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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>The trade-off triangle — sensitivity · range · bandwidth · Sensing 2026</title>
<!--
Sensing 2026 · Pillar-1 hub · Anchor 1 (the trade-off triangle: sensitivity · dynamic range · bandwidth).
Single self-contained file: NO external requests, works fully offline, inline-SVG widget (no canvas,
resolution-independent). Reconstruction four-beat: The question · Try · Notice · Explain · Connect.
The widget: drag a handle P inside an equilateral triangle; P's barycentric weights (wS,wD,wB) sum to
1 — "the budget you cannot exceed". Real sensors are plotted at fixed barycentric spots. A target zone
near the BANDWIDTH corner encodes the design exercise (10 nT @ 1 kHz). Spec tables + magnetometer table
fold into "Notice this" as evidence; the Claim/Evidence/Assumption/Limitation/Decision exercise follows.
Order-of-magnitude values from the 2025 Sensing I notes.
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<div class="wrap">
<header class="top">
<a class="home" href="workshop.html">← Sensing 2026 hub</a>
<div style="margin-top:6px"><span class="tag">Anchor 1 · core · the trade-off triangle</span></div>
<h1>You can't have it all</h1>
<p class="sub">Sensitivity, dynamic range and bandwidth pull against each other. A sensor is a
<i>choice of corner</i>, not a wish-list — and in this toy model the three weights sum to one fixed
<i>design budget</i>. (Real hardware moves the whole triangle, not just the point inside it — see
<a href="#explain">Explain</a>.)</p>
</header>
<div class="stage">
<section class="widget" aria-label="Interactive: drag the budget around the trade-off triangle">
<span class="wtitle">The instrument — drag the budget P around the triangle</span>
<div class="presets" id="presets">
<span class="hint" style="font-weight:700;color:var(--ink)">Jump to:</span>
<button class="preset" data-bs="0.34,0.33,0.33">Balanced</button>
<button class="preset" data-bs="0.80,0.13,0.07">Sensitivity corner</button>
<button class="preset" data-bs="0.10,0.80,0.10">Range corner</button>
<button class="preset" data-bs="0.10,0.10,0.80">Bandwidth corner</button>
<button class="preset" data-bs="target">▶ The exercise target (10 nT @ 1 kHz)</button>
</div>
<div class="tri2col">
<figure>
<svg class="trisvg" id="tri" viewBox="0 0 520 480" role="img"
aria-label="Equilateral trade-off triangle with a draggable handle P">
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stroke-width="2" stroke-dasharray="6 5" opacity="0.55"></polygon>
<text id="targetLbl" x="0" y="0" font-size="13" font-weight="700" fill="#991b1b"
text-anchor="middle">target: high bandwidth</text>
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<g id="sensors"></g>
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<g id="cornerLabels" font-weight="800" font-size="16"></g>
<!-- draggable handle P -->
<g id="handle">
<circle id="pdot" r="13" fill="#15212e" stroke="#fff" stroke-width="3"></circle>
<text id="plabel" font-size="13" font-weight="800" fill="#fff" text-anchor="middle"
dominant-baseline="central">P</text>
</g>
</svg>
<figcaption>Drag <b>P</b> (mouse or touch). Corner = pure focus; centre = an even split.
