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Tutoring abilities

If you connect a standard digital multimeter to measure a tiny resistance change, lead resistance and internal meter inaccuracies can ruin your measurement. How can we measure unknown resistance with parts-per-million accuracy without relying on the exact calibration of a meter?

In 1833, Samuel Hunter Christie invented a diamond-shaped circuit to compare resistances, which Sir Charles Wheatstone later popularized in 1843 for testing telegraph lines.

This circuit, called a **Wheatstone bridge**, uses a **null measurement technique** — a method where you balance two circuit paths until zero current flows across a detector, making the measurement independent of meter calibration.

<!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="1:1">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      width: 100%;
      font-family: system-ui, -apple-system, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 12px;
    }

    .bridge-card {
      width: 100%;
      max-width: 440px;
      background: #ffffff;
      border-radius: 16px;
      padding: 16px;
      box-shadow: 0 4px 16px rgba(0, 0, 0, 0.05);
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }

    .status-badge {
      font-size: 13px;
      font-weight: 700;
      padding: 6px 14px;
      border-radius: 20px;
      transition: background 0.3s ease, color 0.3s ease;
      text-align: center;
    }
    .status-unbalanced {
      background: #fffbeb;
      color: #d97706;
      border: 1px solid #fef3c7;
    }
    .status-balanced {
      background: #ecfdf5;
      color: #059669;
      border: 1px solid #a7f3d0;
    }

    svg {
      width: 100%;
      height: auto;
      max-height: 250px;
      overflow: visible;
    }

    .wire {
      stroke: #94a3b8;
      stroke-width: 2.5;
      stroke-linecap: round;
      stroke-linejoin: round;
      fill: none;
    }

    .wire-active {
      stroke: #22b7ff;
      transition: stroke 0.3s ease;
    }

    .node-label {
      font-size: 11px;
      font-weight: 800;
      fill: #1e2945;
      text-anchor: middle;
      dominant-baseline: central;
    }

    .comp-box {
      fill: #ffffff;
      stroke: #cbd5e1;
      stroke-width: 1.5;
      rx: 4;
    }
    .comp-box.highlight {
      stroke: #22b7ff;
      fill: #f0f9ff;
    }
    .comp-box.target {
      stroke: #f59e0b;
      fill: #fffbeb;
    }

    .val-text {
      font-size: 10px;
      font-weight: 700;
      fill: #1e2945;
      text-anchor: middle;
      dominant-baseline: central;
    }

    .voltage-tag {
      font-size: 9.5px;
      font-weight: 700;
      fill: #2563eb;
      text-anchor: middle;
    }

    .needle {
      stroke: #ef4444;
      stroke-width: 2;
      stroke-linecap: round;
      transform-origin: 200px 145px;
      transition: transform 0.2s cubic-bezier(0.2, 0, 0.2, 1);
    }

    .ctrl-wrap {
      width: 100%;
      display: flex;
      flex-direction: column;
      gap: 6px;
      padding: 0 8px;
    }

    .ctrl-header {
      display: flex;
      justify-content: space-between;
      align-items: center;
      font-size: 12px;
      font-weight: 700;
      color: #64748b;
    }

    .ctrl-val {
      font-size: 14px;
      font-weight: 800;
      color: #22b7ff;
    }

    input[type="range"] {
      width: 100%;
      height: 6px;
      border-radius: 3px;
      background: #e2e8f0;
      outline: none;
      -webkit-appearance: none;
      appearance: none;
    }
    input[type="range"]::-webkit-slider-thumb {
      -webkit-appearance: none;
      appearance: none;
      width: 20px;
      height: 20px;
      border-radius: 50%;
      background: #22b7ff;
      cursor: pointer;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.4);
      border: 2px solid #ffffff;
    }

    .preset-btns {
      display: flex;
      gap: 6px;
      width: 100%;
    }

    .preset-btn {
      flex: 1;
      padding: 6px 0;
      font-size: 11px;
      font-weight: 600;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      background: #ffffff;
      color: #475569;
      cursor: pointer;
      transition: all 0.2s ease;
    }
    .preset-btn:hover {
      background: #f1f5f9;
    }
    .preset-btn.active {
      background: #e0f2fe;
      border-color: #22b7ff;
      color: #0284c7;
    }
  </style>
</head>
<body>

