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—
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" /><meta name="description" content="Tags Position Sensors Reference (Andrew Fleming, 2013) 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 Calibration and nonlinearity Usually quoted as a percentage of the fill-scale range (FSR):
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\begin{equation} \text{mapping error (%)} = \pm 100 \frac{\max{}|e_m(v)|}{\text{FSR}} \end{equation}
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With \(e_m(v)\) is the mapping error." />
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<header class="post-header">
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<h1 class="post-title">A review of nanometer resolution position sensors: operation and performance</h1>
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</header>
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<div class="post-toc" id="post-toc">
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<h2 class="post-toc-title">Contents</h2>
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<div class="post-toc-content">
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<nav id="TableOfContents">
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<ul>
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<li><a href="#sensor-characteristics">Sensor Characteristics</a>
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<ul>
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<li><a href="#calibration-and-nonlinearity">Calibration and nonlinearity</a></li>
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<li><a href="#drift-and-stability">Drift and Stability</a></li>
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<li><a href="#bandwidth">Bandwidth</a></li>
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<li><a href="#noise">Noise</a></li>
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<li><a href="#resolution">Resolution</a></li>
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</ul>
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</li>
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<li><a href="#comparison-and-summary">Comparison and summary</a></li>
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<div class="post-content">
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<dl>
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<dt>Tags</dt>
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<dd><a href="/zettels/position_sensors/">Position Sensors</a></dd>
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<dt>Reference</dt>
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<dd><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></dd>
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<dt>Author(s)</dt>
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<dd>Fleming, A. J.</dd>
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<dt>Year</dt>
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<dd>2013</dd>
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</dl>
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<ul>
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<li>Define concise performance metric and provide expressions for errors sources (non-linearity, drift, noise)</li>
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<li>Review current position sensor technologies and compare their performance</li>
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</ul>
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<h2 id="sensor-characteristics">Sensor Characteristics</h2>
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<h3 id="calibration-and-nonlinearity">Calibration and nonlinearity</h3>
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<p>Usually quoted as a percentage of the fill-scale range (FSR):</p>
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<p>\begin{equation}
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\text{mapping error (%)} = \pm 100 \frac{\max{}|e_m(v)|}{\text{FSR}}
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\end{equation}</p>
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<p>With \(e_m(v)\) is the mapping error.</p>
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<p><a id="org18802f9"></a></p>
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<figure>
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<img src="/ox-hugo/fleming13_mapping_error.png"
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alt="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."/> <figcaption>
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<p>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.</p>
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</figcaption>
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</figure>
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<h3 id="drift-and-stability">Drift and Stability</h3>
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<p>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.</p>
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<p><a id="org65fb6f9"></a></p>
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<figure>
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<img src="/ox-hugo/fleming13_drift_stability.png"
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alt="Figure 2: The worst case range of a linear mapping function \(f_a(v)\) for a given error in sensitivity and offset."/> <figcaption>
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<p>Figure 2: The worst case range of a linear mapping function \(f_a(v)\) for a given error in sensitivity and offset.</p>
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</figcaption>
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</figure>
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<h3 id="bandwidth">Bandwidth</h3>
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<p>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 \(3,dB\).</p>
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<p>Although the bandwidth specification is useful for predicting the resolution of sensor, it reveals very little about the measurement errors caused by sensor dynamics.</p>
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<p>The frequency domain position error is</p>
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<p>\begin{equation}
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\begin{aligned}
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e_{bw}(s) &= x(s) - v(s) \\\<br>
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&= x(s) (1 - P(s))
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\end{aligned}
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\end{equation}</p>
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<p>If the actual position is a sinewave of peak amplitude \(A = \text{FSR}/2\):</p>
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<p>\begin{equation}
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\begin{aligned}
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e_{bw} &= \pm \frac{\text{FSR}}{2} |1 - P(s)| \\\<br>
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&\approx \pm A n \frac{f}{f_c}
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\end{aligned}
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\end{equation}</p>
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<p>with \(n\) is the low pass filter order corresponding to the sensor dynamics and \(f_c\) is the measurement bandwidth.</p>
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<p>Thus, the sensor bandwidth must be significantly higher than the operating frequency if dynamic errors are to be avoided.</p>
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<h3 id="noise">Noise</h3>
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<p>In addition to the actual position signal, all sensors produce some additive measurement noise.
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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.
