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lossless reciprocal three port network

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tarigmusa

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reciprocal lossless network

prove the following decoupling theorem for me...

[for any lossless reciprocal three port network , one port (say port 3) can be terminated in a reactance so that the other two ports (ports 1 and 2)are decoupled (no power will flow from port 1 to port 2 or from port 2 to port 1)]
 

not true

This does not seem possible. Let us do a thought experiment with a parallel line directional coupler with the reflected port terminated and the other three ports taken to the outside world. Port 1 is the input, port 2 is the direct out, and port 3 is the coupled out. If you put 50 +j0 on port 1 and put power into port 2, it all goes to port 1 and none to port 3. Port 2 and 3 are isolated.

If you put any other impedance on port 1, and send power into port 2, there will be reflections and so some of the reflected power (now forward power relative to the coupler) comes out port 3. Therefore ports 2 and 3 are not isolated.
 

it is difficult to me.if the ports can be varied, the problem is equal to prove:
|SjkSii-SijSik|=|Sjk|,(i,j,k=1,2,3) should exist a solution.
but it is easy to find one counterexample just let s12<>0,s13<>0,s23==0.s11,s22,s33<>0
 

I know the thread is toooooooooo old but I found the solution

Attached !
 

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  • 60457511-Decoupling-TheoremHW7.pdf
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