  <div class="bridge-card">
    <div id="statusBadge" class="status-badge status-unbalanced">
      Unbalanced: V<sub>B</sub> ≠ V<sub>D</sub> (I<sub>G</sub> ≠ 0)
    </div>

    <svg viewBox="0 0 400 240">
      
      <path class="wire" d="M 200,16 L 200,32" />
      <path class="wire" d="M 200,218 L 200,234" />
      
      
      <circle cx="200" cy="32" r="3.5" fill="#1e2945" />
      <text x="200" y="24" class="val-text" fill="#64748b">V_in = 10V (A)</text>
      
      <circle cx="200" cy="218" r="3.5" fill="#1e2945" />
      <text x="200" y="232" class="val-text" fill="#64748b">0V (C)</text>

      
      
      <path class="wire" d="M 200,32 L 80,125 L 200,218" />
      
      <path class="wire" d="M 200,32 L 320,125 L 200,218" />

      
      <path class="wire" d="M 80,125 L 160,125" />
      <path class="wire" d="M 240,125 L 320,125" />

      
      <g transform="translate(140, 78.5) rotate(-37.8)">
        <rect x="-24" y="-9" width="48" height="18" class="comp-box" />
        <text x="0" y="0" class="val-text">R₁: 100Ω</text>
      </g>

      
      <g transform="translate(260, 78.5) rotate(37.8)">
        <rect x="-24" y="-9" width="48" height="18" class="comp-box" />
        <text x="0" y="0" class="val-text">R₂: 200Ω</text>
      </g>

      
      <g transform="translate(140, 171.5) rotate(37.8)">
        <rect x="-24" y="-9" width="48" height="18" class="comp-box highlight" />
        
        <path d="M -18,12 L 18,-12" stroke="#22b7ff" stroke-width="1.5" />
        <path d="M 12,-12 L 18,-12 L 18,-6" fill="none" stroke="#22b7ff" stroke-width="1.5" />
        <text id="r3Label" x="0" y="0" class="val-text">R₃: 150Ω</text>
      </g>

      
      <g transform="translate(260, 171.5) rotate(-37.8)">
        <rect x="-24" y="-9" width="48" height="18" class="comp-box target" />
        <text x="0" y="0" class="val-text">Rx: 300Ω</text>
      </g>

      
      <circle cx="80" cy="125" r="5" fill="#22b7ff" />
      <text x="64" y="125" class="node-label">B</text>
      <text id="vbLabel" x="64" y="142" class="voltage-tag">6.0V</text>

      
      <circle cx="320" cy="125" r="5" fill="#22b7ff" />
      <text x="336" y="125" class="node-label">D</text>
      <text id="vdLabel" x="336" y="142" class="voltage-tag">6.0V</text>

      
      <circle cx="200" cy="125" r="26" fill="#ffffff" stroke="#1e2945" stroke-width="2" />
      
      
      <path d="M 182,120 A 20 20 0 0 1 218,120" fill="none" stroke="#cbd5e1" stroke-width="1.5" />
      <line x1="200" y1="105" x2="200" y2="109" stroke="#10b981" stroke-width="2" />
      <line x1="184" y1="117" x2="188" y2="119" stroke="#94a3b8" stroke-width="1.5" />
      <line x1="216" y1="117" x2="212" y2="119" stroke="#94a3b8" stroke-width="1.5" />
      
      <text x="200" y="101" font-size="7.5" font-weight="800" fill="#10b981" text-anchor="middle">0</text>
      <text x="200" y="138" font-size="8.5" font-weight="800" fill="#64748b" text-anchor="middle">G</text>

      
      <line id="meterNeedle" x1="200" y1="140" x2="200" y2="108" class="needle" />
    </svg>

    <div class="ctrl-wrap">
      <div class="ctrl-header">
        <span>Variable Resistor (R₃)</span>
        <span id="ctrlVal" class="ctrl-val">150 Ω</span>
      </div>
      <input type="range" id="r3Slider" min="50" max="250" value="100" step="5">
      
      <div class="preset-btns">
        <button class="preset-btn" id="btnLow" onclick="setR3(75)">Low (75Ω)</button>
        <button class="preset-btn" id="btnBal" onclick="setR3(150)">Balance (150Ω)</button>
        <button class="preset-btn" id="btnHigh" onclick="setR3(225)">High (225Ω)</button>
      </div>
    </div>
  </div>

