126 lines
5.9 KiB
Markdown
126 lines
5.9 KiB
Markdown
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title = "Analog to Digital Converters"
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author = ["Dehaeze Thomas"]
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keywords = ["electronics"]
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draft = false
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category = "equipment"
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Tags
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: [Electronics]({{< relref "electronics.md" >}})
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## Types of Analog to Digital Converters {#types-of-analog-to-digital-converters}
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<https://dewesoft.com/daq/types-of-adc-converters>
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- Delta Sigma (<a href="#citeproc_bib_item_1">Baker 2011</a>)
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- Successive Approximation
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## Power Spectral Density of the Quantization Noise {#power-spectral-density-of-the-quantization-noise}
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This analysis is taken from [here](https://www.allaboutcircuits.com/technical-articles/quantization-nois-amplitude-quantization-error-analog-to-digital-converters/).
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Let's note:
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- \\(q = \frac{\Delta V}{2^n}\\) the quantization in [V], which is the corresponding value in [V] of the least significant bit
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- \\(\Delta V\\) is the full range of the ADC in [V]
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- \\(n\\) is the number of ADC's bits
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- \\(f\_s\\) is the sample frequency in [Hz]
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Let's suppose that the ADC is ideal and the only noise comes from the quantization error.
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Interestingly, the noise amplitude is uniformly distributed.
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The quantization noise can take a value between \\(\pm q/2\\), and the probability density function is constant in this range (i.e., it’s a uniform distribution).
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Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 < e < q/2\\) (Fig. [1](#figure--fig:probability-density-function-adc)).
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<a id="figure--fig:probability-density-function-adc"></a>
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{{< figure src="/ox-hugo/probability_density_function_adc.png" caption="<span class=\"figure-number\">Figure 1: </span>Probability density function \\(p(e)\\) of the ADC error \\(e\\)" >}}
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Now, we can calculate the time average power of the quantization noise as
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\begin{equation}
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P\_q = \int\_{-q/2}^{q/2} e^2 p(e) de = \frac{q^2}{12}
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\end{equation}
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The other important parameter of a noise source is the power spectral density (PSD), which indicates how the noise power spreads in different frequency bands.
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To find the power spectral density, we need to calculate the Fourier transform of the autocorrelation function of the noise.
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Assuming that the noise samples are not correlated with one another, we can approximate the autocorrelation function with a delta function in the time domain.
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Since the Fourier transform of a delta function is equal to one, the **power spectral density will be frequency independent**.
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Therefore, the quantization noise is white noise with total power equal to \\(P\_q = \frac{q^2}{12}\\).
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Thus, the two-sided PSD (from \\(\frac{-f\_s}{2}\\) to \\(\frac{f\_s}{2}\\)), we should divide the noise power \\(P\_q\\) by \\(f\_s\\):
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\begin{equation}
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\int\_{-f\_s/2}^{f\_s/2} \Gamma(f) d f = f\_s \Gamma = \frac{q^2}{12}
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\end{equation}
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<div class="important">
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Finally, the Power Spectral Density of the quantization noise of an ADC is equal to:
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\begin{equation}
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\begin{aligned}
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\Gamma &= \frac{q^2}{12 f\_s} \\\\
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&= \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 f\_s} \text{ in } \left[ \frac{V^2}{Hz} \right]
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\end{aligned}
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\end{equation}
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</div>
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<div class="exampl">
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Let's take a 18bits ADC with a range of +/-10V and a sample frequency of 10kHz.
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The quantization is:
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\\[ q = \frac{20}{2^{18}} = 0.000076 \ [V] = 76 \ [\mu V] \\]
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\\[ \Gamma\_Q = \frac{q^2}{12 f\_N} = 4.85 \cdot 10^{-14} \quad [V^2/Hz] \\]
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</div>
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{{< youtube b9lxtOJj3yU >}}
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Also see (<a href="#citeproc_bib_item_2">Kester 2005</a>).
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## Link between required dynamic range and effective number of bits {#link-between-required-dynamic-range-and-effective-number-of-bits}
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<a id="figure--fig:dynamic-range-enob"></a>
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{{< figure src="/ox-hugo/dynamic_range_enob.png" caption="<span class=\"figure-number\">Figure 2: </span>Relation between Dynamic range and required number of bits (effective)" >}}
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## Oversampling {#oversampling}
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(<a href="#citeproc_bib_item_3">Lab 2013</a>)
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## Sigma Delta ADC {#sigma-delta-adc}
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From (<a href="#citeproc_bib_item_4">Schmidt, Schitter, and Rankers 2020</a>):
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> The low cost and excellent linearity properties of the Sigma-Delta ADC have replaced other ADC types in many measurement and registration systems, especially where storage of data is more important than real-time measurement.
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> This has typically been the case in audio recording and reproduction.
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> The reason why this principle is less applied with real-time measurements is the time delay between the bitstream representing the actual value and the availability of the corresponding value after the decimation filter.
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> The resulting **latency** amounts with a low cost sigma-delta ADC approximately **twenty times the sampling period of the decimated digital output**.
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Therefore, even though there are sigma-delta ADC with high precision and sampling rate, they add large latency (i.e. time delay) that are very problematic for feedback systems.
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> The SAR-ADC (Successive approximation ADCs) is still the mostly applied type for data-acquisition and feedback systems because of its single sample latency.
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<https://www.crystalinstruments.com/antialiasing-filter-and-phase-match>
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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 class="csl-entry"><a id="citeproc_bib_item_1"></a>Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” <i>Analog Applications</i> 7.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Kester, Walt. 2005. “Taking the Mystery out of the Infamous Formula, $snr = 6.02 N + 1.76 Db$, and Why You Should Care.”</div>
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<div class="csl-entry"><a id="citeproc_bib_item_3"></a>Lab, Silicon. 2013. “Improving the ADC Resolution by Oversampling and Averaging.” Silicon Laboratories.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_4"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. <i>The Design of High Performance Mechatronics - Third Revised Edition</i>. Ios Press.</div>
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</div>
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