{"id":606,"date":"2023-02-15T18:42:25","date_gmt":"2023-02-15T17:42:25","guid":{"rendered":"https:\/\/alma-consulting.eu\/?p=606"},"modified":"2023-03-04T17:02:32","modified_gmt":"2023-03-04T16:02:32","slug":"working-wonders-with-dynamic-instruments-ep01-resolution-limits","status":"publish","type":"post","link":"https:\/\/alma-consulting.eu\/index.php\/2023\/02\/15\/working-wonders-with-dynamic-instruments-ep01-resolution-limits\/","title":{"rendered":"Working wonders with dynamic instruments &#8211; Ep 01: Resolution limits"},"content":{"rendered":"\n<p> This post comes as an answer to the question I perhaps heard most often when troubleshooting optomechanical devices. When faced with a anything other than an optical sensing device (i.e., an <em>inertial <\/em>sensor<em>)<\/em>, the optomechanical specialist immediately (and rightfully) wonders:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><em>What is this sensor\u2019s (instrument, setup) resolution?<\/em><\/p>\n\n\n\n<p>As it is, this is the perfect example of a question that takes 5 words to ask but requires a relatively lengthy discussion to answer (except for the obvious <em>\u201cit depends\u201d <\/em>which indeed is only optimal in terms of conciseness). Reading what follows will hopefully provide a satisfying answer for anyone interested in the question.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Inertial Sensors<\/h2>\n\n\n\n<p>There are many virtues that make inertia sensors (either velocimeter or accelerometers) attractive. To name only a few, they are compact, relatively economical, have excellent linearity. Importantly, since they are inertia based, they do not require any external reference, making it possible to know exactly which part is moving, because they are delivering a signal corresponding tochanges<em> in position<\/em> of the attached object (contrary to optical based, or capacitive based sensors etc which deliver a signal correlated to changes<em> in separation length <\/em>between two objects). Also, they can provide quantities along any direction, and come into a variety of flavor, that is, with or without integrated electronics, the latter being fit for survival in harsh environments, possibly enduring years of operation with high-energy photons exposure.<\/p>\n\n\n\n<p>Now, an immediate question is, how to convert information produced as velocities or acceleration into lengths? There would be integration (single or double) involved, which inevitably poses important limitations:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Only changes in position are detected, and since the reference (initial value) of position is unknown, so obviously only variation about a time-averaged position can be produced.<\/li><li>Because of first principles, DC (i.e. constant) components cannot be measured (nor do they need to, unless one wishes to double-check local gravity). The lower limit is dictated by the coil\/spring natural frequency (in the case of a velocimeter) or by the crystal leak resistance (in case of a piezoelectric accelerometer).<\/li><li>Integration involves accumulating signals, but so does errors: hence integrating \u2018forever\u2019, i.e. without any high-pass filter, will result in erroneous drifts that will soon render the integrated signals indecipherable. We will come back to this point in a minute.<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Resolution and Usable Bandwidth<\/strong><\/h2>\n\n\n\n<p>Now, letting aside for the moment the discussion about the sensor\u2019s usable bandwidth, we need to recognize that inertial sensors have no \u201cthreshold\u201d effect, hence their effective resolution will be dictated by their intrinsic noise only. An excellent discussion is made in [1], which I will not repeat here.