RF & Microwave PCB Design, Coplanar Waveguide (CPW) Elliptic Impedance & Delay
Coplanar Waveguide (CPW) RF PCB Impedance Z0 Formula
Calculate CPW Elliptic Modulus k0 = w/(w + 2g), Half-Space Effective Permittivity (1 + eps_r)/2, Hilberg Characteristic Impedance Z0 (Ohms), and Phase Velocity (mm/ns).
On an unbacked Coplanar Waveguide, Z0 depends ONLY on trace-to-gap aspect ratio w/(w+2g)—independent of substrate thickness!
Unlike microstrip (whose `50`-Ohm trace width scales linearly with dielectric core height `h`), an unbacked **Coplanar Waveguide (`CPW`)** places coplanar top-layer ground planes on BOTH sides of Center Conductor Width `w` (mm) separated by Symmetric Coplanar Air/Mask Slot Gaps `g` (mm) atop thick Relative Permittivity `epsilon_r`. Conformal Schwarz-Christoffel mapping (Wen / Hilberg closed-form elliptic integral ratio for `0 < k_0 <= 0.707`) shows half the electric flux travels in air (`epsilon = 1`) and half inside the thick dielectric (`epsilon_r`), locking **Effective Permittivity `epsilon_{eff} = (1 + epsilon_r) / 2`**, **Elliptic Aspect Modulus `k_0 = w / (w + 2 * g)`** (`k’_0 = SQRT(1 - k_0^2)`), **Elliptic Ratio `K(k_0) / K(k’_0) = PI() / LN(2 * (1 + SQRT(k’_0)) / (1 - SQRT(k’_0)))`**, **Characteristic Impedance `Z_0 = (30 * PI()) / (SQRT(epsilon_{eff}) * (K / K’))`** (`Ohms`), and **Quasi-TEM Phase Velocity `v_p = 299.792458 / SQRT(epsilon_{eff})`** (`mm/ns`)!
Verify on an FR-4 / Rogers Hybrid `2.4` GHz RF Front-End Launch—Center_Strip_Width `w` = 1.05 mm (B2), Symmetric_Side_Gap `g` = 0.23 mm (B3 -> Elliptic Modulus `k_0 = 0.6954` in E2), and Substrate_Permittivity `epsilon_r` = 4.35 (B4): Effective Permittivity equals `2.675` (E3), CPW Impedance hits `58.50` Ohms (E4), and Phase Velocity runs at `183.30` mm/ns (E5)!
| Unbacked CPW RF Transmission-Line KPI (w=1.05 mm, g=0.23 vs 0.48 mm, eps_r=4.35) | Formula | Verified Output |
|---|---|---|
| Elliptic Modulus k0 | Width_w_mm / (Width_w_mm + 2 * Gap_g_mm) | 0.6954 (0.5224 @ wide gap g=0.48 mm) |
| Effective Permittivity eps_eff | (1 + Substrate_Permittivity_eps_r) / 2 | 2.675 |
| CPW Characteristic Impedance Z0 | (30 * PI()) / (SQRT(eps_eff) * (K/Kp)) | 58.50 Ohms (71.84 Ohms @ g=0.48 mm) |
| Quasi-TEM Phase Velocity (mm/ns) | 299.792458 / SQRT(eps_eff) | 183.30 mm/ns (5.46 ps/mm delay) |
Hilberg Conformal-Elliptic CPW Impedance & Phase Velocity Formula (0 < k0 <= 0.707)
Validate `Strip_Width w > 0.05` mm, `Side_Gap g > 0.05` mm, Hilberg lower-branch regime `w / (w + 2 * g) <= 0.707`, and `Substrate_eps_r` in `[1.0, 25.0]`! Try writing the nested `SQRT(1 - k_0^2)` elliptic ratio yourself first; if your parentheses mismatch inside `LN(...)`, ask the in-app AI assistant for a step-by-step `LET` decomposition hint.
=LET(wt_m, B2, gp_m, B3, er_s, B4, kz_m, wt_m / (wt_m + 2 * gp_m), ef_p, (1 + er_s) / 2, IF(OR(wt_m <= 0.05, gp_m <= 0.05, kz_m > 0.707, er_s < 1, er_s > 25), "Invalid CPW Geometry", ROUND(299.792458 / SQRT(ef_p), 2)))Over-Etched 0.48 mm Coplanar Ground Gap Impedance Mismatch Check (Set Gap g B3 = 0.48 mm)
Widen the coplanar ground-plane clearance gap from `0.23` to `0.48` mm (`B3 = 0.48`): Elliptic Modulus E2 drops to `0.5224` and CPW Impedance E4 jumps from `58.50` to `71.84` Ohms—causing an immediate return-loss reflection spike on a `50`-Ohm VNA port!
=IF(ISTEXT(E5), E5, LET(kz_m, B2 / (B2 + 2 * B3), kp_m, SQRT(1 - (kz_m ^ 2)), kk_r, PI() / LN((2 * (1 + SQRT(kp_m))) / (1 - SQRT(kp_m))), ef_p, (1 + B4) / 2, ROUND((30 * PI()) / (SQRT(ef_p) * kk_r), 2)))Try the example
Use the original values or change them to test your understanding.