  <script>
    const slider = document.getElementById('r3Slider');
    const r3Label = document.getElementById('r3Label');
    const ctrlVal = document.getElementById('ctrlVal');
    const vbLabel = document.getElementById('vbLabel');
    const vdLabel = document.getElementById('vdLabel');
    const meterNeedle = document.getElementById('meterNeedle');
    const statusBadge = document.getElementById('statusBadge');

    const R1 = 100;
    const R2 = 200;
    const Rx = 300;
    const Vin = 10;

    function updateBridge() {
      const R3 = parseFloat(slider.value);
      r3Label.textContent = `R₃: ${R3}Ω`;
      ctrlVal.textContent = `${R3} Ω`;

      // Potential divider voltages
      const VB = Vin * (R3 / (R1 + R3));
      const VD = Vin * (Rx / (R2 + Rx)); // fixed at 10 * (300/500) = 6.0V

      vbLabel.textContent = `${VB.toFixed(2)}V`;
      vdLabel.textContent = `${VD.toFixed(2)}V`;

      // Voltage difference drives deflection
      const diff = VB - VD;
      // Max angle +/- 45 deg
      const angle = Math.max(-45, Math.min(45, diff * 35));
      meterNeedle.style.transform = `rotate(${angle}deg)`;

      // Check balance (within 0.05V margin)
      const isBalanced = Math.abs(diff) < 0.02;
      
      if (isBalanced) {
        statusBadge.className = 'status-badge status-balanced';
        statusBadge.innerHTML = 'Balanced: V<sub>B</sub> = V<sub>D</sub> (Null: I<sub>G</sub> = 0)';
      } else {
        statusBadge.className = 'status-badge status-unbalanced';
        statusBadge.innerHTML = `Unbalanced: V<sub>B</sub> ${diff > 0 ? '>' : '<'} V<sub>D</sub> (I<sub>G</sub> ≠ 0)`;
      }

      // Update preset buttons state
      document.getElementById('btnLow').classList.toggle('active', R3 === 75);
      document.getElementById('btnBal').classList.toggle('active', R3 === 150);
      document.getElementById('btnHigh').classList.toggle('active', R3 === 225);
    }

    function setR3(val) {
      slider.value = val;
      updateBridge();
    }

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

How do the voltages at nodes B and D relate to each other when no current passes through the central detector?

A Wheatstone bridge consists of two parallel **voltage dividers** — circuits of two series resistors that split a total input voltage into a smaller output voltage proportional to their resistances.

The potential at node B relative to ground node C is given by $V_B = V_s \left(\frac{R_2}{R_1 + R_2}\right)$, and the potential at node D is $V_D = V_s \left(\frac{R_4}{R_3 + R_4}\right)$.

The bridge reaches **balance** when the detector reads zero volts ($V_G = V_B - V_D = 0$), meaning both nodes have identical electrical potentials: $V_B = V_D$.

<!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="1:1">
  <title>Balanced Bridge Circuit</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      user-select: none;
      -webkit-user-select: none;
    }

    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
    }

    .circuit-container {
      position: relative;
      width: 100%;
      max-width: 440px;
      aspect-ratio: 1.15;
    }

    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }

    /* Circuit styling */
    .wire {
      stroke: #94a3b8;
      stroke-width: 2.5;
      stroke-linecap: round;
      stroke-linejoin: round;
      fill: none;
    }

    .bridge-wire {
      stroke: #cbd5e1;
      stroke-width: 2;
      stroke-dasharray: 4 3;
    }

    .node {
      fill: #1e2945;
      stroke: #fff;
      stroke-width: 2;
    }

    .node-active {
      transition: fill 0.2s ease;
    }

    .resistor-box {
      fill: #ffffff;
      stroke: #4f46e5;
      stroke-width: 2;
      rx: 4;
    }

    .resistor-box.variable {
      stroke: #0284c7;
    }

    .resistor-box.unknown {
      stroke: #e11d48;
    }

    .label-title {
      font-size: 11px;
      font-weight: 700;
      fill: #1e2945;
      text-anchor: middle;
      font-family: inherit;
    }

    .label-sub {
      font-size: 10px;
      fill: #64748b;
      text-anchor: middle;
      font-family: inherit;
    }

    .potential-tag {
      font-size: 10px;
      font-weight: 600;
      text-anchor: middle;
      font-family: monospace;
    }

    /* Galvanometer */
    .galvo-bg {
      fill: #ffffff;
      stroke: #1e2945;
      stroke-width: 2;
    }