<\/p>\n\n\n\n<p>For the moment, it suffices to say that noise is by definition a wide-band process, hence before trying to quantify a noise amplitude, one should start by specifying the bandwidth that is of relevance for the problem at hand. Starting with the most simple situation, that is the \u201cwhite noise of the engineer\u201d, the instrumental noise variance simply scales linearly with the bandwidth:<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-3899eeab43728f45cd9ad8a88cb71395_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#105;&#103;&#109;&#97;&#125;&#32;&#94;&#32;&#123;&#50;&#125;&#32;&#61;&#32;&#92;&#105;&#110;&#116;&#95;&#123;&#102;&#95;&#49;&#125;&#94;&#123;&#102;&#95;&#50;&#125;&#32;&#123;&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#40;&#102;&#32;&#41;&#32;&#100;&#102;&#32;&#92;&#115;&#105;&#109;&#101;&#113;&#32;&#123;&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#32;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#40;&#102;&#95;&#50;&#45;&#102;&#95;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"285\" style=\"vertical-align: -9px;\"\/> <\/p>\n\n\n\n<p>with:<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-9aa7b0f7230e0d4254ea6680be527b31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#80;&#104;&#105;&Phi;&#95;&#123;&#110;&#110;&#125;&#40;&#102;&#41;&#58;&#32;&#80;&#111;&#119;&#101;&#114;&#32;&#83;&#112;&#101;&#99;&#116;&#114;&#97;&#108;&#32;&#68;&#101;&#110;&#115;&#105;&#116;&#121;&#32;&#91;&#117;&#110;&#105;&#116;&#32;&#69;&#85;&#94;&#50;&#47;&#72;&#122;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"364\" style=\"vertical-align: -5px;\"\/> <\/p>\n\n\n\n<p>The approximation assumes small variation of PSD about its mean value (i.e. a &#8220;flat&#8221; spectrum).<\/p>\n\n\n\n<p>Inertial sensors, have most of their bandwith dominated by thermal noise (i.e. uniform power spectral density), this means a single value is enough to characterize the instrument noise. This, however, is not exactly true at the lower end (where flicker noise dominates, see [2]).<\/p>\n\n\n\n<p>In practice, the RMS amplitude velocity (or acceleration) noise would scale as the bandwidth, and will be independent of the starting or ending frequencies.<\/p>\n\n\n\n<p>Now, this is not the case anymore if a single integration is needed. Namely, for a single integration the variance would read:<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-09a814154bb78c1f3bd4a717a9465062_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#123;&#115;&#105;&#110;&#103;&#108;&#101;&#32;&#105;&#110;&#116;&#101;&#103;&#114;&#97;&#116;&#105;&#111;&#110;&#125;&#125;&#32;&#94;&#32;&#123;&#50;&#125;&#32;&#61;&#32;&#92;&#105;&#110;&#116;&#95;&#123;&#102;&#95;&#49;&#125;&#94;&#123;&#102;&#95;&#50;&#125;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#40;&#102;&#32;&#41;&#125;&#123;&#52;&#32;&#92;&#112;&#105;&#94;&#50;&#32;&#102;&#94;&#50;&#125;&#32;&#100;&#102;&#32;&#92;&#115;&#105;&#109;&#101;&#113;&#32;&#123;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#125;&#123;&#52;&#32;&#92;&#112;&#105;&#94;&#50;&#125;&#32;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#102;&#95;&#49;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#102;&#95;&#50;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"408\" style=\"vertical-align: -16px;\"\/><\/p>\n\n\n\n<p>Completely opposite to the case without integration, the variance is dominated by the low frequency content. In practical terms, there are orders of magnitude separating lowermost and uppermost frequencies so that only the former is of relevance, and the previous expression can be simplified into<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-ded1447dd32ac2565d85acb509c5e478_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#123;&#115;&#105;&#110;&#103;&#108;&#101;&#32;&#105;&#110;&#116;&#101;&#103;&#114;&#97;&#116;&#105;&#111;&#110;&#125;&#125;&#32;&#94;&#32;&#123;&#50;&#125;&#32;&#92;&#115;&#105;&#109;&#101;&#113;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#125;&#123;&#52;&#32;&#92;&#112;&#105;&#94;&#50;&#32;&#102;&#95;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"195\" style=\"vertical-align: -16px;\"\/><\/p>\n\n\n\n<p>This is even more pronounced when one needs to apply a double integration, in which case the variance due