Two loss mechanisms with two different slopes, a copper surface that is not flat, and a dielectric constant that cannot be constant without describing a material in which effects precede their causes.
A signal loses energy on a circuit board in two quite separate ways, and keeping them apart is the single most useful thing in this deck because they scale differently and they are fixed by different purchases.
Conductor loss is ohmic heating in the copper. It grows as the square root of frequency, because the current crowds into a surface layer whose depth falls as $1/\sqrt{f}$, so the effective cross-section shrinks and the resistance rises. Dielectric loss is energy absorbed by the laminate as its molecules are polarised and re-polarised. It grows roughly in proportion to frequency, because the number of polarisation cycles per second does.
Those two exponents mean the mix changes with data rate. At a gigahertz a typical stripline is conductor-dominated and a wider trace is the cheap fix. Past about ten gigahertz on most laminates the dielectric term has overtaken, and no amount of copper helps — the only remedy is a better resin system, which is a procurement decision rather than a layout one.
Fixed by trace width, copper thickness, and — more than most people expect — the roughness of the foil. Improved by making the trace wider, which the escape routing usually forbids, or smoother, which costs money and adhesion.
Fixed by the resin system and the glass, through the loss tangent. Cannot be improved by any layout change at all. This is the term that drove the industry from FR‑4 to the low-loss laminates, and it is why the material choice is made before the stackup.
A changing magnetic field inside a conductor induces eddy currents that oppose the current in the interior and reinforce it at the surface. The result is that current density falls exponentially with depth, with a characteristic skin depth
$$\delta = \sqrt{\frac{2}{\omega\mu\sigma}}.$$
In copper this is about two micrometres at one gigahertz and about six-tenths of a micrometre at ten. Since a half-ounce foil is roughly seventeen micrometres thick, the current is confined to a small fraction of the available copper across the entire band of interest — which is why the thickness of the foil stops mattering to loss above a few hundred megahertz, while its perimeter, and the state of its surface, matter a great deal.
Current does not spread evenly around the perimeter either. It crowds onto the faces looking at the return conductor, because that is what minimises loop inductance — the same principle that put the return current in a narrow band in deck 02. The ratio by which this raises the resistance above the uniform-perimeter estimate is Johnson's proximity factor, and for a tightly coupled differential stripline it is around three.
Copper foil is deliberately roughened so that the resin grips it. The teeth are between a fraction of a micrometre and a couple of micrometres tall, which is the same scale as the skin depth at the frequencies that matter, so the current is forced to follow a longer path than the flat dimension suggests and the resistance rises.
Two models are in general use. Hammerstad fits Morgan's 1949 corrugated-surface results to an arctangent; it is simple, it is in every field solver, and it is bounded — it cannot predict a factor greater than two, however rough the copper. Huray replaces the tooth structure with a cluster of spheres and computes the power each scatters; it has no ceiling. The difference between them is invisible below about five gigahertz and decisive above it.
Neither model is worth using unless it reproduces the worked examples in the literature, so both implementations here are checked against Hall and Heck's, whose numbers come from measured transmission lines rather than from other models.
Hammerstad takes one number, the root-mean-square tooth height. Huray needs a sphere radius and a count, and those come from fitting to a measured surface or a measured insertion loss. That is a real practical cost: the more accurate model is the one you cannot use until you have already measured something. In practice laminate suppliers publish fitted Huray parameters for their foils, and the alternative is to fit them from a test coupon.
A laminate is described to first order by two numbers. The dielectric constant Dk, properly the relative permittivity, sets how much charge the line holds per unit length and therefore both its impedance and its propagation delay. The dissipation factor Df, also called the loss tangent, is the ratio of the imaginary part of the permittivity to the real part, and sets how much energy is absorbed per cycle. Attenuation from this mechanism is
$$\alpha_d = \frac{\pi f \sqrt{\mathrm{Dk}}\,\mathrm{Df}}{c} \ \text{nepers per metre},$$
which is linear in frequency and, importantly, independent of the trace geometry. Widening a trace reduces conductor loss and does nothing at all to dielectric loss. That asymmetry is why the crossover frequency between the two mechanisms is a property of the material and the geometry together, and why it moves down in frequency as traces get narrower.
A quoted Dk and Df are meaningless without the frequency at which they were measured and the method used, because both vary. Older FR‑4 datasheets quote at one megahertz or one gigahertz, where the numbers flatter the material considerably. Comparison between laminates is only fair at a common frequency, which is why the table on the next slide states the measurement frequency for each.
Each curve below is the total loss per inch for one laminate, split into its conductor and dielectric parts, computed for a hundred-and-fifty-micrometre stripline on the foil selected. The crossover point — where the dielectric term overtakes the conductor term — is marked, because it is the frequency above which better copper stops being the answer.
The obvious way to model a laminate is to take the datasheet Dk and Df and hold both fixed across the band. This is what a great many channel models do and it is wrong in a way that matters, because it describes a material that violates causality.
