Classical update of content

content/zettels/signal_to_noise_ratio.md

@@ -7,19 +7,10 @@ draft = false
 Tags
 : [Electronics]({{< relref "electronics" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
 
-From <sup id="3b7899e183dba866e6a6419cf820467f"><a href="#jabben07_mechat" title="@phdthesis{jabben07_mechat,
-  author          = {Jabben, Leon},
-  school          = {Delft University},
-  title           = {Mechatronic design of a magnetically suspended rotating
-                  platform},
-  year            = 2007,
-}">@phdthesis{jabben07_mechat,
-  author          = {Jabben, Leon},
-  school          = {Delft University},
-  title           = {Mechatronic design of a magnetically suspended rotating
-                  platform},
-  year            = 2007,
-}</a></sup> (Section 3.3.2):
+
+## SNR to Noise PSD {#snr-to-noise-psd}
+
+From ([Jabben 2007](#orgd8e3764)) (Section 3.3.2):
 
 > Electronic equipment does most often not come with detailed electric schemes, in which case the PSD should be determined from measurements.
 > In the design phase however, one has to rely on information provided by specification sheets from the manufacturer.
@@ -31,5 +22,88 @@ From <sup id="3b7899e183dba866e6a6419cf820467f"><a href="#jabben07_mechat" title
 > \\[ S\_{snr} = \frac{x\_{fr}^2}{8 f\_c C\_{snr}^2} \\]
 > with \\(x\_{fr}\\) the full range of \\(x\\), and \\(C\_{snr}\\) the SNR.
 
-# Bibliography
-<a id="jabben07_mechat"></a>Jabben, L., *Mechatronic design of a magnetically suspended rotating platform* (Doctoral dissertation) (2007). Delft University, . [↩](#3b7899e183dba866e6a6419cf820467f)
+<div class="examp">
+  <div></div>
+
+Let's take an example.
+
+-   \\(x\_{fr} = 170 V\\)
+-   \\(C\_{snr} = 85 dB\\)
+-   \\(f\_c = 200 Hz\\)
+
+The Power Spectral Density of the output voltage is:
+\\[ S\_{snr} = \frac{170^2}{8 \cdot 200 \cdot {10^{\frac{2 \cdot 85}{20}}}} = 5.7 \cdot 10^{-8}\ V^2/Hz \\]
+
+And the RMS of that noise up to \\(f\_c\\) is:
+\\[ S\_{rms} = \sqrt{S\_{snr} \cdot f\_c} \approx 3.4\ mV \\]
+
+</div>
+
+
+## Convert SNR to Noise RMS value {#convert-snr-to-noise-rms-value}
+
+The RMS value of the noise can be computed from:
+\\[ N\_\text{rms} = 10^{-\frac{S\_{snr}}{20}} S\_\text{rms} \\]
+where \\(S\_{snr}\\) is the SNR in dB and \\(S\_\text{rms}\\) is the RMS value of a sinus taking the full range.
+
+If the full range is \\(\Delta V\\), then:
+\\[ S\_\text{rms} = \frac{\Delta V/2}{\sqrt{2}} \\]
+
+<div class="examp">
+  <div></div>
+
+As an example, let's take a voltage amplifier with a full range of \\(\Delta V = 20V\\) and a SNR of 85dB.
+The RMS value of the noise is then:
+\\[ n\_\text{rms} = 10^{-\frac{S\_{nrs}}{20}} s\_\text{rms} \\]
+
+\\[ n\_\text{rms} = 10^{-\frac{85}{20}} \frac{10}{\sqrt{2}} \approx 0.4 mV\_\text{rms} \\]
+
+</div>
+
+
+## Convert wanted Noise RMS value to required SNR {#convert-wanted-noise-rms-value-to-required-snr}
+
+If the wanted full range and RMS value of the noise are defined, the required SNR can be computed from:
+\\[ S\_{snr} = 20 \log \frac{\text{Signal, rms}}{\text{Noise, rms}} \\]
+
+<div class="examp">
+  <div></div>
+
+Let's say the wanted noise is \\(1 mV, \text{rms}\\) for a full range of \\(20 V\\), the corresponding SNR is:
+\\[ S\_{snr} = 20 \log \frac{\frac{20/2}{\sqrt{2}}}{10^{-3}} \approx 77dB \\]
+
+</div>
+
+
+## Noise Density to RMS noise {#noise-density-to-rms-noise}
+
+From ([Fleming 2010](#org68e05a9)):
+\\[ \text{RMS noise} = \sqrt{2 \times \text{bandwidth}} \times \text{noise density} \\]
+
+If the noise is normally distributed, the RMS value is also the standard deviation \\(\sigma\\).
+The peak to peak amplitude is then approximatively \\(6 \sigma\\).
+
+<div class="exampl">
+  <div></div>
+
+-   noise density = \\(20 pm/\sqrt{Hz}\\)
+-   bandwidth = 100Hz
+
+\\[ \sigma = \sqrt{2 \times 100} \times 20 = 0.28nm RMS \\]
+The peak-to-peak noise will be approximately \\(6 \sigma = 1.7 nm\\)
+
+</div>
+
+
+## Bibliography {#bibliography}
+
+<a id="org68e05a9"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>.
+
+<a id="orgd8e3764"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.
+
+
+## Backlinks {#backlinks}
+
+-   [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
+-   [Voltage Amplifier]({{< relref "voltage_amplifier" >}})
+-   [Voltage Amplifier]({{< relref "voltage_amplifier" >}})