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An operational amplifier (op-amp) straight from the factory has a raw voltage gain well over $100{,}000$, yet trying to use it without modification will ruin almost any signal.

Without control, even microscopic electrical noise gets amplified until the output crashes into the power supply limits and clips.

How do engineers turn this wildly erratic component into a precise, ultra-stable tool?

In 1927, Bell Labs engineer Harold Black was riding a ferry to work when he realized that feeding part of an amplifier's output back out-of-phase would cancel out internal distortion.

This principle is **negative feedback**: returning a fraction of the output voltage ($V_{out}$) to the **inverting input** (marked with a $-$ sign) so it subtracts from the input.

<!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, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }

    .controls-row {
      display: flex;
      gap: 8px;
      width: 100%;
      max-width: 480px;
      justify-content: center;
      margin-bottom: 6px;
    }

    .mode-btn {
      flex: 1;
      padding: 8px 12px;
      font-size: 13px;
      font-weight: 600;
      border-radius: 8px;
      border: 1px solid #d1d5db;
      background: #ffffff;
      color: #4b5563;
      cursor: pointer;
      transition: all 0.2s ease;
      text-align: center;
    }

    .mode-btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
      box-shadow: 0 2px 6px rgba(37, 99, 235, 0.25);
    }

    .circuit-container {
      width: 100%;
      max-width: 520px;
      position: relative;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e5e7eb;
      padding: 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
    }

    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }

    .slider-card {
      width: 100%;
      max-width: 480px;
      background: #ffffff;
      border-radius: 10px;
      border: 1px solid #e5e7eb;
      padding: 10px 14px;
      margin-top: 8px;
      display: flex;
      align-items: center;
      gap: 12px;
    }

    .slider-label {
      font-size: 13px;
      font-weight: 700;
      white-space: nowrap;
      min-width: 68px;
    }

    input[type=range] {
      flex: 1;
      accent-color: #2563eb;
      cursor: pointer;
    }

    .status-badge {
      font-size: 12px;
      font-weight: 700;
      padding: 4px 8px;
      border-radius: 6px;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }
    .status-stable { background: #dcfce7; color: #166534; }
    .status-sat { background: #fee2e2; color: #991b1b; }

    .pulse-line {
      stroke-dasharray: 6, 6;
      animation: dash 1s linear infinite;
    }
    @keyframes dash {
      to { stroke-dashoffset: -12; }
    }
  </style>
</head>
<body>

  
  <div class="controls-row">
    <button class="mode-btn active" id="btnFeedback" onclick="setMode(true)">Negative Feedback</button>
    <button class="mode-btn" id="btnOpenLoop" onclick="setMode(false)">Open Loop (No Feedback)</button>
  </div>

  
  <div class="circuit-container">
    <svg viewBox="0 0 460 210">
      <defs>
        <marker id="arrow" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
          <path d="M 0 1 L 8 5 L 0 9 z" fill="#2563eb" />
        </marker>
        <marker id="arrow-gray" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
          <path d="M 0 1 L 8 5 L 0 9 z" fill="#9ca3af" />
        </marker>
      </defs>

      
      <polygon points="170,45 170,165 290,105" fill="#f0fdf4" stroke="#16a34a" stroke-width="2.5" id="opampBody" />
      
      
      <text x="180" y="80" font-size="16" font-weight="bold" fill="#dc2626">−</text>
      <text x="180" y="145" font-size="16" font-weight="bold" fill="#2563eb">+</text>

      
      <line x1="40" y1="140" x2="170" y2="140" stroke="#2563eb" stroke-width="2.5" />
      <circle cx="40" cy="140" r="4" fill="#2563eb" />
      <text x="40" y="165" font-size="13" font-weight="bold" fill="#2563eb" text-anchor="middle">V_in: <tspan id="txtVin">+1.5V</tspan></text>

      
      <line x1="120" y1="75" x2="170" y2="75" stroke="#4b5563" stroke-width="2.5" />
      <circle cx="120" cy="75" r="3.5" fill="#4b5563" />
      <text x="115" y="65" font-size="12" font-weight="bold" fill="#dc2626" text-anchor="end">V_−: <tspan id="txtVminus">+1.5V</tspan></text>