to instrumental noise reads:<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-b2e31712a9855357bd205225901ac58e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#123;&#100;&#111;&#117;&#98;&#108;&#101;&#32;&#105;&#110;&#116;&#101;&#103;&#114;&#97;&#116;&#105;&#111;&#110;&#125;&#125;&#32;&#94;&#32;&#123;&#50;&#125;&#32;&#61;&#32;&#92;&#105;&#110;&#116;&#95;&#123;&#102;&#95;&#49;&#125;&#94;&#123;&#102;&#95;&#50;&#125;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#40;&#102;&#32;&#41;&#125;&#123;&#49;&#54;&#32;&#92;&#112;&#105;&#94;&#52;&#32;&#102;&#94;&#52;&#125;&#32;&#100;&#102;&#32;&#92;&#115;&#105;&#109;&#101;&#113;&#32;&#123;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#125;&#123;&#52;&#56;&#32;&#92;&#112;&#105;&#94;&#52;&#125;&#32;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#102;&#95;&#49;&#94;&#51;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#102;&#95;&#50;&#94;&#51;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"419\" style=\"vertical-align: -18px;\"\/><\/p>\n\n\n\n<p>where the approximation again only holds if the noise spectrum is flat. By an argument similar to the previous one, this expression can further degenerate into:<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-76a6abb0cb4545ca9a5861ef7222e576_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#123;&#100;&#111;&#117;&#98;&#108;&#101;&#32;&#105;&#110;&#116;&#101;&#103;&#114;&#97;&#116;&#105;&#111;&#110;&#125;&#125;&#32;&#94;&#32;&#123;&#50;&#125;&#32;&#92;&#115;&#105;&#109;&#101;&#113;&#32;&#123;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#125;&#123;&#52;&#56;&#32;&#92;&#112;&#105;&#94;&#52;&#32;&#102;&#95;&#49;&#94;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"207\" style=\"vertical-align: -18px;\"\/><\/p>\n\n\n\n<p>Now, clearly, we have a very strong dependency of intrinsic noise on the selected bandwidth, or more accurately on the lower end of the bandwidth. Dividing the lowermost frequency by a factor of 2 will increase the intrinsic noise by at least a factor of 8 (and in practice, a bit more than that as soon as flicker noise comes in).<\/p>\n\n\n\n<!--nextpage-->\n\n\n\n<h2 class=\"wp-block-heading\">Numbers, please!<\/h2>\n\n\n\n<p>Next, let\u2019s apply the preceding reasoning to actual devices. Reading vendors datasheets can be disconcerting, mostly because of heterogeneity.<\/p>\n\n\n\n<p>As always, it is good to start with the basics. And in this respect old documentation has my favor over the more recent ones.<\/p>\n\n\n\n<p>Maybe the most widespread signal conditioner of all times is the B&amp;K model 2635 charge amplifier. Reading the datasheet [3], the specifications states:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\"><p>INHERENT NOISE (2 Hz to 22 kHz):<br>5  10-3 pC referred to input with maximum sensitivity and 1 nF transducer(*) capacitance<\/p><\/blockquote>\n\n\n\n<p>Equipped with this information, one can easily arrive at a first estimate of spurious displacements due to electronics self-noise. Here, we should start by making an assumption regarding the sensitivity of our sensor. Let\u2019s assume we are using a general purpose, relatively high-sensitivity piezoelectric accelerometer such as the B&amp;K 4370 or its counterpart 4381. Those sensors deliver 10pC\/m\/s\u00b2, so that the inherent noise reads&nbsp;(in acceleration units) <\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-d73121bbe82e1c02c1718a06e3ddde5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#123;&#97;&#99;&#99;&#101;&#108;&#101;&#114;&#97;&#116;&#105;&#111;&#110;&#125;&#61;&#32;&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#123;&#99;&#104;&#97;&#114;&#103;&#101;&#125;&#47;&#32;&#83;&#101;&#110;&#115;&#105;&#116;&#105;&#118;&#105;&#116;&#121;&#61;&#32;&#53;&#92;&#44;&#32;&#49;&#48;&#94;&#123;&#45;&#52;&#125;&#109;&#47;&#115;&#94;&#50;&#32;&#82;&#77;&#83;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"416\" style=\"vertical-align: -6px;\"\/> <\/p>\n\n\n\n<p>And the equivalent acceleration PSD