186 lines
8.6 KiB
Markdown
186 lines
8.6 KiB
Markdown
|
+++
|
|||
|
|
title = "A review of nanometer resolution position sensors: operation and performance"
|
|||
|
|
author = ["Thomas Dehaeze"]
|
|||
|
|
draft = false
|
|||
|
|
+++
|
|||
|
|
|
|||
|
|
Tags
|
|||
|
|
: [Position Sensors]({{< relref "position_sensors" >}})
|
|||
|
|
|
|||
|
|
Reference
|
|||
|
|
: <sup id="3fb5b61524290e36d639a4fac65703d0"><a href="#fleming13_review_nanom_resol_posit_sensor" title="Andrew Fleming, A Review of Nanometer Resolution Position Sensors: Operation and Performance, {Sensors and Actuators A: Physical}, v(nil), 106-126 (2013).">(Andrew Fleming, 2013)</a></sup>
|
|||
|
|
|
|||
|
|
Author(s)
|
|||
|
|
: Fleming, A. J.
|
|||
|
|
|
|||
|
|
Year
|
|||
|
|
: 2013
|
|||
|
|
|
|||
|
|
- Define concise performance metric and provide expressions for errors sources (non-linearity, drift, noise)
|
|||
|
|
- Review current position sensor technologies and compare their performance
|
|||
|
|
|
|||
|
|
|
|||
|
|
## Sensor Characteristics {#sensor-characteristics}
|
|||
|
|
|
|||
|
|
|
|||
|
|
### Calibration and nonlinearity {#calibration-and-nonlinearity}
|
|||
|
|
|
|||
|
|
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}}
|
|||
|
|
\end{equation}
|
|||
|
|
|
|||
|
|
With \\(e\_m(v)\\) is the mapping error.
|
|||
|
|
|
|||
|
|
<a id="org64f54e9"></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." >}}
|
|||
|
|
|
|||
|
|
|
|||
|
|
### 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="org81ab6a9"></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." >}}
|
|||
|
|
|
|||
|
|
|
|||
|
|
### 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\\).
|
|||
|
|
|
|||
|
|
Although the bandwidth specification is useful for predicting the resolution of sensor, it reveals very little about the measurement errors caused by sensor dynamics.
|
|||
|
|
|
|||
|
|
The frequency domain position error is
|
|||
|
|
|
|||
|
|
\begin{equation}
|
|||
|
|
\begin{aligned}
|
|||
|
|
e\_{bw}(s) &= x(s) - v(s) \\\\\\
|
|||
|
|
&= x(s) (1 - P(s))
|
|||
|
|
\end{aligned}
|
|||
|
|
\end{equation}
|
|||
|
|
|
|||
|
|
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)| \\\\\\
|
|||
|
|
&\approx \pm A n \frac{f}{f\_c}
|
|||
|
|
\end{aligned}
|
|||
|
|
\end{equation}
|
|||
|
|
|
|||
|
|
with \\(n\\) is the low pass filter order corresponding to the sensor dynamics and \\(f\_c\\) is the measurement bandwidth.
|
|||
|
|
|
|||
|
|
Thus, the sensor bandwidth must be significantly higher than the operating frequency if dynamic errors are to be avoided.
|
|||
|
|
|
|||
|
|
|
|||
|
|
### Noise {#noise}
|
|||
|
|
|
|||
|
|
In addition to the actual position signal, all sensors produce some additive measurement noise.
|
|||
|
|
In many types of sensor, the majority of noise arises from the thermal noise in resistors and the voltage and current noise in conditioning circuit transistors.
|
|||
|
|
These noise processes can usually be approximated by a Gaussian random process.<br />
|
|||
|
|
|
|||
|
|
A Gaussian random process is usually described by its autocorrelation function or its Power Spectral Density.
|
|||
|
|
|
|||
|
|
The autocorrelation function of a random process \\(\mathcal{X}\\) is
|
|||
|
|
|
|||
|
|
\begin{equation}
|
|||
|
|
R\_{\mathcal{X}}(\tau) = E[\mathcal{X}(t)\mathcal{X}(t + \tau)]
|
|||
|
|
\end{equation}
|
|||
|
|
|
|||
|
|
where \\(E\\) is the expected value operator.
