Advanced

Tutor model

i62 💰:9.4

i52 💰:7.9

🏆 i67 💰:62.5

i70 💰:125.0

i68 💰:125.0

🏆 i66 💰:75.0

i65 💰:125.0

i63 💰:62.5

i61 💰:75.0

i60 💰:9.4

i60 💰:37.5

i58 💰:62.5

i57 💰:22.5

i57 💰:37.5

i56 💰:45.0

⚡ i55 💰:11.25

i55 💰:37.5

i54 💰:62.5

i52 💰:35.0

i51 💰:37.5

i49 💰:15.0

i49 💰:62.5

i48 💰:11.3

i47 💰:7.5

i44 💰:25.0

i43 💰:37.5

i42 💰:187.5

i40 💰:5.0

i39 💰:100.0

i38 💰:37.5

i37 💰:11.0

i36 💰:187.5

i34 💰:12.5

i33 💰:37.5

i30 💰:1.0

i29 💰:1.5

i28 💰:150.0

i23 💰:20.0

i20 💰:4.0

i18 💰:1.0

i16 💰:25.0

i13 💰:1.5

Keep session private

What would you like to learn?

Override for the tutor's lesson content generation calls. Auto uses the system default (Claude Sonnet 5 for lesson text, Opus 5 for diagrams).

Lesson generation model

System default

i58 💰:18.8

i31 💰:11.0

i25 💰:31.3

Tutoring abilities

If you build an op-amp circuit with a gain of 100 to amplify high-frequency signals, you might find that your output voltage unexpectedly collapses to almost zero.

Op-amps cannot deliver maximum amplification across all frequencies because internal compensation limits their speed to ensure circuit stability.

In the 1930s, Hendrik Bode at Bell Labs studied feedback amplifiers and demonstrated that adding a controlled high-frequency roll-off prevents circuits from oscillating out of control.

An op-amp's **open-loop gain** ($A_{OL}$) is its raw amplification without feedback, while the **gain-bandwidth product (GBP)** is the constant value representing the product of gain and frequency along the standard $-20\text{ dB/decade}$ roll-off slope.

<!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;
      -webkit-font-smoothing: antialiased;
      user-select: none;
    }

    body {
      width: 100%;
      min-height: 100%;
      background: #ffffff;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      padding: 12px 16px;
    }

    .controls {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 6px;
      margin-bottom: 10px;
      width: 100%;
      max-width: 580px;
    }

    .btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      padding: 6px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #1e2945;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .btn:hover {
      background: #e2e8f0;
    }

    .btn.active {
      background: #22b7ff;
      border-color: #009be5;
      color: #ffffff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }

    .graph-wrapper {
      width: 100%;
      max-width: 580px;
      position: relative;
    }

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

    .grid-line {
      stroke: #e2e8f0;
      stroke-width: 1;
    }

    .axis-line {
      stroke: #64748b;
      stroke-width: 1.5;
    }

    .tick-label {
      font-size: 11px;
      fill: #64748b;
      font-weight: 500;
    }

    .axis-title {
      font-size: 12px;
      fill: #1e2945;
      font-weight: 700;
    }

    .ol-curve {
      fill: none;
      stroke: #22b7ff;
      stroke-width: 3.5;
      stroke-linecap: round;
      stroke-linejoin: round;
    }

    .slope-label {
      font-size: 11px;
      fill: #0284c7;
      font-weight: 700;
      letter-spacing: 0.5px;
    }

    .cl-band {
      fill: rgba(34, 183, 255, 0.12);
      transition: all 0.3s ease;
    }

    .cl-line {
      stroke: #f59e0b;
      stroke-width: 2.5;
      stroke-dasharray: 4 3;
      transition: all 0.3s ease;
    }

    .info-bar {
      margin-top: 8px;
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      max-width: 540px;
      padding: 8px 14px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
    }

    .info-stat {
      display: flex;
      gap: 6px;
    }

    .info-label {
      color: #64748b;
      font-weight: 500;
    }

    .info-val {
      color: #1e2945;
      font-weight: 700;
    }
  </style>
</head>
<body>

