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content/article/fleming13_review_nanom_resol_posit_sensor.md
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title = "A review of nanometer resolution position sensors: operation and performance"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Position Sensors]({{< relref "position_sensors" >}})
: [Position Sensors]({{< relref "position_sensors.md" >}})
Reference
: ([Fleming 2013](#org687716f))
: (<a href="#citeproc_bib_item_1">Fleming 2013</a>)
Author(s)
: Fleming, A. J.
@@ -28,28 +28,28 @@ Year
Usually quoted as a percentage of the fill-scale range (FSR):
\begin{equation}
\text{mapping error (\%)} = \pm 100 \frac{\max{}|e\_m(v)|}{\text{FSR}}
\text{mapping error (\\%)} = \pm 100 \frac{\max{}|e\_m(v)|}{\text{FSR}}
\end{equation}
With \\(e\_m(v)\\) is the mapping error.
<a id="org0a1d321"></a>
<a id="figure--fig:mapping-error"></a>
{{< figure src="/ox-hugo/fleming13_mapping_error.png" caption="Figure 1: The actual position versus the output voltage of a position sensor. The calibration function \\(f\_{cal}(v)\\) is an approximation of the sensor mapping function \\(f\_a(v)\\) where \\(v\\) is the voltage resulting from a displacement \\(x\\). \\(e\_m(v)\\) is the residual error." >}}
{{< figure src="/ox-hugo/fleming13_mapping_error.png" caption="<span class=\"figure-number\">Figure 1: </span>The actual position versus the output voltage of a position sensor. The calibration function \\(f\_{cal}(v)\\) is an approximation of the sensor mapping function \\(f\_a(v)\\) where \\(v\\) is the voltage resulting from a displacement \\(x\\). \\(e\_m(v)\\) is the residual error." >}}
### Drift and Stability {#drift-and-stability}
If the shape of the mapping function actually varies with time, the maximum error due to drift must be evaluated by finding the worst-case mapping error.
<a id="orgc781e90"></a>
<a id="figure--fig:drift-stability"></a>
{{< figure src="/ox-hugo/fleming13_drift_stability.png" caption="Figure 2: The worst case range of a linear mapping function \\(f\_a(v)\\) for a given error in sensitivity and offset." >}}
{{< figure src="/ox-hugo/fleming13_drift_stability.png" caption="<span class=\"figure-number\">Figure 2: </span>The worst case range of a linear mapping function \\(f\_a(v)\\) for a given error in sensitivity and offset." >}}
### Bandwidth {#bandwidth}
The bandwidth of a position sensor is the frequency at which the magnitude of the transfer function \\(P(s) = v(s)/x(s)\\) drops by \\(3\,dB\\).
The bandwidth of a position sensor is the frequency at which the magnitude of the transfer function \\(P(s) = v(s)/x(s)\\) drops by \\(3\\,dB\\).
Although the bandwidth specification is useful for predicting the resolution of sensor, it reveals very little about the measurement errors caused by sensor dynamics.
@@ -57,7 +57,7 @@ The frequency domain position error is
\begin{equation}
\begin{aligned}
e\_{bw}(s) &= x(s) - v(s) \\\\\\
e\_{bw}(s) &= x(s) - v(s) \\\\
&= x(s) (1 - P(s))
\end{aligned}
\end{equation}
@@ -66,7 +66,7 @@ If the actual position is a sinewave of peak amplitude \\(A = \text{FSR}/2\\):
\begin{equation}
\begin{aligned}
e\_{bw} &= \pm \frac{\text{FSR}}{2} |1 - P(s)| \\\\\\
e\_{bw} &= \pm \frac{\text{FSR}}{2} |1 - P(s)| \\\\
&\approx \pm A n \frac{f}{f\_c}
\end{aligned}
\end{equation}
@@ -143,15 +143,15 @@ To characterize the resolution, we use the probability that the measured value i
If the measurement noise is approximately Gaussian, the resolution can be quantified by the standard deviation \\(\sigma\\) (RMS value).
The empirical rule states that there is a \\(99.7\%\\) probability that a sample of a Gaussian random process lie within \\(\pm 3 \sigma\\).