    .galvo-needle {
      stroke: #ef4444;
      stroke-width: 2;
      stroke-linecap: round;
      transform-origin: 220px 145px;
      transition: transform 0.15s ease-out;
    }

    .balanced .galvo-needle {
      stroke: #10b981;
    }

    /* Controls & Calculation Cards */
    .bottom-panel {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      gap: 10px;
    }

    .control-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      background: #ffffff;
      padding: 8px 14px;
      border-radius: 10px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
      border: 1px solid #e2e8f0;
      gap: 12px;
    }

    .control-label {
      font-size: 13px;
      font-weight: 700;
      color: #0284c7;
      white-space: nowrap;
    }

    input[type="range"] {
      flex: 1;
      accent-color: #0284c7;
      cursor: pointer;
      height: 6px;
    }

    .status-card {
      display: flex;
      justify-content: space-between;
      align-items: center;
      background: #ffffff;
      padding: 10px 14px;
      border-radius: 10px;
      border: 1px solid #e2e8f0;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
    }

    .badge {
      display: inline-flex;
      align-items: center;
      gap: 6px;
      padding: 4px 10px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 700;
      letter-spacing: 0.3px;
    }

    .badge.unbalanced {
      background: #fee2e2;
      color: #b91c1c;
    }

    .badge.balanced {
      background: #d1fae5;
      color: #047857;
    }

    .equation {
      font-family: monospace;
      font-size: 13px;
      font-weight: 600;
      color: #334155;
    }

    .equation strong {
      color: #047857;
    }

    .preset-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      padding: 3px 8px;
      border-radius: 5px;
      font-size: 11px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
    }
    .preset-btn:hover {
      background: #e2e8f0;
    }
  </style>
</head>
<body>

  
  <div class="circuit-container" id="circuitWrap">
    <svg viewBox="0 0 440 290">
      
      <path class="wire" d="M 40 40 L 220 40 L 400 40" />
      <path class="wire" d="M 40 250 L 220 250 L 400 250" />
      <path class="wire" d="M 40 40 L 40 125 M 40 165 L 40 250" />
      
      
      <line x1="25" y1="125" x2="55" y2="125" stroke="#1e2945" stroke-width="3" stroke-linecap="round" />
      <line x1="32" y1="135" x2="48" y2="135" stroke="#1e2945" stroke-width="1.5" stroke-linecap="round" />
      <line x1="25" y1="145" x2="55" y2="145" stroke="#1e2945" stroke-width="3" stroke-linecap="round" />
      <line x1="32" y1="155" x2="48" y2="155" stroke="#1e2945" stroke-width="1.5" stroke-linecap="round" />
      <text x="18" y="144" class="label-title" style="font-size: 12px;">12V</text>
      <text x="40" y="32" class="potential-tag" fill="#4f46e5">12.0V</text>
      <text x="40" y="268" class="potential-tag" fill="#64748b">0.0V</text>

      
      
      <path class="wire" d="M 220 40 L 130 145" />
      
      <path class="wire" d="M 130 145 L 220 250" />
      
      <path class="wire" d="M 220 40 L 310 145" />
      
      <path class="wire" d="M 310 145 L 220 250" />

      
      <line class="wire bridge-wire" x1="130" y1="145" x2="310" y2="145" />

      
      <g transform="translate(155, 78) rotate(49.4)">
        <rect class="resistor-box" x="-24" y="-10" width="48" height="20" />
      </g>
      <text x="145" y="76" class="label-title">R₁</text>
      <text x="145" y="90" class="label-sub">100 Ω</text>

      
      <g transform="translate(155, 212) rotate(-49.4)">
        <rect class="resistor-box variable" x="-24" y="-10" width="48" height="20" />
        
        <line x1="-20" y1="16" x2="20" y2="-16" stroke="#0284c7" stroke-width="1.5" />
        <polyline points="14,-16 20,-16 20,-10" fill="none" stroke="#0284c7" stroke-width="1.5" stroke-linecap="round" />
      </g>
      <text x="145" y="206" class="label-title" fill="#0284c7">R₂ (var)</text>
      <text id="r2Label" x="145" y="220" class="label-sub" font-weight="bold" fill="#0284c7">200 Ω</text>

      
      <g transform="translate(285, 78) rotate(-49.4)">
        <rect class="resistor-box" x="-24" y="-10" width="48" height="20" />
      </g>
      <text x="295" y="76" class="label-title">R₃</text>
      <text x="295" y="90" class="label-sub">150 Ω</text>

