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If you need an amplifier to boost an audio signal by exactly negative ten, you might think you can just hook up any two resistors that give a 10-to-1 ratio. But if you choose $10\,\Omega$ and $100\,\Omega$, your signal source will crash, and if you choose $10\,\text{M}\Omega$ and $100\,\text{M}\Omega$, your circuit will drown in background noise.

Designing an inverting amplifier requires balancing **voltage gain**, **input impedance**, and the practical reality of standard resistor series.

How do these two resistors actually set the gain and input resistance? Let's trace the signal through the circuit.

In 1941, Bell Labs engineer Karl Swartzel designed the first operational amplifier summing circuit to perform real-time math for anti-aircraft gun directors during World War II.

An **inverting amplifier** connects the input signal through an input resistor $R_{in}$ to the inverting input ($-$), while a feedback resistor $R_f$ connects that same node to the output.

<!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: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
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    .schematic-box {
      width: 100%;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 4px;
      display: flex;
      justify-content: center;
      align-items: center;
    }

    svg {
      width: 100%;
      max-height: 190px;
      overflow: visible;
    }

    .math-panel {
      display: flex;
      flex-wrap: wrap;
      justify-content: space-around;
      align-items: center;
      gap: 8px;
      background: #f0f9ff;
      border: 1px solid #bae6fd;
      border-radius: 10px;
      padding: 10px 12px;
    }

    .calc-block {
      display: flex;
      align-items: center;
      gap: 6px;
      font-size: 15px;
      font-weight: 600;
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    .fraction {
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      line-height: 1.1;
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    .fraction .num {
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      padding-bottom: 1px;
    }

    .fraction .den {
      padding-top: 1px;
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    .val-rf { color: #0284c7; }
    .val-rin { color: #d97706; }
    .val-gain { color: #dc2626; font-size: 17px; }
    .val-out { color: #7c3aed; font-size: 17px; }

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    .control-card {
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      padding: 8px 10px;
      display: flex;
      flex-direction: column;
      gap: 4px;
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    .control-head {
      display: flex;
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      font-size: 13px;
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<body>

  
  <div class="schematic-box">
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      <text x="217" y="93" font-size="16" font-weight="700" fill="#dc2626">−</text>
      <text x="217" y="137" font-size="16" font-weight="700" fill="#64748b">+</text>

      
      <line x1="210" y1="132" x2="175" y2="132" stroke="#1e2945" stroke-width="2" />
      <use href="#gnd" x="175" y="132" />
      <text x="145" y="152" font-size="11" font-weight="600" fill="#64748b">0 V</text>

      
      
      <circle cx="35" cy="88" r="4" fill="#ffffff" stroke="#1e2945" stroke-width="2" />
      <text x="12" y="92" font-size="12" font-weight="700" fill="#1e2945">V<tspan font-size="9" dy="2">in</tspan></text>
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      <rect x="65" y="78" width="55" height="20" fill="#fef3c7" stroke="#d97706" stroke-width="2" rx="3" />
      <text id="svgRin" x="92" y="92" font-size="11" font-weight="700" fill="#b45309" text-anchor="middle">10kΩ</text>
      <text x="92" y="72" font-size="11" font-weight="700" fill="#d97706" text-anchor="middle">R<tspan font-size="9" dy="2">in</tspan></text>

      
      <line x1="120" y1="88" x2="210" y2="88" stroke="#1e2945" stroke-width="2" />
      <circle cx="155" cy="88" r="3.5" fill="#0284c7" />
      <text x="155" y="105" font-size="10" font-weight="600" fill="#0284c7" text-anchor="middle">Virtual 0V</text>

      
      <line x1="155" y1="88" x2="155" y2="35" stroke="#1e2945" stroke-width="2" />
      <line x1="155" y1="35" x2="185" y2="35" stroke="#1e2945" stroke-width="2" />
      
      
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      <text id="svgRf" x="215" y="39" font-size="11" font-weight="700" fill="#0369a1" text-anchor="middle">50kΩ</text>
      <text x="215" y="19" font-size="11" font-weight="700" fill="#0284c7" text-anchor="middle">R<tspan font-size="9" dy="2">f</tspan></text>

      <line x1="245" y1="35" x2="315" y2="35" stroke="#1e2945" stroke-width="2" />
      <line x1="315" y1="35" x2="315" y2="110" stroke="#1e2945" stroke-width="2" />

