Coplanar waveguide impedance: CPW and grounded CPW, the elliptic-integral formulas, stitching, and a worked example
A coplanar waveguide is a strip with ground copper on the same layer either side of it, separated by a gap. The gap sets the impedance, so the strip can stay the width of the pad it feeds, and the ground is always a via-free hop away, which is why RF boards use it for antenna feeds, connector launches and anything above a gigahertz. With a plane underneath as well it is a grounded coplanar waveguide, the usual form on a multilayer board. This calculator gives Z₀ and εeff for both from the strip width, the gap, the dielectric height, εr and the copper weight using the conformal-mapping formulas, the delay, capacitance and inductance per length, the gap and the width for a target, the wavelength at a frequency and the comparisons with the other kind and a plain microstrip, with the cross-section and its field drawn to scale.
How to use the coplanar waveguide calculator
- Choose grounded or no plane in the menu row, with the substrate (it fills in εr) and the copper weight.
- Type the strip width, the gap to the pour on each side, and the dielectric height to the plane below (or the substrate's thickness without one).
- Add a target Z₀ for the gap at this width and the width at this gap, and a frequency for the wavelength and the stitching pitch. The Gap table tab sweeps the gap.
CPW and grounded CPW
A plain coplanar waveguide (CPW) has no copper under the strip: the field goes from the strip across the two gaps to the pours, half through the air and half through the dielectric, so Z₀ depends almost only on the ratio w/(w+2s) and hardly at all on the substrate thickness. On a board with a plane below, the plane takes a share of the field too, which lowers the impedance and brings εeff closer to εr: a grounded coplanar waveguide (GCPW, or conductor-backed CPW). With the pours far away the GCPW turns into a microstrip; with them close, the gap dominates. The calculator shows both kinds and the plain microstrip for every geometry so you can see which copper is doing the work.
The formulas
CPW: Z₀ = (30π/√εeff) · K′(k₁)/K(k₁) GCPW: Z₀ = 60π / (√εeff · (K(k₁)/K′(k₁) + K(k₃)/K′(k₃)))
k₁ = w/(w+2s) describes the top copper; k₂ = sinh(πw/4h)/sinh(π(w+2s)/4h) and k₃ = tanh(πw/4h)/tanh(π(w+2s)/4h) are the same geometry seen through a dielectric of height h, without and with a plane. K and K′ are complete elliptic integrals of the first kind, and their ratio is what conformal mapping leaves of the geometry; Hilberg's closed form gives it to better than 0.1 %. εeff is 1 + (εr−1)/2 · (K(k₂)/K′(k₂))/(K(k₁)/K′(k₁)) for CPW and (1 + εr q)/(1 + q) with q = (K(k₃)/K′(k₃))/(K(k₁)/K′(k₁)) for GCPW. The derivations are Wen's (1969) and Ghione and Naldi's (1987), and they assume pours much wider than the gap and thin copper; the copper's thickness is allowed for by Gupta's correction, which widens the strip and narrows the gaps by (1.25t/π)(1 + ln(4πw/t)). Accuracy is a few percent; fabs use a field solver.
Pours and stitching
The pours only act as ground if they are ground everywhere along the line. On a GCPW stitch them to the plane with a row of vias along each gap, spaced closer than a twentieth of the wavelength at the highest frequency, and more densely at bends, launches and the ends; unstitched pours resonate and the line's loss and impedance wander. The pours should extend several gaps beyond the line (five is a safe rule), with no other traces in them near the strip. For plain CPW, tie the two pours together at the ends and across any bend with a wire or a jumper, and keep the opposite side of the board clear under the line.
When to use it
Antenna feeds and chip-antenna matching, SMA and U.FL launches, RF front ends, and any line above a gigahertz on a two-layer board, where a 50 Ω microstrip would be 3 mm wide but a GCPW with a 0.3 mm gap can be 1 mm. Also wherever shunt components need a nearby ground. For long digital buses a microstrip or stripline is simpler; coplanar lines cost board area for their pours.
Your waveguide, step by step
- Copper: Δ = 44.1 µm, so w' = 0.34 mm and s' = 0.16 mm; k₁ = 0.525, k₃ = 0.885.
- Result: εeff = 3.056, Z₀ = 50.5 Ω; without the plane 78.8 Ω, as a microstrip 55.8 Ω.
- Wave: 0.57 c, 5.83 ps/mm; λ = 71.4 mm at 2.4 GHz, stitch every 3.57 mm or less.
- For 50.0 Ω: gap 0.19 mm at this width, or width 0.31 mm at this gap.
Worked example: a GCPW on a 0.2 mm prepreg
A 0.3 mm strip with 0.2 mm gaps on 0.2 mm of FR-4 (εr 4.3) over a plane, ½ oz copper. The copper correction is small, 44.2 µm; k₁ = 0.525 and k₃ = 0.885, so q = 1.65: the plane takes most of the field, εeff = 3.06 and Z₀ = 50.5 Ω. The same strip as a plain microstrip on that prepreg would be 55.8 Ω, so the pours have pulled only a few ohms off it; without the plane it would be 78.8 Ω. For exactly 50 Ω the gap would have to close to 0.19 mm at this width, or the strip widen to 0.31 mm at this gap. At 2.4 GHz the wavelength on the line is 71.4 mm, so the stitching vias should sit no more than 3.57 mm apart. On a two-layer 1.6 mm board the same 50 Ω takes a 1.00 mm strip with 0.30 mm gaps: 53.6 Ω.
Questions
Does the pour have to be on both sides?
Yes, and symmetrically: a pour on one side only makes an asymmetric line with a different impedance and a tendency to radiate. Keep both gaps equal.
How close can the gap be?
What the fab etches cleanly, usually 0.1–0.15 mm, and the solder-mask registration; tighter gaps make Z₀ very sensitive to etching. The calculator flags gaps under 0.1 mm.
Why does my field solver give a different number?
Finite pour width, solder mask, the copper's trapezoidal etch profile and the plating all shift Z₀ by a few percent, which these closed forms cannot see. Use them to get close and the fab's solver to finish.
Can a coplanar line carry a differential pair?
Yes, as a coplanar differential pair with pours outside both traces; that needs coupled-line formulas beyond this tool.