PCB Transmission-Line Loss Mechanisms

Why this plot goes up and an S-parameter goes down Both are decibels, and they are not the same quantity. What is plotted above is the attenuation constant α — a rate, in dB per inch — and a rate of loss is a positive number that grows with frequency. Insertion loss is the result of applying that rate over a length: S₂₁ = −α × length, which is why an SDD21 plot heads downwards into negative decibels while this one climbs. The giveaway is the unit: anything quoted “per inch” is a slope, not a channel.

Where energy goes when a signal travels

Send a fast edge down a short trace and it arrives looking much like it left. Send the same edge down a 12-inch trace and it arrives visibly slower — the sharp corners have been rounded off, the rise time has grown, and the eye diagram has closed. The signal is not just smaller; it is smeared. That distinction is why loss is a signal-integrity problem, not just an amplitude problem: an amplifier can recover amplitude, but it cannot un-smear an edge that has already lost its high-frequency content.

The smearing happens because a PCB trace does not attenuate all frequencies equally. High frequencies — the ones that make edges sharp — are attenuated far more than low frequencies. After a long run, the low-frequency body of the signal arrives nearly intact but the fast transitions that carry timing information have been selectively destroyed. Understanding where that energy goes, and which mechanism is responsible, tells you whether the fix is a wider trace, a smoother copper foil, or a better laminate.

A signal moving down a trace loses energy in exactly two places, and it is worth being concrete about both before any formulas appear.

Some of it is lost in the copper. Copper is a very good conductor but not a perfect one, so current flowing through it dissipates power as heat — ordinary I²R.

The rest is lost in the insulator. The laminate between the trace and the plane is full of molecules that respond to an electric field by twisting slightly to align with it. Reversing the field makes them twist back. That motion is not free — it is internal friction, and it turns signal energy into heat.

Neither of these is surprising on its own. What makes loss interesting is that the two scale with frequency in different ways, so their relative importance changes as you go up in band — and the fix you should reach for changes with it.

Conductor loss: √f

At DC, current fills the conductor. As frequency rises, the magnetic field inside the copper opposes it and current is pushed to the surface. The depth it occupies is the skin depth, and it shrinks as the inverse square root of frequency: 2.06 µm at 1 GHz for copper, 0.65 µm at 10 GHz. Less usable cross-section means more resistance, and because the depth goes as 1/√f, the resistance goes as √f.

δ = √( ρ / π f µ ) → δ_Cu ≈ 2.06 µm / √( f in GHz ) At 10 GHz the current is running in the outermost 0.65 µm of the trace — thinner than the copper's own surface texture, which is the reason roughness matters at all.

Dielectric loss: f

The laminate absorbs energy too. Every cycle, the polar molecules in the resin are dragged back and forth and lag slightly behind the field; that lag is the loss tangent, D_f (also written tanδ). Because the loss is per cycle, the total scales linearly with frequency:

α_d [dB/inch] ≈ 2.3 · f[GHz] · D_f · √D_k A mid-loss laminate (D_f = 0.008, D_k = 3.8) gives ≈ 0.36 dB/inch at 10 GHz from dielectric alone. Switching to a low-loss build at D_f = 0.003 cuts that to 0.14 dB/inch — the single biggest lever on a long channel.

Because one term grows as √f and the other as f, they cross — and where they cross depends almost entirely on D_f. Setting the two expressions equal gives √f = k_c / (2.3 · D_f · √D_k), so on standard FR-4 at D_f = 0.020 the crossover lands near 3 GHz, while on a mid-loss laminate at 0.008 it is up around 20 GHz. Below the crossover the trace geometry owns the budget; above it the material does. Drag D_f in the panel and watch the crossing move.

Follow that through and the conclusion is the opposite of the usual folklore. Widening a trace attacks the conductor term, so it helps precisely where conductor loss dominates — below the crossover. On lossy FR-4 the crossover is already down at 3 GHz, so a bus running near or above it is dielectric-limited and extra width buys little; the fix is the laminate. On a low-loss build the crossover is up around 20 GHz, so a 32 GT/s channel working at 16 GHz Nyquist is still conductor-limited — and there geometry is genuinely back on the table. Good material does not retire the trace width question; it is what puts it back.

The catch is that width is not a free variable: it sets impedance too. Going wider at fixed impedance means moving further from the reference plane, which costs crosstalk isolation and return-path tightness. That trade, not the loss arithmetic, is usually what decides it.

Roughness: the multiplier

Copper foil is deliberately roughened so the laminate bonds to it. At low frequencies that texture is irrelevant, because the current is flowing through the full thickness. Once the skin depth drops below the roughness profile, the current has to follow the contour instead of a straight line — a longer path through the same copper, and therefore more loss. It can add up to a factor of two over the smooth-copper prediction in the worst pairings.

Mind which roughness number the model wants. Foil datasheets quote R_z, a peak-to-valley measure. A Hammerstad-style correction is written in terms of an RMS roughness R_q, and the two are not the same quantity — their ratio depends on the surface's profile shape, so there is no universal conversion factor to apply. Feed R_z into a model expecting R_q and the correction comes out too large. Use the parameter the model asks for, from a source that measured it, and treat the panel above as showing the shape of the roughness effect rather than a calibrated prediction for your stackup.