  <div class="controls">
    <button class="btn active" id="btn-ol" onclick="setMode('open')">Open-Loop</button>
    <button class="btn" id="btn-g100" onclick="setMode('g100')">Gain = 100 (40 dB)</button>
    <button class="btn" id="btn-g10" onclick="setMode('g10')">Gain = 10 (20 dB)</button>
    <button class="btn" id="btn-g1" onclick="setMode('g1')">Unity Gain = 1 (0 dB)</button>
  </div>

  <div class="graph-wrapper">
    <svg viewBox="0 0 540 330">
      <defs>
        <marker id="arrow" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
          <path d="M 0 1.5 L 8 5 L 0 8.5 z" fill="#64748b"/>
        </marker>
      </defs>

      
      
      <line class="grid-line" x1="65" y1="35" x2="505" y2="35"/>
      <line class="grid-line" x1="65" y1="83" x2="505" y2="83"/>
      <line class="grid-line" x1="65" y1="131" x2="505" y2="131"/>
      <line class="grid-line" x1="65" y1="179" x2="505" y2="179"/>
      <line class="grid-line" x1="65" y1="227" x2="505" y2="227"/>
      <line class="grid-line" x1="65" y1="275" x2="505" y2="275"/>

      
      
      <line class="grid-line" x1="65" y1="35" x2="65" y2="275"/>
      <line class="grid-line" x1="138" y1="35" x2="138" y2="275"/>
      <line class="grid-line" x1="212" y1="35" x2="212" y2="275"/>
      <line class="grid-line" x1="285" y1="35" x2="285" y2="275"/>
      <line class="grid-line" x1="358" y1="35" x2="358" y2="275"/>
      <line class="grid-line" x1="432" y1="35" x2="432" y2="275"/>
      <line class="grid-line" x1="505" y1="35" x2="505" y2="275"/>

      
      <rect id="cl-rect" class="cl-band" x="65" y="35" width="0" height="240" opacity="0"/>

      
      <line id="cl-gain-line" class="cl-line" x1="65" y1="35" x2="65" y2="35" opacity="0"/>
      <line id="cl-cutoff-line" class="cl-line" x1="65" y1="35" x2="65" y2="275" opacity="0"/>

      
      <line class="axis-line" x1="65" y1="275" x2="515" y2="275" marker-end="url(#arrow)"/>
      <line class="axis-line" x1="65" y1="275" x2="65" y2="20" marker-end="url(#arrow)"/>

      
      <text class="tick-label" x="55" y="39" text-anchor="end">100</text>
      <text class="tick-label" x="55" y="87" text-anchor="end">80</text>
      <text class="tick-label" x="55" y="135" text-anchor="end">60</text>
      <text class="tick-label" x="55" y="183" text-anchor="end">40</text>
      <text class="tick-label" x="55" y="231" text-anchor="end">20</text>
      <text class="tick-label" x="55" y="279" text-anchor="end">0 dB</text>
      <text class="axis-title" x="20" y="15" transform="rotate(-90 20,15)" text-anchor="end">Gain (dB)</text>

      
      <text class="tick-label" x="65" y="293" text-anchor="middle">10 Hz</text>
      <text class="tick-label" x="138" y="293" text-anchor="middle">100 Hz</text>
      <text class="tick-label" x="212" y="293" text-anchor="middle">1 kHz</text>
      <text class="tick-label" x="285" y="293" text-anchor="middle">10 kHz</text>
      <text class="tick-label" x="358" y="293" text-anchor="middle">100 kHz</text>
      <text class="tick-label" x="432" y="293" text-anchor="middle">1 MHz</text>
      <text class="tick-label" x="505" y="293" text-anchor="middle">10 MHz</text>
      <text class="axis-title" x="505" y="315" text-anchor="end">Frequency (log scale)</text>

      
      
      <path class="ol-curve" d="M 50 35 L 65 35 L 432 275"/>

      
      <text class="slope-label" x="260" y="140" transform="rotate(33 260,140)">-20 dB / decade</text>

      
      
      <circle cx="65" cy="35" r="4.5" fill="#1e2945"/>
      <text x="75" y="28" font-size="12" font-weight="700" fill="#1e2945">DC Open-Loop Gain A₀ (100 dB)</text>

