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  <title>ABCD Matrix Cascade — Two-Port Network Multiplier | RF Toolbox</title>
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<h3>ABCD Matrix Cascade</h3>
<div class="tool-concept">
  <p>The ABCD (chain) matrix makes cascaded two-ports trivial: the matrix of a chain is simply the product of the individual matrices, in order. This tool builds the ABCD matrix for each element you list — a series impedance, a shunt admittance, an ideal transmission line, or an ideal transformer — multiplies them, and converts the result to S-parameters at your reference impedance. It reports the reciprocity check (AD − BC = 1) as an internal consistency test.</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; Element syntax ▸</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:2.0;overflow-x:auto;">series Z: \(\begin{bmatrix}1&amp;Z\\0&amp;1\end{bmatrix}\) &nbsp; shunt Y: \(\begin{bmatrix}1&amp;0\\Y&amp;1\end{bmatrix}\) &nbsp; line: \(\begin{bmatrix}\cos\theta&amp;jZ_0\sin\theta\\ j\sin\theta/Z_0&amp;\cos\theta\end{bmatrix}\) &nbsp; xfmr: \(\begin{bmatrix}n&amp;0\\0&amp;1/n\end{bmatrix}\)<br>\(S_{21}=\dfrac{2}{A+B/Z_0+CZ_0+D},\ S_{11}=\dfrac{A+B/Z_0-CZ_0-D}{A+B/Z_0+CZ_0+D}\)</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;white-space:nowrap;vertical-align:top;"><code>Zs R X</code></td><td style="padding:3px 0;">Series impedance Z = R + jX (Ω).</td></tr>
    <tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;white-space:nowrap;vertical-align:top;"><code>Yp G B</code></td><td style="padding:3px 0;">Shunt admittance Y = G + jB (S).</td></tr>
    <tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;white-space:nowrap;vertical-align:top;"><code>TL Zc deg</code></td><td style="padding:3px 0;">Ideal line: characteristic impedance Zc (Ω), electrical length (degrees).</td></tr>
    <tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;white-space:nowrap;vertical-align:top;"><code>XFMR n</code></td><td style="padding:3px 0;">Ideal transformer, turns ratio n:1.</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>D. M. Pozar, <em>Microwave Engineering</em>, 4th ed., §4.4 (Tables 4.1, 4.2).</div>
</details>
</div>
<div class="sec-lbl">Inputs</div>
<div class="inp-grid">
  <div class="inp-row"><label>Z<sub>0</sub> ref</label><div class="inp-inline"><input type="text" id="ab-z0" placeholder="e.g. 50" data-tip="Reference impedance for S-parameters (Ω)."><span class="inp-units">Ω</span></div><span class="inp-hint">For S-par</span></div>
</div>
<div class="sec-lbl" style="margin-top:8px;">Elements (one per line, port 1 → port 2)</div>
<textarea id="ab-net" rows="5" style="width:100%;box-sizing:border-box;font-family:monospace;font-size:13px;padding:8px;border:1px solid #d0c8f0;border-radius:4px;" placeholder="Zs 0 50&#10;Yp 0 0.02&#10;TL 70.7 90"></textarea>
<div style="display:flex;gap:8px;align-items:center;margin:12px 0 6px;"><button class="calc-btn" id="btn" onclick="abCalc()">Calculate</button> <button class="example-btn" onclick="abExample()" title="Series L, shunt C, quarter-wave line">Load Example</button></div>
<div id="error" class="err-msg" style="color:#c0392b;font-size:12px;margin-bottom:6px;min-height:16px;"></div>
<div class="sec-lbl">Total ABCD matrix</div>
<div id="ab-abcd" style="overflow-x:auto;"></div>
<div class="sec-lbl">S-parameters</div>
<div class="res-grid">
  <div class="res-card"><h4>Reflection / transmission</h4>
    <div class="res-row"><span class="res-lbl">S<sub>11</sub></span><span class="res-val" id="ab-s11">—</span></div>
    <div class="res-row"><span class="res-lbl">S<sub>21</sub></span><span class="res-val" id="ab-s21">—</span></div>
    <div class="res-row"><span class="res-lbl">|S<sub>21</sub>| (insertion)</span><span class="res-val" id="ab-s21db">—</span></div>
  </div>
  <div class="res-card"><h4>Consistency</h4>
    <div class="res-row"><span class="res-lbl">S<sub>22</sub></span><span class="res-val" id="ab-s22">—</span></div>
    <div class="res-row"><span class="res-lbl">|S<sub>11</sub>| (return loss)</span><span class="res-val" id="ab-s11db">—</span></div>
    <div class="res-row"><span class="res-lbl">det = AD − BC</span><span class="res-val" id="ab-det">—</span></div>
  </div>
</div>

<div class="sec-lbl">Cascade</div>
<div class="schematic-box"><div id="tool-diagram"></div></div>
<script>
function drawDiagram() {
  if (typeof abStages === 'undefined' || !abStages) return;
  var st = abStages, N = st.length, W=540, H=110, y=55, x0=40;
  var seg = Math.min(110, (W - 80) / Math.max(N, 1));
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  s+='<rect width="'+W+'" height="'+H+'" fill="#faf9ff" rx="4"/>';
  s+='<line x1="'+x0+'" y1="'+y+'" x2="'+(x0 + N*seg)+'" y2="'+y+'" stroke="#5533aa" stroke-width="1.4"/>';
  for (var i = 0; i < N; i++) {
    var x = x0 + i * seg + seg/2;
    s+='<rect x="'+(x-30)+'" y="'+(y-16)+'" width="60" height="32" rx="3" fill="#fff" stroke="#AA77FF" stroke-width="1.4"/>';
    s+='<text x="'+x+'" y="'+(y+4)+'" text-anchor="middle" font-size="9" fill="#7a4fd0">'+st[i]+'</text>';
  }
  s+='<circle cx="'+x0+'" cy="'+y+'" r="3" fill="#5533aa"/><circle cx="'+(x0+N*seg)+'" cy="'+y+'" r="3" fill="#5533aa"/>';
  s+='<text x="'+W/2+'" y="18" text-anchor="middle" font-size="10" fill="#5533aa" font-weight="bold">'+N+'-element ABCD cascade</text>';
  document.getElementById('tool-diagram').innerHTML=s+'</svg>';
}
</script>
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