78 lines
2.9 KiB
Markdown
78 lines
2.9 KiB
Markdown
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title = "Electronic Noise"
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author = ["Dehaeze Thomas"]
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draft = false
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Tags
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: [Electronics]({{< relref "electronics.md" >}}), [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio.md" >}})
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## Thermal (Johnson) Noise {#thermal--johnson--noise}
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Thermal noise is generated by the thermal agitation of the electrons inside the electrical conductor.
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Its Power Spectral Density is equal to:
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\begin{equation}
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S\_T \approx 4 k T \text{Re}(Z(f)) \quad [V^2/Hz]
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\end{equation}
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with:
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with \\(k = 1.38 \cdot 10^{-23} \\,[J/K]\\) the Boltzmann's constant, \\(T\\) the temperature [K] and \\(Z(f)\\) the frequency dependent impedance of the system.
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This noise can be modeled as a voltage source in series with the system impedance.
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| Resistance | PSD \\([V^2 / Hz]\\) | ASD \\([V/\sqrt{Hz}]\\) | RMS (1kHz) | RMS (10kHz) |
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|-----------------|--------------------------|--------------------------|------------|-------------|
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| \\(1 \Omega\\) | \\(1.6 \cdot 10^{-20}\\) | \\(1.2 \cdot 10^{-10}\\) | 4nV | 130nV |
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| \\(1 k\Omega\\) | \\(1.6 \cdot 10^{-17}\\) | \\(4 \cdot 10^{-9}\\) | 130nV | 4uV |
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| \\(1 M\Omega\\) | \\(1.6 \cdot 10^{-14}\\) | \\(1.2 \cdot 10^{-7}\\) | 4uV | 130uV |
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## Shot Noise {#shot-noise}
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Seen with junctions in a transistor.
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It has a white spectral density:
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\begin{equation}
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S\_S = 2 q\_e i\_{dc} \ [A^2/Hz]
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\end{equation}
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with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\\, [C]\\)), \\(i\_{dc}\\) the average current [A].
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<div class="exampl">
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A current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\\,[A]\\) from zero up to one kHz.
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</div>
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## Excess Noise (or \\(1/f\\) noise) {#excess-noise--or-1-f-noise}
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It results from fluctuating conductivity due to imperfect contact between two materials.
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The PSD of excess noise increases when the frequency decreases:
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\\[ S\_E = \frac{K\_f}{f^\alpha}\ [V^2/Hz] \\]
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where \\(K\_f\\) is dependent on the average voltage drop over the resistor and the index \\(\alpha\\) is usually between 0.8 and 1.4, and often set to unity for approximate calculation.
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## Noise of Amplifiers {#noise-of-amplifiers}
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The noise of amplifiers can be modelled as shown in Figure [1](#figure--fig:electronic-amplifier-noise).
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<a id="figure--fig:electronic-amplifier-noise"></a>
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{{< figure src="/ox-hugo/electronic_amplifier_noise.png" caption="<span class=\"figure-number\">Figure 1: </span>Amplifier noise model" >}}
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The identification of this noise is a two steps process:
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1. The amplifier input is short-circuited such that only \\(V^2(f)\\) has an impact on the output.
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The output noise is measured and \\(V^2\\) in \\([V^2/Hz]\\) is identified
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2. The amplifier input is open-circuited such that only \\(I^2(f)\\) has an impact on the output.
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The output noise is measured and \\(I^2(f)\\) in \\([A^2/Hz]\\) is identified.
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## Bibliography {#bibliography}
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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</div>
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