Isolator

An isolator is a one-way two-port, transporting incoming waves lossless from the input (port 1) to the output (port 2), but dissipating all waves flowing into the output. The ideal isolator with reference impedances $Z_1$ (input) and $Z_2$ (output) is determined by the following Z parameters (for DC and AC simulation).

$$Z_{11} = Z_1 \qquad Z_{12} = 0$$ (9.88)

$$Z_{21} = 2\cdot\sqrt{Z_1\cdot Z_2} \qquad Z_{22} = Z_2$$ (9.89)

A more useful MNA representation is with Y parameters.

$$(\underline{Y}) = \begin{pmatrix}\dfrac{1}{Z_1} & 0 \\ \dfrac{-2}{\sqrt{Z_1\cdot Z_2}} & \dfrac{1}{Z_2} \end{pmatrix}$$ (9.90)

Isolator with reference impedance $Z_1$ (input) and $Z_2$ (output) and temperature $T$:

$$(\underline{C}_Y) = 4\cdot k\cdot T\cdot \begin{pmatrix}\dfrac{1}{Z_1} & 0 \\ \dfrac{-2}{\sqrt{Z_1\cdot Z_2}} & \dfrac{1}{Z_2} \\ \end{pmatrix}$$ (9.91)

With the reference impedance of the input $Z_1$ and the one of the output $Z_2$, the scattering parameters of an ideal isolator writes as follows.

$$S_{11} = \frac{Z_1-Z_0}{Z_1+Z_0}$$ (9.92)

$$S_{12} = 0$$ (9.93)

$$S_{22} = \frac{Z_2-Z_0}{Z_2+Z_0}$$ (9.94)

$$S_{21} = \sqrt{1-(S_{11})^2}\cdot\sqrt{1-(S_{22})^2}$$ (9.95)

Being on temperature $T$, the noise wave correlation matrix of an ideal isolator with reference impedance $Z_1$ and $Z_2$ (input and output) writes as follows.

$$(\underline{C}) = \frac{4\cdot k\cdot T\cdot Z_0}{(Z_1+Z_0)^2}\cd... ...Z_1}{Z_0+Z_2} & Z_2\cdot\left(\dfrac{Z_1-Z_0}{Z_2+Z_0}\right)^2\\ \end{pmatrix}$$ (9.96)