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Your smartphone battery and 3D printer nozzle share a critical safety problem: if they get too hot, they fail catastrophically. How does a digital circuit measure that rising heat in real time?

The answer is an **NTC thermistor**, an electronic component whose electrical **resistance** (the opposition to current flow, measured in ohms, $\Omega$) drops rapidly as temperature increases.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Metal Conductor vs NTC Thermistor</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; font-family: system-ui, -apple-system, sans-serif; }
    body {
      background: #f8fafc;
      color: #1e2945;
      padding: 10px;
      display: flex;
      flex-direction: column;
      gap: 8px;
      user-select: none;
      max-width: 760px;
      margin: 0 auto;
    }

    .controls {
      display: flex;
      align-items: center;
      justify-content: space-between;
      background: #ffffff;
      padding: 8px 14px;
      border-radius: 8px;
      border: 1px solid #e2e8f0;
      gap: 12px;
    }
    .slider-wrap {
      display: flex;
      align-items: center;
      flex: 1;
      gap: 10px;
    }
    .temp-label {
      font-size: 13px;
      font-weight: 700;
      color: #1e2945;
      white-space: nowrap;
    }
    input[type=range] {
      flex: 1;
      accent-color: #ff5722;
      cursor: pointer;
      height: 6px;
    }
    .temp-badge {
      font-size: 13px;
      font-weight: 800;
      padding: 3px 10px;
      border-radius: 6px;
      min-width: 64px;
      text-align: center;
      border: 1px solid transparent;
      transition: background 0.3s, color 0.3s, border-color 0.3s;
    }

    .grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 8px;
    }

    @media (max-width: 520px) {
      .grid { grid-template-columns: 1fr; }
    }

    .panel {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 10px;
      display: flex;
      flex-direction: column;
      gap: 6px;
    }

    .panel-header {
      display: flex;
      justify-content: space-between;
      align-items: center;
    }
    .panel-title {
      font-size: 12px;
      font-weight: 700;
      color: #1e2945;
      text-transform: uppercase;
      letter-spacing: 0.3px;
    }
    .meter {
      font-size: 13px;
      font-weight: 800;
      font-family: ui-monospace, monospace;
      padding: 2px 7px;
      background: #f1f5f9;
      border-radius: 4px;
      border: 1px solid #cbd5e1;
      color: #1e2945;
    }

    .lattice-box {
      position: relative;
      height: 95px;
      background: #0f172a;
      border-radius: 6px;
      overflow: hidden;
    }
    canvas {
      width: 100%;
      height: 100%;
      display: block;
    }

    .caption {
      font-size: 10.5px;
      color: #64748b;
      line-height: 1.25;
      min-height: 26px;
    }
    .caption strong {
      color: #1e2945;
    }

    .graph-box {
      height: 75px;
      position: relative;
      background: #fafbfc;
      border: 1px solid #f1f5f9;
      border-radius: 6px;
      padding: 2px;
    }
    svg {
      width: 100%;
      height: 100%;
      display: block;
    }
  </style>
</head>
<body>

  <div class="controls">
    <div class="slider-wrap">
      <span class="temp-label">Temperature:</span>
      <input type="range" id="tempSlider" min="0" max="100" value="20" step="1">
    </div>
    <div id="tempVal" class="temp-badge">20°C</div>
  </div>

  <div class="grid">
    
    <div class="panel">
      <div class="panel-header">
        <span class="panel-title">Metal Conductor</span>
        <span class="meter" id="metalMeter">10.8 Ω</span>
      </div>
      <div class="lattice-box">
        <canvas id="metalCanvas"></canvas>
      </div>
      <div class="caption">
        <strong>Vibration obstruction:</strong> Carrier count is constant. Lattice vibrations scatter electrons, slightly increasing resistance.
      </div>
      <div class="graph-box">
        <svg id="metalGraph" viewBox="0 0 220 75"></svg>
      </div>
    </div>

    
    <div class="panel">
      <div class="panel-header">
        <span class="panel-title">NTC Thermistor</span>
        <span class="meter" id="ntcMeter" style="color:#0284c7;">35.0 kΩ</span>
      </div>
      <div class="lattice-box">
        <canvas id="ntcCanvas"></canvas>
      </div>
      <div class="caption">
        <strong>Carrier generation:</strong> Thermal energy frees bound valence electrons, causing an exponential drop in resistance.
      </div>
      <div class="graph-box">
        <svg id="ntcGraph" viewBox="0 0 220 75"></svg>
      </div>
    </div>
  </div>

  <script>
    const slider = document.getElementById('tempSlider');
    const tempVal = document.getElementById('tempVal');
    const metalMeter = document.getElementById('metalMeter');
    const ntcMeter = document.getElementById('ntcMeter');

    const metalCanvas = document.getElementById('metalCanvas');
    const ntcCanvas = document.getElementById('ntcCanvas');
    const ctxM = metalCanvas.getContext('2d');
    const ctxN = ntcCanvas.getContext('2d');

    function resizeCanvases() {
      const dpr = window.devicePixelRatio || 1;
      [metalCanvas, ntcCanvas].forEach(c => {
        c.width = c.clientWidth * dpr;
        c.height = c.clientHeight * dpr;
      });
    }
    window.addEventListener('resize', () => { resizeCanvases(); drawGraphs(); });
    resizeCanvases();

    let temp = 20;

    // Metal electrons
    const metalElectrons = [];
    for (let i = 0; i < 22; i++) {
      metalElectrons.push({
        x: Math.random() * 220,
        y: Math.random() * 95,
        vx: (Math.random() - 0.5) * 28,
        vy: (Math.random() - 0.5) * 28
      });
    }