      
      <line x1="280" y1="110" x2="355" y2="110" stroke="#1e2945" stroke-width="2" />
      <circle cx="315" cy="110" r="3.5" fill="#1e2945" />
      <circle cx="355" cy="110" r="4" fill="#ffffff" stroke="#7c3aed" stroke-width="2" />
      <text x="365" y="114" font-size="12" font-weight="700" fill="#7c3aed">V<tspan font-size="9" dy="2">out</tspan></text>

      
      <g transform="translate(25, 120)">
        <rect width="60" height="34" rx="5" fill="#ffffff" stroke="#cbd5e1" />
        <path d="M 5,22 Q 17,8 30,22 T 55,22" fill="none" stroke="#2563eb" stroke-width="1.8" />
        <text x="30" y="31" font-size="9" font-weight="700" fill="#2563eb" text-anchor="middle">+1.0 V</text>
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      <g transform="translate(325, 125)">
        <rect width="65" height="34" rx="5" fill="#ffffff" stroke="#cbd5e1" />
        <path id="svgOutWave" d="M 5,12 Q 17,26 32,12 T 60,12" fill="none" stroke="#7c3aed" stroke-width="2" />
        <text id="svgOutVal" x="32" y="31" font-size="9" font-weight="700" fill="#7c3aed" text-anchor="middle">-5.0 V</text>
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    </svg>
  </div>

  
  <div class="math-panel">
    <div class="calc-block">
      <span>Gain (G) = −</span>
      <div class="fraction">
        <span class="num val-rf" id="mathRf">50 kΩ</span>
        <span class="den val-rin" id="mathRin">10 kΩ</span>
      </div>
      <span>= <span class="val-gain" id="mathGain">−5.0×</span></span>
    </div>

    <div class="calc-block">
      <span>V<sub style="font-size:10px">out</sub> = G × (+1V) =</span>
      <span class="val-out" id="mathVout">−5.0 V</span>
    </div>
  </div>

  
  <div class="controls-grid">
    <div class="control-card">
      <div class="control-head">
        <span style="color:#d97706;">R<sub>in</sub> (Input)</span>
        <span id="txtRin" style="color:#b45309;">10 kΩ</span>
      </div>
      <input type="range" id="sliderRin" min="5" max="50" step="5" value="10">
    </div>

    <div class="control-card">
      <div class="control-head">
        <span style="color:#0284c7;">R<sub>f</sub> (Feedback)</span>
        <span id="txtRf" style="color:#0369a1;">50 kΩ</span>
      </div>
      <input type="range" id="sliderRf" min="10" max="100" step="5" value="50">
    </div>
  </div>

  
  <div class="presets-row">
    <button class="preset-btn" data-gain="2">Target Gain −2×</button>
    <button class="preset-btn active" data-gain="5">Target Gain −5×</button>
    <button class="preset-btn" data-gain="10">Target Gain −10×</button>
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  <script>
    const sliderRin = document.getElementById('sliderRin');
    const sliderRf = document.getElementById('sliderRf');
    const txtRin = document.getElementById('txtRin');
    const txtRf = document.getElementById('txtRf');
    const svgRin = document.getElementById('svgRin');
    const svgRf = document.getElementById('svgRf');
    const mathRin = document.getElementById('mathRin');
    const mathRf = document.getElementById('mathRf');
    const mathGain = document.getElementById('mathGain');
    const mathVout = document.getElementById('mathVout');
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    const presetBtns = document.querySelectorAll('.preset-btn');

    function update() {
      const rin = parseFloat(sliderRin.value);
      const rf = parseFloat(sliderRf.value);
      const gain = -(rf / rin);
      const vout = gain * 1.0;

      txtRin.textContent = rin + ' kΩ';
      txtRf.textContent = rf + ' kΩ';
      svgRin.textContent = rin + 'kΩ';
      svgRf.textContent = rf + 'kΩ';

      mathRin.textContent = rin + ' kΩ';
      mathRf.textContent = rf + ' kΩ';
      
      const gainStr = Number.isInteger(gain) ? gain.toString() : gain.toFixed(1);
      const voutStr = Number.isInteger(vout) ? vout.toString() : vout.toFixed(1);

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      mathVout.textContent = voutStr + ' V';
      svgOutVal.textContent = voutStr + ' V';

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          btn.classList.remove('active');
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    }

    sliderRin.addEventListener('input', update);
    sliderRf.addEventListener('input', update);

    presetBtns.forEach(btn => {
      btn.addEventListener('click', () => {
        const g = parseFloat(btn.dataset.gain);
        sliderRin.value = 10;
        sliderRf.value = Math.min(100, 10 * g);
        update();
      });
    });

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

Because negative feedback drives the difference between inputs to zero and the non-inverting input is grounded, the inverting terminal is held at a **virtual ground** ($0\,\text{V}$).