      
      <line x1="290" y1="105" x2="410" y2="105" stroke="#2563eb" stroke-width="2.5" id="outWire" />
      <circle cx="410" cy="105" r="4" fill="#2563eb" id="outNode" />
      <text x="410" y="130" font-size="13" font-weight="bold" fill="#2563eb" text-anchor="middle" id="txtVoutLabel">V_out: <tspan id="txtVout">+1.5V</tspan></text>

      
      <path id="feedbackPath" d="M 350,105 L 350,25 L 120,25 L 120,75" fill="none" stroke="#2563eb" stroke-width="2.5" stroke-linejoin="round" class="pulse-line" marker-end="url(#arrow)" />
      
      
      <g id="feedbackIndicator" transform="translate(235, 18)">
        <rect x="-45" y="-12" width="90" height="22" rx="4" fill="#eff6ff" stroke="#2563eb" stroke-width="1.5"/>
        <text x="0" y="3" font-size="11" font-weight="bold" fill="#1e40af" text-anchor="middle">Feedback −</text>
      </g>
    </svg>
  </div>

  
  <div class="slider-card">
    <span class="slider-label">Input V_in</span>
    <input type="range" id="vinRange" min="-5" max="5" step="0.5" value="1.5" oninput="updateCircuit()">
    <span class="status-badge status-stable" id="statusBadge">Controlled</span>
  </div>

  <script>
    let hasFeedback = true;

    function setMode(fb) {
      hasFeedback = fb;
      document.getElementById('btnFeedback').classList.toggle('active', fb);
      document.getElementById('btnOpenLoop').classList.toggle('active', !fb);
      updateCircuit();
    }

    function updateCircuit() {
      const vin = parseFloat(document.getElementById('vinRange').value);
      const vMinusTxt = document.getElementById('txtVminus');
      const vInTxt = document.getElementById('txtVin');
      const vOutTxt = document.getElementById('txtVout');
      const fbPath = document.getElementById('feedbackPath');
      const fbInd = document.getElementById('feedbackIndicator');
      const statusBadge = document.getElementById('statusBadge');
      const opamp = document.getElementById('opampBody');
      const outWire = document.getElementById('outWire');
      const outNode = document.getElementById('outNode');
      const txtVoutLabel = document.getElementById('txtVoutLabel');

      vInTxt.textContent = (vin >= 0 ? '+' : '') + vin.toFixed(1) + 'V';

      if (hasFeedback) {
        // Negative feedback: V_out follows V_in, V_- matches V_+
        const vout = vin;
        vMinusTxt.textContent = (vout >= 0 ? '+' : '') + vout.toFixed(1) + 'V';
        vOutTxt.textContent = (vout >= 0 ? '+' : '') + vout.toFixed(1) + 'V';

        fbPath.setAttribute('stroke', '#2563eb');
        fbPath.setAttribute('class', 'pulse-line');
        fbPath.setAttribute('marker-end', 'url(#arrow)');
        fbInd.style.display = 'block';

        opamp.setAttribute('fill', '#f0fdf4');
        opamp.setAttribute('stroke', '#16a34a');
        outWire.setAttribute('stroke', '#2563eb');
        outNode.setAttribute('fill', '#2563eb');
        txtVoutLabel.setAttribute('fill', '#2563eb');

        statusBadge.className = 'status-badge status-stable';
        statusBadge.textContent = 'Stable Linear';
      } else {
        // Open loop: V_- is floating (0V), massive gain causes immediate saturation (+12V or -12V)
        vMinusTxt.textContent = '0.0V';
        let vout = 0;
        if (vin > 0.05) vout = 12.0;
        else if (vin < -0.05) vout = -12.0;
        else vout = 0.0;

        vOutTxt.textContent = (vout >= 0 ? '+' : '') + vout.toFixed(1) + 'V (Rail)';

        fbPath.setAttribute('stroke', '#e5e7eb');
        fbPath.removeAttribute('class');
        fbPath.removeAttribute('marker-end');
        fbInd.style.display = 'none';

        opamp.setAttribute('fill', '#fef2f2');
        opamp.setAttribute('stroke', '#dc2626');
        outWire.setAttribute('stroke', '#dc2626');
        outNode.setAttribute('fill', '#dc2626');
        txtVoutLabel.setAttribute('fill', '#dc2626');

        statusBadge.className = 'status-badge status-sat';
        statusBadge.textContent = vout === 0 ? 'Zero' : 'Saturated';
      }
    }

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

Why would we deliberately throw away massive amounts of raw amplification?

