williammathieu.github.io/CoupledResonators.html at main · WilliamMathieu/williammathieu.github.io · GitHub

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  <title>Coupled LC Resonators Calculator — Bandwidth &amp; Mode Splitting | RF Toolbox</title>
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<h3>Coupled LC Resonators</h3>
<div class="tool-concept">
  <p>Two identical LC tanks coupled by mutual inductance M split into two resonant modes: an even mode (lower frequency f₋) and an odd mode (upper frequency f₊). The coupling coefficient k = M/L determines the mode splitting and passband bandwidth. Critical coupling (k = 1/Q) maximises the transfer response with a flat passband.</p>
  <details style="margin-top:8px;">
  <summary style="cursor:pointer;font-size:11px;color:#AA77FF;font-weight:bold;letter-spacing:0.05em;text-transform:uppercase;user-select:none;">Equations &amp; Parameters ▸</summary>
  <div style="background:#f0eeff;border-left:3px solid #AA77FF;padding:8px 12px 6px;border-radius:0 4px 4px 0;margin:6px 0 10px;font-size:12.5px;line-height:2.2;overflow-x:auto;">
    \(f_{\pm} = \dfrac{f_0}{\sqrt{1\mp k}} \quad \text{BW} \approx k\cdot f_0 \quad k_{\rm crit} = \dfrac{1}{Q}\)
  </div>
  <table style="font-size:12px;border-collapse:collapse;margin-top:6px;">
    <tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;">k</td><td>Coupling coefficient (dimensionless, 0 &lt; k &lt; 1). Enter directly or derive from M.</td></tr>
    <tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;">M</td><td>Mutual inductance between the two coils. k = M/L for identical resonators.</td></tr>
    <tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;">Q</td><td>Unloaded quality factor of each resonator; set R = 0 for lossless (Q → ∞).</td></tr>
    <tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;">k<sub>crit</sub></td><td>Critical coupling: flat-top passband at k = k_crit = 1/Q.</td></tr>
  </table>
  <div style="margin-top:8px;padding-top:6px;border-top:1px solid #e0d8ff;font-size:11.5px;color:#666;"><span style="font-weight:bold;color:#AA77FF;">Reference: </span>G. Matthaei et al., <em>Microwave Filters, Impedance-Matching Networks, and Coupling Structures</em> (Artech House, 1980), §8</div>
</details>
</div>

<div class="sec-lbl">Resonator Parameters</div>
<div class="inp-grid">
  <div class="inp-row">
    <label>Inductance, <span class="var">L</span></label>
    <div class="inp-inline">
      <input type="text" id="cr-L" placeholder="value" data-tip="Inductance of each resonator (identical tanks assumed).">
      <select id="cr-L-unit">
        <option value="1e-9">nH</option>
        <option value="1e-6" selected>µH</option>
        <option value="1e-3">mH</option>
      </select>
    </div>
  </div>
  <div class="inp-row">
    <label>Capacitance, <span class="var">C</span></label>
    <div class="inp-inline">
      <input type="text" id="cr-C" placeholder="value" data-tip="Capacitance of each resonator.">
      <select id="cr-C-unit">
        <option value="1e-12" selected>pF</option>
        <option value="1e-9">nF</option>
        <option value="1e-6">µF</option>
      </select>
    </div>
  </div>
  <div class="inp-row">
    <label>Series resistance, <span class="var">R</span> <span style="color:#888;font-size:11px;">(loss, optional)</span></label>
    <div class="inp-inline">
      <input type="text" id="cr-R" placeholder="0 = lossless" data-tip="Series resistance of each inductor coil. Set to 0 or leave blank for a lossless resonator.">
      <span style="padding:0 8px;line-height:32px;color:#888;font-size:13px;">Ω</span>
    </div>
  </div>
</div>

<div class="sec-lbl">Coupling (enter k <em>or</em> M)</div>
<div class="inp-grid">
  <div class="inp-row">
    <label>Coupling coefficient, <span class="var">k</span></label>
    <div class="inp-inline">
      <input type="text" id="cr-k" placeholder="e.g. 0.05" data-tip="Dimensionless coupling coefficient (0 < k < 1). Typical bandpass filter: 0.01–0.1.">
    </div>
  </div>
  <div class="inp-row">
    <label>Mutual inductance, <span class="var">M</span> <span style="color:#888;font-size:11px;">(alternative)</span></label>
    <div class="inp-inline">
      <input type="text" id="cr-M" placeholder="or enter M" data-tip="Mutual inductance between the two coils. Used to compute k = M/L if k is not given.">
      <select id="cr-M-unit">
        <option value="1e-9">nH</option>
        <option value="1e-6" selected>µH</option>
        <option value="1e-3">mH</option>
      </select>
    </div>
  </div>
</div>