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These noise processes can usually be approximated by a Gaussian random process.<br /></p>
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<p>A Gaussian random process is usually described by its autocorrelation function or its Power Spectral Density.</p>
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<p>The autocorrelation function of a random process \(\mathcal{X}\) is</p>
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<p>\begin{equation}
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R_{\mathcal{X}}(\tau) = E[\mathcal{X}(t)\mathcal{X}(t + \tau)]
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\end{equation}</p>
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<p>where \(E\) is the expected value operator.</p>
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<p>The variance of the process is equal to \(R_\mathcal{X}(0)\) and is the expected value of the varying part squared:</p>
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<p>\begin{equation}
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\text{Var} \mathcal{X} = E \left[ (\mathcal{X} - E[\mathcal{X}])^2 \right]
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\end{equation}</p>
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<p>The standard deviation \(\sigma\) is the square root of the variance:</p>
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<p>\begin{equation}
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\sigma_\mathcal{X} = \sqrt{\text{Var} \mathcal{X}}
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\end{equation}</p>
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<p>The standard deviation is also the Root Mean Square (RMS) value of a zero-mean random process.</p>
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<p>The Power Spectral Density \(S_\mathcal{X}(f)\) of a random process represents the distribution of power (or variance) across frequency \(f\).</p>
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<p>For example, if the random process under consideration was measured in volts, the power spectral density would have the units of \(V^2/\text{Hz}\).</p>
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<p>The Power Spectral Density can be obtained from the autocorrelation function from the Wiener-Khinchin relation:</p>
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<p>\begin{equation}
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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
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\end{equation}</p>
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<p>If the power Spectral Density is known, the variance of the generating process can be found from the area under the curve:</p>
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||||
<p>\begin{equation}
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\sigma_\mathcal{X}^2 = E[\mathcal{X}^2(t)] = R_\mathcal{X}(0) = \int_0^\infty S_\mathcal{X}(f) df
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\end{equation}</p>
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<p>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.
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It is related to the power spectral density by a square root:</p>
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<p>\begin{equation}
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\text{spectral density} = \sqrt{S_\mathcal{X}(f)}
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\end{equation}</p>
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<p>The units of \(\sqrt{S_\mathcal{X}(f)}\) are \(\text{units}/\sqrt{Hz}\).</p>
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<p>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.</p>
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<h3 id="resolution">Resolution</h3>
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<p>The random noise of a position sensor causes an uncertainty in the measured position.
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If the distance between two measured locations is smaller than the uncertainty, it is possible to mistake one point for the other.</p>
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<p>To characterize the resolution, we use the probability that the measured value is within a certain error bound.</p>
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<p>If the measurement noise is approximately Gaussian, the resolution can be quantified by the standard deviation \(\sigma\) (RMS value).</p>
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<p>The empirical rule states that there is a \(99.7%\) probability that a sample of a Gaussian random process lie within \(\pm 3 \sigma\).
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This if we define the resolution as \(\delta = 6 \sigma\), we will referred to as the \(6\sigma\text{-resolution}\).</p>
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<p>Another important parameter that must be specified when quoting resolution is the sensor bandwidth.
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There is usually a trade-off between bandwidth and resolution (figure <a href="#org954f29f">3</a>).</p>
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<p><a id="org954f29f"></a></p>
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<figure>
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<img src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png"
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alt="Figure 3: The resolution versus banwidth of a position sensor."/> <figcaption>
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<p>Figure 3: The resolution versus banwidth of a position sensor.</p>
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</figcaption>
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</figure>
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<p>Many type of sensor have a limited full-scale-range (FSR) and tend to have an approximated proportional relationship between the resolution and range.
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As a result, it is convenient to consider the ratio of resolution to the FSR, or equivalently, the dynamic range (DNR).