    // NTC Lattice & Carrier pool
    const semiElectrons = [];
    for (let i = 0; i < 50; i++) {
      semiElectrons.push({
        x: Math.random() * 220,
        y: Math.random() * 95,
        vx: (Math.random() - 0.5) * 24,
        vy: (Math.random() - 0.5) * 24,
        threshold: 2 + (i / 50) * 96,
        phase: Math.random() * Math.PI * 2
      });
    }

    // Metal resistance calculation (linear rise)
    function getMetalR(T) {
      return 10.0 + 0.04 * T;
    }

    // NTC resistance calculation (exponential decay R = R0 * exp(B * (1/T - 1/T0)))
    function getNtcR(T) {
      const Tk = T + 273.15;
      const T0 = 293.15; // 20°C
      const R0 = 35.0;   // 35 kOhm at 20°C
      const B = 2100;
      return R0 * Math.exp(B * (1 / Tk - 1 / T0));
    }

    function drawGraphs() {
      const mSvg = document.getElementById('metalGraph');
      const nSvg = document.getElementById('ntcGraph');

      // Plot layout: viewBox 0 0 220 75
      // Left margin 32, right margin 12, top margin 8, bottom margin 18
      const x0 = 34, x1 = 208;
      const yTop = 10, yBottom = 58;

      // X coordinate for current T
      const curX = x0 + (temp / 100) * (x1 - x0);

      // --- Metal Graph (Linear modest rise from 10 to 14 Ohm) ---
      const mR_min = 9.0, mR_max = 15.0;
      const mY0 = yBottom - ((getMetalR(0) - mR_min) / (mR_max - mR_min)) * (yBottom - yTop);
      const mY100 = yBottom - ((getMetalR(100) - mR_min) / (mR_max - mR_min)) * (yBottom - yTop);
      const curMy = yBottom - ((getMetalR(temp) - mR_min) / (mR_max - mR_min)) * (yBottom - yTop);

      mSvg.innerHTML = `
        <line x1="${x0}" y1="${yBottom}" x2="${x1}" y2="${yBottom}" stroke="#cbd5e1" stroke-width="1"/>
        <line x1="${x0}" y1="${yTop}" x2="${x0}" y2="${yBottom}" stroke="#cbd5e1" stroke-width="1"/>
        <text x="${x1}" y="70" font-size="8.5" fill="#64748b" text-anchor="end">Temp (°C)</text>
        <text x="10" y="16" font-size="8.5" fill="#64748b">R (Ω)</text>
        <text x="28" y="${yBottom + 2}" font-size="7.5" fill="#94a3b8" text-anchor="end">10</text>
        <text x="28" y="${yTop + 5}" font-size="7.5" fill="#94a3b8" text-anchor="end">14</text>
        <path d="M ${x0} ${mY0.toFixed(1)} L ${x1} ${mY100.toFixed(1)}" fill="none" stroke="#64748b" stroke-width="2.2" stroke-linecap="round"/>
        <circle cx="${curX.toFixed(1)}" cy="${curMy.toFixed(1)}" r="4" fill="#ff5722" stroke="#ffffff" stroke-width="1.5"/>
      `;

      // --- NTC Graph (True steep exponential drop from ~100k to ~2.5k) ---
      const nR_max = 100.0;
      const nR_min = 0.0;
      const pts = [];
      for (let t = 0; t <= 100; t += 2) {
        const x = x0 + (t / 100) * (x1 - x0);
        const rVal = Math.min(nR_max, getNtcR(t));
        const y = yBottom - ((rVal - nR_min) / (nR_max - nR_min)) * (yBottom - yTop);
        pts.push(`${x.toFixed(1)},${y.toFixed(1)}`);
      }

      const curNr = Math.min(nR_max, getNtcR(temp));
      const curNy = yBottom - ((curNr - nR_min) / (nR_max - nR_min)) * (yBottom - yTop);

      nSvg.innerHTML = `
        <line x1="${x0}" y1="${yBottom}" x2="${x1}" y2="${yBottom}" stroke="#cbd5e1" stroke-width="1"/>
        <line x1="${x0}" y1="${yTop}" x2="${x0}" y2="${yBottom}" stroke="#cbd5e1" stroke-width="1"/>
        <text x="${x1}" y="70" font-size="8.5" fill="#64748b" text-anchor="end">Temp (°C)</text>
        <text x="4" y="16" font-size="8.5" fill="#64748b">R (kΩ)</text>
        <text x="28" y="${yBottom + 2}" font-size="7.5" fill="#94a3b8" text-anchor="end">0</text>
        <text x="28" y="${yTop + 5}" font-size="7.5" fill="#94a3b8" text-anchor="end">100</text>
        <path d="M ${pts.join(' L ')}" fill="none" stroke="#0284c7" stroke-width="2.2" stroke-linecap="round"/>
        <circle cx="${curX.toFixed(1)}" cy="${curNy.toFixed(1)}" r="4" fill="#ff5722" stroke="#ffffff" stroke-width="1.5"/>
      `;
    }

    function update() {
      temp = parseFloat(slider.value);
      tempVal.textContent = `${temp}°C`;

      // Thermal badge color interpolator
      const ratio = temp / 100;
      if (ratio < 0.3) {
        tempVal.style.background = '#e0f2fe';
        tempVal.style.color = '#0284c7';
        tempVal.style.borderColor = '#bae6fd';
      } else if (ratio < 0.7) {
        tempVal.style.background = '#fef3c7';
        tempVal.style.color = '#d97706';
        tempVal.style.borderColor = '#fde68a';
      } else {
        tempVal.style.background = '#fee2e2';
        tempVal.style.color = '#dc2626';
        tempVal.style.borderColor = '#fecaca';
      }

      metalMeter.textContent = getMetalR(temp).toFixed(1) + ' Ω';
      const nR = getNtcR(temp);
      ntcMeter.textContent = nR >= 10 ? (nR.toFixed(1) + ' kΩ') : (nR >= 1 ? nR.toFixed(2) + ' kΩ' : (nR * 1000).toFixed(0) + ' Ω');

      drawGraphs();
    }

    slider.addEventListener('input', update);
    update();