      
      <g transform="translate(285, 212) rotate(49.4)">
        <rect class="resistor-box unknown" x="-24" y="-10" width="48" height="20" />
      </g>
      <text x="295" y="206" class="label-title" fill="#e11d48">Rₓ (unknown)</text>
      <text id="rxLabel" x="295" y="220" class="label-sub" font-weight="bold" fill="#e11d48">?</text>

      
      <circle class="galvo-bg" cx="220" cy="145" r="28" />
      
      <line x1="220" y1="120" x2="220" y2="124" stroke="#64748b" stroke-width="1.5" />
      <line x1="202" y1="124" x2="205" y2="127" stroke="#94a3b8" stroke-width="1" />
      <line x1="238" y1="124" x2="235" y2="127" stroke="#94a3b8" stroke-width="1" />
      
      <line id="galvoNeedle" class="galvo-needle" x1="220" y1="145" x2="220" y2="122" />
      <circle cx="220" cy="145" r="3.5" fill="#1e2945" />
      <text x="220" y="161" class="label-title" style="font-size: 10px;">VG</text>

      
      
      <circle class="node" cx="220" cy="40" r="4" />
      <text x="220" y="30" class="potential-tag" fill="#4f46e5">Node A</text>

      
      <circle class="node" cx="220" cy="250" r="4" />
      <text x="220" y="270" class="potential-tag" fill="#64748b">Node C</text>

      
      <circle id="nodeB" class="node node-active" cx="130" cy="145" r="5" fill="#4f46e5" />
      <text x="96" y="142" class="potential-tag" fill="#1e2945">Node B</text>
      <text id="vbText" x="96" y="156" class="potential-tag" fill="#4f46e5">8.00V</text>

      
      <circle id="nodeD" class="node node-active" cx="310" cy="145" r="5" fill="#4f46e5" />
      <text x="344" y="142" class="potential-tag" fill="#1e2945">Node D</text>
      <text id="vdText" x="344" y="156" class="potential-tag" fill="#4f46e5">8.00V</text>
    </svg>
  </div>

  
  <div class="bottom-panel">
    <div class="control-row">
      <span class="control-label">Adjust R₂</span>
      <input type="range" id="r2Slider" min="50" max="400" step="5" value="120" />
      <button class="preset-btn" id="balanceBtn">Balance</button>
    </div>

    <div class="status-card">
      <div id="statusBadge" class="badge unbalanced">
        <span id="badgeIcon">✕</span>
        <span id="badgeText">VG ≠ 0V</span>
      </div>
      <div class="equation" id="calcText">
        Rₓ = (R₂ / R₁) × R₃ = <strong>---</strong>
      </div>
    </div>
  </div>

  <script>
    const VS = 12.0;
    const R1 = 100.0;
    const R3 = 150.0;
    const RX_ACTUAL = 300.0; // Target unknown resistance

    const r2Slider = document.getElementById('r2Slider');
    const balanceBtn = document.getElementById('balanceBtn');
    const r2Label = document.getElementById('r2Label');
    const rxLabel = document.getElementById('rxLabel');
    const vbText = document.getElementById('vbText');
    const vdText = document.getElementById('vdText');
    const galvoNeedle = document.getElementById('galvoNeedle');
    const circuitWrap = document.getElementById('circuitWrap');
    const statusBadge = document.getElementById('statusBadge');
    const badgeIcon = document.getElementById('badgeIcon');
    const badgeText = document.getElementById('badgeText');
    const calcText = document.getElementById('calcText');
    const nodeB = document.getElementById('nodeB');
    const nodeD = document.getElementById('nodeD');

    function update() {
      const R2 = parseFloat(r2Slider.value);
      r2Label.textContent = `${Math.round(R2)} Ω`;

      // Potential dividers
      // VB is potential across R2 (from ground C)
      const VB = VS * (R2 / (R1 + R2));
      // VD is potential across Rx (from ground C)
      const VD = VS * (RX_ACTUAL / (R3 + RX_ACTUAL));

      const VG = VB - VD;

      vbText.textContent = `${VB.toFixed(2)}V`;
      vdText.textContent = `${VD.toFixed(2)}V`;

