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Full changelogEngine 1.26.4

AC Resistance, Skin Effect & Trace Loss

PCB & Copper

The AC resistance of a real trace, the loss it produces, and the frequency where the laminate takes over from the copper.

Inputs

Board profile

Fills Copper weight and Dielectric constant (Dk) from your board profile.

1 oz/ft² of finished copper is 34.8 µm
oz
For stripline, the plane-to-plane spacing
From the laminate datasheet
Hz
10 GHz
Hz
5 GHz
From the foil datasheet or the fab. Rz and Ra are not RMS.
From the channel budget, after connectors, vias and package
dB
Temperature
°C

Results

1 passed
Loss over the routed length IL
3.079dB
Pass

Checked: loss over 254 mm at 5 GHz ≤ the 10.0 dB allowance, at the rough end of the band
Attenuation at the stated frequency α
0.3079dB/in
AC resistance per unit length Rac
57.85Ω/m
Crossover, conductor equals dielectric f×
All results (10)
Skin depth δ
934.6nm
DC resistance per unit length Rdc
2.477Ω/m
Roughness factor K
Conductor loss over this length
Dielectric loss over this length
Loss over this length, smooth copper
Reach for the allowance
Frequency at which the copper is two skin depths thick
Characteristic impedance used Z0
48.18Ω
Effective permittivity used εeff
2.784
conductor, with roughnessconductor, smoothdielectrictotal

Accuracy

Source
  • Wheeler
  • Hammerstad and Jensen
  • Hammerstad and Bekkadal

Wheeler, Formulas for the Skin Effect, Proceedings of the IRE the equation

Hammerstad and Jensen, Accurate Models for Microstrip Computer-Aided Design the equation

Hammerstad and Bekkadal, A Microstrip Handbook, University of Trondheim fitted to measurements

Valid range
  • Dielectric constant (Dk) 1 to 20

Dielectric constant (Dk) The impedance model was fitted for ordinary laminate permittivities. Outside that range it is not reliable enough to differentiate.

Precision
Dielectric loss is exact for stripline.

For microstrip the √ε_eff form leaves out the filling factor, so it overstates dielectric loss somewhat, on the safe side. Conductor loss uses Wheeler’s rule, which reproduces the exact coaxial result. Foil roughness and the loss tangent at your frequency move the answer most, so roughness is shown as a band and the verdict uses its rough end.

Most of the uncertainty comes from Foil roughness, RMS. Tighten that first.

Verified against

6 independent cases. See the working.

This is a design aid. The engineer remains responsible for the design and for checking the standard itself.

Parameter sweep

Vary one input over a range and see the answer and verdict at each step, as a table and a curve.Pro

Worst-case corners

Put a tolerance on each input and get the worst-case band around the answer.Pro

Give at least one input a tolerance above zero.

Cross-section

The principle

Trace loss has two parts. Copper loss rises with the square root of frequency as the skin depth shrinks; dielectric loss rises linearly. The frequency where they cross decides the fix: below it, a wider trace and smoother foil; above it, a better laminate.

Skin depth
δ=ρπfμ0\delta = \sqrt{\frac{\rho}{\pi f \mu_0}}
2.09 µm in annealed copper at 1 GHz (ρ = 1.724 × 10-8 Ω·m).
Surface resistance
Rs=πfμ0ρ=ρδR_s = \sqrt{\pi f \mu_0 \rho} = \frac{\rho}{\delta}
Wheeler’s incremental inductance rule
R=Rsμ0 ∂L∂nR = \frac{R_s}{\mu_0}\,\frac{\partial L}{\partial n}
L is the external inductance per unit length, and n is how far every conductor surface recedes into its own conductor. For coax it gives Rs/(2πa) + Rs/(2πb) exactly, so the ground return is included.
Conductor attenuation
αc=K R2Z0\alpha_c = \frac{K\,R}{2 Z_0}
nepers per metre; multiply by 8.6859 for decibels.
Hammerstad roughness correction
K=1+2πarctan⁡(1.4(Δδ)2)K = 1 + \frac{2}{\pi}\arctan\left(1.4\left(\frac{\Delta}{\delta}\right)^{2}\right)
Dielectric attenuation
αd=πfεeff tan⁡δc0\alpha_d = \frac{\pi f \sqrt{\varepsilon_{eff}}\,\tan\delta}{c_0}
Equivalently 2.31 dB/in per GHz × √εeff × tan δ.
The DC blend
R(f)=Rdc2+Rac2R(f) = \sqrt{R_{dc}^{2} + R_{ac}^{2}}
Only smooths the curve through the transition; read nothing from its shape there.
  • δ\deltaskin depth
  • RsR_ssurface resistance, ohms per square
  • Δ\Deltafoil roughness, RMS
  • KKHammerstad roughness factor, between 1 and 2
  • αc\alpha_cconductor attenuation
  • αd\alpha_ddielectric attenuation
  • εeff\varepsilon_{eff}effective permittivity of the line
  • Z0Z_0characteristic impedance, from the same model as the impedance page
More detail

A simple strip formula, ρ/(δw), misses the current crowding to the edges and underside of the trace, and the loss in the return plane. Wheeler’s rule captures both.

Roughness. Hammerstad needs only the RMS roughness and caps the effect at 2×. The Huray model needs parameters fabs rarely publish, and helps only once fitted to measured S-parameters for that foil.

Not included: reflections at via stubs, connectors, AC coupling capacitors and package transitions; fibre weave; crosstalk. This is a matched, uniform line.

References: Wheeler, incremental inductance rule; Hammerstad and Jensen, impedance model; Hammerstad and Bekkadal, roughness correction. For a link standard’s insertion-loss limit, consult that standard.

Engine version ⁨1.26.4⁩