    // Smooth deterministic animation loop
    let lastTimestamp = performance.now();

    function render(timestamp) {
      const dt = Math.min((timestamp - lastTimestamp) / 1000, 0.1);
      lastTimestamp = timestamp;
      const tSec = timestamp / 1000;

      const wM = metalCanvas.width;
      const hM = metalCanvas.height;
      if (wM === 0 || hM === 0) {
        requestAnimationFrame(render);
        return;
      }
      const scale = wM / 220;

      // --- 1. Render Metal Conductor ---
      ctxM.clearRect(0, 0, wM, hM);
      const vibAmp = (0.4 + (temp / 100) * 2.2) * scale;
      const vibFreq = 5 + (temp / 100) * 4; // Gentle oscillation frequency

      // Metal lattice ions
      let ionIndex = 0;
      for (let gx = 25; gx <= 200; gx += 42) {
        for (let gy = 20; gy <= 80; gy += 38) {
          ionIndex++;
          const phaseX = ionIndex * 1.7;
          const phaseY = ionIndex * 2.9;
          const ox = Math.sin(tSec * vibFreq + phaseX) * vibAmp;
          const oy = Math.cos(tSec * vibFreq + phaseY) * vibAmp;
          const cx = gx * scale + ox;
          const cy = gy * scale + oy;

          ctxM.beginPath();
          ctxM.arc(cx, cy, 6.5 * scale, 0, Math.PI * 2);
          ctxM.fillStyle = '#334155';
          ctxM.fill();

          ctxM.fillStyle = '#94a3b8';
          ctxM.font = `bold ${7 * scale}px sans-serif`;
          ctxM.textAlign = 'center';
          ctxM.textBaseline = 'middle';
          ctxM.fillText('+', cx, cy);
        }
      }

      // Metal free conduction electrons (constant number, moving smoothly)
      ctxM.fillStyle = '#22b7ff';
      metalElectrons.forEach(e => {
        const speedScale = 1.0 + (temp / 100) * 0.4;
        e.x += e.vx * dt * speedScale;
        e.y += e.vy * dt * speedScale;

        if (e.x < 2) e.x = 218;
        if (e.x > 218) e.x = 2;
        if (e.y < 2) e.y = 93;
        if (e.y > 93) e.y = 2;

        ctxM.beginPath();
        ctxM.arc(e.x * scale, e.y * scale, 2.3 * scale, 0, Math.PI * 2);
        ctxM.fill();
      });

      // --- 2. Render NTC Thermistor Lattice & Carriers ---
      ctxN.clearRect(0, 0, wM, hM);

      // Semiconductor covalent bonds
      ctxN.strokeStyle = '#1e293b';
      ctxN.lineWidth = 1.2 * scale;
      for (let gx = 25; gx <= 200; gx += 42) {
        ctxN.beginPath();
        ctxN.moveTo(gx * scale, 0);
        ctxN.lineTo(gx * scale, hM);
        ctxN.stroke();
      }
      for (let gy = 20; gy <= 80; gy += 38) {
        ctxN.beginPath();
        ctxN.moveTo(0, gy * scale);
        ctxN.lineTo(wM, gy * scale);
        ctxN.stroke();
      }

      // Semiconductor core atoms
      for (let gx = 25; gx <= 200; gx += 42) {
        for (let gy = 20; gy <= 80; gy += 38) {
          ctxN.beginPath();
          ctxN.arc(gx * scale, gy * scale, 5.5 * scale, 0, Math.PI * 2);
          ctxN.fillStyle = '#0f172a';
          ctxN.fill();
          ctxN.strokeStyle = '#475569';
          ctxN.lineWidth = 1.2 * scale;
          ctxN.stroke();
        }
      }

      // Thermistor charge carriers (bound vs liberated)
      semiElectrons.forEach(e => {
        const isFree = e.threshold <= temp;
        if (isFree) {
          e.x += e.vx * dt * 1.2;
          e.y += e.vy * dt * 1.2;
          if (e.x < 2) e.x = 218;
          if (e.x > 218) e.x = 2;
          if (e.y < 2) e.y = 93;
          if (e.y > 93) e.y = 2;

          ctxN.beginPath();
          ctxN.arc(e.x * scale, e.y * scale, 2.5 * scale, 0, Math.PI * 2);
          ctxN.fillStyle = '#22b7ff';
          ctxN.shadowColor = '#22b7ff';
          ctxN.shadowBlur = 4 * scale;
          ctxN.fill();
          ctxN.shadowBlur = 0;
        } else {
          // Bound electrons vibrating smoothly near atomic sites
          const gridCol = Math.floor(e.threshold / 20) % 5;
          const gridRow = Math.floor(e.threshold / 4) % 3;
          const baseGx = (25 + gridCol * 42);
          const baseGy = (20 + gridRow * 38);
          const orbitR = 8.5;
          const bx = baseGx + Math.cos(e.phase + tSec * 1.2) * orbitR;
          const by = baseGy + Math.sin(e.phase + tSec * 1.2) * orbitR;

          ctxN.beginPath();
          ctxN.arc(bx * scale, by * scale, 1.4 * scale, 0, Math.PI * 2);
          ctxN.fillStyle = '#334155';
          ctxN.fill();
        }
      });

      requestAnimationFrame(render);
    }

    requestAnimationFrame(render);
  </script>
</body>
</html>

In 1833, Michael Faraday first discovered this effect while studying silver sulfide, noticing that unlike metals, its resistance plummeted when heated because thermal energy frees bound electrons into the conduction band.

Because NTC stands for **negative temperature coefficient**, heating the device creates a negative change in resistance. But microcontrollers and comparators cannot read resistance directly—they only measure voltage.