<b>Coloured dots</b> = the seven magnetometers in the table below (colour = best corner);
<span style="color:#9aa7b4">grey dots</span> = other sensor types, for scale.</figcaption>
</figure>
<div>
<div class="bars" aria-hidden="false">
<div class="bar"><span class="nm" style="color:var(--S)">Sensitivity</span>
<span class="track"><span class="fill" id="fS" style="background:var(--S);width:33%"></span></span>
<span class="pc" id="pS">33%</span></div>
<div class="bar"><span class="nm" style="color:var(--D)">Dynamic range</span>
<span class="track"><span class="fill" id="fD" style="background:var(--D);width:33%"></span></span>
<span class="pc" id="pD">33%</span></div>
<div class="bar"><span class="nm" style="color:var(--B)">Bandwidth</span>
<span class="track"><span class="fill" id="fB" style="background:var(--B);width:33%"></span></span>
<span class="pc" id="pB">34%</span></div>
<div class="note sum">Σ = <span id="sumv">100</span>% — a fixed <i>toy</i> budget. You re-allocate; only the hardware sets the total.</div>
</div>
<div class="readout" id="liveline" style="font-size:16px"></div>
</div>
</div>
</section><!-- /widget -->
<section class="coach" aria-label="Guided steps — Try, Notice, Explain, Connect">
<div class="stepper" role="tablist" aria-label="Lesson steps">
<button class="step" role="tab" id="tab-try" aria-controls="try" aria-selected="true">1 · Try</button>
<button class="step" role="tab" id="tab-notice" aria-controls="notice" aria-selected="false" tabindex="-1">2 · Notice</button>
<button class="step" role="tab" id="tab-explain" aria-controls="explain" aria-selected="false" tabindex="-1">3 · Explain</button>
<button class="step" role="tab" id="tab-connect" aria-controls="connect" aria-selected="false" tabindex="-1">4 · Connect</button>
</div>
<div class="panels">
<div class="beatpanel is-active" id="try" role="tabpanel" aria-labelledby="tab-try" tabindex="-1">
<h2><span class="beat">Try this</span> Drag the budget around the triangle</h2>
<p class="q"><b>The question:</b> why can't one detector be exquisitely <b>sensitive</b>, cover a huge
<b>dynamic range</b>, <i>and</i> respond at high <b>bandwidth</b> all at once? Drag the handle
<b>P</b> above and watch the three weights — they always add to 100%. Push toward one corner and the
other two collapse: that is the proof you can't max all three.</p>
<p class="hint">Got a feel for it? Step to <b>2 · Notice</b> →</p>
</div>
<div class="beatpanel" id="notice" role="tabpanel" aria-labelledby="tab-notice" tabindex="-1">
<h2><span class="beat">Notice this</span> The three always sum to 100%</h2>
<p class="readout" id="readout">…</p>
<p class="note">The thing to notice: every weight you pour into one corner is drained straight out of
the other two. The full <b>seven-magnetometer comparison</b> — the evidence for the exercise — is in
the table <b>below the triangle</b>.</p>
</div><!-- /#notice -->
<div class="beatpanel" id="explain" role="tabpanel" aria-labelledby="tab-explain" tabindex="-1">
<h2><span class="beat">Explain this</span> A fixed budget you can only re-allocate</h2>
<p>Wherever P sits, the three weights sum to one. Slide P toward the <span class="pill B">Bandwidth</span>
corner and watch <span class="pill S">Sensitivity</span> and <span class="pill D">Dynamic range</span>
drain away in lock-step. <b>Why is the total conserved — why can't a clever design just buy back the
corners it spent?</b></p>
<details>
<summary>Reveal a one-paragraph answer</summary>
<p>The three corners draw on the <b>same physical pool</b>: a finite count of quanta (photons, atoms,
spins) read out in a finite time. Spend that pool on resolving a <i>tiny</i> change and you have used
up the headroom for a <i>large</i> one (range) and the speed of repeated reads (bandwidth). You can
move the allocation — that is engineering — but you cannot create more total without changing the
hardware (more atoms, longer integration, lower temperature), which moves the <i>whole</i> triangle,
not P inside it. Real sensors are points pinned to a corner because their physics already chose the
split for them.</p>
</details>
<details>
