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+++ title = "Electronic Noise" author = ["Dehaeze Thomas"] draft = false +++
- Tags
- [Electronics]({{< relref "electronics.md" >}}), [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio.md" >}})
Thermal (Johnson) Noise
Thermal noise is generated by the thermal agitation of the electrons inside the electrical conductor. Its Power Spectral Density is equal to:
\begin{equation} S_T \approx 4 k T \text{Re}(Z(f)) \quad [V^2/Hz] \end{equation}
with: 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.
This noise can be modeled as a voltage source in series with the system impedance.
| Resistance | PSD \([V^2 / Hz]\) | ASD \([V/\sqrt{Hz}]\) | RMS (1kHz) | RMS (10kHz) |
|---|---|---|---|---|
| \(1 \Omega\) | \(1.6 \cdot 10^{-20}\) | \(1.2 \cdot 10^{-10}\) | 4nV | 130nV |
| \(1 k\Omega\) | \(1.6 \cdot 10^{-17}\) | \(4 \cdot 10^{-9}\) | 130nV | 4uV |
| \(1 M\Omega\) | \(1.6 \cdot 10^{-14}\) | \(1.2 \cdot 10^{-7}\) | 4uV | 130uV |
Shot Noise
Seen with junctions in a transistor. It has a white spectral density:
\begin{equation} S_S = 2 q_e i_{dc} \ [A^2/Hz] \end{equation}
with \(q_e\) the electronic charge (\(1.6 \cdot 10^{-19}\, [C]\)), \(i_{dc}\) the average current [A].
A current of 1 A will introduce noise with a STD of \(10 \cdot 10^{-9}\,[A]\) from zero up to one kHz.
Excess Noise (or \(1/f\) noise)
It results from fluctuating conductivity due to imperfect contact between two materials. The PSD of excess noise increases when the frequency decreases: \[ S_E = \frac{K_f}{f^\alpha}\ [V^2/Hz] \] 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.
Noise of Amplifiers
The noise of amplifiers can be modelled as shown in Figure 1.
{{< figure src="/ox-hugo/electronic_amplifier_noise.png" caption="<span class="figure-number">Figure 1: Amplifier noise model" >}}
The identification of this noise is a two steps process:
- The amplifier input is short-circuited such that only \(V^2(f)\) has an impact on the output. The output noise is measured and \(V^2\) in \([V^2/Hz]\) is identified
- The amplifier input is open-circuited such that only \(I^2(f)\) has an impact on the output. The output noise is measured and \(I^2(f)\) in \([A^2/Hz]\) is identified.