How can we convert this dropping resistance into a clean, measurable voltage signal?

A **potential divider** (or voltage divider) is a circuit with two resistors in series that splits an input voltage ($V_{in}$) across them in proportion to their resistance values.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; font-family: system-ui, -apple-system, sans-serif; }
    body {
      background: #ffffff;
      color: #1e293b;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      padding: 12px;
      min-height: 100%;
      user-select: none;
    }

    .eq-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      font-size: 17px;
      font-weight: 700;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 6px 14px;
      margin-bottom: 8px;
    }

    .val-vout { color: #0284c7; background: #e0f2fe; padding: 2px 6px; border-radius: 4px; }
    .val-vin { color: #475569; }
    .val-r1 { color: #d97706; background: #fef3c7; padding: 1px 5px; border-radius: 4px; }
    .val-r2 { color: #059669; background: #d1fae5; padding: 1px 5px; border-radius: 4px; }

    .fraction {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      font-size: 14px;
      line-height: 1.1;
      padding: 0 2px;
    }
    .fraction .top { border-bottom: 2px solid #334155; padding-bottom: 1px; }
    .fraction .bot { padding-top: 1px; }

    .main-stage {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 20px;
      width: 100%;
      max-width: 480px;
      flex: 1;
    }

    /* Circuit SVG */
    .circuit-svg {
      width: 180px;
      height: 240px;
      overflow: visible;
    }

    /* Bar Graph Column */
    .meter-col {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 6px;
    }

    .bar-track {
      position: relative;
      width: 36px;
      height: 180px;
      background: #f1f5f9;
      border: 2px solid #cbd5e1;
      border-radius: 6px;
      overflow: hidden;
      display: flex;
      align-items: flex-end;
    }

    .bar-fill {
      width: 100%;
      background: linear-gradient(180deg, #38bdf8, #0284c7);
      transition: height 0.15s ease-out;
      border-radius: 0 0 4px 4px;
    }

    .meter-scale {
      position: absolute;
      right: -32px;
      top: 0;
      height: 100%;
      display: flex;
      flex-direction: column;
      justify-content: space-between;
      font-size: 11px;
      font-weight: 600;
      color: #64748b;
      pointer-events: none;
    }

    .meter-label {
      font-size: 15px;
      font-weight: 700;
      color: #0284c7;
      min-width: 60px;
      text-align: center;
    }

    /* Controls */
    .control-panel {
      width: 100%;
      max-width: 440px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 14px;
      display: flex;
      flex-direction: column;
      gap: 6px;
    }

    .slider-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 10px;
      font-size: 13px;
      font-weight: 600;
    }

    input[type=range] {
      flex: 1;
      accent-color: #059669;
      cursor: pointer;
      height: 6px;
    }

    .preset-row {
      display: flex;
      justify-content: center;
      gap: 8px;
    }

    .btn-preset {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 5px;
      padding: 3px 10px;
      font-size: 11px;
      font-weight: 600;
      color: #334155;
      cursor: pointer;
    }
    .btn-preset:active {
      background: #e2e8f0;
    }
  </style>
</head>
<body>

  
  <div class="eq-bar">
    <span class="val-vout" id="eqVout">V_out</span>
    <span>=</span>
    <span class="val-vin">5V</span>
    <span>×</span>
    <div class="fraction">
      <span class="top"><span class="val-r2" id="eqR2Top">R₂</span></span>
      <span class="bot"><span class="val-r1">R₁</span> + <span class="val-r2" id="eqR2Bot">R₂</span></span>
    </div>
  </div>

  
  <div class="main-stage">
    
    
    <svg class="circuit-svg" viewBox="0 0 180 240">
      
      <line x1="40" y1="20" x2="120" y2="20" stroke="#dc2626" stroke-width="3" stroke-linecap="round"/>
      <text x="80" y="14" font-size="12" font-weight="700" fill="#dc2626" text-anchor="middle">V_in = +5V</text>

      
      <line x1="80" y1="20" x2="80" y2="46" stroke="#334155" stroke-width="2.5"/>
      <line x1="80" y1="94" x2="80" y2="136" stroke="#334155" stroke-width="2.5"/>
      <line x1="80" y1="184" x2="80" y2="210" stroke="#334155" stroke-width="2.5"/>

      
      <rect x="65" y="46" width="30" height="48" rx="3" fill="#fef3c7" stroke="#d97706" stroke-width="2.5"/>
      <text x="80" y="75" font-size="12" font-weight="800" fill="#b45309" text-anchor="middle">R₁</text>
      <text x="102" y="75" font-size="10" font-weight="600" fill="#78350f" text-anchor="start">10kΩ</text>

      
      <circle cx="80" cy="115" r="4.5" fill="#0284c7"/>
      <line x1="80" y1="115" x2="148" y2="115" stroke="#0284c7" stroke-width="2.5" stroke-dasharray="3,2"/>
      <circle cx="148" cy="115" r="3.5" fill="#0284c7"/>
      <text x="156" y="119" font-size="13" font-weight="800" fill="#0284c7">V_out</text>

      
      <rect id="r2Box" x="65" y="136" width="30" height="48" rx="3" fill="#d1fae5" stroke="#059669" stroke-width="2.5"/>
      <text x="80" y="165" font-size="12" font-weight="800" fill="#047857" text-anchor="middle">R₂</text>
      <text id="r2ValLabel" x="102" y="165" font-size="10" font-weight="600" fill="#065f46" text-anchor="start">10kΩ</text>

      
      <line x1="60" y1="210" x2="100" y2="210" stroke="#475569" stroke-width="2.5"/>
      <line x1="68" y1="215" x2="92" y2="215" stroke="#475569" stroke-width="2"/>
      <line x1="74" y1="220" x2="86" y2="220" stroke="#475569" stroke-width="1.5"/>
      <text x="80" y="233" font-size="11" font-weight="700" fill="#64748b" text-anchor="middle">0V (GND)</text>
    </svg>