Since an ideal op-amp draws zero input current ($I_- = 0$), all current through $R_{in}$ must flow entirely through $R_f$:

$$\frac{V_{in} - 0}{R_{in}} = \frac{0 - V_{out}}{R_f} \implies A_v = \frac{V_{out}}{V_{in}} = -\frac{R_f}{R_{in}}$$

The **input impedance** ($Z_{in}$)—the effective resistance seen by the signal source—is simply equal to $R_{in}$, because the other side of $R_{in}$ sits at $0\,\text{V}$.

Now that we have the gain equation $A_v = -R_f / R_{in}$, what boundaries decide whether a resistor value is safe or dangerous to use?

## Design Constraints for Resistor Selection

If $R_{in}$ is too small (e.g., under $1\,\text{k}\Omega$), it creates **source loading** by drawing excessive current from the prior stage, causing its voltage to drop.

Conversely, if $R_f$ is too large (e.g., over $1\,\text{M}\Omega$), thermal noise increases significantly and stray circuit capacitance causes unwanted phase lag and instability.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      min-height: 100%;
    }

    .main-grid {
      display: grid;
      grid-template-columns: 1fr;
      gap: 10px;
      width: 100%;
    }

    .circuit-panel {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 12px;
      position: relative;
    }

    .svg-container {
      width: 100%;
      height: 150px;
    }

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

    .controls-panel {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
    }

    .slider-row {
      display: grid;
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      align-items: center;
      gap: 10px;
      font-size: 13px;
      font-weight: 600;
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    .val-badge {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      padding: 2px 6px;
      border-radius: 6px;
      font-family: ui-monospace, monospace;
      text-align: right;
      font-size: 12px;
      color: #0f172a;
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    input[type="range"] {
      width: 100%;
      accent-color: #0284c7;
      cursor: pointer;
      height: 6px;
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    .calc-banner {
      display: flex;
      justify-content: space-between;
      align-items: center;
      padding: 8px 12px;
      background: #f0f9ff;
      border: 1px solid #bae6fd;
      border-radius: 8px;
      font-size: 14px;
      font-weight: 600;
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      gap: 4px;
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    .status-ok { background: #dcfce7; color: #166534; border: 1px solid #bbf7d0; }
    .status-warn { background: #fef3c7; color: #92400e; border: 1px solid #fde68a; }

    .waves-panel {
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      border-radius: 12px;
      padding: 8px 12px;
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    .wave-legend {
      position: absolute;
      top: 6px;
      right: 10px;
      display: flex;
      gap: 12px;
      font-size: 11px;
      font-weight: 600;
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    .leg-in { color: #0284c7; }
    .leg-out { color: #8b5cf6; }

    .wave-svg {
      width: 100%;
      height: 100%;
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  </style>
</head>
<body>

  <div class="circuit-panel">
    <div class="svg-container">
      <svg viewBox="0 0 400 150">
        
        <text x="15" y="68" font-size="12" font-weight="bold" fill="#0284c7">Vin</text>
        <circle cx="45" cy="65" r="3.5" fill="#0284c7" />
        <line x1="45" y1="65" x2="80" y2="65" stroke="#64748b" stroke-width="2" />
        
        
        <rect x="80" y="55" width="45" height="20" fill="#ffffff" stroke="#0284c7" stroke-width="2" rx="3" />
        <text id="rinTextSvg" x="102" y="69" font-size="10" font-weight="bold" fill="#0284c7" text-anchor="middle">Rin</text>
        
        
        <line x1="125" y1="65" x2="185" y2="65" stroke="#64748b" stroke-width="2" />
        <circle cx="160" cy="65" r="3" fill="#64748b" />
        
