Noise sources

To implement the frequency dependencies of all common noise PSDs the following equation can be used.

$$PSD = \frac{PSD_0}{a+b\cdot f^c}$$ (9.144)

Where $f$ is frequency and $a$, $b$, $c$ are the parameters. The following PSDs appear in electric devices.

white noise (thermal noise, shot noise):	$a=0$, $b=1$, $c=0$
pink noise (flicker noise):	$a=0$, $b=1$, $c=1$
Lorentzian PSD (generation-recombination noise):	$a=1$, $b=1/f_c^2$, $c=2$

Noise current source

Noise current source with a current power spectral density of $cPSD$:

$$(\underline{C}_Y) = cPSD \cdot \begin{pmatrix}1 & -1 \\ -1 & 1 \\ \end{pmatrix}$$ (9.145)

The MNA matrix entries for DC and AC analysis are all zero.

The noise wave correlation matrix of a noise current source with current power spectral density $cPSD$ and its S parameter matrix write as follows.

$$(\underline{C}) = cPSD\cdot Z_0\cdot \begin{pmatrix}1 & -1\\ -1 &... ...{pmatrix} \qquad (\underline{S}) = \begin{pmatrix}1 & 0\\ 0 & 1\\ \end{pmatrix}$$ (9.146)

Noise voltage source

A noise voltage source (voltage power spectral density $vPSD$) cannot be modeled with the noise current matrix. That is why one has to use a noise current source (current power spectral density $cPSD$) connected to a gyrator (transimpedance $R$) satisfying the equation

$$vPSD = cPSD \cdot R^2$$ (9.147)

Figure 9.7 shows an example.

Figure 9.7: noise voltage source (left-hand side) and its equivalent circuit (right-hand side)

The MNA matrix entries of the above construct (gyrator ratio $R=1$) is similiar to a voltage source with zero voltage.

$$\begin{bmatrix}.&.& -1\\ .&.& 1\\ 1 & -1 & 0 \end{bmatrix} \cdot ... ...matrix}I_{1}\\ I_{2}\\ 0 \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0 \end{bmatrix}$$ (9.148)

The appropriate noise current correlation matrix yields:

$$(\underline{C}_Y) = cPSD \cdot \begin{pmatrix}0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 1\\ \end{pmatrix}$$ (9.149)

The noise wave correlation matrix of a noise voltage source with voltage power spectral density $vPSD$ and its S parameter matrix write as follows.

$$(\underline{C}) = \frac{vPSD}{4\cdot Z_0}\cdot \begin{pmatrix}1 &... ...{pmatrix} \qquad (\underline{S}) = \begin{pmatrix}0 & 1\\ 1 & 0\\ \end{pmatrix}$$ (9.150)

Correlated noise sources

For two correlated noise current sources the (normalized) correlation coefficient $K$ must be known (with $\vert K\vert=0\dots 1$). If the first noise source has the current power spectral density $S_{i1}$ and is connected to node 1 and 2, and if furthermore the second noise source has the spectral density $S_{i2}$ and is connected to node 3 and 4, then the correlation matrix writes:

$$(\underline{C}_Y) = \begin{pmatrix}S_{i1} & -S_{i1} & K\cdot\sqrt... ...t S_{i2}} & K\cdot\sqrt{S_{i1}\cdot S_{i2}} & -S_{i2} & S_{i2} \\ \end{pmatrix}$$ (9.151)

The MNA matrix entries for DC and AC analysis are all zero.

The noise wave correlation matrix of two correlated noise current sources with current power spectral densities $S_{i1}$ and $S_{i2}$ and correlation coefficient $K$ writes as follows.

$$(\underline{C}) = Z_0\cdot \begin{pmatrix}S_{i1} & -S_{i1} & K\cd... ...t S_{i2}} & K\cdot\sqrt{S_{i1}\cdot S_{i2}} & -S_{i2} & S_{i2} \\ \end{pmatrix}$$ (9.152)

$$(\underline{S}) = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$$ (9.153)

For two correlated noise voltage sources two extra rows and columns are needed in the MNA matrix:

$$\begin{bmatrix}.&.&.&.& -1 & 0 \\ .&.&.&.& 1 & 0 \\ .&.&.&.& 0 & ... ..._{4}\\ 0\\ 0 \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 0 \end{bmatrix}$$ (9.154)

The appropriate noise current correlation matrix (with the noise voltage power spectral densities $S_{v1}$ and $S_{v2}$ and the correlation coefficient $K$) yields:

$$(\underline{C}_Y) = \begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 ... ...v2}}\\ 0 & 0 & 0 & 0 & K\cdot\sqrt{S_{v1}\cdot S_{v2}} & S_{v2}\\ \end{pmatrix}$$ (9.155)

The noise wave correlation matrix of two correlated noise voltage sources with voltage power spectral densities $S_{v1}$ and $S_{v2}$ and correlation coefficient $K$ and its S parameter matrix write as follows.

$$(\underline{C}) = \frac{1}{4\cdot Z_0}\cdot \begin{pmatrix}S_{v1}... ...t S_{v2}} & K\cdot\sqrt{S_{v1}\cdot S_{v2}} & -S_{v2} & S_{v2} \\ \end{pmatrix}$$ (9.156)

$$(\underline{S}) = \begin{pmatrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix}$$ (9.157)

If a noise current source (ports 1 and 2) and a noise voltage source (ports 3 and 4) are correlated, the MNA matrix entries are as follows.

$$\begin{bmatrix}.&.&.&.& 0 \\ .&.&.&.& 0 \\ .&.&.&.& -1 \\ .&.&.&.... ..._{3}\\ I_{4}\\ 0 \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0 \end{bmatrix}$$ (9.158)

The appropriate noise current correlation matrix (with the noise power spectral densities $S_{i1}$ and $S_{v2}$ and the correlation coefficient $K$) yields:

$$(\underline{C}_Y) = \begin{pmatrix}S_{i1} & -S_{i1} & 0 & 0 & K\c... ... & 0 & 0\\ K\cdot\sqrt{S_{i1}\cdot S_{v2}} & 0 & 0 & 0 & S_{v2}\\ \end{pmatrix}$$ (9.159)

Note: Because the gyrator factor (It is unity.) has been omitted in the above matrix the units are not correct. This can be ignored.

The noise wave correlation matrix of one correlated noise current source $S_{i1}$ and one noise voltage source $S_{v2}$ with correlation coefficient $K$ writes as follows.

$$(\underline{C}) = \begin{pmatrix}Z_0\cdot S_{i1} & -Z_0\cdot S_{i... ...2\cdot\sqrt{S_{i1}\cdot S_{v2}} & -S_{v2}/4/Z_0 & S_{v2}/4/Z_0 \\ \end{pmatrix}$$ (9.160)

$$(\underline{S}) = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix}$$ (9.161)