    
    <div class="meter-col">
      <div class="bar-track">
        <div class="bar-fill" id="barFill" style="height: 50%;"></div>
        <div class="meter-scale">
          <span>5V</span>
          <span>2.5V</span>
          <span>0V</span>
        </div>
      </div>
      <div class="meter-label" id="voutText">2.50 V</div>
    </div>

  </div>

  
  <div class="control-panel">
    <div class="slider-row">
      <span style="color:#059669;">R₂ Resistance:</span>
      <input type="range" id="r2Slider" min="0" max="40" step="0.5" value="10">
      <span id="sliderVal" style="min-width: 48px; text-align: right; color:#059669;">10 kΩ</span>
    </div>
    <div class="preset-row">
      <button class="btn-preset" onclick="setR2(0)">R₂ = 0Ω</button>
      <button class="btn-preset" onclick="setR2(10)">R₂ = R₁</button>
      <button class="btn-preset" onclick="setR2(30)">R₂ = 3×R₁</button>
    </div>
  </div>

  <script>
    const R1 = 10; // 10 kΩ fixed
    const Vin = 5; // 5 Volts

    const r2Slider = document.getElementById('r2Slider');
    const r2ValLabel = document.getElementById('r2ValLabel');
    const sliderVal = document.getElementById('sliderVal');
    const barFill = document.getElementById('barFill');
    const voutText = document.getElementById('voutText');
    const eqVout = document.getElementById('eqVout');

    function update() {
      const r2 = parseFloat(r2Slider.value);
      const vout = Vin * (r2 / (R1 + r2));
      const percentage = (vout / Vin) * 100;

      r2ValLabel.textContent = r2 + 'kΩ';
      sliderVal.textContent = r2 + ' kΩ';
      barFill.style.height = percentage + '%';
      voutText.textContent = vout.toFixed(2) + ' V';
      eqVout.textContent = vout.toFixed(2) + 'V';
    }

    function setR2(val) {
      r2Slider.value = val;
      update();
    }

    r2Slider.addEventListener('input', update);
    update();
  </script>
</body>
</html>

By Ohm's law ($V = I \cdot R$), both resistors share the identical series current $I = \frac{V_{in}}{R_1 + R_2}$, making the voltage across the lower resistor equal to $V_{out} = V_{in} \cdot \frac{R_2}{R_1 + R_2}$.

If we replace one of these fixed resistors with our thermistor, which position should it go in: the top or the bottom?

The placement of the thermistor ($R_{th}$) determines whether the output voltage increases or decreases when the environment warms up.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
      color: #1e2945;
      background: #ffffff;
      padding: 10px;
      user-select: none;
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    .slider-wrap {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      margin-bottom: 12px;
      padding: 8px 14px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
    }
    .slider-label {
      font-size: 14px;
      font-weight: 600;
      min-width: 75px;
    }
    .temp-val {
      font-size: 15px;
      font-weight: 700;
      min-width: 55px;
      text-align: right;
      color: #0284c7;
    }
    input[type=range] {
      flex: 1;
      max-width: 240px;
      accent-color: #0284c7;
      cursor: pointer;
    }

    .circuits-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
    }

    .circuit-panel {
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 10px 8px 8px;
      background: #ffffff;
      display: flex;
      flex-direction: column;
      align-items: center;
      position: relative;
    }

    .panel-title {
      font-size: 13px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      margin-bottom: 6px;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .high-title { color: #dc2626; }
    .frost-title { color: #0284c7; }

    svg {
      width: 100%;
      max-width: 240px;
      height: auto;
      overflow: visible;
    }

    .schem-wire {
      stroke: #64748b;
      stroke-width: 2.5;
      fill: none;
      stroke-linecap: round;
      stroke-linejoin: round;
    }
    .schem-node {
      fill: #1e2945;
    }
    .schem-label {
      font-size: 11px;
      font-weight: 600;
      fill: #475569;
    }
    .val-label {
      font-size: 10px;
      font-weight: 700;
      fill: #0f172a;
    }
    .rail-text {
      font-size: 12px;
      font-weight: 700;
      fill: #1e2945;
    }

    .meter-box {
      fill: #f1f5f9;
      stroke: #cbd5e1;
      stroke-width: 1.5;
      rx: 4;
    }
    .meter-text {
      font-size: 13px;
      font-weight: 800;
      font-family: ui-monospace, monospace;
      fill: #0f172a;
    }

    .alarm-badge {
      margin-top: 6px;
      font-size: 11px;
      font-weight: 700;
      padding: 3px 8px;
      border-radius: 4px;
      display: inline-flex;
      align-items: center;
      gap: 4px;
      transition: background-color 0.2s, color 0.2s, opacity 0.2s;
    }
    .alarm-active-hot {
      background: #fee2e2;
      color: #b91c1c;
      box-shadow: 0 0 0 1px #f87171;
    }
    .alarm-active-cold {
      background: #e0f2fe;
      color: #0369a1;
      box-shadow: 0 0 0 1px #38bdf8;
    }
    .alarm-off {
      background: #f1f5f9;
      color: #94a3b8;
      box-shadow: 0 0 0 1px #e2e8f0;
    }
  </style>
</head>
<body>

  <div class="slider-wrap">
    <span class="slider-label">🌡️ Temp:</span>
    <input type="range" id="tempSlider" min="-10" max="80" value="25" step="1">
    <span class="temp-val" id="tempText">25°C</span>
  </div>

  <div class="circuits-grid">
    
    <div class="circuit-panel">
      <div class="panel-title high-title">🔥 High-T Alarm</div>
      <svg viewBox="0 0 200 240">
        
        <line x1="30" y1="20" x2="110" y2="20" class="schem-wire" stroke="#dc2626" />
        <text x="15" y="24" class="rail-text" text-anchor="middle">+5V</text>