        
        <polygon points="185,45 185,115 255,80" fill="#ffffff" stroke="#1e2945" stroke-width="2" stroke-linejoin="round" />
        
        
        <text x="190" y="69" font-size="13" font-weight="bold" fill="#1e2945">−</text>
        <text x="190" y="104" font-size="12" font-weight="bold" fill="#1e2945">+</text>
        
        
        <line x1="170" y1="100" x2="185" y2="100" stroke="#64748b" stroke-width="2" />
        <line x1="170" y1="100" x2="170" y2="115" stroke="#64748b" stroke-width="2" />
        
        <line x1="160" y1="115" x2="180" y2="115" stroke="#64748b" stroke-width="2" />
        <line x1="164" y1="119" x2="176" y2="119" stroke="#64748b" stroke-width="1.5" />
        <line x1="167" y1="123" x2="173" y2="123" stroke="#64748b" stroke-width="1.5" />

        
        <line x1="160" y1="65" x2="160" y2="25" stroke="#64748b" stroke-width="2" />
        <line x1="160" y1="25" x2="200" y2="25" stroke="#64748b" stroke-width="2" />
        
        
        <rect x="200" y="15" width="50" height="20" fill="#ffffff" stroke="#8b5cf6" stroke-width="2" rx="3" />
        <text id="rfTextSvg" x="225" y="29" font-size="10" font-weight="bold" fill="#8b5cf6" text-anchor="middle">Rf</text>
        
        <line x1="250" y1="25" x2="290" y2="25" stroke="#64748b" stroke-width="2" />
        <line x1="290" y1="25" x2="290" y2="80" stroke="#64748b" stroke-width="2" />
        
        
        <line x1="255" y1="80" x2="350" y2="80" stroke="#64748b" stroke-width="2" />
        <circle cx="290" cy="80" r="3" fill="#64748b" />
        <circle cx="350" cy="80" r="3.5" fill="#8b5cf6" />
        <text x="358" y="84" font-size="12" font-weight="bold" fill="#8b5cf6">Vout</text>
      </svg>
    </div>
  </div>

  <div class="controls-panel">
    <div class="slider-row">
      <span>R<sub>in</sub></span>
      <input type="range" id="rinSlider" min="10" max="100" step="5" value="20">
      <span class="val-badge" id="rinVal">20 kΩ</span>
    </div>

    <div class="slider-row">
      <span>R<sub>f</sub></span>
      <input type="range" id="rfSlider" min="20" max="1400" step="20" value="100">
      <span class="val-badge" id="rfVal">100 kΩ</span>
    </div>

    <div class="calc-banner">
      <span>A<sub>v</sub> = −(R<sub>f</sub> / R<sub>in</sub>)</span>
      <span class="gain-val" id="gainVal">−5.0×</span>
    </div>
    
    <div style="display: flex; justify-content: flex-end;">
      <div id="statusBadge" class="status-badge status-ok">✓ Standard Range</div>
    </div>
  </div>

  <div class="waves-panel">
    <div class="wave-legend">
      <span class="leg-in">● V<sub>in</sub> (1V)</span>
      <span class="leg-out" id="voutLeg">● V<sub>out</sub> (−5V)</span>
    </div>
    <svg class="wave-svg" viewBox="0 0 360 80">
      <line x1="10" y1="40" x2="350" y2="40" stroke="#e2e8f0" stroke-width="1.5" stroke-dasharray="3,3" />
      <path id="vinWave" fill="none" stroke="#0284c7" stroke-width="2" />
      <path id="voutWave" fill="none" stroke="#8b5cf6" stroke-width="2.5" />
    </svg>
  </div>

  <script>
    const rinSlider = document.getElementById('rinSlider');
    const rfSlider = document.getElementById('rfSlider');
    const rinVal = document.getElementById('rinVal');
    const rfVal = document.getElementById('rfVal');
    const rinTextSvg = document.getElementById('rinTextSvg');
    const rfTextSvg = document.getElementById('rfTextSvg');
    const gainVal = document.getElementById('gainVal');
    const statusBadge = document.getElementById('statusBadge');
    const voutLeg = document.getElementById('voutLeg');
    const vinWave = document.getElementById('vinWave');
    const voutWave = document.getElementById('voutWave');

    function formatR(kOhms) {
      if (kOhms >= 1000) {
        return (kOhms / 1000).toFixed(1) + ' MΩ';
      }
      return kOhms + ' kΩ';
    }

    function renderWaves(gain) {
      const width = 340;
      const startX = 10;
      const centerY = 40;
      const cycles = 2;
      const vInAmp = 12; // visual px for 1V
      
