Coplanar Waveguide Calculator (CPW & GCPW)

Coplanar waveguide (CPW) puts the signal conductor and its return grounds on the same copper layer, separated by two narrow gaps. That single change fixes several problems microstrip has at high frequency: the return current sits micrometres away instead of a substrate thickness away, the fields stay tightly bound to the gaps, and impedance stops depending on how thick the laminate happens to be.

The closed-form solution comes from conformal mapping. Transforming the cross-section turns the awkward three-conductor geometry into a parallel-plate capacitor, and the whole impedance falls out of one ratio of complete elliptic integrals:

where $W$ is the centre conductor width, $G$ is the gap to each coplanar ground, and $k' = \sqrt{1-k^2}$. Notice what is missing: substrate height appears nowhere. For an ideal CPW on an infinitely thick substrate, impedance depends only on the ratio $W/(W+2G)$ — scale the whole cross-section up or down and $Z_0$ does not move.

Grounded CPW

In practice nobody builds on an infinitely thick substrate, and most designs add a ground plane on the far side to carry heat and provide a solid reference for the rest of the board. That is grounded CPW, also called GCPW or conductor-backed CPW. The backing plane pulls some field down into the substrate, which raises the effective permittivity and lowers impedance:

with $k_3 = \tanh(\pi W/4h)/\tanh(\pi(W+2G)/4h)$. As the substrate gets thicker, $k_3 \to k$, $q \to 1$, and the expression collapses back to $\varepsilon_{r,eff} = (1+\varepsilon_r)/2$ — the ungrounded CPW answer. That limit is a useful sanity check on any CPW tool.

Copper thickness matters twice

Finite metal thickness is not a second-order detail in CPW. The sidewalls widen the strip and narrow the gap, which lowers $Z_0$ — that much is standard. Less often modelled is what it does to $\varepsilon_{r,eff}$: the strip sidewall faces the ground sidewall across the gap, above the substrate surface, so that added capacitance is in air. It raises the air and substrate capacitances by the same absolute amount and therefore pulls $\varepsilon_{r,eff}$ down.

The effect is large. At 1 oz copper on a 0.3 mm gap it is worth about 3.5%, and at 2 oz on a 0.15 mm gap over 13%. A zero-thickness formula misses all of it, and the error propagates straight into propagation delay and guided wavelength.

The regime that matters

Grounded CPW only behaves like CPW while the coplanar grounds dominate. Once the gap grows much larger than the substrate height, most field lines terminate on the backing plane instead, and the structure quietly turns into microstrip with some useless copper either side. Keep the gap comparable to or smaller than the substrate height to stay in the CPW regime.

The other practical requirement is stitching. The coplanar grounds and the backing plane form a parallel-plate cavity, and if they are only connected at the board edges that cavity will resonate somewhere in band. Rows of vias down both sides of the line short the two grounds together and push the cavity modes above the operating frequency.