The **open-loop gain** ($A_{OL}$) is the op-amp's internal gain with no feedback, while the **closed-loop gain** ($A_{CL}$) is the total gain of the circuit with feedback applied.

When a fraction $\beta$ (the **feedback factor**) feeds back to subtract from the input, the differential voltage entering the op-amp becomes $V_d = V_{in} - \beta V_{out}$.

Since $V_{out} = A_{OL} V_d$, substituting $V_d$ gives: $$V_{out} = A_{OL}(V_{in} - \beta V_{out})$$

Expanding and rearranging the terms gives: $$V_{out}(1 + A_{OL}\beta) = A_{OL}V_{in} \implies A_{CL} = \frac{V_{out}}{V_{in}} = \frac{A_{OL}}{1 + A_{OL}\beta}$$

Because $A_{OL}$ is massive (typically $10^5$ to $10^6$), the product $A_{OL}\beta$ is vastly greater than $1$, meaning $1 + A_{OL}\beta \approx A_{OL}\beta$.

Canceling $A_{OL}$ yields: $$A_{CL} \approx \frac{1}{\beta}$$

<!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: #f8f9fa;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }
    .wrapper {
      width: 100%;
      max-width: 540px;
      display: flex;
      flex-direction: column;
      gap: 12px;
    }
    .diagram-svg {
      width: 100%;
      height: auto;
      background: #ffffff;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.05);
      border: 1px solid #e2e8f0;
    }
    .controls {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
      background: #ffffff;
      padding: 12px 14px;
      border-radius: 12px;
      border: 1px solid #e2e8f0;
      box-shadow: 0 2px 8px rgba(0,0,0,0.05);
    }
    .control-group {
      display: flex;
      flex-direction: column;
      gap: 4px;
    }
    .control-header {
      display: flex;
      justify-content: space-between;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
    }
    .val-badge {
      color: #2563eb;
      font-family: ui-monospace, monospace;
      font-weight: 700;
    }
    input[type="range"] {
      width: 100%;
      accent-color: #2563eb;
      cursor: pointer;
    }
    .results {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
    }
    .stat-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px 12px;
      text-align: center;
      box-shadow: 0 1px 4px rgba(0,0,0,0.04);
    }
    .stat-label {
      font-size: 12px;
      font-weight: 600;
      color: #64748b;
    }
    .stat-val {
      font-size: 18px;
      font-weight: 800;
      font-family: ui-monospace, monospace;
      margin-top: 2px;
    }
    .stat-exact { color: #16a34a; }
    .stat-approx { color: #2563eb; }
    .sig-flow {
      stroke-dasharray: 6;
      animation: flow 1.2s linear infinite;
    }
    @keyframes flow {
      to { stroke-dashoffset: -12; }
    }
  </style>
</head>
<body>
<div class="wrapper">
  
  <svg class="diagram-svg" viewBox="0 0 540 220">
    <defs>
      
      <marker id="arrow" viewBox="0 0 10 10" refX="8" refY="5" markerWidth="6" markerHeight="6" orient="auto">
        <path d="M 0 1 L 10 5 L 0 9 z" fill="#2563eb" />
      </marker>
      <marker id="arrow-green" viewBox="0 0 10 10" refX="8" refY="5" markerWidth="6" markerHeight="6" orient="auto">
        <path d="M 0 1 L 10 5 L 0 9 z" fill="#16a34a" />
      </marker>
    </defs>

    
    
    <line x1="25" y1="70" x2="88" y2="70" stroke="#2563eb" stroke-width="2.5" marker-end="url(#arrow)" class="sig-flow" />
    <text x="28" y="58" font-size="13" font-weight="700" fill="#1e2945">V_in</text>

    
    <circle cx="102" cy="70" r="14" fill="#ffffff" stroke="#2563eb" stroke-width="2" />
    <text x="97" y="65" font-size="11" font-weight="700" fill="#16a34a">+</text>
    <text x="98" y="81" font-size="12" font-weight="700" fill="#dc2626">−</text>

