Characteristic impedance Z₀ and effective permittivity εeff for coplanar waveguide (CPW) and grounded coplanar waveguide (CPWG / GCPW). Uses Wadell's elliptic-integral formulation with Hilberg's approximation for K(k)/K(k′).
Coplanar waveguide carries the signal on a center conductor flanked by two coplanar ground strips on the same surface. Adding a bottom ground plane (CPWG) gives a richer return-path geometry and better isolation, at the cost of a more complex impedance solution.
The closed-form solution uses the ratio of complete elliptic integrals K(k)/K(k′), where k is a geometric parameter encoding the conductor and gap widths. This calculator uses Hilberg's approximation, which has accuracy better than 8 ppm across the full range of k — far more than typical PCB tolerances will see.
For CPWG, the effective permittivity is computed by combining the air-side and substrate-side capacitances using a correction factor that depends on the substrate height H. As H → ∞, the result asymptotes to conventional CPW (εeff = (εr + 1)/2).
B.C. Wadell, Transmission Line Design Handbook, Artech House, 1991, Chapter 3 (Coplanar Lines).