Scattering matrix of a circulator from its admittance matrix

Question

According to wikipedia, one can get the S pararemeters from an impedance matrix $Y$ as follows:
$S = (mathbb{1}_N - sqrt{z} Y sqrt{z})(mathbb{1}_N + sqrt{z} Y sqrt{z})^{-1}$ where $mathbb{1}_N$ is an identity matrix and $sqrt{z}$ is a diagonal matrix having the square root of the characteristic impedance $Z_{0n}$ on its diagonal.
Now, lets take a circulator where each port has the same input impedance: it has admittance matrix
begin{equation} Y = frac{1}{Z_0}begin{pmatrix}
0 & 1 & -1
-1 & 0 & 1 
1 & -1 & 0
end{pmatrix}
end{equation}
If I try to perform the above calculation to convert to S parameters, however, we are immediately faced with a problem: the product of the identity matrix and the admittance matrix is zero. So we just end up with the identity matrix. But the result should be
begin{equation} S = begin{pmatrix}
0 & 1 & 0
0 & 0 & 1 
1 & 0 & 0
end{pmatrix}
end{equation}
There must be an error in my calculation, but I do not see what it is

hyportnex · Accepted Answer

It can be shown, see [1], that every passive circuit has a scattering matrix. It is not true that every circuit has an admittance (or impedance) matrix, one such example is the circulator. It is true that if an admittance matrix exists, that is one for which the port currents are a linear function of the port voltages,  $mathbf{j}= mathbf{Y} mathbf {v}$, then one can always write that $mathbf{Y}=(mathbf {I} - mathbf{S})cdot (mathbf {I} +mathbf{S})^{-1}$ where $mathbf{I}$ is the identity matrix, but here for a circulator the scattering matrix is $$begin{equation} mathbf{S} = begin{pmatrix}
0 & 1 & 0
0 & 0 & 1 
1 & 0 & 0
end{pmatrix}
end{equation}$$ but the matrix $mathbf{I}-mathbf{S}$ is singular, its determinant is $0$, hence the admittance matrix is also singular, i.e., the port currents $mathbf{j}$ do not determine the port voltages $mathbf{v}$.
[1] Youla, Castriota, Carlin: "Bounded Real Scattering Matrices and the Foundations of Linear Passive Network Theory", IRE TRANSACTIONS ON CIRCUIT THEORY, March 1959

Scattering matrix of a circulator from its admittance matrix

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