    
    <line x1="116" y1="70" x2="198" y2="70" stroke="#2563eb" stroke-width="2.5" marker-end="url(#arrow)" class="sig-flow" />
    <text x="142" y="60" font-size="12" font-weight="600" fill="#64748b">ε = V_in − V_f</text>

    
    <rect x="200" y="42" width="130" height="56" rx="8" fill="#eff6ff" stroke="#2563eb" stroke-width="2" />
    <text x="265" y="66" font-size="13" font-weight="800" fill="#1e2945" text-anchor="middle">A_OL</text>
    <text id="aol-sub" x="265" y="84" font-size="11" font-weight="600" fill="#2563eb" text-anchor="middle">100,000</text>

    
    <line x1="330" y1="70" x2="510" y2="70" stroke="#2563eb" stroke-width="2.5" marker-end="url(#arrow)" class="sig-flow" />
    <text x="495" y="58" font-size="13" font-weight="700" fill="#1e2945">V_out</text>

    
    <circle cx="430" cy="70" r="4" fill="#2563eb" />
    <line x1="430" y1="70" x2="430" y2="160" stroke="#16a34a" stroke-width="2.5" />
    <line x1="430" y1="160" x2="332" y2="160" stroke="#16a34a" stroke-width="2.5" marker-end="url(#arrow-green)" class="sig-flow" />

    
    <rect x="200" y="132" width="130" height="56" rx="8" fill="#f0fdf4" stroke="#16a34a" stroke-width="2" />
    <text x="265" y="156" font-size="13" font-weight="800" fill="#1e2945" text-anchor="middle">Feedback β</text>
    <text id="beta-sub" x="265" y="174" font-size="11" font-weight="600" fill="#16a34a" text-anchor="middle">0.100</text>

    
    <line x1="200" y1="160" x2="102" y2="160" stroke="#16a34a" stroke-width="2.5" />
    <line x1="102" y1="160" x2="102" y2="86" stroke="#16a34a" stroke-width="2.5" marker-end="url(#arrow-green)" class="sig-flow" />
    <text x="110" y="145" font-size="12" font-weight="600" fill="#16a34a">V_f = β·V_out</text>
  </svg>

  
  <div class="controls">
    <div class="control-group">
      <div class="control-header">
        <span>Open-Loop A_OL</span>
        <span class="val-badge" id="aol-text">100,000</span>
      </div>
      <input type="range" id="aol-slider" min="3" max="6" step="0.05" value="5">
    </div>
    <div class="control-group">
      <div class="control-header">
        <span>Feedback β</span>
        <span class="val-badge" id="beta-text">0.10</span>
      </div>
      <input type="range" id="beta-slider" min="0.01" max="0.5" step="0.01" value="0.10">
    </div>
  </div>

  
  <div class="results">
    <div class="stat-card">
      <div class="stat-label">A_CL (Exact: A_OL / (1 + β·A_OL))</div>
      <div class="stat-val stat-exact" id="exact-val">9.999</div>
    </div>
    <div class="stat-card">
      <div class="stat-label">A_CL (Ideal Limit: 1 / β)</div>
      <div class="stat-val stat-approx" id="approx-val">10.000</div>
    </div>
  </div>
</div>

<script>
  const aolSlider = document.getElementById('aol-slider');
  const betaSlider = document.getElementById('beta-slider');
  const aolText = document.getElementById('aol-text');
  const betaText = document.getElementById('beta-text');
  const aolSub = document.getElementById('aol-sub');
  const betaSub = document.getElementById('beta-sub');
  const exactVal = document.getElementById('exact-val');
  const approxVal = document.getElementById('approx-val');

  function update() {
    const aol = Math.round(Math.pow(10, parseFloat(aolSlider.value)));
    const beta = parseFloat(betaSlider.value);

    aolText.textContent = aol.toLocaleString();
    aolSub.textContent = aol.toLocaleString();

    betaText.textContent = beta.toFixed(2);
    betaSub.textContent = beta.toFixed(3);

    const aclExact = aol / (1 + beta * aol);
    const aclApprox = 1 / beta;

    exactVal.textContent = aclExact >= 100 ? aclExact.toFixed(1) : aclExact.toFixed(3);
    approxVal.textContent = aclApprox >= 100 ? aclApprox.toFixed(1) : aclApprox.toFixed(3);
  }

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

We traded astronomical gain for rock-solid stability, but what happens to the range of frequencies the amplifier can process?