<summary>Show the math — barycentric coordinates & the conserved budget</summary>
<p>Put the three corners at vertices <b>v<sub>S</sub>, v<sub>D</sub>, v<sub>B</sub></b>. Any point P
inside the triangle has a unique <b>barycentric coordinate</b>
P = w<sub>S</sub>v<sub>S</sub> + w<sub>D</sub>v<sub>D</sub> + w<sub>B</sub>v<sub>B</sub> with
<b>w<sub>S</sub> + w<sub>D</sub> + w<sub>B</sub> = 1</b> and every w ≥ 0. Each weight is the
<i>relative area</i> of the sub-triangle opposite that corner (divide P-to-edge distance by the
triangle's height). At a vertex one weight is 1 and the others are 0 — the visual proof you cannot
max all three. The constraint Σw = 1 is the conserved quantity: the readout's percentages are just
100·w.</p>
<p><b>Same shape elsewhere on the hub:</b></p>
<ul>
<li><b>Squeezing (anchor 5):</b> the quantum uncertainty <i>area</i> Δx·Δp is conserved — squeeze
one quadrature and the conjugate one must widen. Same conservation law, a different pair of
corners.</li>
<li><b>The Allan / GHZ "bathtubs" (anchors 2 & 4):</b> averaging buys sensitivity as 1/√N only
until drift takes over; entanglement buys 1/N only until decoherence does — a resource that
<b>stops paying past a point</b>. Here, weight you pour into one corner stops paying because it
is drained straight out of the others.</li>
</ul>
</details>
</div><!-- /#explain -->
<div class="beatpanel connect" id="connect" role="tabpanel" aria-labelledby="tab-connect" tabindex="-1">
<h2><span class="beat">Now connect it</span> Pick a corner, defend it</h2>
<p>You need to <b>detect a 10 nT magnetic field at 1 kHz</b>. Drag P into the red <span class="pill tgt">target
zone</span> near the bandwidth corner — see how much sensitivity and range you spend to get there. Then
read the <b>seven-magnetometer table below the triangle</b>, pick <b>two</b> candidates, and decide
between them.</p>
<p>Fill the judgement (Claim · Evidence · Assumption · Limitation · Decision):</p>
<table class="b1">
<tr><td>Claim</td><td>(which technology, in one line)</td></tr>
<tr><td>Evidence</td><td>where each sits on sensitivity / range / <b>bandwidth</b> for the 10 nT @ 1 kHz target</td></tr>
<tr><td>Assumption</td><td>what about the task you're taking for granted (field strength stable? single-shot? environment?)</td></tr>
<tr><td>Limitation</td><td>which corner your choice gives up — and whether that matters here</td></tr>
<tr><td>Decision</td><td>your pick, and what would change it</td></tr>
</table>
<p style="margin-top:12px"><b>What good looks like:</b></p>
<ul>
<li>Names the <b>1 kHz bandwidth</b> as the binding constraint — not sensitivity (10 nT is a large
field, so every row <i>except the Hall probe</i> has ample sensitivity). The pick is decided by
which rows actually <i>clear</i> 1 kHz and at what practical cost (room-T vs cryogenics, size,
integration) — which is why SERF/OPM drops out and NV-diamond stands out.</li>
<li>Uses the triangle as evidence — not "sensor X is better" but "X trades range for the bandwidth
I need".</li>
<li>States one thing that would flip the decision (e.g. if the signal were DC, the slow sensitive
option wins).</li>
</ul>
<p style="margin-top:12px"><b><a href="logbook.html">Logbook</a> (one entry):</b> Tried · Stuck (where/how long) · Hub resource
used · <i>Changed my mind because…</i> · Still unclear.</p>
<h3 style="margin-top:18px">Where this goes next</h3>
<ul>
<li><b>What sets the sensitivity corner?</b> Noise — how low you can push it, and when averaging
stops helping, is <a href="allan-psd-explorer.html">anchor 2 (noise & Allan)</a>.</li>
<li><b>The ultimate sensitivity corner</b> is the <a href="anchor-3-sql.html">standard quantum limit
(anchor 3)</a>; beating it without breaking the budget is what <a href="squeezing-sql.html">squeezing
(anchor 5)</a> and the
<a href="quantum-ladder.html">quantum ladder (anchor 4)</a> are about.</li>
</ul>
<div class="check">
<b>Exercise check.</b> Drag P to the target zone. In one sentence: which magnetometer would you
actually build for 10 nT @ 1 kHz, which corner did you spend to get there, and what single change to
the task would make the opposite corner win?