      // Galvanometer deflection (-40 deg to +40 deg)
      const maxDeflection = 40;
      const angle = Math.max(-maxDeflection, Math.min(maxDeflection, VG * 25));
      galvoNeedle.style.transform = `rotate(${angle}deg)`;

      const isBalanced = Math.abs(VG) < 0.05;

      if (isBalanced) {
        circuitWrap.classList.add('balanced');
        statusBadge.className = 'badge balanced';
        badgeIcon.textContent = '✓';
        badgeText.textContent = 'VG = 0.00V';

        const rxCalculated = (R2 * R3) / R1;
        rxLabel.textContent = `${Math.round(rxCalculated)} Ω`;
        rxLabel.setAttribute('fill', '#047857');
        calcText.innerHTML = `Rₓ = (${Math.round(R2)} / 100) × 150 = <strong>${Math.round(rxCalculated)} Ω</strong>`;

        nodeB.setAttribute('fill', '#10b981');
        nodeD.setAttribute('fill', '#10b981');
      } else {
        circuitWrap.classList.remove('balanced');
        statusBadge.className = 'badge unbalanced';
        badgeIcon.textContent = '✦';
        badgeText.textContent = `VG = ${VG > 0 ? '+' : ''}${VG.toFixed(2)}V`;

        rxLabel.textContent = '?';
        rxLabel.setAttribute('fill', '#e11d48');
        calcText.innerHTML = `Rₓ = (${Math.round(R2)} / 100) × 150 = <span style="color:#94a3b8">?</span>`;

        nodeB.setAttribute('fill', '#4f46e5');
        nodeD.setAttribute('fill', '#4f46e5');
      }
    }

    r2Slider.addEventListener('input', update);
    balanceBtn.addEventListener('click', () => {
      // Balanced condition: R2 = (R1 * Rx) / R3 = (100 * 300) / 150 = 200
      r2Slider.value = 200;
      update();
    });

    // Initial render
    update();
  </script>
</body>
</html>

Notice that the supply voltage $V_s$ cancels out completely. This means bridge balance depends strictly on resistance ratios, never on battery fluctuations.

How do we rearrange this ratio to find the exact value of an unknown test resistor?

If we place an unknown resistor $R_x$ in position $R_4$, we can solve the balance equation directly: $R_x = R_3 \left(\frac{R_2}{R_1}\right)$.

Here, $R_1$ and $R_2$ form the **ratio arm** that sets a multiplier (like $1:1$, $1:10$, or $1:100$), while $R_3$ is a calibrated adjustable precision resistor (such as a decade box).

<!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="1:1">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      width: 100%;
      font-family: system-ui, -apple-system, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      padding: 12px;
      gap: 10px;
    }

    .main-stage {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 8px;
    }

    /* SVG Bridge */
    .bridge-svg {
      width: 100%;
      max-width: 340px;
      height: auto;
      overflow: visible;
    }

    .resistor-box {
      cursor: pointer;
      transition: filter 0.2s ease;
    }
    .resistor-box:hover {
      filter: brightness(0.95);
    }

    .wire {
      stroke: #64748b;
      stroke-width: 2.5;
      stroke-linecap: round;
      stroke-linejoin: round;
      fill: none;
    }

    .cross-pair-a {
      stroke: #0284c7;
      stroke-width: 2;
      stroke-dasharray: 4 3;
    }

    .cross-pair-b {
      stroke: #d97706;
      stroke-width: 2;
      stroke-dasharray: 4 3;
    }

    /* Controls & Readout */
    .panel {
      width: 100%;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 14px;
      display: flex;
      flex-direction: column;
      gap: 8px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.04);
    }

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

    .slider-container {
      display: flex;
      align-items: center;
      gap: 10px;
    }

    input[type="range"] {
      flex: 1;
      accent-color: #0284c7;
      height: 6px;
      cursor: pointer;
    }

    .math-display {
      display: flex;
      justify-content: center;
      align-items: center;
      gap: 6px;
      font-size: 13px;
      font-family: ui-monospace, SFMono-Regular, Menlo, monospace;
      padding: 6px 8px;
      border-radius: 8px;
      background: #f1f5f9;
      transition: background 0.3s ease;
    }

    .math-display.balanced {
      background: #dcfce7;
      color: #15803d;
      font-weight: 700;
    }

    .badge-a { color: #0284c7; font-weight: 700; }
    .badge-b { color: #d97706; font-weight: 700; }

    .btn-group {
      display: flex;
      gap: 6px;
    }

    .btn {
      padding: 4px 8px;
      font-size: 12px;
      font-weight: 600;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      background: #ffffff;
      color: #334155;
      cursor: pointer;
    }

