Resistor

For DC and AC simulation an ideal resistor with resistance $R$ yields:

$$Y = \dfrac{1}{R} \cdot \begin{pmatrix}1 & -1 \\ -1 & 1 \\ \end{pmatrix}$$ (9.3)

The noise correlation matrix at temperature $T$ yields:

$$(\underline{C}_Y) = \frac{4\cdot k\cdot T}{R} \cdot \begin{pmatrix}1 & -1 \\ -1 & 1 \\ \end{pmatrix}$$ (9.4)

The scattering parameters normalized to impedance $Z_0$ writes as follows.

$$S_{11} = S_{22} = \frac{R}{2\cdot Z_0+R} \\$$ (9.5)

$$S_{12} = S_{21} = 1-S_{11} = \frac{2\cdot Z_0}{2\cdot Z_0+R}$$ (9.6)

Being on temperature $T$, the noise wave correlation matrix writes as follows.

$$(\underline{C}) = k\cdot T\cdot\frac{4\cdot R\cdot Z_0}{(2\cdot Z_0+R)^2}\cdot \begin{pmatrix}1 & -1\\ -1 & 1\\ \end{pmatrix}$$ (9.7)

The noise wave correlation matrix of a parallel resistor with resistance $R$ writes as follows.

$$(\underline{C}) = k\cdot T\cdot\frac{4\cdot R\cdot Z_0}{(2\cdot R+Z_0)^2}\cdot \begin{pmatrix}1 & 1\\ 1 & 1\\ \end{pmatrix}$$ (9.8)

The noise wave correlation matrix of a grounded resistor with resistance $R$ is a matrix consisting of one element and writes as follows.

$$(\underline{C}) = k\cdot T\cdot\frac{4\cdot R\cdot Z_0}{(R+Z_0)^2}$$ (9.9)