      // Visual scale for Vout (clipped visually if too large)
      const maxVisualAmp = 34;
      const vOutAmp = Math.min(Math.abs(gain) * (vInAmp / 3), maxVisualAmp);

      let dIn = `M ${startX} ${centerY}`;
      let dOut = `M ${startX} ${centerY}`;

      for (let x = 0; x <= width; x += 2) {
        const rad = (x / width) * cycles * 2 * Math.PI;
        const yIn = centerY - Math.sin(rad) * vInAmp;
        // Inverted polarity: sin is negated
        const yOut = centerY + Math.sin(rad) * vOutAmp;

        dIn += ` L ${startX + x} ${yIn.toFixed(2)}`;
        dOut += ` L ${startX + x} ${yOut.toFixed(2)}`;
      }

      vinWave.setAttribute('d', dIn);
      voutWave.setAttribute('d', dOut);
    }

    function update() {
      const rin = parseFloat(rinSlider.value);
      const rf = parseFloat(rfSlider.value);

      const rinStr = formatR(rin);
      const rfStr = formatR(rf);

      rinVal.textContent = rinStr;
      rfVal.textContent = rfStr;
      rinTextSvg.textContent = rinStr;
      rfTextSvg.textContent = rfStr;

      const gain = -(rf / rin);
      gainVal.textContent = gain.toFixed(1) + '×';
      voutLeg.textContent = `● Vout (${(gain).toFixed(1)}V)`;

      if (rf > 1000) {
        statusBadge.className = 'status-badge status-warn';
        statusBadge.innerHTML = '⚠️ Rf > 1 MΩ (Thermal Noise / Stray C)';
      } else {
        statusBadge.className = 'status-badge status-ok';
        statusBadge.innerHTML = '✓ Safe Standard Range';
      }

      renderWaves(gain);
    }

    rinSlider.addEventListener('input', update);
    rfSlider.addEventListener('input', update);

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

As a standard rule of thumb for general-purpose circuits, keep $R_{in}$ between $10\,\text{k}\Omega$ and $100\,\text{k}\Omega$, and keep $R_f$ below $1\,\text{M}\Omega$.

You can calculate any theoretical resistance you want on paper, but can you actually buy those exact numbers from a supplier?

Resistors are manufactured in logarithmic standard series; the **E24 series** provides 24 standard values per decade with a $5\%$ tolerance.

Each value is spaced by a factor of roughly $\sqrt[24]{10} \approx 1.10$, ensuring overlapping tolerance bands without wasteful duplicate coverage.

<!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">
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      height: 190px;
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      bottom: 4px;
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      transform: translateX(-50%);
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      position: absolute;
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      height: 110px;
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      transition: background-color 0.2s ease, opacity 0.2s ease;
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      position: absolute;
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      width: 100%;
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    }

    .readout-title {
      font-size: 11px;
      font-weight: 600;
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      letter-spacing: 0.5px;
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      font-size: 16px;
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      color: #0f172a;
    }

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      color: #0284c7;
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    .mult-btn.selected {
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      margin-top: 4px;
    }

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      width: 28px;
      height: 28px;
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      border: 1px solid #cbd5e1;
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      font-weight: bold;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
    }
  </style>
</head>
<body>

  <div class="header-row">
    <span>E24 Decade Band (±5% Overlap)</span>
    <span class="badge" id="factorBadge">10<sup>1/24</sup> ≈ 1.10</span>
  </div>

  <div class="chart-container" id="chart">
    <div class="log-axis"></div>
    <span class="axis-label" style="left: 20px;">1.0</span>
    <span class="axis-label" style="left: 36%;">2.2</span>
    <span class="axis-label" style="left: 67%;">4.7</span>
    <span class="axis-label" style="left: calc(100% - 20px);">10.0</span>
    <div id="bands"></div>
  </div>

