coplanar waveguide calculator - Aaron Graves, PhDude Replica

CPW Impedance Calculator (Quasi-Static)

Use this tool to estimate characteristic impedance, effective dielectric constant, phase velocity, and guided wavelength for a coplanar waveguide (CPW).

Center trace width, W (mm)

Gap to ground, S (mm)

Substrate thickness, H (mm)

Relative permittivity, εr

Frequency (GHz, optional for wavelength)

Model assumptions: ideal conductor, low-loss dielectric, quasi-TEM CPW approximation using complete elliptic integrals.

If you design RF or microwave circuits on printed circuit boards, CPW can be one of the most practical transmission line options. It keeps the signal and return conductors on the same layer, which simplifies probing, tuning, and integration with active components. This calculator gives a quick first-pass estimate before you run a full-wave EM simulation.

What this coplanar waveguide calculator computes

Characteristic impedance (Z₀) in ohms
Effective dielectric constant (ε_eff)
Phase velocity (v_p) in m/s
Guided wavelength (λ_g) at your selected frequency

These values are especially useful for line matching (for example, targeting 50 Ω), electrical length calculations, and layout sanity checks.

Input definitions

Geometry terms

W: center conductor width
S: gap between center conductor and each adjacent ground rail
H: substrate thickness

Material and operating point

εr: substrate relative dielectric constant (e.g., FR-4 is often around 4.1–4.6 depending on frequency and vendor)
Frequency: used only for wavelength and phase-based length estimates

Core equations used

The calculator uses a conformal-mapping-based quasi-static CPW model. In compact form:

k = W / (W + 2S) k1 = sinh(πW / 4H) / sinh(π(W + 2S) / 4H) q = [K(k') / K(k)] · [K(k1) / K(k1')] εeff = 1 + ((εr - 1)/2) · q Z0 = (30π / √εeff) · [K(k') / K(k)]

Where K(·) is the complete elliptic integral of the first kind, and k′ means √(1−k²). The page computes K numerically using a stable arithmetic-geometric-mean method.

Practical design workflow

Start with your board stack-up (H, εr, copper style from your fab notes).
Pick a target impedance (commonly 50 Ω single-ended).
Adjust W and S until calculator output is close to target.
Check guided wavelength at your RF band to size stubs and resonant sections.
Finalize with a 2.5D/3D EM solver and include launch/via transitions.

Quick interpretation tips

Increasing W generally lowers impedance.
Increasing S generally raises impedance.
Higher εr usually increases field confinement and lowers phase velocity.
At fixed frequency, higher ε_eff means shorter guided wavelength.

Limitations you should know

This calculator is ideal for initial sizing but does not include every real-world effect. For production-grade accuracy, verify with EM simulation and measurement because results are affected by:

Finite conductor thickness and surface roughness
Solder mask and nearby metal structures
Via fences and ground discontinuities
Frequency-dependent dielectric dispersion/loss tangent
Connector/launch parasitics

Bottom line

A CPW calculator is a fast way to move from concept to a workable RF layout. Use it to narrow your geometry quickly, then close the gap with EM validation and test coupons. That two-step approach is usually the fastest route to predictable impedance and repeatable RF performance.

Educational calculator only; values are estimates.