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  <title>Coplanar Waveguide (CPW) Calculator — Z₀ & εeff | RF Toolbox</title>
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<h3>Coplanar Waveguide (CPW) Calculator</h3>
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  <p>Coplanar waveguide has the signal trace and ground planes on the same side of the substrate. This allows easy shunt connections without vias and is preferred for MMICs and millimetre-wave PCBs. Uses elliptic integral conformal mapping for accurate results.</p>
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  <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:2.0;overflow-x:auto;">\(Z_0 = \dfrac{30\pi}{\sqrt{\varepsilon_{eff}}}\dfrac{K(k^\prime)}{K(k)},\quad k=\dfrac{w}{w+2s},\quad \varepsilon_{eff}=1+\dfrac{\varepsilon_r-1}{2}\dfrac{K(k_1^\prime)/K(k_1)}{K(k^\prime)/K(k)}\)</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;">w</td><td style="padding:3px 0;">Signal trace width (mm).</td></tr><tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;white-space:nowrap;vertical-align:top;">s</td><td style="padding:3px 0;">Gap between trace and ground plane (mm).</td></tr><tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;white-space:nowrap;vertical-align:top;">h</td><td style="padding:3px 0;">Substrate height (mm). Infinite assumed if h >> w+2s.</td></tr><tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;white-space:nowrap;vertical-align:top;">εᵣ</td><td style="padding:3px 0;">Substrate dielectric constant.</td></tr><tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;white-space:nowrap;vertical-align:top;">Z₀</td><td style="padding:3px 0;">Characteristic impedance = 30π/√εeff · K(k′)/K(k).</td></tr><tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;white-space:nowrap;vertical-align:top;">εeff</td><td style="padding:3px 0;">Effective permittivity via Schnieder/Hammerstad conformal mapping.</td></tr></table>

<div style="margin-top:10px;padding-top:8px;border-top:1px solid #e0d8ff;"><div style="font-size:10px;font-weight:bold;color:#AA77FF;letter-spacing:0.07em;text-transform:uppercase;margin-bottom:4px;">Physical constants used</div><table style="font-size:12px;border-collapse:collapse;"><tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;">c</td><td>Speed of light = 2.998×10⁸ m/s</td></tr>
          <tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;">µ₀</td><td>Permeability of free space = 4π×10⁻⁷ H/m ≈ 1.2566×10⁻⁶ H/m</td></tr>
          <tr><td style="padding:3px 14px 3px 0;font-weight:bold;color:#5533aa;">ε₀</td><td>Permittivity of free space = 8.854×10⁻¹² F/m</td></tr></table></div>
<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>M. Schneider, "Microstrip lines for microwave integrated circuits," <em>Bell Syst. Tech. J.</em> 48 (1969); M. Pozar, <em>Microwave Engineering</em>, §3.8 (Wiley, 2012)</div>
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<div class="sec-lbl">Inputs</div>
<div class="inp-grid"><div class="inp-row"><label>Trace width, <span class="var">w</span></label><div class="inp-inline"><input type="text" id="cpw-w" value="1.0" ><span class='inp-units'>mm</span></div><span class="inp-hint">signal conductor width</span></div><div class="inp-row"><label>Gap width, <span class="var">s</span></label><div class="inp-inline"><input type="text" id="cpw-s" value="0.3" ><span class='inp-units'>mm</span></div><span class="inp-hint">trace edge to ground</span></div><div class="inp-row"><label>Substrate height, <span class="var">h</span></label><div class="inp-inline"><input type="text" id="cpw-h" value="0.5" ><span class='inp-units'>mm</span></div><span class="inp-hint">trace to bottom ground (if any)</span></div><div class="inp-row"><label>Dielectric constant, <span class="var">εᵣ</span></label><div class="inp-inline"><input type="text" id="cpw-er" value="9.9" ></div><span class="inp-hint">alumina=9.9, GaAs=12.9, Si=11.9, FR4=4.4</span></div><div class="inp-row"><label>Has bottom ground plane</label><select id='cpw-bg'><option value='1'>Yes (GCPW)</option><option value='0'>No (CPW on thick substrate)</option></select></div></div>
<div style="display:flex;gap:8px;align-items:center;margin:12px 0 6px;"><button class="calc-btn" id="cpw-btn">Calculate</button> <button class="example-btn" onclick="loadExample({&quot;cpw-w&quot;: &quot;1.0&quot;, &quot;cpw-s&quot;: &quot;0.3&quot;, &quot;cpw-h&quot;: &quot;0.5&quot;, &quot;cpw-er&quot;: &quot;9.9&quot;, &quot;cpw-bg&quot;: &quot;1&quot;})" title="1 mm trace, 0.3 mm gap on alumina">Load Example</button></div>
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<div class="sec-lbl">Results</div>
<div class="res-grid"><div class="res-card"><h4>Electrical</h4><div class="res-row"><span class="res-lbl">Characteristic impedance, Z₀</span><span class="res-val" id="cpw-z0">—</span></div><div class="res-row"><span class="res-lbl">Effective εᵣ</span><span class="res-val" id="cpw-eeff">—</span></div><div class="res-row"><span class="res-lbl">Phase velocity, vₚ</span><span class="res-val" id="cpw-vp">—</span></div><div class="res-row"><span class="res-lbl">Guide wavelength at 10 GHz</span><span class="res-val" id="cpw-lam">—</span></div></div></div>
<div class="sec-lbl">Diagram</div><div class="schematic-box"><div id="tool-diagram"></div></div>

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