</div>
</div><!-- /#connect -->
</div><!-- /.panels -->
<div class="stepnav">
<button id="prevBeat" disabled>← Back</button>
<span class="stepcount" id="stepCount">Step 1 of 4</span>
<button id="nextBeat">Next →</button>
</div>
</section><!-- /.coach -->
</div><!-- /.stage -->
<section aria-label="The evidence — seven magnetometers">
<h2 style="font-size:19px">Seven magnetometers — the evidence for the exercise</h2>
<p class="tri" style="margin-top:0">The dots highlighted in the triangle above are these seven (greyed
dots are non-magnetometers, for scale). • <b>Sensitivity</b> — smallest field you can resolve (noise
floor). • <b>Dynamic range</b> — smallest to largest before it saturates. • <b>Bandwidth</b> — how fast
a change you can follow.</p>
<table>
<thead><tr><th>Technology</th><th>Sensitivity (noise floor)</th><th>Dynamic range</th><th>Bandwidth</th><th>Best corner</th></tr></thead>
<tbody>
<tr><td class="s"><span style="color:#d97706">●</span> Hall<span class="nf"> probe</span></td><td>~0.1–1 µT/√Hz</td><td>mT – tens of T</td><td>DC – MHz</td><td><span class="pill D">range</span></td></tr>
<tr><td class="s"><span style="color:#7c3aed">●</span> MR<span class="nf"> · magnetoresistive (AMR/GMR/TMR)</span></td><td>~0.1–10 nT/√Hz</td><td>nT – mT</td><td>DC – MHz</td><td><span class="pill B">bandwidth</span></td></tr>
<tr><td class="s"><span style="color:#7c3aed">●</span> Coil<span class="nf"> · search / induction coil</span></td><td>~fT–pT/√Hz (rises at low f)</td><td>wide</td><td>AC only, ~Hz – MHz</td><td><span class="pill B">bandwidth</span></td></tr>
<tr><td class="s"><span style="color:#d97706">●</span> FGM<span class="nf"> · fluxgate</span></td><td>~5–10 pT/√Hz</td><td>±~100 µT (Earth field)</td><td>DC – ~1 kHz</td><td><span class="pill D">range</span></td></tr>
<tr><td class="s"><span style="color:#7c3aed">●</span> NV<span class="nf"> · NV-diamond</span></td><td>~1 pT – nT/√Hz</td><td>µT – T (vector)</td><td>DC – MHz+</td><td><span class="pill B">bandwidth</span></td></tr>
<tr><td class="s"><span style="color:#2563eb">●</span> SQUID</td><td>~1–10 fT/√Hz</td><td>wide (flux-locked loop)</td><td>DC – kHz+</td><td><span class="pill S">sensitivity</span></td></tr>
<tr><td class="s"><span style="color:#2563eb">●</span> OPM<span class="nf"> · SERF / optically-pumped</span></td><td>~1–15 fT/√Hz</td><td>small (near-zero field)</td><td>DC – ~100–200 Hz</td><td><span class="pill S">sensitivity</span></td></tr>
</tbody>
</table>
<p class="note">Order-of-magnitude, after Bennett <i>et al.</i>,
<a href="https://doi.org/10.3390/s21165568" target="_blank" rel="noopener noreferrer">Sensors 21, 5568 (2021)</a>
(open: <a href="https://arxiv.org/abs/2106.15843" target="_blank" rel="noopener noreferrer">arXiv:2106.15843</a>);
the conventional sensors after Lenz & Edelstein,
<a href="https://doi.org/10.1109/JSEN.2006.874493" target="_blank" rel="noopener noreferrer">IEEE Sens. J. 6, 631 (2006)</a>.