The reason is the Kramers–Kronig relations. For any linear system that cannot respond before it is excited, the real and imaginary parts of its response function are not independent: each is the Hilbert transform of the other. A medium with loss is therefore obliged to have a permittivity whose real part varies with frequency, and specifically to have a Dk that falls as frequency rises. Holding Dk flat while admitting a nonzero Df asserts an impossibility.
The consequence is not abstract. Transform such a model into the time domain and energy arrives before the wavefront — the response begins before the stimulus. The model below computes exactly that for ten inches of a low-loss laminate.
The standard repair is the Djordjević–Sarkar model, sometimes called the wideband Debye model. A single Debye relaxation gives a loss peak rather than the flat loss tangent real laminates show, so the trick is to superpose a continuum of relaxations spread between two frequency limits. That integral has a closed form:
$$\varepsilon(\omega) = \varepsilon_\infty + \frac{\Delta\varepsilon}{\ln(m_2/m_1)} \ln\!\left(\frac{m_2 + j\omega}{m_1 + j\omega}\right).$$
Between the two limits the imaginary part is nearly constant, which is what a datasheet's flat Df describes, and the real part falls logarithmically, which is what causality demands. The two are not separate design choices; fixing one fixes the other.
The downward slope in Dk is not a defect to be flattened. It is small — a couple of hundredths per decade on a modern laminate — but it is the visible signature of the loss, and a model that removes it has removed the consistency between the two halves of the permittivity.
The claim on the previous two slides can be tested rather than asserted. Take the imaginary part of each model's permittivity, reconstruct the real part from it by the Kramers–Kronig integral, and compare with the real part the model actually has. A causal model reproduces itself; a non-causal one does not.
The same relations, and the same contour-integral argument behind them, are treated at length in Kramers–Kronig Relations. Deck 16 applies the identical test to a measured channel rather than to a material model, where it detects a different failure: a data set whose magnitude has been edited without the dispersion that must accompany it.
A laminate is not a uniform dielectric. It is woven glass yarn, with a relative permittivity near six, embedded in resin near three. The windows between the yarn bundles are wide enough that one leg of a differential pair can sit over glass while the other sits over resin, and the two then propagate at measurably different speeds. The pair arrives skewed, and skew converts differential signal into common mode.
What matters is the spread in effective permittivity seen by the trace, not the raw contrast between the materials: the field spans several yarn pitches and several plies, so most of the contrast averages away. That spread has been measured — Hall and Heck report 0.23 across sixty-four parallel microstrips on 2116 cloth — and that measurement, rather than a mixing rule, anchors the model here.
The standard mitigation is to route the pair at an angle to the weave so that each trace crosses glass and resin alternately and the difference averages out. How much angle is enough is usually given as a rule of thumb; it can be computed instead.
A trace at angle $\theta$ over a length $\ell$ traverses $\ell\sin\theta/p$ yarn pitches laterally, and a periodic modulation sampled over $N$ periods averages down by roughly $1/\pi N$. The model reports that envelope rather than the exact expression, because the exact expression has nulls at particular angles and a real board does not hold its angle to a fraction of a degree across a panel.
A few degrees over a few inches is worth more than an order of magnitude. Free, if it is planned; awkward if the connectors are already placed.
Mechanically flattened yarn fills the windows and cuts the modulation at source. Costs money on the laminate and needs no layout change at all.
Deliberate small jogs achieve the averaging without rotating the whole board. Ugly, effective, and the usual answer when the stackup is fixed.
| Quantity | Expression | Worth remembering |
|---|---|---|
| Skin depth | $\delta=\sqrt{2/\omega\mu\sigma}$ | 2.1 µm at 1 GHz in copper, 0.66 µm at 10 GHz |
| Conductor loss | $\alpha_c \propto \sqrt{f}$ | Fixed by perimeter, proximity and roughness; improved by wider traces |
| Dielectric loss | $\alpha_d=\pi f\sqrt{\mathrm{Dk}}\,\mathrm{Df}/c$ | Linear in frequency and independent of geometry |
| Hammerstad | $1+\frac{2}{\pi}\arctan\!\left(1.4(\Delta/\delta)^2\right)$ | Bounded at 2; adequate below about 5 GHz |
| Huray | Sphere-scattering power ratio | Unbounded; needs a fitted radius and count |
| Djordjević–Sarkar | $\varepsilon_\infty+\frac{\Delta\varepsilon}{\ln(m_2/m_1)}\ln\frac{m_2+j\omega}{m_1+j\omega}$ | Flat Df, logarithmically falling Dk, and causal |
| Weave skew | $\ell\left(\sqrt{\varepsilon_1}-\sqrt{\varepsilon_2}\right)/c$ | About 5 ps/inch worst case on 2116 cloth |
| Skew to mode conversion | $|S_{cd21}|=|\sin(\pi f \tau)|$ | Complete at half a period of skew |
Deck 3 of eleven in Signal Integrity & High-Speed Digital Design. Every figure on this page is computed by si_models/deck03 and embedded as data; nothing is typed in by hand.
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