      
      <line x1="65" y1="35" x2="65" y2="48" stroke="#1e2945" stroke-width="1.5"/>
      <text x="75" y="52" font-size="11" font-weight="600" fill="#64748b">Break f_b (10 Hz)</text>

      
      <circle cx="432" cy="275" r="4.5" fill="#22b7ff"/>
      <text x="432" y="258" font-size="12" font-weight="700" fill="#0284c7" text-anchor="middle">f_T = GBP (1 MHz)</text>
      <text x="432" y="270" font-size="10" font-weight="600" fill="#64748b" text-anchor="middle">Unity Gain (0 dB)</text>

      
      <circle id="probe-dot" cx="65" cy="35" r="6" fill="#f59e0b" stroke="#ffffff" stroke-width="2" opacity="0"/>
    </svg>
  </div>

  <div class="info-bar">
    <div class="info-stat">
      <span class="info-label">Gain × Bandwidth:</span>
      <span class="info-val" id="gbp-val">100,000 × 10 Hz = 1 MHz</span>
    </div>
    <div class="info-stat">
      <span class="info-label">Limit:</span>
      <span class="info-val" id="limit-val">f_max = 10 Hz</span>
    </div>
  </div>

  <script>
    const modes = {
      open: {
        btn: 'btn-ol',
        gainDb: 100,
        gainLinear: '100,000',
        bw: '10 Hz',
        x: 65,
        y: 35,
        gbp: '100,000 × 10 Hz = 1 MHz',
        limit: 'f_max = 10 Hz',
        showClosed: false
      },
      g100: {
        btn: 'btn-g100',
        gainDb: 40,
        gainLinear: '100',
        bw: '10 kHz',
        x: 285,
        y: 179,
        gbp: '100 × 10 kHz = 1 MHz',
        limit: 'f_max = 10 kHz',
        showClosed: true
      },
      g10: {
        btn: 'btn-g10',
        gainDb: 20,
        gainLinear: '10',
        bw: '100 kHz',
        x: 358,
        y: 227,
        gbp: '10 × 100 kHz = 1 MHz',
        limit: 'f_max = 100 kHz',
        showClosed: true
      },
      g1: {
        btn: 'btn-g1',
        gainDb: 0,
        gainLinear: '1',
        bw: '1 MHz',
        x: 432,
        y: 275,
        gbp: '1 × 1 MHz = 1 MHz',
        limit: 'f_max = 1 MHz',
        showClosed: true
      }
    };

    function setMode(key) {
      const m = modes[key];
      if (!m) return;

      // Update button styles
      document.querySelectorAll('.btn').forEach(b => b.classList.remove('active'));
      document.getElementById(m.btn).classList.add('active');

      // Update Info
      document.getElementById('gbp-val').textContent = m.gbp;
      document.getElementById('limit-val').textContent = m.limit;

      // SVG Elements
      const rect = document.getElementById('cl-rect');
      const gLine = document.getElementById('cl-gain-line');
      const cLine = document.getElementById('cl-cutoff-line');
      const dot = document.getElementById('probe-dot');

      if (m.showClosed) {
        rect.setAttribute('x', '65');
        rect.setAttribute('y', m.y);
        rect.setAttribute('width', m.x - 65);
        rect.setAttribute('height', 275 - m.y);
        rect.style.opacity = '1';

        gLine.setAttribute('x1', '65');
        gLine.setAttribute('y1', m.y);
        gLine.setAttribute('x2', m.x);
        gLine.setAttribute('y2', m.y);
        gLine.style.opacity = '1';

        cLine.setAttribute('x1', m.x);
        cLine.setAttribute('y1', m.y);
        cLine.setAttribute('x2', m.x);
        cLine.setAttribute('y2', 275);
        cLine.style.opacity = '1';

        dot.setAttribute('cx', m.x);
        dot.setAttribute('cy', m.y);
        dot.style.opacity = '1';
      } else {
        rect.style.opacity = '0';
        gLine.style.opacity = '0';
        cLine.style.opacity = '0';
        dot.style.opacity = '0';
      }
    }
  </script>
</body>
</html>

Because the slope falls at a constant rate, any decrease in closed-loop gain produces an equal proportional increase in available bandwidth: $$\text{GBP} = A_{CL} \times f_{-3\text{dB}}$$

The **$-3\text{ dB}$ cut-off frequency** ($f_{-3\text{dB}}$) is the upper frequency limit where the closed-loop voltage gain drops to $1/\sqrt{2} \approx 0.707$ (or $-3\text{ dB}$) of its mid-band value.