<div style="display:flex;gap:8px;align-items:center;flex-wrap:wrap;margin:12px 0 6px;">
  <button class="calc-btn" id="cr-btn">Calculate</button>
  <button class="example-btn" onclick="loadExample({'cr-L':'10','cr-L-unit':'1e-6','cr-C':'25','cr-C-unit':'1e-12','cr-R':'5','cr-k':'0.05'})" title="10 µH, 25 pF, R=5 Ω, k=0.05 — slightly over-coupled">IF transformer (10.7 MHz)</button>
  <button class="example-btn" onclick="loadExample({'cr-L':'1','cr-L-unit':'1e-6','cr-C':'1','cr-C-unit':'1e-12','cr-R':'2','cr-k':'0.02'})" title="1 µH, 1 pF, R=2 Ω, k=0.02">UHF resonator pair</button>
  <button class="example-btn" onclick="loadExample({'cr-L':'100','cr-L-unit':'1e-6','cr-C':'100','cr-C-unit':'1e-12','cr-R':'0','cr-k':'0.1'})" title="Lossless, k=0.1">Lossless k=0.1</button>
</div>

<div id="cr-error" class="err-msg" style="color:#c0392b;font-size:12px;margin-bottom:6px;min-height:16px;"></div>
<div class="sec-lbl">Results</div>
<div id="cr-results" class="res-grid"></div>

<div class="sec-lbl">Diagram</div>
<div class="schematic-box"><div id="tool-diagram"></div></div>
<script>
function drawDiagram(r) {
  var W = 480, H = 160;
  var s = '<svg viewBox="0 0 ' + W + ' ' + H + '" width="' + W + '" height="' + H + '" xmlns="http://www.w3.org/2000/svg" style="font-family:monospace;max-width:100%;">';
  s += '<rect width="' + W + '" height="' + H + '" fill="#faf9ff" rx="4"/>';

  // Draw two tank circuits separated by coupling symbol
  // Left tank: L1 + C1 parallel
  var lx1 = 90, lx2 = 370, cy = 70;

  function drawTank(cx) {
    // Inductor coil (left side)
    var lx = cx - 22, ltop = cy - 30, lbot = cy + 30;
    var arcR = 7;
    var arcPath = 'M' + lx + ',' + lbot;
    for (var i = 0; i < 3; i++) {
      var ay = lbot - i * 2*arcR - arcR;
      arcPath += ' A' + arcR + ',' + arcR + ' 0 0,1 ' + lx + ',' + (ay - arcR);
    }
    arcPath += ' L' + lx + ',' + ltop;
    s += '<path d="' + arcPath + '" fill="none" stroke="#5533aa" stroke-width="2" stroke-linecap="round"/>';
    // Capacitor (right side)
    var rx = cx + 22;
    s += '<line x1="' + rx + '" y1="' + ltop + '" x2="' + rx + '" y2="' + (cy-8) + '" stroke="#5533aa" stroke-width="2"/>';
    s += '<line x1="' + (rx-12) + '" y1="' + (cy-8) + '" x2="' + (rx+12) + '" y2="' + (cy-8) + '" stroke="#5533aa" stroke-width="2.5"/>';
    s += '<line x1="' + (rx-12) + '" y1="' + (cy+8) + '" x2="' + (rx+12) + '" y2="' + (cy+8) + '" stroke="#5533aa" stroke-width="2.5"/>';
    s += '<line x1="' + rx + '" y1="' + (cy+8) + '" x2="' + rx + '" y2="' + lbot + '" stroke="#5533aa" stroke-width="2"/>';
    // Top/bottom wires connecting L and C
    s += '<line x1="' + lx + '" y1="' + ltop + '" x2="' + rx + '" y2="' + ltop + '" stroke="#5533aa" stroke-width="2"/>';
    s += '<line x1="' + lx + '" y1="' + lbot + '" x2="' + rx + '" y2="' + lbot + '" stroke="#5533aa" stroke-width="2"/>';
  }

  drawTank(lx1);
  drawTank(lx2);

  // Coupling M arrow between the two tanks
  var mx = (lx1 + lx2) / 2;
  s += '<text x="' + mx + '" y="' + (cy + 5) + '" text-anchor="middle" font-size="13" fill="#AA77FF" font-weight="bold">M</text>';
  s += '<line x1="' + (mx-18) + '" y1="' + cy + '" x2="' + (lx1+24) + '" y2="' + cy + '" stroke="#AA77FF" stroke-width="1.5" stroke-dasharray="4,3"/>';
  s += '<line x1="' + (mx+18) + '" y1="' + cy + '" x2="' + (lx2-24) + '" y2="' + cy + '" stroke="#AA77FF" stroke-width="1.5" stroke-dasharray="4,3"/>';

  // Frequency annotations
  s += '<text x="' + lx1 + '" y="140" text-anchor="middle" font-size="10" fill="#5533aa">f₀ = ' + engFmt(r.f0,'Hz') + '</text>';
  s += '<text x="' + mx + '" y="' + (H-10) + '" text-anchor="middle" font-size="10" fill="#AA77FF">f₋ = ' + engFmt(r.f_minus,'Hz') + '   f₊ = ' + engFmt(r.f_plus,'Hz') + '</text>';
  s += '<text x="' + lx2 + '" y="140" text-anchor="middle" font-size="10" fill="#5533aa">k = ' + r.k.toFixed(4) + '</text>';

  s += '</svg>';
  document.getElementById('tool-diagram').innerHTML = s;
}
</script>
<script src="CoupledResonators.js"></script>
</article>
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