Transmission Lines

Conductor Surface Roughness

Q: At what frequency does conductor surface roughness start to matter?

Roughness becomes significant when the RMS roughness Rq approaches the skin depth. A useful rule of thumb is that the roughness correction stays below 10 percent until Rq is roughly equal to the skin depth, then climbs toward an asymptotic doubling of resistance as Rq exceeds about twice the skin depth. For copper, skin depth is about 2.1 micrometers at 1 GHz, 0.66 micrometers at 10 GHz, and 0.21 micrometers at 100 GHz. A typical electrodeposited copper foil with Rq near 0.5 to 1 micrometer therefore has almost no effect at 1 GHz but can add 30 to 80 percent to conductor loss by 30 to 60 GHz. For millimeter-wave designs above 20 GHz, very low profile foils with Rq below 0.5 micrometers are normally specified.

Q: What is the difference between the Hammerstad and Huray roughness models?

The Hammerstad model multiplies the smooth conductor loss by a correction factor K equal to 1 plus (2 divided by pi) times the arctangent of 1.4 times (Rq divided by skin depth) squared. It is simple and accurate up to about 5 to 10 GHz but saturates at a factor of 2, so it underpredicts loss for rough copper at millimeter-wave frequencies. The Huray snowball model treats the roughness as a distribution of conducting spheres and scales loss with the ratio of total sphere surface area to the flat tile area. It tracks measured data well to 50 GHz and beyond but requires surface metrology to extract sphere radius and density, which are often supplied as Huray-Bracken parameters in laminate datasheets.

Q: How do I extract roughness parameters for my electromagnetic simulator?

Most field solvers accept either a single RMS roughness value with a chosen model, or pre-fit Huray-Bracken coefficients. For a quick analysis, take the Rq value from the foil datasheet and select the Cannonball-Huray or modified Hammerstad option. For accurate millimeter-wave correlation, run a back-drilled or stripline test coupon, measure insertion loss versus frequency, and use the Delta-L or short-pulse-propagation method to separate conductor loss from dielectric loss. Fit the roughness model so the extracted conductor loss matches measurement, then reuse those coefficients across the laminate family. Always verify which surface is treated, since the drum side and matte side of rolled foil differ substantially.

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Every plated or laminated metal trace has a microscopic peak-and-valley topography, quantified by an RMS height R_q, that the RF current must follow once it is confined to the thin skin depth near the conductor wall. As frequency rises and the skin depth shrinks below the texture height, the effective current path lengthens, the surface resistance climbs, and conductor loss can nearly double relative to an idealized smooth surface. This is the single most important parasitic in low-loss microwave laminates and millimeter-wave waveguide above roughly 20 GHz.

Category: Transmission Lines

Typical R_q: 0.3 to 2 μm

Max loss penalty: ≈ 2× smooth

Why Rough Copper Raises Insertion Loss

At DC and low frequencies, current fills the entire cross-section of a conductor and the surface texture is irrelevant. As frequency increases, the current crowds into a sheet of thickness equal to the skin depth δ. For copper, δ is about 2.1 μm at 1 GHz, 0.66 μm at 10 GHz, and only 0.21 μm at 100 GHz. Once δ becomes comparable to or smaller than the surface roughness height, the current is forced to flow up and over the copper nodules instead of across a flat plane. The longer path and the extra surface area between the peaks both add resistance, which appears directly as additional conductor loss and degraded insertion loss in the transmission line.

Surface roughness is not an accident; it is engineered into copper foil to improve adhesion. Electrodeposited (ED) foil is intentionally treated on the matte side so the resin can grip the teeth, while rolled-annealed (RA) foil is much smoother. Foil suppliers therefore offer a profile ladder, from standard treatment down through reverse-treated, very-low-profile (VLP), and ultra-low-profile (HVLP) grades. The roughness number that matters for loss is the RMS value R_q (often reported as R_q or S_q), although datasheets frequently also quote the peak-to-valley R_z, which runs roughly four to six times R_q.