  <div class="readout-card">
    <div class="readout-col">
      <span class="readout-title">Min (-5%)</span>
      <span class="readout-val" id="minVal">950 Ω</span>
    </div>
    <div class="readout-col">
      <span class="readout-title">Nominal</span>
      <span class="readout-val active-val" id="nomVal">1.0 kΩ</span>
    </div>
    <div class="readout-col">
      <span class="readout-title">Max (+5%)</span>
      <span class="readout-val" id="maxVal">1.05 kΩ</span>
    </div>
  </div>

  <div class="multiplier-row">
    <button class="mult-btn" onclick="setMult(1, 'Ω')">×1</button>
    <button class="mult-btn" onclick="setMult(10, 'Ω')">×10</button>
    <button class="mult-btn" onclick="setMult(100, 'Ω')">×100</button>
    <button class="mult-btn selected" onclick="setMult(1000, 'kΩ')">×1k</button>
    <button class="mult-btn" onclick="setMult(10000, 'kΩ')">×10k</button>
    <button class="mult-btn" onclick="setMult(100000, 'kΩ')">×100k</button>
  </div>

  <div class="stepper-row">
    <button class="step-btn" onclick="stepVal(-1)">‹</button>
    <span style="font-size: 12px; color: #64748b; font-weight: 500;">Tap/Drag band or step</span>
    <button class="step-btn" onclick="stepVal(1)">›</button>
  </div>

  <script>
    const E24 = [1.0, 1.1, 1.2, 1.3, 1.5, 1.6, 1.8, 2.0, 2.2, 2.4, 2.7, 3.0, 3.3, 3.6, 3.9, 4.3, 4.7, 5.1, 5.6, 6.2, 6.8, 7.5, 8.2, 9.1, 10.0];
    let selectedIdx = 0;
    let multiplier = 1000;
    let unitPrefix = 'kΩ';

    const bandsEl = document.getElementById('bands');
    const minValEl = document.getElementById('minVal');
    const nomValEl = document.getElementById('nomVal');
    const maxValEl = document.getElementById('maxVal');
    const chartEl = document.getElementById('chart');

    function logPos(val) {
      // maps 1.0 -> 0%, 10.0 -> 100%
      return (Math.log10(val) / Math.log10(10)) * 100;
    }

    function renderBands() {
      bandsEl.innerHTML = '';
      E24.forEach((val, idx) => {
        const minVal = val * 0.95;
        const maxVal = val * 1.05;
        const leftPct = logPos(minVal);
        const rightPct = logPos(maxVal);
        const centerPct = logPos(val);

        const band = document.createElement('div');
        band.className = 'band-item';
        
        // Alternate heights for staggered visual separation
        const h = 40 + (idx % 3) * 26;
        band.style.height = `${h}px`;
        band.style.left = `calc(20px + (100% - 40px) * ${leftPct / 100})`;
        band.style.width = `calc((100% - 40px) * ${(rightPct - leftPct) / 100})`;
        
        const isSel = idx === selectedIdx;
        band.style.backgroundColor = isSel ? 'rgba(2, 132, 199, 0.45)' : ((idx % 2 === 0) ? 'rgba(148, 163, 184, 0.25)' : 'rgba(203, 213, 225, 0.35)');
        band.style.border = isSel ? '1.5px solid #0284c7' : '1px solid #cbd5e1';

        const line = document.createElement('div');
        line.className = 'band-center-line';
        line.style.left = `calc(20px + (100% - 40px) * ${centerPct / 100})`;
        line.style.height = `${h + 8}px`;
        line.style.background = isSel ? '#0284c7' : '#94a3b8';

        bandsEl.appendChild(band);
        bandsEl.appendChild(line);
      });
      updateReadout();
    }

    function formatR(ohms) {
      if (ohms >= 1e6) return (ohms / 1e6).toFixed(2).replace(/\.00$/, '') + ' MΩ';
      if (ohms >= 1e3) return (ohms / 1e3).toFixed(2).replace(/\.00$/, '') + ' kΩ';
      return ohms.toFixed(1).replace(/\.0$/, '') + ' Ω';
    }

    function updateReadout() {
      const val = E24[selectedIdx];
      const actualNom = val * multiplier;
      const actualMin = actualNom * 0.95;
      const actualMax = actualNom * 1.05;

      nomValEl.textContent = formatR(actualNom);
      minValEl.textContent = formatR(actualMin);
      maxValEl.textContent = formatR(actualMax);
    }