This is why foil grade appears in high-speed stackup specs at all. Standard foil has a peak profile around 5–7 µm; very-low-profile and hyper-very-low-profile foils bring it under 2 µm and about 1 µm respectively. At 10 GHz, with a 0.65 µm skin depth, that difference is the whole argument. A simulation that models loss from D_k and D_f alone and ignores roughness will be optimistic, and it will be optimistic in exactly the band you cared about.

Why this decides how your board behaves

Loss is rarely a problem of amplitude. A 20 dB channel delivers a tenth of the voltage, and a receiver with a bit of gain does not care much about that. What it cares about is frequency-dependent loss, and the damage it does:

The edge arrives slower than it left. The high-frequency content that made the edge sharp is what got attenuated most, so the rise time at the receiver is longer than at the driver. The two are genuinely different numbers on the same net.
That slower edge spills into the next bit. Once the smear is comparable to a unit interval, the voltage at the sampling instant depends on which bits came before — which is ISI, and it is the dominant impairment on almost every long channel.
It gets worse twice over when you raise the data rate. Nyquist moves up, where the loss is higher, and the bit period shrinks, so the same tail in picoseconds covers more bits. Both push the same way.
Loss is also the thing that saves you from reflections. A long lossy channel attenuates an echo twice — out and back — so the worst ringing is usually on short mismatched nets, not long ones.

What to do about it

First, find out which term dominates in your band. This is the decision that determines everything else, and the panel above answers it directly: set your D_f and read where the crossover lands relative to your Nyquist frequency. Below the crossover, geometry is the lever. Above it, material is.

If you are conductor-limited, go wider — and budget what that costs. Width attacks the conductor term directly. But width sets impedance too, so holding impedance while widening means moving further from the reference plane, which costs crosstalk isolation and loosens the return path. That trade, not the loss arithmetic, is usually what decides it.

If you are dielectric-limited, change the laminate. Nothing about the routing will help. Going from D_f = 0.020 to 0.008 roughly halves the dielectric term, and to 0.003 halves it again. This is a bill-of-materials decision made at stackup time, which is why loss has to be estimated before layout rather than discovered after it.

Specify the foil, not just the resin. Roughness is a multiplier on the conductor term, and a low-loss resin on rough foil throws away much of what you paid for.

And check whether you can simply go shorter. Loss is proportional to length, with no threshold and no cleverness available. Two inches saved by a floorplan change is two inches of budget that never has to be equalised.

Using this to find a fault

Measured loss higher than simulated, and the gap grows with frequency. Almost always roughness — either not modelled, or modelled with the wrong parameter. Check whether the model wanted R_q and was given R_z.
Measured loss higher than simulated by a constant offset. Not the trace. Look at connectors, launches and the fixture — a flat offset is a fixed number of discontinuities, not a distributed effect.
Loss that varies board to board on the same design. Laminate lot variation is real: D_k and D_f both drift with the glass-to-resin ratio. Ask for the lot data before redesigning anything.
An eye that closes far more than the loss number suggests. Loss alone rarely closes an eye; loss plus a reflection does. Look at the return loss in the same band before adding equalisation.

Go deeper — why √f and f, and what the two-term fit is really for

Where the √f comes from. The current in a conductor generates a magnetic field, and that field induces eddy currents which oppose the flow in the centre of the conductor more than at the surface. The result is that current is confined to a surface layer whose thickness is the skin depth, δ = √(ρ/πfµ). Resistance is inversely proportional to the conducting cross-section, and the cross-section is proportional to δ, so R ∝ 1/δ ∝ √f. That is the entire derivation — the square root comes from the square root in the skin depth, and nothing else.

Where the linear term comes from. Dielectric loss is a fixed fraction of the energy stored per cycle. The number of cycles per second is f, so the power lost is proportional to f. Expressed as an attenuation per unit length it is linear in frequency, with D_f as the constant of proportionality. D_f is itself mildly frequency-dependent in real materials, which is one reason measured and modelled curves diverge slowly rather than exactly tracking.

What the two-term fit is for. Channel models across this site use α(f) = k_c√f + k_df, and it is worth being clear about its status. It is a fit, not a derivation from geometry — it captures the two physical scalings and rolls everything else, including roughness and the mild dispersion of D_f, into the two coefficients. That is why it works well across a decade and less well across four, and why the honest way to use it is to fit it to measured or solved data for your stackup rather than to predict from datasheet numbers.

Causality is not optional. A loss curve alone does not define a channel. A filter that attenuates without the corresponding phase shift is not physical — it would produce output before its input. The channel model in the labs builds a minimum-phase response so that the attenuation and the delay belong to each other, and this matters in practice: a magnitude-only channel model produces a plausible-looking eye that is wrong in exactly the way that makes equalisation look better than it is.

One last practical note on D_k. It appears in the dielectric loss expression, but its bigger effect is on velocity — v = c/√D_k — and therefore on delay and on every length-matching calculation. A laminate change made for loss reasons changes your timing too, and the two reviews are usually done by different people.

In the real world

Loss is the most predictable impairment on this site, which is both its virtue and its trap. It is predictable enough that budgets are built on it years before hardware exists — and that means the most informative failures are often those where the prediction used an unchecked input. A laminate substituted at the last minute, a foil grade that was never specified, a D_f quoted at 1 GHz and used at 14.

The habit worth building: whenever you are handed a loss number, ask at what frequency, on what stackup, and measured or modelled? A dB-per-inch figure with no frequency attached is not a specification, and the three most common ways a channel budget goes wrong are all contained in that question.