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It's simple. Circulator is device which has a peculiar property that each terminal is connected only to the next clockwise terminal.
Consider a 3 port circulator with ports 1,2,3 ( arranged in anti clock wise). As there are 3 ports, order of scattering matrix is 3 X 3.
[S]ij i= input applied at port i,
j= output taken from port j
when input is applied at port 1, we get output from port 3 only. i.e,s11=0,s12=0
when input is applied at port 2, we get output from port 1 only.
when input is applied at port 3, we get ouput from port 2 only.
hence, scattering matrix is
0 0 s13
S = s21 0 0
0 s32 0
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