    function setMult(m, prefix) {
      multiplier = m;
      unitPrefix = prefix;
      document.querySelectorAll('.mult-btn').forEach(btn => {
        btn.classList.toggle('selected', btn.textContent === (m >= 1000 ? `×${m/1000}k` : `×${m}`));
      });
      updateReadout();
    }

    function stepVal(dir) {
      selectedIdx = Math.max(0, Math.min(E24.length - 1, selectedIdx + dir));
      renderBands();
    }

    function handlePointer(e) {
      const rect = chartEl.getBoundingClientRect();
      const x = (e.clientX || (e.touches && e.touches[0].clientX)) - rect.left - 20;
      const w = rect.width - 40;
      const ratio = Math.max(0, Math.min(1, x / w));
      const logVal = Math.pow(10, ratio);

      let closestIdx = 0;
      let minDiff = Infinity;
      E24.forEach((v, i) => {
        const diff = Math.abs(v - logVal);
        if (diff < minDiff) {
          minDiff = diff;
          closestIdx = i;
        }
      });

      if (closestIdx !== selectedIdx) {
        selectedIdx = closestIdx;
        renderBands();
      }
    }

    let isDown = false;
    chartEl.addEventListener('pointerdown', (e) => { isDown = true; handlePointer(e); });
    window.addEventListener('pointermove', (e) => { if (isDown) handlePointer(e); });
    window.addEventListener('pointerup', () => { isDown = false; });

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

Any standard resistor is one of these 24 base numbers multiplied by a power of ten, such as $4.7\,\text{k}\Omega$, $47\,\text{k}\Omega$, or $470\,\text{k}\Omega$.

How do we put these constraints together to design a circuit for a specific target gain?

## Worked Example: Designing for Target Gain

Suppose you need an inverting amplifier with a voltage gain of $A_v = -7.5$ and an input impedance of at least $Z_{in} \ge 10\,\text{k}\Omega$.

First, satisfy the input impedance specification by choosing the smallest allowable standard value for $R_{in}$:

Next, calculate the required feedback resistance using $|A_v| = R_f / R_{in}$:

$$R_f = |A_v| \times R_{in} = 7.5 \times 10\,\text{k}\Omega = 75\,\text{k}\Omega$$

Because $7.5$ is an exact base value in the E24 series, $R_f = 75\,\text{k}\Omega$ is an available standard resistor, delivering the exact target gain of $-7.5$.

What happens when your required gain produces a value that is not available in the E24 series?

Suppose your target gain is $A_v = -5.3$ with $Z_{in} \ge 10\,\text{k}\Omega$. Setting $R_{in} = 10\,\text{k}\Omega$ gives an ideal $R_f = 53\,\text{k}\Omega$, which is not in the E24 series.

The nearest standard E24 values are $51\,\text{k}\Omega$ and $56\,\text{k}\Omega$. We can check the actual gain and percentage error for each option.

<!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">
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      background: #ffffff;
      border-radius: 8px;
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      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 2px;
      transition: all 0.2s ease;
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      border-color: #0284c7;
      background: #f0f9ff;
      color: #0284c7;
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      background: #e2e8f0;
      border-radius: 6px;
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      top: -6px;
      bottom: -6px;
      width: 2px;
      background: #0f172a;
      z-index: 2;
    }
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      font-weight: 700;
      color: #0f172a;
    }
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      height: 100%;
      border-radius: 6px;
      transition: width 0.3s ease, background-color 0.3s ease;
    }
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</head>
<body>

  <div class="controls">
    <button class="btn-opt active" id="btn51" onclick="selectResistor(51)">
      <span>R<sub>f</sub> = 51 kΩ</span>
      <span class="tag tag-good">3.8% Error ✓</span>
    </button>
    <button class="btn-opt" id="btn56" onclick="selectResistor(56)">
      <span>R<sub>f</sub> = 56 kΩ</span>
      <span class="tag tag-warn">5.7% Error</span>
    </button>
  </div>

  <div class="circuit-box">
    <svg viewBox="0 0 360 170">
      
      <line x1="20" y1="90" x2="60" y2="90" stroke="#64748b" stroke-width="2" />
      <circle cx="20" cy="90" r="3" fill="#64748b" />
      <text x="18" y="80" font-size="11" font-weight="600" fill="#64748b">V_in</text>