reads:<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-8dd47d7d867260e59497e79df3472fe3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#40;&#53;&#94;&#44;&#49;&#48;&#123;&#45;&#52;&#125;&#41;&#94;&#50;&#125;&#123;&#40;&#50;&#50;&#46;&#49;&#48;&#94;&#51;&#41;&#125;&#61;&#49;&#46;&#49;&#51;&#92;&#44;&#49;&#48;&#94;&#123;&#45;&#49;&#49;&#125;&#40;&#109;&#47;&#115;&#94;&#50;&#41;&#94;&#50;&#47;&#72;&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"330\" style=\"vertical-align: -17px;\"\/><\/p>\n\n\n\n<p>Now, the corresponding displacement can be estimated as:<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-8d2a0a2bed87ce1b9f81d38676b504de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#123;&#100;&#111;&#117;&#98;&#108;&#101;&#32;&#105;&#110;&#116;&#101;&#103;&#114;&#97;&#116;&#105;&#111;&#110;&#125;&#32;&#61;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#125;&#123;&#52;&#56;&#32;&#92;&#112;&#105;&#94;&#52;&#32;&#102;&#95;&#49;&#94;&#51;&#125;&#125;&#61;&#49;&#55;&#92;&#44;&#49;&#48;&#94;&#123;&#45;&#57;&#125;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"314\" style=\"vertical-align: -23px;\"\/><\/p>\n\n\n\n<p>This is a pretty impressive result, if you ask me, but of course relies on a wild guess from our side that is&nbsp;: the inherent noise has a sufficiently flat spectrum for our simplified formulas to be applied.<\/p>\n\n\n\n<p>(*) incidentally, this introduces the notion of charge capacitance. For the sake of simplicity, we will ignore this aspect for the moment and come back to it in a second article. The interested readers can start to dig into the subject by reading Enrico Rubiola\u2019s lectures on instruments [3], and in particular the description of thermal noise across a capacitor.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Practical Methods and Results<\/strong><\/h2>\n\n\n\n<p>In practical terms, the preceding discussion is interesting but doesn\u2019t allow for the quantitative determination of actual devices&#8217; instrumental noise. Furthermore, other (extraneous) effects exist so that the environment does contribute to the instrument performance (not in a positive way), and those additionnal effects can only be quantified through testing.<\/p>\n\n\n\n<p>Obviously, one could wish for conducting measurements in a perfectly clean environment (quiet \u00ab&nbsp;room&nbsp;\u00bb), with controlled external influences, and sufficiently small micro-seismic background noise. This condition can of course be obtained in specific laboratories, such as the Provence based Laboratoire Souterrain \u00e0 Bas Bruit (LSBB) [4], but we said we wanted <em>practical<\/em> methods, right&nbsp;?<\/p>\n\n\n\n<p>To start with, we will use here two fairly simple (but independent) approaches&nbsp;:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>the corrected difference method<\/li><li>the dummy sensor method<\/li><\/ol>\n\n\n\n<p>The first method assumes that one has a pair of completely identical sensors, with footprint small enough so that they can be placed side-by-side and measure essentially the same signals. By subtracting the measured signals, all what should be left should be noise (assuming both channels electronics are at least nominally identical). This error signal is multiplied by  0.707, to account for the fact that being independent, the two noise sources will add quadratically, hence the term \u201ccorrected\u201d.<\/p>\n\n\n\n<p>The second method assumes we can replace our transducer with an electrically equivalent but vibration insensitive component. In our case, to replace our piezoelectric accelerometers, we will use a 1nF capacitor (off the shelf available from Virtins Technology, see [4]).<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><img decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/WWDI-2sensorsMethod.jpg\" alt=\"\" style=\"width: 300px;\"><\/td><td><img decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/WWDI-DummySensorMethod.jpg\" alt=\"\" style=\"width: 300px;\"><\/td><\/tr><\/tbody><\/table><figcaption>left: two sensors method right: dummy sensor method<\/figcaption><\/figure>\n\n\n\n<p>Both methods have obvious limits, which for the sake of conciseness we will not address here.