Due to internal compensation capacitors, an op-amp's open-loop gain begins dropping at very low frequencies—often starting around $10\text{ Hz}$.

The **gain-bandwidth product** (**GBWP**) is a constant figure of merit for any given op-amp, defined as: $$\text{GBWP} = \text{Gain} \times \text{Bandwidth}$$

Because this product stays constant, slashing voltage gain with negative feedback forces the **bandwidth** (the operational frequency range) to expand by the exact same proportion.

<!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: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      padding: 12px;
      min-height: 100%;
      user-select: none;
    }

    .controls {
      display: flex;
      flex-wrap: wrap;
      gap: 6px;
      justify-content: center;
      width: 100%;
      max-width: 520px;
    }

    .btn {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 6px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
      transition: background 0.2s, color 0.2s, border-color 0.2s;
    }
    .btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
    }

    .slider-row {
      display: flex;
      align-items: center;
      gap: 10px;
      width: 100%;
      max-width: 480px;
      margin-top: 4px;
    }
    .slider-row span {
      font-size: 12px;
      font-weight: 600;
      color: #64748b;
    }
    input[type=range] {
      flex: 1;
      accent-color: #2563eb;
      cursor: pointer;
      height: 6px;
    }

    .svg-container {
      width: 100%;
      max-width: 540px;
      background: #ffffff;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      padding: 8px;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
    }

    .metric-bar {
      display: flex;
      justify-content: space-around;
      width: 100%;
      max-width: 520px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px 12px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.04);
    }
    .metric {
      text-align: center;
    }
    .metric-label {
      font-size: 11px;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      font-weight: 700;
      color: #64748b;
    }
    .metric-val {
      font-size: 15px;
      font-weight: 800;
    }
    .val-gain { color: #2563eb; }
    .val-bw { color: #16a34a; }
    .val-gbw { color: #d97706; }
  </style>
</head>
<body>

  
  <div class="controls">
    <button class="btn" onclick="setPreset(5)">Open Loop</button>
    <button class="btn active" onclick="setPreset(3)">Gain ×1,000</button>
    <button class="btn" onclick="setPreset(2)">Gain ×100</button>
    <button class="btn" onclick="setPreset(1)">Gain ×10</button>
    <button class="btn" onclick="setPreset(0)">Unity (×1)</button>
  </div>

  <div class="slider-row">
    <span>Gain</span>
    <input type="range" id="gainSlider" min="0" max="5" step="0.05" value="3" oninput="updateFromSlider(this.value)">
    <span>Bandwidth</span>
  </div>

  
  <div class="svg-container">
    <svg viewBox="0 0 520 250">
      <defs>
        <linearGradient id="bwFill" x1="0" y1="0" x2="0" y2="1">
          <stop offset="0%" stop-color="#16a34a" stop-opacity="0.18"/>
          <stop offset="100%" stop-color="#16a34a" stop-opacity="0.02"/>
        </linearGradient>
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        <line x1="55" y1="63" x2="495" y2="63"/>
        <line x1="55" y1="101" x2="495" y2="101"/>
        <line x1="55" y1="139" x2="495" y2="139"/>
        <line x1="55" y1="177" x2="495" y2="177"/>
        <line x1="55" y1="215" x2="495" y2="215"/>

        <line x1="55" y1="25" x2="55" y2="215"/>
        <line x1="128" y1="25" x2="128" y2="215"/>
        <line x1="201" y1="25" x2="201" y2="215"/>
        <line x1="275" y1="25" x2="275" y2="215"/>
        <line x1="348" y1="25" x2="348" y2="215"/>
        <line x1="421" y1="25" x2="421" y2="215"/>
        <line x1="495" y1="25" x2="495" y2="215"/>
      </g>

      
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      <line x1="55" y1="215" x2="495" y2="215" stroke="#94a3b8" stroke-width="1.5"/>
      <line x1="55" y1="25" x2="55" y2="215" stroke="#94a3b8" stroke-width="1.5"/>