      
      <rect x="60" y="80" width="40" height="20" fill="#f1f5f9" stroke="#0284c7" stroke-width="2" rx="3" />
      <text x="80" y="94" font-size="10" font-weight="700" fill="#0284c7" text-anchor="middle">10 kΩ</text>

      
      <line x1="100" y1="90" x2="160" y2="90" stroke="#64748b" stroke-width="2" />
      <circle cx="130" cy="90" r="3" fill="#64748b" />

      
      <line x1="130" y1="90" x2="130" y2="30" stroke="#64748b" stroke-width="2" />
      <line x1="130" y1="30" x2="180" y2="30" stroke="#64748b" stroke-width="2" />
      
      
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      <text id="rfLabel" x="205" y="34" font-size="11" font-weight="700" fill="#0284c7" text-anchor="middle">51 kΩ</text>

      <line x1="230" y1="30" x2="290" y2="30" stroke="#64748b" stroke-width="2" />
      <line x1="290" y1="30" x2="290" y2="105" stroke="#64748b" stroke-width="2" />

      
      <line x1="130" y1="120" x2="160" y2="120" stroke="#64748b" stroke-width="2" />
      <line x1="130" y1="120" x2="130" y2="140" stroke="#64748b" stroke-width="2" />
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      <text x="168" y="94" font-size="14" font-weight="700" fill="#64748b">−</text>
      <text x="168" y="124" font-size="14" font-weight="700" fill="#64748b">+</text>

      
      <line x1="230" y1="105" x2="330" y2="105" stroke="#64748b" stroke-width="2" />
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    </svg>
  </div>

  <div class="metrics-grid">
    <div class="metric">
      <div class="metric-val" id="gainVal">−5.1</div>
      <div class="metric-lbl">Actual Gain</div>
    </div>
    <div class="metric">
      <div class="metric-val" id="errorVal" style="color: #15803d;">3.8%</div>
      <div class="metric-lbl">TOLERANCE ERROR</div>
    </div>
  </div>

  <div class="bar-container">
    <div style="display: flex; justify-content: space-between; font-size: 11px; font-weight: 600; color: #64748b;">
      <span>Gain Comparison</span>
      <span>Target: −5.3</span>
    </div>
    <div class="bar-track">
      <div class="target-line" style="left: 70%;">
        <span class="target-label">−5.3</span>
      </div>
      <div class="bar-fill" id="barFill" style="width: 65%; background-color: #22c55e;"></div>
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Using $R_f = 51\,\text{k}\Omega$ with $R_{in} = 10\,\text{k}\Omega$ yields an actual gain of $-5.1$ ($3.8\%$ error), which is well within standard $5\%$ component tolerances.

A common mistake is assuming that an inverting op-amp circuit inherits the op-amp's internal infinite input impedance; remember that the source directly sees $R_{in}$, never infinity.

When choosing pairs, testing different base values for $R_{in}$ like $12\,\text{k}\Omega$, $15\,\text{k}\Omega$, or $22\,\text{k}\Omega$ often uncovers an E24 combination with even lower gain error.

electronics

Calculating Resistor Values for Specific Inverting Voltage Gains. Select and calculate appropriate standard resistor values from the E24 series to achieve a target inverting voltage gain while maintaining acceptable input impedance.

Operational Amplifiers: Inverting and Non-Inverting Configurations

The Inverting Operational Amplifier Circuit and Gain Equations

Operational Amplifiers: Inverting Amplifier Analysis and Resistor Sizing

To achieve mastery in Year 13 A-Level Electronics (e.g., Eduqas/WJEC ELEC-8.6), instruction should follow a two-tier pedagogical sequence: 1) Circuit Analysis Foundations (deriving the virtual earth concept from Kirchhoff's Current Law and operational amplifier boundary conditions to establish $Z_{in} = R_{in}$ and $G = -R_f/R_{in}$), followed by 2) Practical Engineering Synthesis (applying real-world constraints including discrete E24 preferred value tables, source loading limits, and upper resistor thresholds to minimize noise). Emphasize having students compute gain error tolerances when standard resistor ratios do not perfectly match theoretical continuous calculations.

Calculating Resistor Values for Specific Inverting Voltage Gains

The Inverting Topology

Design Constraints for Resistor Selection