Abbreviations & how each one works: the <a href="#ref">reference list below</a>.</p>
<p class="note"><b>For 10 nT @ 1 kHz:</b> 10 nT is a <i>large</i> field, so sensitivity is <b>not</b> the
binding constraint — only the <b>Hall probe</b> (µT floor) is too coarse to see it. <b>1 kHz bandwidth
is</b>: it rules out <b>SERF/OPM</b> (fades by ~200 Hz) and strains the fluxgate. Which room-temperature
rows actually clear 1 kHz with field to spare? Deciding that is the <b>Connect</b> step (4).</p>
</section>
<section id="ref" aria-label="Reference — abbreviations and operating principles">
<h2 style="font-size:19px">Reference — how each magnetometer works <span style="font-weight:400;color:var(--muted);font-size:15px">(lookup only, not part of the exercise)</span></h2>
<h3>Operating principle — one line each</h3>
<ul>
<li><b>Hall</b> (Hall probe) — a field deflects the current in a conductor, giving a transverse (Hall) voltage. Cheap, robust, integrated — but coarse.</li>
<li><b>MR</b> (magnetoresistive — AMR / GMR / TMR) — a thin-film resistance changes with field (anisotropic / giant / tunnelling magnetoresistance). Tiny, low-power, fast.</li>
<li><b>Coil</b> (search / induction coil) — a changing field induces a coil voltage (Faraday, V ∝ dB/dt). AC only — blind to DC — but very sensitive at high frequency.</li>
<li><b>FGM</b> (fluxgate) — a driven ferromagnetic core saturates asymmetrically in an external field; its even harmonics read the field. The room-temperature DC workhorse.</li>
<li><b>NV</b> (NV-diamond) — a nitrogen-vacancy spin in diamond, optically read out (ODMR); the Zeeman shift gives a vector field at room temperature, at µm scale.</li>
<li><b>SQUID</b> — a superconducting loop with Josephson junctions counts flux quanta. The most sensitive of all — but needs cryogenics.</li>
<li><b>OPM</b> (SERF) — optically-pumped atomic vapour; in the spin-exchange-relaxation-free regime near zero field, alkali spins reach femtotesla sensitivity in a narrow, slow, low-field window.</li>
</ul>
<h3>Abbreviations</h3>
<p class="note">The seven graph/table labels: <b>Hall</b> Hall probe · <b>MR</b> magnetoresistive
(AMR / GMR / TMR = anisotropic / giant / tunnelling) · <b>Coil</b> search / induction coil ·
<b>FGM</b> fluxgate magnetometer · <b>NV</b> nitrogen-vacancy (centre in diamond) ·
<b>SQUID</b> superconducting quantum interference device · <b>OPM</b> optically-pumped magnetometer.
Also: <b>SERF</b> spin-exchange-relaxation-free · <b>ODMR</b> optically-detected magnetic resonance ·
<b>/√Hz</b> per-root-hertz (amplitude noise density) · <b>T</b> tesla (1 T = 10⁹ nT).</p>
<h3>The same triangle beyond magnetometers</h3>
<p class="note">It governs sensors at every scale — the clock buys 10¹⁸ sensitivity by surrendering range and speed:</p>
<table>
<thead><tr><th>Sensor</th><th>Sensitivity</th><th>Dynamic range</th><th>Bandwidth</th><th>Corner</th></tr></thead>
<tbody>
<tr><td class="s">Human eye (dark-adapted rod)</td><td>≈ a single photon</td><td>~10⁶ (with adaptation)</td><td>~10–60 Hz</td><td>sensitive & wide, but <b>slow</b></td></tr>
<tr><td class="s">Smartphone MEMS accelerometer</td><td>~10⁻³ m/s²</td><td>0 – 80 m/s² (≈ ±8 g)</td><td>< 100 Hz</td><td>wide & fast, modest <b>sensitivity</b></td></tr>
<tr><td class="s">Trapped-ion optical clock</td><td>~1 part in 10¹⁸</td><td>one transition ± few MHz</td><td>~1 reading / hour</td><td>ultimate sensitivity, <b>narrow & slow</b></td></tr>
</tbody>
</table>
<p class="note">From the 2025 <i>Sensing I</i> notes.</p>
</section>
<footer>
Single-file, offline, no tracking. Order-of-magnitude values from the 2025 <i>Sensing I</i> notes
(University of Freiburg). Barycentric-budget framing after the standard sensitivity / dynamic-range /
bandwidth trade-off; magnetometer figures: fluxgate / SQUID / SERF-OPM / NV-diamond noise-floor and
bandwidth ranges as commonly tabulated.
</footer>
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{name:'clock', b:[0.90,0.05,0.05], note:'10¹⁸, narrow & slow', mag:false},
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