Hello,
I will be grateful if you could see attachment first. Based on the symmetry (Follow the "For example" in the first page up to end), I need to know why S13 is not equal to S31 (or S41 and S14 , ...).
Thanks
Symmetry and reciprocity are close but not equivalent concepts. But the question is somewhat incomprehensible - in your example, the case of a mutual circuit is described, where direct and reverse inclusion is equivalent. Are you wondering in what cases this is violated?
Dear John Jin ,
Mathematically the condition of symmetry of multiport networks is that
the Sii,s are equal and Sij= Sji.
This is on the mathematical level.
In the implementation level in the T- network every port has an eqaul impedance to the common branch. In this way the circuit will be symmetrical by construction. In the pi networks the parallel admittance to the ports myst must be eqaul and there will be a common series elements to every port.
These may be just thoughts rather than exhaustive answer.
Best wishes
You are asking about the middle example, which is the only one where S31 is not equal to S13.
In this example it is because port 1 sees 50 ohms and 200 ohms leading off from it, and port 3 sees 50 ohms and 100 ohms leading off from it. You might expect that the impedances beyond this might even things out, but it is not certain, or even probable that this will happen. When there are losses, then the effect of further components, on things like the match, is reduced.
Just dont forget that when calculating S-parameters for this kind of networks or structures you have to assume that ALL ports including the generator port are terminated with their characteristic impedance (often 50 Ohm but not always and it may also differ from one port to another of the same network), and as already said above: dont mix up symmetry and reciprocity; a network can be asymmetric but still be reciprocal
As I understand:
1- Symmetry and reciprocity of the network are two different things. Mathematically speaking (in terms of network parameters):
> reciprocity means: Sjk = Skj for the reciprocal ports k and j.
> symmetry means: Skk = Sjj for the symmetric ports k and j.
They do not imply each other.
A lossless 3-port network can be:
reciprocal asymmetric network:like a lossless power divider
or, non-reciprocal symmetric network like a lossless circulator
2- A source of ambiguity is that: a reciprocal network achieves ST = S, which means that S is a "symmetric matrix" for non-reciprocal networks.
3- In the examples of your attached file: the network is assumed to have special kind of symmetry which goes beyond the above definition: it is said literally that two symmetric ports, say 1 & 2, if exchanged in position, lead to a congruent network. As explained in the example, "congruent" means you can exchange the 1&2 in any parameter you want. So this will lead directly to the result that:
S12 = S21 for port 1&2
S34 = S43 for ports 3&4
But as you see, it has nothing to do with the reciprocity of any other couple of ports.
Just one comment on 3 ports
There is NO lossless symmetric abd reciprocal 3 port which is impedance matched on all ports
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