<\/p>\n\n\n\n<p>A specific setup has been used, as follows:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>the Device Under Test (DUT) here was a charge\/voltage amplifier from MMF (model M72B3), with 3 inputs<ul><li>Inputs 1 and 2 were used for the corrected method: connected to B&amp;K 4381 sensors, (sensitivities in agreement within less than 1%). Both sensors are magnetically mounted side-by- onto a 20mm thick steel plate. Connexions are made using low-noise cables (UA0038 from B&amp;K).<\/li><li>Input 3 has been used for the dummy sensor method: This capacitor is directly connected into the conditionner then fitted with a 50 Ohm termination resistor. At room temperature, such a resistor has a thermal noise slightly below 1nV\/sqrt(Hz), so largely negligible for our purpose.<\/li><li>All channels have a gain of 100, a 3Hz (analog) high-pass filter is activated.<\/li><\/ul><\/li><li>all signals are measured thanks to an ultra low-noise DAQ: Data Translation model DT9824A, (readout noise below 0.3microVolt\/sqrt(Hz) )<\/li><li>the setup was placed in a quiet room (acoustic noise level below 40dB(A) with minimal electrical and mechanical perturbations (basement room, the optic table resting against a wall)). Temperature was between 12 and 13 degrees Celsius, and relative humidity between 50 and 55%.<\/li><\/ul>\n\n\n\n<p>Let\u2019s start by examining the results in terms of acceleration. Recorded time-histories are shown on Figure 1.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"757\" height=\"1024\" src=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/image-757x1024.png\" alt=\"\" class=\"wp-image-647\" srcset=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/image-757x1024.png 757w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/image-222x300.png 222w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/image-768x1039.png 768w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/image-1135x1536.png 1135w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/image-1514x2048.png 1514w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/image-1200x1623.png 1200w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/image.png 1831w\" sizes=\"auto, (max-width: 757px) 100vw, 757px\" \/><figcaption>Figure 1: Acceleration signals time-histories using both corrected difference (channels 1 and 2) and dummy sensor (channel 3) methods.<\/figcaption><\/figure>\n\n\n\n<p>Clearly, the sensors pick up more signal than the intrinsic noise. This is probably more readable in the frequency domain, as show on Figure 2. Here, we restrict ourselves to the 1-200Hz frequency band, which largely covers the useful applications. We use a rectangular window, since we essentially are dealing with wide-band noise. The frequency resolution is set to 0.5Hz, and the duration of each short-term spectrum is 20s.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA1_2023-02-05@21h51mn-757x1024.png\" alt=\"\" class=\"wp-image-651\" width=\"607\" height=\"821\" srcset=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA1_2023-02-05@21h51mn-757x1024.png 757w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA1_2023-02-05@21h51mn-222x300.png 222w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA1_2023-02-05@21h51mn-768x1039.png 768w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA1_2023-02-05@21h51mn-1135x1536.png 1135w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA1_2023-02-05@21h51mn-1514x2048.png 1514w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA1_2023-02-05@21h51mn-1200x1623.png 1200w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA1_2023-02-05@21h51mn.png 1831w\" sizes=\"auto, (max-width: 607px) 100vw, 607px\" \/><figcaption>Figure 2:  acceleration Amplitude Spectral Densities using both corrected difference (channels 1 and 2) and dummy sensor (channel 3) methods.