        <line x1="30" y1="220" x2="110" y2="220" class="schem-wire" stroke="#0f172a" />
        <text x="15" y="224" class="rail-text" text-anchor="middle">0V</text>

        
        <line x1="70" y1="20" x2="70" y2="45" class="schem-wire" />
        
        
        <g transform="translate(52, 45)">
          <rect id="therm1-body" x="0" y="0" width="36" height="50" rx="3" fill="#fed7aa" stroke="#ea580c" stroke-width="2" />
          <line x1="-5" y1="52" x2="41" y2="-2" stroke="#ea580c" stroke-width="2" stroke-linecap="round" />
          <line x1="-5" y1="52" x2="-5" y2="44" stroke="#ea580c" stroke-width="2" stroke-linecap="round" />
        </g>
        <text x="100" y="65" class="schem-label">R<tspan dy="2" font-size="8">th</tspan></text>
        <text x="100" y="82" id="rth1-val" class="val-label">10.0 kΩ</text>

        <line x1="70" y1="95" x2="70" y2="140" class="schem-wire" />
        <circle cx="70" cy="118" r="3.5" class="schem-node" />

        
        <line x1="70" y1="118" x2="135" y2="118" class="schem-wire" />
        <line x1="135" y1="118" x2="135" y2="135" class="schem-wire" />
        
        
        <rect x="105" y="135" width="60" height="40" class="meter-box" />
        <text x="135" y="152" text-anchor="middle" class="schem-label">V<tspan dy="2" font-size="8">out</tspan></text>
        <text x="135" y="168" id="vout1-text" text-anchor="middle" class="meter-text">2.50V</text>
        <line x1="135" y1="175" x2="135" y2="220" class="schem-wire" stroke-dasharray="3 3" />
        <circle cx="135" cy="220" r="3" class="schem-node" />

        
        <g transform="translate(52, 140)">
          <rect x="0" y="0" width="36" height="50" rx="3" fill="#f8fafc" stroke="#64748b" stroke-width="2" />
        </g>
        <text x="10" y="160" class="schem-label">R<tspan dy="2" font-size="8">fixed</tspan></text>
        <text x="10" y="177" class="val-label">10.0 kΩ</text>

        <line x1="70" y1="190" x2="70" y2="220" class="schem-wire" />
      </svg>
      <div id="alarm1-badge" class="alarm-badge alarm-off">🔔 Alarm Standby</div>
    </div>

    
    <div class="circuit-panel">
      <div class="panel-title frost-title">❄️ Frost Alarm</div>
      <svg viewBox="0 0 200 240">
        
        <line x1="30" y1="20" x2="110" y2="20" class="schem-wire" stroke="#dc2626" />
        <text x="15" y="24" class="rail-text" text-anchor="middle">+5V</text>

        <line x1="30" y1="220" x2="110" y2="220" class="schem-wire" stroke="#0f172a" />
        <text x="15" y="224" class="rail-text" text-anchor="middle">0V</text>

        
        <line x1="70" y1="20" x2="70" y2="45" class="schem-wire" />
        
        
        <g transform="translate(52, 45)">
          <rect x="0" y="0" width="36" height="50" rx="3" fill="#f8fafc" stroke="#64748b" stroke-width="2" />
        </g>
        <text x="100" y="65" class="schem-label">R<tspan dy="2" font-size="8">fixed</tspan></text>
        <text x="100" y="82" class="val-label">10.0 kΩ</text>

        <line x1="70" y1="95" x2="70" y2="140" class="schem-wire" />
        <circle cx="70" cy="118" r="3.5" class="schem-node" />

        
        <line x1="70" y1="118" x2="135" y2="118" class="schem-wire" />
        <line x1="135" y1="118" x2="135" y2="135" class="schem-wire" />
        
        
        <rect x="105" y="135" width="60" height="40" class="meter-box" />
        <text x="135" y="152" text-anchor="middle" class="schem-label">V<tspan dy="2" font-size="8">out</tspan></text>
        <text x="135" y="168" id="vout2-text" text-anchor="middle" class="meter-text">2.50V</text>
        <line x1="135" y1="175" x2="135" y2="220" class="schem-wire" stroke-dasharray="3 3" />
        <circle cx="135" cy="220" r="3" class="schem-node" />

        
        <g transform="translate(52, 140)">
          <rect id="therm2-body" x="0" y="0" width="36" height="50" rx="3" fill="#fed7aa" stroke="#ea580c" stroke-width="2" />
          <line x1="-5" y1="52" x2="41" y2="-2" stroke="#ea580c" stroke-width="2" stroke-linecap="round" />
          <line x1="-5" y1="52" x2="-5" y2="44" stroke="#ea580c" stroke-width="2" stroke-linecap="round" />
        </g>
        <text x="10" y="160" class="schem-label">R<tspan dy="2" font-size="8">th</tspan></text>
        <text x="10" y="177" id="rth2-val" class="val-label">10.0 kΩ</text>

        <line x1="70" y1="190" x2="70" y2="220" class="schem-wire" />
      </svg>
      <div id="alarm2-badge" class="alarm-badge alarm-off">🔔 Alarm Standby</div>
    </div>
  </div>

