"Symbolic noise analysis"

Symbolic noise analysis

Symbolic noise analysis results

Detector-referred noise spectrum

$$S_{out}=\frac{4 R_{p}^{2} R_{s} T k}{\left(R_{p} + R_{s}\right)^{2}} + \frac{4 R_{p} R_{s}^{2} T k \left(f + f_{\ell}\right)}{f \left(R_{p} + R_{s}\right)^{2}}\, \mathrm{\left[\frac{V^2}{Hz}\right]}$$

Source-referred noise spectrum

$$S_{in}=4 R_{s} T k + \frac{4 R_{s}^{2} T k \left(f + f_{\ell}\right)}{R_{p} f}\, \mathrm{\left[\frac{V^2}{Hz}\right]}$$

Contributions of individual noise sources

Noise source: I_noise_R2
Spectral density:$\frac{4 T k \left(1 + \frac{f_{\ell}}{f}\right)}{R_{p}}$$\,\mathrm{\left[\frac{A^2}{Hz}\right]}$
Detector-referred:$\frac{4 R_{p} R_{s}^{2} T k \left(f + f_{\ell}\right)}{f \left(R_{p} + R_{s}\right)^{2}}$$\,\mathrm{\left[\frac{V^2}{Hz}\right]}$
Source-referred:$\frac{4 R_{s}^{2} T k \left(f + f_{\ell}\right)}{R_{p} f}$$\,\mathrm{\left[\frac{V^2}{Hz}\right]}$
Noise source: V1
Spectral density:$4 R_{s} T k$$\,\mathrm{\left[\frac{V^2}{Hz}\right]}$
Detector-referred:$\frac{4 R_{p}^{2} R_{s} T k}{\left(R_{p} + R_{s}\right)^{2}}$$\,\mathrm{\left[\frac{V^2}{Hz}\right]}$
Source-referred:$4 R_{s} T k$$\,\mathrm{\left[\frac{V^2}{Hz}\right]}$

Noise figure

The noise figure of a circuit is defined as the ratio of the signal-to-noise at the output of the circuit and the signal-to-noise ratio of its input signal.

Alternatively, the noise figure can be defined as the as the ratio of the total source referred noise power and the noise power of the signal source, or the ratio of the total detector-referred noise power and the contribution of the noise of the signal source to the detector-referred noise power.

The variance $P_{onoise}\,\left[ V^2 \right]$ of the total detector referred noise is:

\begin{equation} P_{onoise}=\frac{k \left(4 R_{p} R_{s}^{2} T f_{\ell} \log{\left(f_{max} \right)} + T f_{max} \left(4 R_{p}^{2} R_{s} + 4 R_{p} R_{s}^{2}\right)\right)}{R_{p}^{2} + 2 R_{p} R_{s} + R_{s}^{2}} - \frac{k \left(4 R_{p} R_{s}^{2} T f_{\ell} \log{\left(f_{min} \right)} + T f_{min} \left(4 R_{p}^{2} R_{s} + 4 R_{p} R_{s}^{2}\right)\right)}{R_{p}^{2} + 2 R_{p} R_{s} + R_{s}^{2}}\,\left[ \mathrm{\frac{V^{2}}{Hz}}\right] \end{equation}

The contribution $P_{onoise,source}\,\left[ V^2 \right]$ of the source to $P_{onoise}\,\left[ V^2 \right]$ is:

\begin{equation} P_{onoise}=\frac{4 R_{p}^{2} R_{s} T k \left(f_{max} - f_{min}\right)}{\left(R_{p} + R_{s}\right)^{2}}\,\left[ \mathrm{\frac{V^{2}}{Hz}}\right] \end{equation}

Hence, the noise figure is obtained as

\begin{equation} F=4.343 \log{\left(\frac{R_{p} f_{max} - R_{p} f_{min} + R_{s} f_{\ell} \log{\left(f_{max} \right)} - R_{s} f_{\ell} \log{\left(f_{min} \right)} + R_{s} f_{max} - R_{s} f_{min}}{R_{p} \left(f_{max} - f_{min}\right)} \right)} \end{equation}

Detector referred noise spectrum

The spectral density of the total output noise can be written as

\begin{equation} S_{out}=\frac{4 R_{p} R_{s} T k \left(R_{p} f + R_{s} \left(f + f_{\ell}\right)\right)}{f \left(R_{p} + R_{s}\right)^{2}} \end{equation}

Go to Noise-Figure-$R_p$_index

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Last project update: 2023-06-12 11:49:08