<\/figcaption><\/figure>\n\n\n\n<p>On figure 2, I have superimposed the Amplitude Noise Spectra (ASD, unit: mm\/s\u00b2\/sqrt(Hz)) of the three inputs:<\/p>\n\n\n\n<p>&#8211; inputs 1 and 2 are completely identical, and exhibit clear response peaks, undoubtedly coming from mechanical resonances amplifying micro-seismic background noise, whereas<\/p>\n\n\n\n<p>&#8211; input 3 exhibits a smooth spectrum, typical of electronic noise,<\/p>\n\n\n\n<p>Those finding are completely in line with our understanding. We can perform double integration, to yield Figure 3. Interestingly, the prominent peaks visible in Figure 2 around 32Hz are compressed by double integration, and the 3 signals have very close RMS displacement values.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"757\" height=\"1024\" src=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA2_2023-02-05@21h51mn-757x1024.png\" alt=\"\" class=\"wp-image-653\" srcset=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA2_2023-02-05@21h51mn-757x1024.png 757w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA2_2023-02-05@21h51mn-222x300.png 222w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA2_2023-02-05@21h51mn-768x1039.png 768w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA2_2023-02-05@21h51mn-1135x1536.png 1135w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA2_2023-02-05@21h51mn-1514x2048.png 1514w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA2_2023-02-05@21h51mn-1200x1623.png 1200w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA2_2023-02-05@21h51mn.png 1831w\" sizes=\"auto, (max-width: 757px) 100vw, 757px\" \/><figcaption>Figure 3: displacement Amplitude Spectral Densities using both corrected difference (channels 1 and 2) and dummy sensor (channel 3) methods.<\/figcaption><\/figure>\n\n\n\n<p>Lastly, by subtracting (in the time domain) signals 1 and 2, and again performing double integration (in the frequency domain), one arrives at Figure 4. The resonance peaks are effectively eliminated (canceled), and both methods do yield essentially the same result, here an inherent noise in the range of 200nm RMS value.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"757\" height=\"1024\" src=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA3_2023-02-05@21h51mn-757x1024.png\" alt=\"\" class=\"wp-image-655\" srcset=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA3_2023-02-05@21h51mn-757x1024.png 757w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA3_2023-02-05@21h51mn-222x300.png 222w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA3_2023-02-05@21h51mn-768x1039.png 768w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA3_2023-02-05@21h51mn-1135x1536.png 1135w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA3_2023-02-05@21h51mn-1514x2048.png 1514w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA3_2023-02-05@21h51mn-1200x1623.png 1200w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/RTDA3_2023-02-05@21h51mn.png 1831w\" sizes=\"auto, (max-width: 757px) 100vw, 757px\" \/><figcaption>Figure 4: displacement Amplitude Spectral Densities using both corrected difference (trace 1) and dummy sensor (trace 2) methods.<\/figcaption><\/figure>\n\n\n\n<p>Now, how does this compare to the levels expected <em>without<\/em> actually performing the test? Based on its vendor datasheet, the M72 series have an inherent noise of respectively 3fC RMS for bandwidth 1 to 10 kHz [5]. This in turn implies an inherent charge noise with ASD of 3.10<sup>-2<\/sup> fc\/sqrt(Hz) (referred at input). To convert it to acceleration, we need to account for the sensor sensitivity (10pC\/m\/s\u00b2), yielding an equivalent acceleration ASD of 3.10<sup>-6<\/sup>m\/s<sup>2<\/sup>\/sqrt(Hz).