How do we apply this relationship to determine the operational limit of a practical amplifier?

For a non-inverting op-amp configuration, the closed-loop voltage gain is $A_{CL} = 1 + \frac{R_f}{R_1}$, where $R_f$ is the feedback resistor and $R_1$ connects the inverting terminal to ground.

Suppose you design an amplifier using an op-amp with a rated $\text{GBP} = 4\text{ MHz}$, using resistors $R_f = 39\text{ k}\Omega$ and $R_1 = 1\text{ k}\Omega$.

First, calculate the closed-loop gain: $$A_{CL} = 1 + \frac{39\text{ k}\Omega}{1\text{ k}\Omega} = 40$$

Next, divide the gain-bandwidth product by this gain to find the cut-off frequency: $$f_{-3\text{dB}} = \frac{\text{GBP}}{A_{CL}} = \frac{4\text{ MHz}}{40} = 100\text{ kHz}$$

<!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: #ffffff;
      color: #1e2945;
      padding: 14px;
      display: flex;
      flex-direction: column;
      gap: 12px;
      user-select: none;
    }
    
    .controls {
      display: flex;
      flex-direction: column;
      gap: 6px;
    }
    
    .slider-track-wrap {
      position: relative;
      padding: 0 8px;
    }
    
    input[type="range"] {
      width: 100%;
      accent-color: #22b7ff;
      cursor: pointer;
      height: 6px;
    }
    
    .slider-ticks {
      display: flex;
      justify-content: space-between;
      padding: 0 4px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
    }
    
    .slider-ticks span {
      cursor: pointer;
    }
    .slider-ticks span.active {
      color: #0284c7;
      font-weight: 700;
    }

    .meters {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
    }
    
    .meter-card {
      background: #f1f5f9;
      border-radius: 8px;
      padding: 10px 12px;
      display: flex;
      flex-direction: column;
      gap: 2px;
      border: 1px solid #e2e8f0;
    }
    
    .meter-label {
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }
    
    .meter-value {
      font-size: 22px;
      font-weight: 700;
      color: #1e2945;
      display: flex;
      align-items: baseline;
      gap: 6px;
    }
    
    .meter-sub {
      font-size: 14px;
      font-weight: 500;
      color: #0284c7;
    }

    .graph-wrap {
      width: 100%;
      background: #f8fafc;
      border-radius: 8px;
      border: 1px solid #e2e8f0;
      padding: 6px 8px 4px 4px;
    }
    
    svg {
      width: 100%;
      height: auto;
      display: block;
    }
    
    .axis-line { stroke: #cbd5e1; stroke-width: 1.5; }
    .grid-line { stroke: #e2e8f0; stroke-width: 1; stroke-dasharray: 3 3; }
    .gbp-line { stroke: #94a3b8; stroke-width: 1.5; stroke-dasharray: 4 4; }
    .curve-fill { fill: rgba(34, 183, 255, 0.12); }
    .curve-stroke { fill: none; stroke: #22b7ff; stroke-width: 2.5; stroke-linecap: round; }
    .cutoff-line { stroke: #0284c7; stroke-width: 1.5; stroke-dasharray: 3 3; }
    .cutoff-dot { fill: #0284c7; stroke: #ffffff; stroke-width: 2; }
    .axis-text { font-size: 11px; fill: #64748b; font-family: system-ui, sans-serif; }
    .tag-text { font-size: 11px; fill: #0284c7; font-weight: 600; font-family: system-ui, sans-serif; }
    .limit-text { font-size: 10px; fill: #94a3b8; font-weight: 600; font-family: system-ui, sans-serif; }
  </style>
</head>
<body>