      
      <text x="48" y="29" text-anchor="end" font-size="10" fill="#64748b">10⁵</text>
      <text x="48" y="67" text-anchor="end" font-size="10" fill="#64748b">10⁴</text>
      <text x="48" y="105" text-anchor="end" font-size="10" fill="#64748b">10³</text>
      <text x="48" y="143" text-anchor="end" font-size="10" fill="#64748b">10²</text>
      <text x="48" y="181" text-anchor="end" font-size="10" fill="#64748b">10¹</text>
      <text x="48" y="218" text-anchor="end" font-size="10" fill="#64748b">1</text>

      
      <text x="55" y="232" text-anchor="middle" font-size="10" fill="#64748b">1 Hz</text>
      <text x="128" y="232" text-anchor="middle" font-size="10" fill="#64748b">10 Hz</text>
      <text x="201" y="232" text-anchor="middle" font-size="10" fill="#64748b">100 Hz</text>
      <text x="275" y="232" text-anchor="middle" font-size="10" fill="#64748b">1 kHz</text>
      <text x="348" y="232" text-anchor="middle" font-size="10" fill="#64748b">10 kHz</text>
      <text x="421" y="232" text-anchor="middle" font-size="10" fill="#64748b">100 kHz</text>
      <text x="495" y="232" text-anchor="middle" font-size="10" fill="#64748b">1 MHz</text>

      
      <path d="M 55,25 L 128,25 L 495,215" fill="none" stroke="#cbd5e1" stroke-width="2.5" stroke-dasharray="4,4"/>

      
      <path id="curvePath" d="M 55,101 L 275,101 L 495,215" fill="none" stroke="#2563eb" stroke-width="3" stroke-linecap="round"/>

      
      <line id="bwDropLine" x1="275" y1="101" x2="275" y2="215" stroke="#16a34a" stroke-width="1.5" stroke-dasharray="3,3"/>
      <circle id="cornerDot" cx="275" cy="101" r="5" fill="#16a34a" stroke="#ffffff" stroke-width="2"/>

      
      <text id="gainIndicator" x="65" y="93" font-size="12" font-weight="700" fill="#2563eb">Gain = 1,000</text>
      <text id="bwIndicator" x="280" y="205" font-size="11" font-weight="700" fill="#16a34a">1 kHz</text>
    </svg>
  </div>

  
  <div class="metric-bar">
    <div class="metric">
      <div class="metric-label">Gain (Aᵥ)</div>
      <div class="metric-val val-gain" id="gainDisplay">1,000</div>
    </div>
    <div class="metric">
      <div class="metric-label">× Bandwidth</div>
      <div class="metric-val val-bw" id="bwDisplay">1 kHz</div>
    </div>
    <div class="metric">
      <div class="metric-label">= Constant GBW</div>
      <div class="metric-val val-gbw">1 MHz</div>
    </div>
  </div>

  <script>
    const slider = document.getElementById('gainSlider');
    const buttons = document.querySelectorAll('.controls .btn');
    const curvePath = document.getElementById('curvePath');
    const shadeArea = document.getElementById('shadeArea');
    const cornerDot = document.getElementById('cornerDot');
    const bwDropLine = document.getElementById('bwDropLine');
    const gainIndicator = document.getElementById('gainIndicator');
    const bwIndicator = document.getElementById('bwIndicator');
    const gainDisplay = document.getElementById('gainDisplay');
    const bwDisplay = document.getElementById('bwDisplay');

    function render(k) {
      // k is exponent of gain (0 to 5)
      // Gain = 10^k
      // Cutoff Freq = 10^(6 - k) Hz
      const xStart = 55;
      const xEnd = 495;
      const yTop = 25;
      const yBottom = 215;

      const plotW = xEnd - xStart; // 440
      const plotH = yBottom - yTop; // 190

      // Calculate corner coordinate
      let cornerLogF = 6 - k;
      let cornerX = xStart + (cornerLogF / 6) * plotW;
      let gainY = yBottom - (k / 5) * plotH;

      if (k >= 5) {
        // Open-loop plateau ends at 10 Hz (logF = 1)
        cornerX = xStart + (1 / 6) * plotW;
        gainY = yTop;
      }

      // Update Path & Shading
      curvePath.setAttribute('d', `M ${xStart},${gainY} L ${cornerX},${gainY} L ${xEnd},${yBottom}`);
      shadeArea.setAttribute('points', `${xStart},${gainY} ${cornerX},${gainY} ${xEnd},${yBottom} ${xStart},${yBottom}`);