  <script>
    const slider = document.getElementById('tempSlider');
    const tempText = document.getElementById('tempText');
    const rth1Val = document.getElementById('rth1-val');
    const rth2Val = document.getElementById('rth2-val');
    const vout1Text = document.getElementById('vout1-text');
    const vout2Text = document.getElementById('vout2-text');
    const therm1Body = document.getElementById('therm1-body');
    const therm2Body = document.getElementById('therm2-body');
    const alarm1Badge = document.getElementById('alarm1-badge');
    const alarm2Badge = document.getElementById('alarm2-badge');

    const Vin = 5.0;
    const R_fixed = 10.0; // kΩ
    const R0 = 10.0; // kΩ at 25°C
    const Beta = 3400; // NTC Beta constant

    function calcRth(tempC) {
      const T_k = tempC + 273.15;
      const T0_k = 298.15;
      return R0 * Math.exp(Beta * (1 / T_k - 1 / T0_k));
    }

    function getHeatColor(tempC) {
      // -10 (cold blue) to 80 (hot red)
      const ratio = Math.max(0, Math.min(1, (tempC - (-10)) / 90));
      if (ratio < 0.38) {
        // Cold ice blue to neutral
        return '#bae6fd';
      } else if (ratio < 0.65) {
        // Mild / warm amber
        return '#fed7aa';
      } else {
        // Hot red/orange
        return '#fca5a5';
      }
    }

    function update() {
      const T = parseFloat(slider.value);
      tempText.textContent = (T > 0 ? '+' : '') + T + '°C';
      
      const rth = calcRth(T);
      const rthFormatted = rth >= 10 ? rth.toFixed(1) + ' kΩ' : rth.toFixed(2) + ' kΩ';
      
      rth1Val.textContent = rthFormatted;
      rth2Val.textContent = rthFormatted;

      // Left circuit: R1 = Rth, R2 = R_fixed. Vout across R_fixed = Vin * (R_fixed / (Rth + R_fixed))
      const vout1 = Vin * (R_fixed / (rth + R_fixed));
      // Right circuit: R1 = R_fixed, R2 = Rth. Vout across Rth = Vin * (rth / (R_fixed + rth))
      const vout2 = Vin * (rth / (R_fixed + rth));

      vout1Text.textContent = vout1.toFixed(2) + 'V';
      vout2Text.textContent = vout2.toFixed(2) + 'V';

      const heatFill = getHeatColor(T);
      therm1Body.setAttribute('fill', heatFill);
      therm2Body.setAttribute('fill', heatFill);

      // High-T Alarm activates if T >= 50°C (Vout1 >= 3.6V)
      if (T >= 50) {
        alarm1Badge.className = 'alarm-badge alarm-active-hot';
        alarm1Badge.innerHTML = '🚨 High-T Triggered!';
      } else {
        alarm1Badge.className = 'alarm-badge alarm-off';
        alarm1Badge.innerHTML = '🔔 Normal State';
      }

      // Frost Alarm activates if T <= 4°C (Vout2 >= 3.6V)
      if (T <= 4) {
        alarm2Badge.className = 'alarm-badge alarm-active-cold';
        alarm2Badge.innerHTML = '❄️ Frost Triggered!';
      } else {
        alarm2Badge.className = 'alarm-badge alarm-off';
        alarm2Badge.innerHTML = '🔔 Normal State';
      }
    }

    slider.addEventListener('input', update);
    update();
  </script>
</body>
</html>

When the thermistor sits in the top position ($R_1$), higher temperature lowers its resistance, pulling $V_{out}$ closer to $V_{in}$, which is perfect for triggering cooling fans.

When the thermistor sits in the bottom position ($R_2$), freezing cold weather increases its resistance, raising $V_{out}$ to trigger heating elements or frost alarms.

Let's put real numbers to this: how do you calculate the exact voltage change across a real thermal transition?

Suppose we build a cooling fan trigger powered by $V_{in} = 9.0\text{ V}$ using a top NTC thermistor ($R_1$) and a fixed $10\text{ k}\Omega$ bottom resistor ($R_2$).

At room temperature ($25^\circ\text{C}$), the thermistor measures $R_1 = 10\text{ k}\Omega$, but when heated to $70^\circ\text{C}$, its resistance drops to $R_1 = 2.0\text{ k}\Omega$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: transparent;
      color: #1e293b;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: flex-start;
      min-height: 100%;
      padding: 8px;
      gap: 10px;
    }

    /* Top Step Navigation */
    .nav-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      width: 100%;
      max-width: 580px;
      flex-wrap: wrap;
    }
    .step-btn {
      padding: 5px 10px;
      font-size: 12px;
      font-weight: 600;
      border-radius: 20px;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      color: #475569;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 4px;
      transition: background 0.15s, color 0.15s, border-color 0.15s;
    }
    .step-btn.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
      box-shadow: 0 1px 4px rgba(2, 132, 199, 0.3);
    }
    .play-btn {
      width: 28px;
      height: 28px;
      border-radius: 50%;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      font-size: 11px;
      color: #1e293b;
      transition: background 0.15s;
    }
    .play-btn:hover {
      background: #f1f5f9;
    }

    /* Main Container (no outer card styling) */
    .stage {
      position: relative;
      width: 100%;
      max-width: 580px;
      display: grid;
      grid-template-columns: 135px 1fr;
      gap: 12px;
      padding: 0;
      background: transparent;
    }
    @media (max-width: 460px) {
      .stage {
        grid-template-columns: 1fr;
        gap: 8px;
      }
      .circuit-col {
        max-height: 110px;
        padding: 4px;
      }
      .circuit-svg {
        max-height: 100px;
      }
    }

    /* Circuit SVG Column */
    .circuit-col {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 6px;
    }
    .circuit-svg {
      width: 100%;
      height: auto;
      max-height: 180px;
    }

    /* Card Content Area */
    .content-col {
      position: relative;
      width: 100%;
      display: flex;
      flex-direction: column;
      min-height: 190px;
    }
    .step-panel {
      display: none;
      flex-direction: column;
      gap: 8px;
      width: 100%;
      animation: fadeIn 0.25s ease-in-out;
    }
    .step-panel.active {
      display: flex;
    }
    @keyframes fadeIn {
      from { opacity: 0; transform: translateY(3px); }
      to { opacity: 1; transform: translateY(0); }
    }

