Attenuator

The ideal attenuator with (power) attenuation $L$ is frequency independent and the model is valid for DC and for AC simulation. It is determined by the following Z parameters.

$$Z_{11} = Z_{22} = Z_{ref}\cdot\frac{L+1}{L-1}$$ (9.72)

$$Z_{12} = Z_{21} = Z_{ref}\cdot\frac{2\cdot\sqrt{L}}{L-1}$$ (9.73)

The Z parameter representation is not very practical as new lines in the MNA matrix have to be added. More useful are the Y parameters.

$$(\underline{Y}) = \frac{1}{Z_{ref}\cdot (L-1)}\cdot \begin{pmatrix}L+1 & -2\cdot\sqrt{L} \\ -2\cdot\sqrt{L} & L+1 \end{pmatrix}$$ (9.74)

Attenuator with (power) attenuation $L$, reference impedance $Z_{ref}$ and temperature $T$:

$$(\underline{C}_Y) = 4\cdot k\cdot T\cdot \textrm{Re}\left(\underl... ... \begin{pmatrix}L+1 & -2\cdot\sqrt{L} \\ -2\cdot\sqrt{L} & L+1 \\ \end{pmatrix}$$ (9.75)

The scattering parameters and noise wave correlation matrix of an ideal attenuator with (power) attenuation $L$ (loss) (or power gain $G=1/L$) in reference to the impedance $Z_{ref}$ writes as follows.

$$S_{11} = S_{22} = \frac{r\cdot(L-1)}{L-r^2} = \frac{r\cdot(1-G)}{1-r^2\cdot G}$$ (9.76)

$$S_{12} = S_{21} = \frac{\sqrt{L}\cdot(1-r^2)}{L-r^2} = \frac{\sqrt{G}\cdot(1-r^2)}{1-r^2\cdot G}$$ (9.77)

$$(\underline{C}) = k\cdot T\cdot\frac{(L-1)\cdot(r^2-1)}{(L-r^2)^2... ...n{pmatrix}-r^2-L & 2\cdot r\sqrt{L}\\ 2\cdot r\sqrt{L} & -r^2-L\\ \end{pmatrix}$$ (9.78)

with

$$r=\frac{Z_{ref}-Z_0}{Z_{ref}+Z_0}$$ (9.79)