    .btn.active {
      background: #e0f2fe;
      border-color: #0284c7;
      color: #0284c7;
    }

    .status-badge {
      padding: 2px 8px;
      border-radius: 12px;
      font-size: 11px;
      font-weight: 700;
      background: #f1f5f9;
      color: #64748b;
    }
    .status-badge.balanced {
      background: #22c55e;
      color: #ffffff;
    }
  </style>
</head>
<body>

  <div class="main-stage">
    
    <svg class="bridge-svg" viewBox="0 0 320 230">
      
      <line x1="100" y1="65" x2="220" y2="165" class="cross-pair-a" opacity="0.45" />
      <line x1="100" y1="165" x2="220" y2="65" class="cross-pair-b" opacity="0.45" />

      
      <path d="M 160 15 L 60 115 L 160 215 L 260 115 Z" class="wire" />
      
      <line x1="60" y1="115" x2="260" y2="115" class="wire" />

      
      <circle cx="160" cy="15" r="4" fill="#1e2945" />
      <circle cx="160" cy="215" r="4" fill="#1e2945" />
      <circle cx="60" cy="115" r="4" fill="#1e2945" />
      <circle cx="260" cy="115" r="4" fill="#1e2945" />

      
      <text x="160" y="10" font-size="11" font-weight="600" text-anchor="middle" fill="#64748b">+V</text>
      <text x="160" y="228" font-size="11" font-weight="600" text-anchor="middle" fill="#64748b">0V</text>

      
      <g transform="translate(85, 45)">
        <rect width="50" height="24" rx="4" fill="#ffffff" stroke="#0284c7" stroke-width="2"/>
        <text x="25" y="12" font-size="10" font-weight="700" fill="#0284c7" text-anchor="middle" dominant-baseline="middle">R₁: 100Ω</text>
      </g>

      
      <g transform="translate(85, 155)">
        <rect width="50" height="24" rx="4" fill="#ffffff" stroke="#d97706" stroke-width="2"/>
        <text id="r2-label" x="25" y="12" font-size="10" font-weight="700" fill="#d97706" text-anchor="middle" dominant-baseline="middle">R₂: 100Ω</text>
      </g>

      
      <g transform="translate(185, 45)">
        <rect width="58" height="24" rx="4" fill="#ffffff" stroke="#d97706" stroke-width="2"/>
        
        <line x1="12" y1="26" x2="46" y2="-2" stroke="#d97706" stroke-width="1.5" />
        <polygon points="46,-2 40,0 44,5" fill="#d97706"/>
        <text id="r3-label" x="29" y="12" font-size="10" font-weight="700" fill="#d97706" text-anchor="middle" dominant-baseline="middle">R₃: 150Ω</text>
      </g>

      
      <g transform="translate(185, 155)">
        <rect width="58" height="24" rx="4" fill="#ffffff" stroke="#0284c7" stroke-width="2"/>
        <text id="rx-label" x="29" y="12" font-size="10" font-weight="700" fill="#0284c7" text-anchor="middle" dominant-baseline="middle">R₄: 250Ω</text>
      </g>

      
      <g transform="translate(160, 115)">
        <circle r="22" fill="#ffffff" stroke="#1e2945" stroke-width="2"/>
        
        <line x1="-12" y1="-8" x2="-8" y2="-4" stroke="#94a3b8" stroke-width="1.5"/>
        <line x1="0" y1="-14" x2="0" y2="-9" stroke="#1e2945" stroke-width="1.5"/>
        <line x1="12" y1="-8" x2="8" y2="-4" stroke="#94a3b8" stroke-width="1.5"/>
        <text x="0" y="12" font-size="9" font-weight="700" text-anchor="middle" fill="#64748b">G</text>
        
        <line id="galvo-needle" x1="0" y1="0" x2="0" y2="-15" stroke="#ef4444" stroke-width="2" stroke-linecap="round" />
        <circle cx="0" cy="0" r="3" fill="#1e2945" />
      </g>
    </svg>

    
    <div class="panel">
      <div class="row-between">
        <span>Standard Arm (R₃)</span>
        <span class="status-badge" id="status-badge">UNBALANCED</span>
      </div>

      <div class="slider-container">
        <input type="range" id="r3-slider" min="50" max="450" step="5" value="150">
        <span id="slider-val" style="min-width: 45px; font-weight: 700; font-size: 13px;">150 Ω</span>
      </div>

