ro - Sqrt(|Xo|^2/Ro^2 + 1) 0
|pt|+ = -|Xo|/Ro + Sqrt(|Xo|^2/Ro^2 + 1) 0

After that,
the P.C.E. is valid at any point
if and only if
the "Reflection Coefficient"
at the terminal load is
lower than or equal to the positive root

0 = |pt| = |pt|+

But

|Xo|/Ro = Tan(|to|)

where tO is the argument of Zo, so

|pt|+ = -Tan(|to|) + Sqrt(Tan(|to|)^2 + 1)

After some trigonometric manipulation
the identity
can be proved

-Tan(|t0|) + Sqrt(Tan(|to|)^2 + 1) =
= Sqrt[(1 + Sin(|to|)/(1 - Sin(|to|)]

Therefore,
the P.C.E. is valid,
at any point,
if and only if
the "Reflection Coefficient"
at the Terminal Load is

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

But this is a relation,
which has been already proved
in the Thread

"Complex Z0 [Corrected]"

QED
I hope,
but,
as any other time
d;^)
your comments are welcomed!

Sincerely,

pez
SV7BAX
[*]
---
e.g. Chipman, 1968, p.138