  <div class="controls">
    <div class="slider-track-wrap">
      <input type="range" id="gainSlider" min="0" max="4" step="1" value="3">
    </div>
    <div class="slider-ticks" id="ticks">
      <span data-idx="0">1×</span>
      <span data-idx="1">4×</span>
      <span data-idx="2">10×</span>
      <span data-idx="3" class="active">40×</span>
      <span data-idx="4">100×</span>
    </div>
  </div>

  <div class="meters">
    <div class="meter-card">
      <div class="meter-label">Gain (A<sub>CL</sub>)</div>
      <div class="meter-value">
        <span id="gainVal">40×</span>
        <span class="meter-sub" id="gainDb">(32 dB)</span>
      </div>
    </div>
    <div class="meter-card">
      <div class="meter-label">Bandwidth (-3 dB)</div>
      <div class="meter-value">
        <span id="bwVal">100 kHz</span>
      </div>
    </div>
  </div>

  <div class="graph-wrap">
    <svg viewBox="0 0 500 210">
      
      <line x1="55" y1="25" x2="475" y2="25" class="grid-line" />
      <line x1="55" y1="62" x2="475" y2="62" class="grid-line" />
      <line x1="55" y1="100" x2="475" y2="100" class="grid-line" />
      <line x1="55" y1="137" x2="475" y2="137" class="grid-line" />
      
      <line x1="195" y1="15" x2="195" y2="175" class="grid-line" />
      <line x1="335" y1="15" x2="335" y2="175" class="grid-line" />
      <line x1="475" y1="15" x2="475" y2="175" class="grid-line" />

      
      <line x1="55" y1="175" x2="475" y2="175" class="axis-line" />
      <line x1="55" y1="15" x2="55" y2="175" class="axis-line" />

      
      <text x="48" y="178" text-anchor="end" class="axis-text">0 dB</text>
      <text x="48" y="103" text-anchor="end" class="axis-text">20 dB</text>
      <text x="48" y="29" text-anchor="end" class="axis-text">40 dB</text>

      
      <text x="55" y="193" text-anchor="middle" class="axis-text">10 kHz</text>
      <text x="195" y="193" text-anchor="middle" class="axis-text">100 kHz</text>
      <text x="335" y="193" text-anchor="middle" class="axis-text">1 MHz</text>
      <text x="475" y="193" text-anchor="middle" class="axis-text">10 MHz</text>

      
      <path d="M 55,25 L 420,175" class="gbp-line" />
      <text x="290" y="70" class="limit-text" transform="rotate(-23 290 70)">GBP = 4 MHz</text>

      
      <path id="curveArea" class="curve-fill" d="" />
      <path id="curvePath" class="curve-stroke" d="" />

      
      <line id="cutoffLineY" x1="0" y1="0" x2="0" y2="175" class="cutoff-line" />
      <circle id="cutoffDot" cx="0" cy="0" r="4.5" class="cutoff-dot" />
      <text id="cutoffTag" x="0" y="0" text-anchor="middle" class="tag-text">fc</text>
    </svg>
  </div>

  <script>
    const data = [
      { gain: 1, db: 0, bwHz: 4000000, bwStr: "4.0 MHz", gStr: "1×", dbStr: "(0 dB)" },
      { gain: 4, db: 12, bwHz: 1000000, bwStr: "1.0 MHz", gStr: "4×", dbStr: "(12 dB)" },
      { gain: 10, db: 20, bwHz: 400000, bwStr: "400 kHz", gStr: "10×", dbStr: "(20 dB)" },
      { gain: 40, db: 32, bwHz: 100000, bwStr: "100 kHz", gStr: "40×", dbStr: "(32 dB)" },
      { gain: 100, db: 40, bwHz: 40000, bwStr: "40 kHz", gStr: "100×", dbStr: "(40 dB)" }
    ];

    const slider = document.getElementById('gainSlider');
    const ticks = document.querySelectorAll('#ticks span');
    const gainVal = document.getElementById('gainVal');
    const gainDb = document.getElementById('gainDb');
    const bwVal = document.getElementById('bwVal');
    
    const curvePath = document.getElementById('curvePath');
    const curveArea = document.getElementById('curveArea');
    const cutoffDot = document.getElementById('cutoffDot');
    const cutoffLineY = document.getElementById('cutoffLineY');
    const cutoffTag = document.getElementById('cutoffTag');