    /* Math & Badges */
    .tag-row {
      display: flex;
      align-items: center;
      gap: 6px;
      flex-wrap: wrap;
    }
    .badge {
      font-size: 11px;
      font-weight: 700;
      padding: 3px 8px;
      border-radius: 6px;
      letter-spacing: 0.2px;
    }
    .badge-blue { background: #e0f2fe; color: #0369a1; }
    .badge-amber { background: #fef3c7; color: #b45309; }
    .badge-red { background: #fee2e2; color: #b91c1c; }
    .badge-green { background: #dcfce7; color: #15803d; }

    .formula-box {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 60px;
    }
    
    /* Clean CSS Math Typography */
    .math-display {
      font-family: "Cambria Math", "Latin Modern Math", "KaTeX_Main", "Times New Roman", serif;
      font-size: 16px;
      display: inline-flex;
      align-items: center;
      justify-content: center;
      gap: 4px;
      color: #0f172a;
      flex-wrap: wrap;
    }
    .fraction {
      display: inline-flex;
      flex-direction: column;
      align-items: center;
      vertical-align: middle;
      padding: 0 4px;
      font-size: 0.9em;
    }
    .numerator {
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  <div class="nav-bar">
    <button class="step-btn active" data-step="0">1. Formula</button>
    <button class="step-btn" data-step="1">2. 25°C (Cold)</button>
    <button class="step-btn" data-step="2">3. 70°C (Hot)</button>
    <button class="step-btn" data-step="3">4. ΔV Swing</button>
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        <text x="50" y="14" text-anchor="middle" font-size="10" font-weight="700" fill="#dc2626">+9 V</text>
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          <span class="badge badge-blue">Divider Formula</span>
          <span class="badge badge-amber">NTC Sensor</span>
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            <span><span class="math-var">V</span><span class="sub">out</span> = 9 V ×</span>
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              <span class="numerator">10 kΩ</span>
              <span class="denominator"><span class="math-var">R</span><span class="sub">th</span> + 10 kΩ</span>
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            <span>0 V</span>
            <span style="color:#0284c7;">Divider Output</span>
            <span>9.0 V</span>
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          <span class="badge badge-blue">T = 25 °C</span>
          <span class="badge badge-blue">R<span class="sub" style="font-size:9px;">th</span> = 10 kΩ</span>
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              <span class="numerator">10</span>
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            <span>0 V</span>
            <span style="color:#0284c7;font-weight:700;">4.50 V (50%)</span>
            <span>9.0 V</span>
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      <div class="step-panel" id="step2">
        <div class="tag-row">
          <span class="badge badge-red">T = 70 °C (Hot)</span>
          <span class="badge badge-amber">R<span class="sub" style="font-size:9px;">th</span> = 2 kΩ</span>
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        <div class="formula-box">
          <div class="math-display">
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            <span>= 9 × <span class="fraction" style="font-size:0.85em;"><span class="numerator">5</span><span class="denominator">6</span></span> = <strong>7.50 V</strong></span>
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        </div>
        <div class="meter-wrap">
          <div class="meter-track">
            <div class="meter-fill" style="width: 83.3%;"></div>
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          <div class="meter-labels">
            <span>0 V</span>
            <span style="color:#10b981;font-weight:700;">7.50 V (83.3%)</span>
            <span>9.0 V</span>
          </div>
        </div>
      </div>

      
      <div class="step-panel" id="step3">
        <div class="tag-row">
          <span class="badge badge-green">Output Voltage Swing</span>
        </div>
        <div class="formula-box">
          <div class="math-display">
            <span>Δ<span class="math-var">V</span><span class="sub">out</span> = <span class="math-var">V</span><span class="sub">out</span>(70°C) − <span class="math-var">V</span><span class="sub">out</span>(25°C)</span>
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          <div class="math-display" style="font-size: 14px; color: #475569; margin-top: 3px;">
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          </div>
        </div>
        <div class="swing-result-box">
          <span style="font-size:12px; font-weight:600; color:#166534;">Signal Range:</span>
          <div class="delta-badge">ΔV = +3.00 V</div>
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Our circuit produces a noticeable $+3.00\text{ V}$ jump, providing a clear threshold voltage that can switch on a transistor or comparator.

What practical mistakes can ruin these calculations in real-world circuits?

## Common Pitfalls: Non-Linearity and Self-Heating

A common mistake is assuming that output voltage scales evenly with temperature—because thermistors have an exponential curve, equal temperature jumps produce shrinking voltage steps at high heat.

Another subtle trap is **self-heating**: if your fixed resistor value is too small, excessive current ($I = V/R$) flows through the thermistor, generating internal electrical heat ($P = I^2 R$) that falsifies your temperature reading.

To achieve the highest sensitivity around your target switching temperature, always choose a fixed resistor ($R_{fixed}$) whose resistance closely matches the thermistor's resistance ($R_{th}$) at that exact temperature.

electronics

Thermistors and Temperature-Sensing Circuits. Investigate how negative temperature coefficient (NTC) thermistors vary in resistance with temperature and design basic potential divider circuits to detect thermal changes.

Potential Dividers and Sensing Circuits - Advanced Problem Solving & Concept Mastery

A-Level D.C. Circuits: Potential Divider Circuits & Input Sensors (Thermistors and LDRs)

Practical Physics & Teaching Guidance: Potential Divider Sensing Circuits

Teaching strategies for A2 Level mastery should combine microscopic solid-state physics explanations (band theory, carrier generation) with macroscopic circuit analysis (potential divider ratio, circuit loading via Thévenin equivalent or parallel resistance models). Encourage students to sketch circuit diagrams with clearly defined input and output nodes, explicitly distinguishing between sensor configurations (output taken across the thermistor vs across the fixed resistor) and analysing the influence of meter/comparator input impedance on voltage measurement linearity.

Thermistors and Temperature-Sensing Circuits

The Potential Divider Interface

Designing for Hot or Cold Detection