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CPW & CPWG

Find width or impedance for a coplanar trace, with optional grounded backing.

CPW / CPWG geometry

Top trace plane to the substrate’s lower surface. Units apply to all geometry dimensions.

Measured edge to edge; this slot stays fixed when solving width.

Use the material value, not effective permittivity.

Calculated trace width w

1.87 mm

Calculated impedance
50 Ω
Effective relative permittivity
2.53
AirGroundGroundSubstrate · εᵣ = 4.3sshAir · no back ground planew
w = 1.87 mm · h = 1.6 mm · s = 0.2 mm.
Cross-section schematic; proportions are illustrative. s is the slot on each side of the signal trace.

Side grounds, air above and below the substrate; no back ground. Zero-thickness, lossless estimate.

More measurements

Electrical length & phase delay

Electrical length
140.29 °
One-way phase delay
159.05 ps
Guided wavelength
76.99 mm

Frequency changes propagation readings, not the quasi-static impedance. Phase delay equals group delay only in this nondispersive model.

Width sensitivity

Only width varies; height and each side slot and material stay fixed. This is not a combined fabrication tolerance.

Width-only sensitivity at fixed geometry and material
TraceWidth (mm)Z₀ (Ω)
Narrower1.78 50.55
Nominal1.87 50
Wider1.97 49.49

Model range & numerical check

Calculation range: 1 ≤ εr ≤ 20; 0.01 ≤ w/h ≤ 10; 0.01 ≤ s/h ≤ 10. These are supported calculation bounds, not manufacturing limits.

Attainable impedance over supported widths: 35.68 Ω to 171.13 Ω.

Width / height
1.17
Velocity factor
0.63
Forward residual
0 Ω

A small residual confirms numerical consistency, not manufacturing accuracy.

Model & assumptions

The signal runs between two side grounds. CPW has air beneath the substrate; CPWG adds a continuous back ground. All ground conductors are assumed to share one reference potential. Width synthesis holds each side slot fixed.

Let R(k) be the complete elliptic-integral ratio K(k)/K(k′), with k′ = √(1−k²). The moduli describe the unbacked and grounded substrate boundaries:

k=ww+2s,ku=sinh⁡(πw/4h)sinh⁡[π(w+2s)/4h],kg=tanh⁡(πw/4h)tanh⁡[π(w+2s)/4h]k=\frac{w}{w+2s},\quad k_u=\frac{\sinh(\pi w/4h)}{\sinh[\pi(w+2s)/4h]},\quad k_g=\frac{\tanh(\pi w/4h)}{\tanh[\pi(w+2s)/4h]}

For CPW, the dielectric filling and impedance are:

εeff=1+εr−12R(ku)R(k),Z0=η04εeffR(k)\varepsilon_{\mathrm{eff}}=1+\frac{\varepsilon_r-1}{2}\frac{R(k_u)}{R(k)},\qquad Z_0=\frac{\eta_0}{4\sqrt{\varepsilon_{\mathrm{eff}}}R(k)}

Adding the ideal back ground changes both filling and impedance:

εeff=1+(εr−1)R(kg)R(k)+R(kg),Z0=η02εeff[R(k)+R(kg)]\varepsilon_{\mathrm{eff}}=1+(\varepsilon_r-1)\frac{R(k_g)}{R(k)+R(k_g)},\qquad Z_0=\frac{\eta_0}{2\sqrt{\varepsilon_{\mathrm{eff}}}[R(k)+R(k_g)]}

These are quasi-static partial-capacitance estimates. The implementation evaluates K using arithmetic–geometric means, with η₀ =376.730313668 Ω. Copper thickness, soldermask, finite grounds, via fences, slotline modes, losses and dispersion are excluded. Ground bonding must be designed separately. Supported calculation bounds are 0.01≤w/h≤10, 0.01≤s/h≤10 and 1≤εᵣ≤20; they are not accuracy guarantees.

Both tools derive propagation from the selected effective permittivity. Frequency and length affect these readings independently of impedance:

λg=cfεeff,τ=ℓεeffc,θ=360∘fτ\lambda_g=\frac{c}{f\sqrt{\varepsilon_{\mathrm{eff}}}},\qquad \tau=\frac{\ell\sqrt{\varepsilon_{\mathrm{eff}}}}{c},\qquad\theta=360^\circ f\tau

Here c=299,792,458 m/s. Phase delay equals group delay only within this nondispersive model.

Independent example: w=1 mm, each slot s=0.2 mm, h=1 mm and εr=4.3 give 58.74 Ω for CPW or 51.74 Ω for CPWG. Numerical residuals are not fabrication tolerances.

SourcesQucs, Coplanar waveguides — conformal mapping (opens in a new tab)NIST SP 330 (2019), The International System of Units (opens in a new tab)

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