    const xMinLog = 4; // 10^4 = 10 kHz
    const xMaxLog = 7; // 10^7 = 10 MHz
    const plotLeft = 55;
    const plotRight = 475;
    const plotWidth = plotRight - plotLeft;
    const plotBottom = 175;
    const plotHeight = 150; // 0 to 40 dB

    function freqToX(f) {
      const logF = Math.log10(f);
      return plotLeft + ((logF - xMinLog) / (xMaxLog - xMinLog)) * plotWidth;
    }

    function dbToY(db) {
      return plotBottom - (db / 40) * plotHeight;
    }

    function update(idx) {
      const d = data[idx];
      
      // Update Texts
      gainVal.textContent = d.gStr;
      gainDb.textContent = d.dbStr;
      bwVal.textContent = d.bwStr;

      // Update Ticks
      ticks.forEach((t, i) => {
        t.classList.toggle('active', i === idx);
      });

      // Calculate Bode plot curve: G(f) = A_CL / sqrt(1 + (f/fc)^2)
      const points = [];
      const steps = 60;
      for (let i = 0; i <= steps; i++) {
        const logF = xMinLog + (i / steps) * (xMaxLog - xMinLog);
        const f = Math.pow(10, logF);
        const g = d.gain / Math.sqrt(1 + Math.pow(f / d.bwHz, 2));
        const db = Math.max(0, 20 * Math.log10(g));
        const x = freqToX(f);
        const y = dbToY(db);
        points.push({ x, y });
      }

      const pathStr = points.map((p, i) => (i === 0 ? `M ${p.x.toFixed(1)},${p.y.toFixed(1)}` : `L ${p.x.toFixed(1)},${p.y.toFixed(1)}`)).join(' ');
      curvePath.setAttribute('d', pathStr);

      const areaStr = `${pathStr} L ${plotRight},${plotBottom} L ${plotLeft},${plotBottom} Z`;
      curveArea.setAttribute('d', areaStr);

      // Cutoff point (-3 dB point)
      const fcX = freqToX(d.bwHz);
      const fcY = dbToY(d.db - 3);

      cutoffDot.setAttribute('cx', fcX.toFixed(1));
      cutoffDot.setAttribute('cy', fcY.toFixed(1));

      cutoffLineY.setAttribute('x1', fcX.toFixed(1));
      cutoffLineY.setAttribute('y1', fcY.toFixed(1));
      cutoffLineY.setAttribute('x2', fcX.toFixed(1));
      cutoffLineY.setAttribute('y2', plotBottom);

      cutoffTag.setAttribute('x', fcX.toFixed(1));
      cutoffTag.setAttribute('y', (fcY - 8).toFixed(1));
      cutoffTag.textContent = d.bwStr;
    }

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

    ticks.forEach(t => {
      t.addEventListener('click', () => {
        const idx = parseInt(t.dataset.idx, 10);
        slider.value = idx;
        update(idx);
      });
    });

    update(3);
  </script>
</body>
</html>

Does an inverting amplifier provide the exact same frequency response for a given gain magnitude?

An inverting amplifier has a signal gain of magnitude $|A_v| = \frac{R_f}{R_1}$, which leads many designers to mistakenly calculate $f_{-3\text{dB}} = \frac{\text{GBP}}{|A_v|}$.

However, the frequency roll-off depends on the loop feedback factor $\beta = \frac{R_1}{R_1 + R_f}$, meaning bandwidth is governed by the **noise gain** ($A_N = 1/\beta = 1 + \frac{R_f}{R_1}$), not the signal gain.

For an inverting amplifier with $R_f = 10\text{ k}\Omega$ and $R_1 = 10\text{ k}\Omega$ powered by a $1\text{ MHz}$ GBP op-amp, the signal gain is $|A_v| = 1$, but the noise gain is $A_N = 1 + \frac{10}{10} = 2$.

