Complex Z0 - Power : A Proof
[Please use fixed-space font]

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

The Missing Step
[Please use fixed-space font]

For any patient reader,
the Missing Step
of the validation of the
Principle of Conservation of Energy
at the Terminal Load,
is now possible to be reproduced here...

[1]
---
Given a Transmission Line with

Zo = Ro + j Xo
Yo = 1/Zo

the following are defined

Vi = V1.Exp[-g.l]
Vr = V2.Exp[+g.l]
V = Vi + Vr

p = Vr/Vi
V = Vi.(1 + p)

Ii = I1.Exp[-g.l]
Ir = I2.Exp[+g.l]
I = Ii + Ir

I1 = Yo.V1
I2 =-Yo.V2
Ii = Yo.Vi
Ir =-Yo.Vr
I = Yo.[Vi - Vr]
I = Yo.Vi.(1 - p)

Z = V/I
Z = Zo.(1 + p)/(1 - p)
p =(Z - Zo)/(Z + Zo)

[2]
---
At any point
the Complex Power is given by

P = V.I* = Vi.(1 + p).Yo*.Vi*.(1 - p*)

where

Yo*= Yo.Yo*/Yo = Zo/|Zo|^2

therefore

P =(|Vi/Zo|^2).Zo.(1 + p).(1 - p*)
P =(|Vi/Zo|^2).Zo.(1 - p* + p -|p|^2)
P =(|Vi/Zo|^2).(Ro + j Xo).[(1 -|p|^2) + (p - p*)]
P =(|Vi/Zo|^2).Ro.[1 + j (Xo/Ro)].[(1 -|p|^2) + (p - p*)]

and since

Vi = V1.Exp[-2.a.l].Exp[-j 2.b.l]
|Vi|^2 = (|V1|^2).Exp[-2.a.l]

we can define

k = (|Vi/Zo|^2).Ro = (|V1/Zo|^2).Ro.Exp[-2.a.l] 0

The Complex Power becomes

P = k.[1 + j (Xo/Ro)].[(1 -|p|^2) + (p - p*)]

where

p - p* = j 2 Im{p}

therefore

P = k.[1 + j (Xo/Ro)].[(1 -|p|^2) + j 2 Im{p}]
P = k.{[(1 -|p|^2) - 2.(Xo/Ro).Im{p}] + j [(Xo/Ro).(1 -|p|^2) + 2 Im{p}]

But

P = PR + j PI
PR = k.{[(1 -|p|^2) - 2.(Xo/Ro).Im{p}]

and

-2.Im{p} = j (p - p*)

Finally,
at any point,
the Average Power is

PR = k.[(1 -|p|^2) + j (Xo/Ro).(p - p*)]

[3]
---
At any point at a distance l from the input

0 = l = L
p =(V2/V1).Exp[2.g.l]

At the Terminal Load

l = L
pt =(V2/V1).Exp[2.g.L]
Z = Zt
pt =(Zt - Zo)/(Zt + Zo)

hence

Wt = PR(p=pt)
Wt = k.T
T = 1 -|pt|^2 + j (Xo/Ro).(pt - pt*)

But we have

|pt|^2 =(|Zt - Zo|^2)/(|Zt + Zo|^2)

pt - pt* = (Zt - Zo)/(Zt + Zo) - (Zt* - Zo*)/(Zt* + Zo*) =
= [(Zt - Zo).(Zt* + Zo*) - (Zt + Zo).(Zt* - Zo*)]/(|Zt + Zo|^2)

Therefore

T.(|Zt + Zo|^2).Ro =
=(|Zt + Zo|^2).Ro - (|Zt - Zo|^2).Ro +
j Xo.[(Zt - Zo).(Zt* + Zo*) - (Zt + Zo).(Zt* - Zo*)]

= Ro.[(Rt + Ro)^2 + (Xt + Xo)^2 - (Rt - Ro)^2 - (Xt - Xo)^2] +
+ j Xo.[Zt.Zt* + Zt.Zo* - Zo.Zt* - Zo.Zo* - Zt.Zt* + Zt.Zo* - Zo.Zt* + Zo.Zo*) =

= Ro.[Rt^2 + Ro^2 + 2.Rt.Ro + Xt^2 + Xo^2 + 2.Xt.Xo -
-Rt^2 - Ro^2 + 2.Rt.Ro - Xt^2 - Xo^2 + 2.Xt.Xo]
+j 2.Xo.(Zt.Zo* - Zo.Zt*)

= 4.Ro.[Rt.Ro + Xt.Xo] + j 2.Xo.(Zt.Zo* - Zo.Zt*)

But

j 2.Xo.(Zt.Zo* - Zo.Zt*) =
j 2.Xo.[(Rt + j Xt).(Ro - j Xo) - (Ro + j Xo)(Rt - j Xt)} =
j 2.Xo.[Rt.Ro - j Rt.Xo + j Xt.Ro + Xt.Xo - Ro.Rt + j Ro.Xt -j Xo.Rt - Xo.Xt]=
j 2.Xo.[ j {-Rt.Xo + Xt.Ro + Ro.Xt - Xo.Rt}]=
-2.Xo.[-Rt.Xo + Xt.Ro + Ro.Xt - Xo.Rt]

Hence

T.(|Zt + Zo|^2).Ro =
4.Ro^2.Rt + 4.Ro.Xt.Xo + 2.Rt.Xo^2 - 2.Xo.Xt.Ro - 2.Xo.Ro.Xt + 2.Xo^2.Rt =
4.Ro^2.Rt + 4.Xo^2.Rt =
4.(|Zo|^2).Rt

Finally we take

T.(|Zt + Zo|^2).Ro = 4.(|Zo|^2).Rt

from which

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

QED, I hope.

And as usually d;^)
"Comments are welcomed"...

Sincerely,

pez
SV7BAX