The empirical rule states that there is a \\(99.7\\%\\) probability that a sample of a Gaussian random process lie within \\(\pm 3 \sigma\\).
This if we define the resolution as \\(\delta = 6 \sigma\\), we will referred to as the \\(6\sigma\text{-resolution}\\).
Another important parameter that must be specified when quoting resolution is the sensor bandwidth.
There is usually a trade-off between bandwidth and resolution (figure [3](#org86a5909)).
There is usually a trade-off between bandwidth and resolution (figure [3](#figure--fig:tradeoff-res-bandwidth)).
<a id="org86a5909"></a>
<a id="figure--fig:tradeoff-res-bandwidth"></a>
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}}
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="<span class=\"figure-number\">Figure 3: </span>The resolution versus banwidth of a position sensor." >}}
Many type of sensor have a limited full-scale-range (FSR) and tend to have an approximated proportional relationship between the resolution and range.
As a result, it is convenient to consider the ratio of resolution to the FSR, or equivalently, the dynamic range (DNR).
@@ -170,19 +170,20 @@ A convenient method for reporting this ratio is in parts-per-million (ppm):
Summary of position sensor characteristics. The dynamic range (DNR) and resolution are approximations based on a full-scale range of \(100\,\mu m\) and a first order bandwidth of \(1\,kHz\)
</div>
| Sensor Type | Range | DNR | Resolution | Max. BW | Accuracy |
|----------------|----------------------------------|---------|------------|----------|-----------|
| Metal foil | \\(10-500\,\mu m\\) | 230 ppm | 23 nm | 1-10 kHz | 1% FSR |
| Piezoresistive | \\(1-500\,\mu m\\) | 5 ppm | 0.5 nm | >100 kHz | 1% FSR |
| Capacitive | \\(10\,\mu m\\) to \\(10\,mm\\) | 24 ppm | 2.4 nm | 100 kHz | 0.1% FSR |
| Electrothermal | \\(10\,\mu m\\) to \\(1\,mm\\) | 100 ppm | 10 nm | 10 kHz | 1% FSR |
| Eddy current | \\(100\,\mu m\\) to \\(80\,mm\\) | 10 ppm | 1 nm | 40 kHz | 0.1% FSR |
| LVDT | \\(0.5-500\,mm\\) | 10 ppm | 5 nm | 1 kHz | 0.25% FSR |
| Interferometer | Meters | | 0.5 nm | >100kHz | 1 ppm FSR |
| Encoder | Meters | | 6 nm | >100kHz | 5 ppm FSR |
| Sensor Type | Range | DNR | Resolution | Max. BW | Accuracy |
|----------------|------------------------------------|---------|------------|-------------|-----------|
| Metal foil | \\(10-500\\,\mu m\\) | 230 ppm | 23 nm | 1-10 kHz | 1% FSR |
| Piezoresistive | \\(1-500\\,\mu m\\) | 5 ppm | 0.5 nm | &gt;100 kHz | 1% FSR |
| Capacitive | \\(10\\,\mu m\\) to \\(10\\,mm\\) | 24 ppm | 2.4 nm | 100 kHz | 0.1% FSR |
| Electrothermal | \\(10\\,\mu m\\) to \\(1\\,mm\\) | 100 ppm | 10 nm | 10 kHz | 1% FSR |
| Eddy current | \\(100\\,\mu m\\) to \\(80\\,mm\\) | 10 ppm | 1 nm | 40 kHz | 0.1% FSR |
| LVDT | \\(0.5-500\\,mm\\) | 10 ppm | 5 nm | 1 kHz | 0.25% FSR |
| Interferometer | Meters | | 0.5 nm | &gt;100kHz | 1 ppm FSR |
| Encoder | Meters | | 6 nm | &gt;100kHz | 5 ppm FSR |
## Bibliography {#bibliography}
<a id="org687716f"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” _Sensors and Actuators a: Physical_ 190 (nil):106–26. <https://doi.org/10.1016/j.sna.2012.10.016>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” <i>Sensors and Actuators a: Physical</i> 190 (nil): 106–26. doi:<a href="https://doi.org/10.1016/j.sna.2012.10.016">10.1016/j.sna.2012.10.016</a>.</div>
</div>