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A convenient method for reporting this ratio is in parts-per-million (ppm):</p>
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<p>\begin{equation}
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\text{DNR}_{\text{ppm}} = 10^6 \frac{\text{full scale range}}{6\sigma\text{-resolution}}
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\end{equation}</p>
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<h2 id="comparison-and-summary">Comparison and summary</h2>
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<p><a id="table--tab:summary-position-sensors"></a></p>
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<div class="table-caption">
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<span class="table-number"><a href="#table--tab:summary-position-sensors">Table 1</a></span>:
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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 \(1\,kHz\)
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</div>
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<table>
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<thead>
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<tr>
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<th>Sensor Type</th>
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<th>Range</th>
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<th>DNR</th>
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<th>Resolution</th>
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<th>Max. BW</th>
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<th>Accuracy</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>Metal foil</td>
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<td>\(10-500,\mu m\)</td>
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<td>230 ppm</td>
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<td>23 nm</td>
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<td>1-10 kHz</td>
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<td>1% FSR</td>
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</tr>
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<tr>
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<td>Piezoresistive</td>
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<td>\(1-500,\mu m\)</td>
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<td>5 ppm</td>
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<td>0.5 nm</td>
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<td>>100 kHz</td>
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<td>1% FSR</td>
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</tr>
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<tr>
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<td>Capacitive</td>
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<td>\(10,\mu m\) to \(10,mm\)</td>
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<td>24 ppm</td>
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<td>2.4 nm</td>
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<td>100 kHz</td>
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<td>0.1% FSR</td>
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</tr>
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<tr>
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<td>Electrothermal</td>
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<td>\(10,\mu m\) to \(1,mm\)</td>
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<td>100 ppm</td>
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<td>10 nm</td>
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<td>10 kHz</td>
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<td>1% FSR</td>
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</tr>
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<tr>
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<td>Eddy current</td>
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||||
<td>\(100,\mu m\) to \(80,mm\)</td>
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||||
<td>10 ppm</td>
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||||
<td>1 nm</td>
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<td>40 kHz</td>
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<td>0.1% FSR</td>
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</tr>
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<tr>
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<td>LVDT</td>
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<td>\(0.5-500,mm\)</td>
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||||
<td>10 ppm</td>
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<td>5 nm</td>
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||||
<td>1 kHz</td>
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<td>0.25% FSR</td>
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</tr>
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||||
<tr>
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<td>Interferometer</td>
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<td>Meters</td>
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<td></td>
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<td>0.5 nm</td>
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<td>>100kHz</td>
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<td>1 ppm FSR</td>
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</tr>
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<tr>
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<td>Encoder</td>
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||||
<td>Meters</td>
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||||
<td></td>
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||||
<td>6 nm</td>
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||||
<td>>100kHz</td>
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<td>5 ppm FSR</td>
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</tr>
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</tbody>
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</table>
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<h1 id="bibliography">Bibliography</h1>
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||||
<p><a id="fleming13_review_nanom_resol_posit_sensor"></a>Fleming, A. J., <em>A review of nanometer resolution position sensors: operation and performance</em>, Sensors and Actuators A: Physical, <em>190(nil)</em>, 106–126 (2013). <a href="http://dx.doi.org/10.1016/j.sna.2012.10.016">http://dx.doi.org/10.1016/j.sna.2012.10.016</a> <a href="#3fb5b61524290e36d639a4fac65703d0">↩</a></p>
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<script src="https://cdn.jsdelivr.net/npm/jquery@3.2.1/dist/jquery.min.js" integrity="sha256-hwg4gsxgFZhOsEEamdOYGBf13FyQuiTwlAQgxVSNgt4=" crossorigin="anonymous"></script>
|
||||
<script src="https://cdn.jsdelivr.net/npm/slideout@1.0.1/dist/slideout.min.js" integrity="sha256-t+zJ/g8/KXIJMjSVQdnibt4dlaDxc9zXr/9oNPeWqdg=" crossorigin="anonymous"></script>
|
||||
|
||||
|
||||
<script type="text/javascript" src="/js/main.4bada4f824623eea2eb7cfd5cf8c1d99c3dd797297e7e8cbc59a41da450bb334.js"></script>
|
||||
|
||||
|
||||
<script type="text/javascript">
|
||||
window.MathJax = {
|
||||
loader: {
|
||||
load: ['[tex]/ams']
|
||||
},
|
||||
tex: {
|
||||
inlineMath: [
|
||||
['$','$'], ['\\(','\\)']
|
||||
],
|
||||
tags: 'ams',
|
||||
packages: {'[+]': ['ams']},
|
||||
}
|
||||
};
|
||||
</script>
|
||||
|
||||
<script async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" crossorigin="anonymous"></script>
|
||||
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/fuse.js/3.2.0/fuse.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mark.js/8.11.1/jquery.mark.min.js"></script>
|
||||
<script type="text/javascript" src="/lib/search/search.js"></script>
|
||||
|
||||
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
Reference in New Issue
Block a user