      // Update Corner Markers
      cornerDot.setAttribute('cx', cornerX);
      cornerDot.setAttribute('cy', gainY);
      bwDropLine.setAttribute('x1', cornerX);
      bwDropLine.setAttribute('y1', gainY);
      bwDropLine.setAttribute('x2', cornerX);
      bwDropLine.setAttribute('y2', yBottom);

      // Value formatting
      const gainVal = Math.round(Math.pow(10, k));
      const bwVal = Math.pow(10, 6 - k);

      let gainText = gainVal >= 1000 ? gainVal.toLocaleString() : gainVal;
      let bwText = bwVal >= 1000000 ? '1 MHz' : bwVal >= 1000 ? (bwVal / 1000) + ' kHz' : Math.round(bwVal) + ' Hz';

      gainIndicator.textContent = `Gain = ${gainText}`;
      gainIndicator.setAttribute('y', Math.max(gainY - 8, 20));

      bwIndicator.textContent = bwText;
      bwIndicator.setAttribute('x', Math.min(cornerX + 6, 430));

      gainDisplay.textContent = gainText;
      bwDisplay.textContent = bwText;
    }

    function updateFromSlider(val) {
      const k = parseFloat(val);
      render(k);
      buttons.forEach(b => b.classList.remove('active'));
    }

    function setPreset(k) {
      slider.value = k;
      render(k);
      buttons.forEach((b, idx) => {
        const targetK = [5, 3, 2, 1, 0][idx];
        if (targetK === k) b.classList.add('active');
        else b.classList.remove('active');
      });
    }

    // Initial state: Gain = 1,000 (k = 3)
    render(3);
  </script>
</body>
</html>

Let's see how this trade-off works in a standard design problem.

## Worked Example: Calculating Bandwidth Expansion

An op-amp has an open-loop gain $A_{OL} = 200{,}000$ and an open-loop cutoff frequency $f_{OL} = 5\text{ Hz}$.

First, find the constant Gain-Bandwidth Product: $$\text{GBWP} = A_{OL} \times f_{OL} = 200{,}000 \times 5\text{ Hz} = 1{,}000{,}000\text{ Hz} = 1\text{ MHz}$$

Next, if negative feedback is added to configure a closed-loop gain of $A_{CL} = 50$, calculate the new bandwidth ($f_{CL}$): $$f_{CL} = \frac{\text{GBWP}}{A_{CL}} = \frac{1{,}000{,}000\text{ Hz}}{50} = 20{,}000\text{ Hz} = 20\text{ kHz}$$

By sacrificing gain from $200{,}000$ down to $50$, the amplifier's usable range broadened from an unusable $5\text{ Hz}$ to cover the entire human audio spectrum ($20\text{ kHz}$).

A common mistake is assuming that configuring an amplifier for higher closed-loop gain makes it universally better.

In reality, increasing your closed-loop gain directly shrinks your circuit's bandwidth and reduces the feedback's ability to suppress distortion at higher frequencies.

Now that you understand the trade-offs of negative feedback, how do we choose specific resistor values to set $\beta$? That leads us straight into designing inverting amplifiers.

electronics

Principles of Negative Feedback in Operational Amplifiers. Explain how negative feedback trades open-loop voltage gain for bandwidth, stability, and predictable closed-loop behaviour.

Feedback and Operational Amplifiers

Operational Amplifier Basics and Frequency Response

Op Amp Bandwidth, Feedback, and Closed-Loop Stability

Teaching Strategy: Introduce the op-amp first as an unconstrained 'monster' with unmanageable open-loop gain ($10^5$) and minimal native bandwidth ($10\text{ Hz}$) that immediately saturates to rails upon applying microvolt signals. Introduce negative feedback as the 'taming mechanism'. Derive $A_{CL} = \frac{A_{OL}}{1 + \beta A_{OL}}$ to show that when $A_{OL}$ is large, internal gain variations drop out of the transfer function, yielding predictable performance dictated exclusively by passive external resistors. Reinforce the concept of Gain-Bandwidth Product conservation using visual log-log Bode plots before moving to inverting amplifier circuit analysis.

Principles of Negative Feedback in Operational Amplifiers

The Power of Negative Feedback

Deriving Closed-Loop Predictability