Because both copper sides of a stripline are roughened and both the trace and the reference plane contribute, a designer must account for every metal interface in the stackup. For a 50-ohm stripline on a low-loss laminate at 28 GHz, switching from standard ED foil (R_q ≈ 1.2 μm) to HVLP foil (R_q ≈ 0.4 μm) can cut total insertion loss by 0.05 to 0.1 dB per centimeter, which is decisive over a long antenna feed network.

Roughness Correction Models

Hammerstad correction factor:
K_H = 1 + (2/π) × arctan[ 1.4 × (R_q/δ)² ] (saturates at K_H → 2)

Copper skin depth:
δ = 1 / √(π × f × μ × σ) ≈ 2.09 / √f(GHz) μm

Huray snowball ratio:
K_Huray = 1 + (A_matte/A_flat) × ∑ [ a_i² / (1 + δ/a_i + δ²/(2a_i²)) ]

Rough conductor loss:
α_c,rough = K × α_c,smooth

Where R_q = RMS roughness, δ = skin depth, f = frequency, μ = permeability, σ = conductivity, a_i = snowball sphere radius. Example: R_q = 1 μm at 28 GHz gives δ ≈ 0.39 μm, R_q/δ ≈ 2.5, K_H ≈ 1.93, so conductor loss is about 93% higher than the smooth value (close to the model ceiling of 2×).

Foil Profile and Roughness Comparison

Foil Type	R_q (RMS)	R_z (peak-valley)	Best Model	Useful Above	Typical Use
Standard ED (STD)	1.0 to 2.0 μm	5 to 10 μm	Hammerstad	< 6 GHz	Digital, low-cost RF
Reverse-treated (RTF)	0.7 to 1.2 μm	3 to 6 μm	Hammerstad / Huray	6 to 20 GHz	Sub-6 GHz radios
Very-low-profile (VLP)	0.4 to 0.7 μm	2 to 4 μm	Huray	10 to 40 GHz	5G mmWave boards
Hyper-VLP (HVLP)	0.2 to 0.4 μm	1 to 2 μm	Huray	20 to 110 GHz	mmWave, 77 GHz radar
Rolled-annealed (RA)	0.1 to 0.3 μm	0.5 to 1.5 μm	Huray	> 40 GHz	Flex, lowest loss

Common Questions

Frequently Asked Questions

At what frequency does conductor surface roughness start to matter?

Roughness matters once R_q approaches the skin depth. The correction stays under 10% while R_q < δ, then climbs toward a 2× resistance asymptote as R_q exceeds about 2δ. Copper δ is 2.1 μm at 1 GHz, 0.66 μm at 10 GHz, and 0.21 μm at 100 GHz, so a 0.5 to 1 μm foil is harmless at 1 GHz but adds 30 to 80% loss by 30 to 60 GHz. Above 20 GHz, specify VLP or HVLP foil with R_q < 0.5 μm.

What is the difference between the Hammerstad and Huray roughness models?

Hammerstad scales smooth loss by K = 1 + (2/π)·arctan[1.4(R_q/δ)²]; it is simple and accurate to roughly 5 to 10 GHz but saturates at 2× and underpredicts rough copper at mmWave. The Huray snowball model represents roughness as conducting spheres and scales loss with the sphere-to-flat surface-area ratio, tracking data well beyond 50 GHz but requiring metrology-derived radius and density (often given as Huray-Bracken parameters).

How do I extract roughness parameters for my electromagnetic simulator?

Field solvers accept either a single R_q with a chosen model or pre-fit Huray-Bracken coefficients. For a quick pass, take R_q from the foil datasheet and pick Cannonball-Huray or modified Hammerstad. For mmWave correlation, build a stripline test coupon, measure insertion loss versus frequency, and use the Delta-L or short-pulse method to separate conductor from dielectric loss, then fit the model so extracted conductor loss matches measurement. Confirm whether the drum side or matte side is treated, since they differ.

Related Terms

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