      <div class="row-between" style="margin-top: 2px;">
        <span style="font-size: 12px; color: #64748b;">Set Unknown (R₄):</span>
        <div class="btn-group">
          <button class="btn" onclick="setRx(150)">150Ω</button>
          <button class="btn active" onclick="setRx(250)">250Ω</button>
          <button class="btn" onclick="setRx(350)">350Ω</button>
        </div>
      </div>

      
      <div class="math-display" id="math-display">
        <span class="badge-a">R₁ · R₄</span>
        <span>=</span>
        <span class="badge-b">R₂ · R₃</span>
        <span id="calc-vals" style="font-size: 11px; margin-left: 4px; color: #64748b;">(25k ≠ 15k)</span>
      </div>
    </div>
  </div>

  <script>
    const R1 = 100;
    const R2 = 100;
    let R4 = 250;
    let R3 = 150;

    const slider = document.getElementById('r3-slider');
    const sliderVal = document.getElementById('slider-val');
    const r3Label = document.getElementById('r3-label');
    const rxLabel = document.getElementById('rx-label');
    const needle = document.getElementById('galvo-needle');
    const statusBadge = document.getElementById('status-badge');
    const mathDisplay = document.getElementById('math-display');
    const calcVals = document.getElementById('calc-vals');
    const buttons = document.querySelectorAll('.btn-group .btn');

    function updateBridge() {
      sliderVal.textContent = R3 + ' Ω';
      r3Label.textContent = 'R₃: ' + R3 + 'Ω';
      rxLabel.textContent = 'R₄: ' + R4 + 'Ω';

      // Cross products: Pair A (R1 * R4), Pair B (R2 * R3)
      const prodA = R1 * R4;
      const prodB = R2 * R3;

      // Needle deflection proportional to bridge balance: (V_left - V_right)
      // V_left = V * R2 / (R1 + R2) = 0.5 * V
      // V_right = V * R4 / (R3 + R4)
      const diff = (prodA - prodB) / 1000;
      const angle = Math.max(-45, Math.min(45, diff * 3));
      needle.setAttribute('transform', `rotate(${angle})`);

      const isBalanced = prodA === prodB;

      if (isBalanced) {
        statusBadge.textContent = 'BALANCED (IG = 0)';
        statusBadge.classList.add('balanced');
        mathDisplay.classList.add('balanced');
        calcVals.textContent = `(${prodA/1000}k = ${prodB/1000}k)`;
      } else {
        statusBadge.textContent = 'UNBALANCED';
        statusBadge.classList.remove('balanced');
        mathDisplay.classList.remove('balanced');
        calcVals.textContent = `(${(prodA/1000).toFixed(1)}k ≠ ${(prodB/1000).toFixed(1)}k)`;
      }
    }

    slider.addEventListener('input', (e) => {
      R3 = parseInt(e.target.value, 10);
      updateBridge();
    });

    function setRx(val) {
      R4 = val;
      buttons.forEach(b => {
        if (parseInt(b.textContent) === val) b.classList.add('active');
        else b.classList.remove('active');
      });
      updateBridge();
    }

    updateBridge();
  </script>
</body>
</html>

A common mistake is pairing opposite resistors incorrectly when setting up the cross-product. Remember the visual rule: **opposite arm products are equal** at balance ($R_1 \times R_4 = R_2 \times R_3$).

What happens when temperature changes or mechanical strain causes tiny, continuous shifts in an unknown resistor during operation?

electronics

Calculating Unknown Resistances Using Balanced Bridge Circuits. Derive and apply the Wheatstone bridge balance condition (R1/R2 = R3/R4) to calculate precision resistive component values.

Bridge Circuits, Kirchhoff's Laws and Potential Dividers

Wheatstone Bridge Circuit and Theory of Operation

DC Bridge Circuits and Transducer Instrumentation

Pedagogical strategy focuses on decomposing the Wheatstone diamond into two parallel potential dividers to anchor prior knowledge before formal algebraic derivation. Emphasize null-deflection as a zero-power measurement technique that eliminates loading effects. Progress directly from abstract calculations to sensor transduction (thermistors and strain gauges) to satisfy A-Level syllabus requirements (ELEC-6.7).

Calculating Unknown Resistances Using Balanced Bridge Circuits

Deriving the Balance Condition

Calculating an Unknown Resistance