Its actual bandwidth is cut in half: $$f_{-3\text{dB}} = \frac{\text{GBP}}{A_N} = \frac{1\text{ MHz}}{2} = 500\text{ kHz}$$

<!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: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #ffffff;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 12px;
      width: 100%;
    }

    .top-bar {
      display: flex;
      justify-content: space-between;
      align-items: center;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 12px;
      font-size: 13px;
      font-weight: 600;
    }
    .gbp-badge {
      background: #e0f2fe;
      color: #0369a1;
      padding: 3px 8px;
      border-radius: 6px;
      font-size: 12px;
    }

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

    .circuit-card {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 10px;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
    }

    .circuit-card.non-inv {
      border-color: #3b82f6;
    }
    .circuit-card.inv {
      border-color: #f59e0b;
    }

    .card-title {
      font-size: 14px;
      font-weight: 700;
      display: flex;
      align-items: center;
      justify-content: space-between;
    }
    .badge {
      font-size: 11px;
      font-weight: 700;
      padding: 2px 6px;
      border-radius: 4px;
    }
    .badge-blue { background: #dbeafe; color: #1d4ed8; }
    .badge-amber { background: #fef3c7; color: #b45309; }

    .metrics-table {
      display: flex;
      flex-direction: column;
      gap: 6px;
      font-size: 12px;
    }
    .metric-row {
      display: flex;
      justify-content: space-between;
      align-items: center;
      padding-bottom: 4px;
      border-bottom: 1px dashed #f1f5f9;
    }
    .metric-label {
      color: #64748b;
    }
    .metric-val {
      font-weight: 600;
      font-family: ui-monospace, monospace;
    }

    .bw-visual {
      display: flex;
      flex-direction: column;
      gap: 4px;
      margin-top: 4px;
    }
    .bw-header {
      display: flex;
      justify-content: space-between;
      font-size: 11px;
      font-weight: 600;
      color: #475569;
    }
    .bar-track {
      width: 100%;
      height: 18px;
      background: #f1f5f9;
      border-radius: 4px;
      overflow: hidden;
      position: relative;
    }
    .bar-fill {
      height: 100%;
      display: flex;
      align-items: center;
      padding-left: 6px;
      font-size: 11px;
      font-weight: 700;
      color: #ffffff;
      transition: width 0.4s ease;
    }
    .fill-full {
      width: 100%;
      background: #3b82f6;
    }
    .fill-half {
      width: 50%;
      background: #f59e0b;
    }

    .trap-box {
      background: #fffbeb;
      border: 1px solid #fef3c7;
      border-left: 4px solid #f59e0b;
      border-radius: 6px;
      padding: 8px 12px;
      font-size: 12px;
      line-height: 1.4;
      color: #92400e;
    }
    .trap-title {
      font-weight: 700;
      display: flex;
      align-items: center;
      gap: 4px;
      margin-bottom: 2px;
    }

    @media (max-width: 440px) {
      .comparison-grid {
        grid-template-columns: 1fr;
      }
    }
  </style>
</head>
<body>

  <div class="top-bar">
    <span>Bandwidth = GBP ÷ Noise Gain</span>
    <span class="gbp-badge">GBP = 1.0 MHz</span>
  </div>

  <div class="comparison-grid">
    
    <div class="circuit-card non-inv">
      <div class="card-title">
        <span>Non-Inverting</span>
        <span class="badge badge-blue">|A<sub>v</sub>| = 1</span>
      </div>

      <div class="metrics-table">
        <div class="metric-row">
          <span class="metric-label">Signal Gain (A<sub>v</sub>)</span>
          <span class="metric-val">+1</span>
        </div>
        <div class="metric-row">
          <span class="metric-label">Noise Gain (G<sub>N</sub>)</span>
          <span class="metric-val">1</span>
        </div>
        <div class="metric-row">
          <span class="metric-label">Cutoff (f<sub>-3dB</sub>)</span>
          <span class="metric-val" style="color:#2563eb;">1.0 MHz</span>
        </div>
      </div>

      <div class="bw-visual">
        <div class="bw-header">
          <span>Bandwidth</span>
          <span>1000 kHz</span>
        </div>
        <div class="bar-track">
          <div class="bar-fill fill-full">100%</div>
        </div>
      </div>
    </div>

