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Reflection Coefficient for unmatched condition

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pancho_hideboo

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If Γin0=0 then Γin1=0=Γin0 for network N.
This is easy to prove.

Even if Γin0=x!=0 then Γin1=x=Γin0 for same network N.
This also can be satisfied.
But this is difficult to prove.

How can I prove this ?
 

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Even if Γin0=x!=0 then Γin1=x=Γin0 for same network N.

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|
 

True for complex value.

Wrong.

magphase.PNG
 

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.
 

If Γin0=0 then Γin1=0=Γin0 for network N.
This is easy to prove.

Even if Γin0=x!=0 then Γin1=x=Γin0 for same network N.
This also can be satisfied.
But this is difficult to prove.

How can I prove this ?

you cannot prove this! i believe its wrong
 

That is true for the reflection factor magnitude, not for the complex value.
Only the magnitudes are equal, the phases depend on the network properties.
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| = |Γmod_on - Γmod_off|

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

in0@state1- Γin0@state2| = |Γin1@state1- Γin1@state2|

Can this relation be satisfied ?
 

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

i used to do scattering parameters using wave theory.
What do you mean by "wave theory" ?

Pseudo Wave Theory ?
Power Wave Theory ?

you can prove above equations using 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.
 
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Not so easy, since reference impedances are different for port1 and port2.
And former is real number, latter is complex number.

it can be done

- - - Updated - - -

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

- - - Updated - - -

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.

- - - Updated - - -

its power wave theory
 

i have a great book to suggest as well but forgot the name!

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.
 

|ΔΓ| is used as metrics of backscatter strength in RFID Tag.
|ΔΓ| = |Γmod_on - Γmod_off|

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

in0@state1- Γin0@state2| = |Γin1@state1- Γin1@state2|

Can this relation be satisfied ?
Could you clarify exactly what you mean by the terms in |Γin0@state1- Γin0@state2| = |Γin1@state1- Γin1@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?
 

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) )

in0@state1| = |Γin1@state1|
in0@state2| = |Γin1@state2|
in0@state1- Γin0@state2| = |Γin1@state1- Γin1@state2|
 
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Then I think the proof looks like:

in0@state1|- |Γin0@state2| = |Γin1@state1|- |Γin1@state2|

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

in0@state1- Γin0@state2| = |Γin1@state1- Γin1@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....
 
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