Reflection Coefficient for unmatched condition

If Γ_in0=0 then Γ_in1=0=Γ_in0 for network N.
This is easy to prove.

Even if Γ_in0=x!=0 then Γ_in1=x=Γ_in0 for same network N.
This also can be satisfied.
But this is difficult to prove.

How can I prove this ?

That is true for the reflection factor magnitude, not for the complex value.

The derivation that I am aware of is power conservation for lossless networks: reflected + transmitted = incident
thus |S11|^2 + |S21|^2 = 1
and for a reciprocal network with S21=S12 we then find |S11| = |S22|

No.
True for complex value.

Wrong.

Only abs(Γ_1in)=abs(Γ_2in) might be satisfied as you say.

I will show example tomorrow.

Only the magnitudes are equal, the phases depend on the network properties.

This is sometimes a useful degree of freedom for adjusting the source impedance on one side.

you cannot prove this! i believe its wrong

You are all correct.
Thanks for correction.

|ΔΓ| is used as metrics of backscatter strength in RFID Tag.
|ΔΓ| = |Γ_{mod_on} - Γ_{mod_off}|

For two unmatched conditions, state1 and state2, I expect following relation.

|Γ_in0@state1- Γ_in0@state2| = |Γ_in1@state1- Γ_in1@state2|

Can this relation be satisfied ?

i used to do scattering parameters using wave theory. you can prove above equations using wave theory.

What do you mean by "wave theory" ?

Pseudo Wave Theory ?
Power Wave Theory ?

Not so easy, since reference impedances are different for port1 and port2.
And former is real number, latter is complex number.

Even if we use unitary matrix nature of S-matrix, proof is not easy.
However I can prove for matched case using unitary matrix nature of S-matrix, since equations are fairly simple in this case.

it can be done



https://en.wikipedia.org/wiki/Scattering_parameters



look at the proof they did for s-parameters wikipedia. i have a great book to suggest as well but forgot the name!

I also have my derivation done but its not with me currently.



its power wave theory

Can you truely understand Power Wave Theory ?

Many people in this forum don't know Power Wave Formulation.

See my appends in the followings.
https://www.edaboard.com/showthread.php?347861
https://www.edaboard.com/showthread.php?378353

yes, I have done derivation of scattering parameters using power wave theory.

Answer my question theoretically.

i will do it over the weekend and post it here. currently busy with projects.

May I ask the name of the book? I study from Pozar, Razavi's and Radmanesh's books about the uWave/RF/Emt, it would be nice to learn a new one.

Could you clarify exactly what you mean by the terms in |Γ_in0@state1- Γ_in0@state2| = |Γ_in1@state1- Γ_in1@state2|?

Is the difference between state1 and state2 defined by a change in the properties of the impedance transformer between the source and load? Or a change in the load impedance?

A change in the load impedance under unchanged reference impedance value.

ZL0=20-j*35

[Example-1]
state1 : ZL=PortZ(4)=ZL0 / 50
state2 : ZL=PortZ(4)=ZL0 / 2

[Example-1]
state1 : ZL=PortZ(4)=1 / ( 50*real(1/ZL0)+j*imag(1/ZL0) )
state2 : ZL=PortZ(4)=1 / ( 2*real(1/ZL0)+j*imag(1/ZL0) )

|Γ_in0@state1| = |Γ_in1@state1|
|Γ_in0@state2| = |Γ_in1@state2|
|Γ_in0@state1- Γ_in0@state2| = |Γ_in1@state1- Γ_in1@state2|

Then I think the proof looks like:

|Γ_in0@state1|- |Γ_in0@state2| = |Γ_in1@state1|- |Γ_in1@state2|

Which is a pretty trivial result. But I don't think that this:

|Γ_in0@state1- Γ_in0@state2| = |Γ_in1@state1- Γ_in1@state2|

Can be true in general.

edit: actually I see that it seems to hold true for test cases, interesting. I'll see about a proof....