    
    <div class="circuit-card inv">
      <div class="card-title">
        <span>Inverting</span>
        <span class="badge badge-amber">|A<sub>v</sub>| = 1</span>
      </div>

      <div class="metrics-table">
        <div class="metric-row">
          <span class="metric-label">Signal Gain (A<sub>v</sub>)</span>
          <span class="metric-val">-1</span>
        </div>
        <div class="metric-row">
          <span class="metric-label">Noise Gain (G<sub>N</sub>)</span>
          <span class="metric-val">1 + 1 = 2</span>
        </div>
        <div class="metric-row">
          <span class="metric-label">Cutoff (f<sub>-3dB</sub>)</span>
          <span class="metric-val" style="color:#d97706;">500 kHz</span>
        </div>
      </div>

      <div class="bw-visual">
        <div class="bw-header">
          <span>Bandwidth</span>
          <span>500 kHz</span>
        </div>
        <div class="bar-track">
          <div class="bar-fill fill-half">50%</div>
        </div>
      </div>
    </div>
  </div>

  <div class="trap-box">
    <div class="trap-title">⚠️ Common Trap</div>
    Inverting amplifiers have <strong>less bandwidth</strong> than non-inverting amplifiers at the same gain magnitude!
  </div>

</body>
</html>

Why does negative feedback create this mathematically exact trade-off between gain and bandwidth?

## Why It Works: The Closed-Loop Derivation

An internally compensated op-amp has an open-loop response modeled by a single dominant pole: $$A(s) = \frac{A_0}{1 + s/\omega_b}$$ where $A_0$ is the DC gain and $\omega_b$ is the open-loop cut-off frequency in radians per second.

Substituting $A(s)$ into the standard closed-loop feedback formula $A_{CL}(s) = \frac{A(s)}{1 + \beta A(s)}$ gives: $$A_{CL}(s) = \frac{\frac{A_0}{1 + s/\omega_b}}{1 + \beta \frac{A_0}{1 + s/\omega_b}} = \frac{A_0}{(1 + \beta A_0) + \frac{s}{\omega_b}}$$

Dividing the numerator and denominator by $(1 + \beta A_0)$ reveals the closed-loop corner frequency: $$A_{CL}(s) = \frac{\frac{A_0}{1 + \beta A_0}}{1 + \frac{s}{\omega_b(1 + \beta A_0)}} \implies \omega_{-3\text{dB}} = \omega_b(1 + \beta A_0)$$.

Since $A_0 \beta \gg 1$ and unity-gain bandwidth is $\omega_T = A_0 \omega_b$, the new bandwidth is $\omega_{-3\text{dB}} \approx \beta \omega_T = \frac{\omega_T}{A_N}$, proving that feedback trades gain for bandwidth in direct proportion.

electronics

Gain-Bandwidth Product and Closed-Loop Frequency Limits. Calculate the -3 dB upper cut-off frequency of closed-loop amplifiers using the op-amp gain-bandwidth product (GBP).

TI Precision Labs – Op Amps: Bandwidth and Gain-Bandwidth Product

Practical Considerations of Operational Amplifiers: Gain-Bandwidth Product

Operational Amplifier Basics and Frequency Response

To develop strong mastery among Year 13 / A2 students: (1) Anchor the lesson by contrasting ideal op-amp parameters ($A_{OL} = \infty$, $BW = \infty$) with real op-amp specifications ($\mu\text{A741}$, TL071). (2) Teach Bode plots as log-log graphs, proving mathematically that a constant slope of $-1$ on a $\log(A)$ vs $\log(f)$ graph corresponds to $A \times f = \text{constant}$. (3) Emphasize linear vs decibel conversions early, ensuring students never calculate $\text{GBP}$ using gain in decibels directly. (4) Use live laboratory oscilloscopes or SPICE simulations (e.g., TINA-TI or CircuitLab) to sweep signal frequencies across the calculated $f_{-3\text{dB}}$ and show the exact $0.707\,V_{peak}$ attenuation and $45^\circ$ phase lag.

Gain-Bandwidth Product and Closed-Loop Frequency Limits

The Gain-Bandwidth Trade-Off

Calculating Non-Inverting Bandwidth