|
|||
|
|
|
|||
|
|
The variance of the process is equal to \\(R\_\mathcal{X}(0)\\) and is the expected value of the varying part squared:
|
|||
|
|
|
|||
|
|
\begin{equation}
|
|||
|
|
\text{Var} \mathcal{X} = E \left[ (\mathcal{X} - E[\mathcal{X}])^2 \right]
|
|||
|
|
\end{equation}
|
|||
|
|
|
|||
|
|
The standard deviation \\(\sigma\\) is the square root of the variance:
|
|||
|
|
|
|||
|
|
\begin{equation}
|
|||
|
|
\sigma\_\mathcal{X} = \sqrt{\text{Var} \mathcal{X}}
|
|||
|
|
\end{equation}
|
|||
|
|
|
|||
|
|
The standard deviation is also the Root Mean Square (RMS) value of a zero-mean random process.
|
|||
|
|
|
|||
|
|
The Power Spectral Density \\(S\_\mathcal{X}(f)\\) of a random process represents the distribution of power (or variance) across frequency \\(f\\).
|
|||
|
|
|
|||
|
|
For example, if the random process under consideration was measured in volts, the power spectral density would have the units of \\(V^2/\text{Hz}\\).
|
|||
|
|
|
|||
|
|
The Power Spectral Density can be obtained from the autocorrelation function from the Wiener-Khinchin relation:
|
|||
|
|
|
|||
|
|
\begin{equation}
|
|||
|
|
S\_{\mathcal{X}} = 2 \mathcal{F}\\{ R\_\mathcal{X}(\tau) \\} = 2 \int\_{-\infty}^{\infty} R\_\mathcal{X}(\tau) e^{-2j\pi f \tau} d\tau
|
|||
|
|
\end{equation}
|
|||
|
|
|
|||
|
|
If the power Spectral Density is known, the variance of the generating process can be found from the area under the curve:
|
|||
|
|
|
|||
|
|
\begin{equation}
|
|||
|
|
\sigma\_\mathcal{X}^2 = E[\mathcal{X}^2(t)] = R\_\mathcal{X}(0) = \int\_0^\infty S\_\mathcal{X}(f) df
|
|||
|
|
\end{equation}
|
|||
|
|
|
|||
|
|
Rather than plotting the frequency distribution of power, it is often convenient to plot the frequency distribution of the standard deviation, which is referred to as the spectral density.
|
|||
|
|
It is related to the power spectral density by a square root:
|
|||
|
|
|
|||
|
|
\begin{equation}
|
|||
|
|
\text{spectral density} = \sqrt{S\_\mathcal{X}(f)}
|
|||
|
|
\end{equation}
|
|||
|
|
|
|||
|
|
The units of \\(\sqrt{S\_\mathcal{X}(f)}\\) are \\(\text{units}/\sqrt{Hz}\\).
|
|||
|
|
|
|||
|
|
The spectral density if preferred in the electronics literature as the RMS value of a noise process can be determined directly from the noise density and effective bandwidth.
|
|||
|
|
|
|||
|
|
|
|||
|
|
### Resolution {#resolution}
|
|||
|
|
|
|||
|
|
The random noise of a position sensor causes an uncertainty in the measured position.
|
|||
|
|
If the distance between two measured locations is smaller than the uncertainty, it is possible to mistake one point for the other.
|
|||
|
|
|
|||
|
|
To characterize the resolution, we use the probability that the measured value is within a certain error bound.
|
|||
|
|
|
|||
|
|
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\\).
|
|||
|
|
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](#orgd8c6776)).
|
|||
|
|
|
|||
|
|
<a id="orgd8c6776"></a>
|
|||
|
|
|
|||
|
|
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: 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).
|
|||
|
|
A convenient method for reporting this ratio is in parts-per-million (ppm):
|
|||
|
|
|
|||
|
|
\begin{equation}
|
|||
|
|
\text{DNR}\_{\text{ppm}} = 10^6 \frac{\text{full scale range}}{6\sigma\text{-resolution}}
|
|||
|
|
\end{equation}
|
|||
|
|
|
|||
|
|
|
|||
|
|
## Comparison and summary {#comparison-and-summary}
|
|||
|
|
|
|||
|
|
<a id="table--tab:summary-position-sensors"></a>
|
|||
|
|
<div class="table-caption">
|
|||
|
|
<span class="table-number"><a href="#table--tab:summary-position-sensors">Table 1</a></span>:
|
|||
|
|
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 |
|
|||
|
|
|
|||
|
|
# Bibliography
|
|||
|
|
<a id="fleming13_review_nanom_resol_posit_sensor"></a>Fleming, A. J., *A review of nanometer resolution position sensors: operation and performance*, Sensors and Actuators A: Physical, *190(nil)*, 106–126 (2013). http://dx.doi.org/10.1016/j.sna.2012.10.016 [↩](#3fb5b61524290e36d639a4fac65703d0)
|