<\/p>\n\n\n\n<p>The \u2018bottom\u2019 value of the acceleration ASD in figure 2 is of about 0.003.mm\/s<sup>2<\/sup>\/sqrt(Hz) , which <em>exactly<\/em> matches the estimated value. The important fact, however, is that below 20Hz an additional contribution appears (besides the &#8220;flat&#8221; thermal noise), which makes our initial estimate in terms of displacement significantly wrong. Since the majority of energy comes from the lower end of the bandwidth, basing our estimate on the \u201caverage\u201d ASD over the entire specified band (1 Hz to 10kHz in that case) would result in a level of about :<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/alma-consulting.eu\/wp-content\/ql-cache\/quicklatex.com-4f4f08725b88b86a87d9cb441276846e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#123;&#100;&#105;&#115;&#112;&#108;&#97;&#99;&#101;&#109;&#101;&#110;&#116;&#125;&#125;&#32;&#92;&#115;&#105;&#109;&#101;&#113;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#32;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#80;&#104;&#105;&#95;&#123;&#110;&#110;&#125;&#125;&#123;&#52;&#56;&#32;&#92;&#112;&#105;&#94;&#52;&#32;&#102;&#95;&#49;&#94;&#51;&#125;&#125;&#125;&#32;&#61;&#32;&#51;&#92;&#44;&#32;&#49;&#48;&#94;&#123;&#45;&#54;&#125;&#32;&#47;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#56;&#32;&#92;&#112;&#105;&#94;&#52;&#32;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#40;&#49;&#46;&#48;&#41;&#94;&#51;&#125;&#61;&#52;&#53;&#32;&#110;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"459\" style=\"vertical-align: -23px;\"\/><\/p>\n\n\n\n<p>Since the actual ASD around 1Hz is about 10 times higher, it is not surprising that the final level is significantly larger (a factor of 4 in this case).<\/p>\n\n\n\n<!--nextpage-->\n\n\n\n<h2 class=\"wp-block-heading\"><strong>In a Nutshell<\/strong><\/h2>\n\n\n\n<p>Dynamic instrumentation uses inertial sensors, which requires to perform either single or double integration of signals. Having excellent linearity and no threshold, their resolution is essentially limited by their inherent noise, which In turn depends prominently on two parameters: firstly, the signal conditioner inherent noise and secondly, the selected lower end of the frequency bandwidth.<\/p>\n\n\n\n<p>While the choice of the starting frequency depends on the problem at hand, the knowledge of the conditioning electronics inherent noise is indispensable, and basing estimates on the overall (wideband) noise level can lead to significant in error in the estimation of the instrument noise. It is therefore mandatory to either obtain more information from the vendor (with all due caution), or better,to run simple tests such as the corrected difference or the dummy sensor test as described above.<\/p>\n\n\n\n<p>Once the spectral content of the noise is known, it is a matter of simple, spreadsheet -type calculation to obtain the required estimate. The most practical chart to be produced looks like the one below: it features the best estimate (i.e., median value, or 50% probability of exceedance) of the displacement standard deviation, as a function of the lower most frequency.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"768\" src=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/NoiseChart-1024x768.png\" alt=\"\" class=\"wp-image-662\" srcset=\"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/NoiseChart-1024x768.png 1024w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/NoiseChart-300x225.png 300w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/NoiseChart-768x576.png 768w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/NoiseChart-1536x1152.png 1536w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/NoiseChart-1200x900.png 1200w, https:\/\/alma-consulting.eu\/wp-content\/uploads\/2023\/02\/NoiseChart.png 1750w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Figure 6: <em>Typical chart for estimation of a setup (sensor + conditionner) noise<\/em><\/figcaption><\/figure>\n\n\n\n<p>On Figure 6, one can readily read the required lower cutoff frequency (high-pass) to be used in order to guarantee that the setup total noise will not exceed a given value. For example, setting 10nm RMS as an acceptable noise level, one simply needs to (mentally) draw a horizontal line at 10nm and reading the intersection with the P50 curve. In this case, the intersection occurs at 7Hz,  so this is the lower frequency limit to be used. It remains to see if this is compatible with the overall system performance, but generally mechanical resonances are well above this value, so even a &#8216;basic&#8217; setup already offers interesting performances.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">What&#8217;s next?<\/h2>\n\n\n\n<p>We have covered the basics. However, in future posts, I will cover two important aspects pertaining to the question, namely:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>environment effects, and<\/li><li>enhanced performance using specific hardware: (low-noise electronics, high-sensitivity sensors)  <\/li><\/ul>\n\n\n\n<p>In the meantime, all questions are very welcome.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">References<\/h2>\n\n\n\n<p>[1] Collette &#8211; ATS\/Note\/2011\/001 (TECH) 2011-01-05 &#8211; Review of sensors for low frequency seismic vibration measurement <\/p>\n\n\n\n<p><a href=\"https:\/\/cds.cern.ch\/record\/1322403\/files\/CERN-ATS-Note-2011-001-TECH.pdf\">https:\/\/cds.cern.ch\/record\/1322403\/files\/CERN-ATS-Note-2011-001-TECH.pdf<\/a><\/p>\n\n\n\n<p>[2] Bruel and Kjaer, Model 2635 datasheet, last retrieved as of Feb 2023 under: <a href=\"https:\/\/www.bksv.com\/-\/media\/literature\/Product-Data\/bp0099.ashx\">https:\/\/www.bksv.com\/-\/media\/literature\/Product-Data\/bp0099.ashx<\/a><\/p>\n\n\n\n<p>[3] Rubiola, Lectures on Instruments, <a href=\"http:\/\/rubiola.org\/\">http:\/\/rubiola.org\/<\/a><\/p>\n\n\n\n<p>[4] Laboratoire Souterrain \u00e0 Bas Bruit, (LSBB) <a href=\"https:\/\/lsbb.cnrs.fr\/\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/lsbb.cnrs.fr\/<\/a><\/p>\n\n\n\n<p>[5] Virtins technologies Charge mode sensors Accessories <a href=\"https:\/\/virtins.com\/Piezo-Sensors.shtml\">https:\/\/virtins.com\/Piezo-Sensors.shtml<\/a><\/p>\n\n\n\n<p>[6] MMF(Metra Mess- und Frequenztechnik ) <a href=\"https:\/\/www.mmf.de\/pdf\/4-1.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/www.mmf.de\/pdf\/4-1.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post comes as an answer to the question I perhaps heard most often when troubleshooting optomechanical devices. When faced with a anything other than an optical sensing device (i.e., an inertial sensor), the optomechanical specialist immediately (and rightfully) wonders: What is this sensor\u2019s (instrument, setup) resolution? As it is, this is the perfect example [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[12],"class_list":["post-606","post","type-post","status-publish","format-standard","hentry","category-technical-literature","tag-accelerometer-geophone-resolution-intrisic-noise-optomechanical-device-stability-charge-amplifier-pyroelectric-effect"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Working wonders with dynamic instruments - Ep 01: Resolution limits - Alma Consulting<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/alma-consulting.eu\/index.php\/2023\/02\/15\/working-wonders-with-dynamic-instruments-ep01-resolution-limits\/\" \/>\n<link rel=\"next\" href=\"https:\/\/alma-consulting.eu\/index.php\/2023\/02\/15\/working-wonders-with-dynamic-instruments-ep01-resolution-limits\/2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Working wonders with dynamic instruments - Ep 01: Resolution limits - Alma Consulting\" \/>\n<meta property=\"og:description\" content=\"This post comes as an answer to the question I perhaps heard most often when troubleshooting optomechanical devices. When faced with a anything other than an optical sensing device (i.e., an inertial sensor), the optomechanical specialist immediately (and rightfully) wonders: What is this sensor\u2019s (instrument, setup) resolution? 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When faced with a anything other than an optical sensing device (i.e., an inertial sensor), the optomechanical specialist immediately (and rightfully) wonders: What is this sensor\u2019s (instrument, setup) resolution? 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