A power delivery network has to be verified, and the quantity to be verified is a milliohm. Every instinct carried over from signal measurement fails here, for reasons worth understanding rather than working around.
Deck 11 established that a modern rail needs an impedance of a few milliohms, and a large accelerator core tens of microhms. Deck 12 established that the answer depends on where the two contacts are. Both of those are claims about a physical board, and a claim about a physical board eventually has to be measured.
The difficulty is not precision in the ordinary sense. It is that every technique familiar from signal measurement is built around a reference impedance of fifty ohms, and a milliohm is four and a half orders of magnitude below it.
A few milliohms of contact or lead resistance is the whole quantity being measured. Any two-wire method measures its own leads.
A reflection measurement of anything far below the reference impedance looks like a short, and the difference from a short is what has to be resolved.
Two cables sharing a ground form a loop whose impedance is comparable with the device, so current intended for the device goes round the outside instead.
Each has a specific remedy, and the three remedies are the content of this deck.
The natural instinct is a one-port measurement: send a wave, measure what comes back, invert $Z = Z_0(1+S_{11})/(1-S_{11})$. It works beautifully down to a few ohms and then stops, and the reason is not noise.
As the impedance falls, $S_{11}$ approaches minus one and the measured quantity becomes a tiny departure from a full reflection. How tiny a departure can be believed is set by the analyser's directivity — the leakage of the incident signal into the reflected channel — which is typically forty decibels. That puts a floor of roughly $2Z_0 \times 10^{-D/20}$ on the impedance that can be resolved at all.
Drive through the device instead. Place the unknown as a shunt between two ports and measure transmission. A shunt admittance $Y$ has the ABCD matrix $\begin{pmatrix}1&0\\Y&1\end{pmatrix}$, from which
$$S_{21} = \frac{2Z}{2Z + Z_0}, \qquad Z = \frac{Z_0}{2}\cdot\frac{S_{21}}{1 - S_{21}}.$$
The key feature is the direction of the inequality. A small impedance produces a small transmitted signal, not a small deviation from a large one. A milliohm transmits about −88 decibels, and an analyser has well over a hundred decibels of dynamic range, so the measurement sits comfortably inside the instrument's capability — and there is no directivity term at all.
The method has one failure mode and it is severe. The two ports share a ground through the cable braids, so there is a second path from one port to the other that does not go through the device. At low frequency the braid impedance is comparable with a milliohm, and the measurement reads the two in parallel.
The error is always in the same direction — the measurement reads lower than the truth, because a parallel path can only reduce the apparent impedance. A power delivery network that measures better than it is, is exactly the wrong error to make.
A common-mode isolator, transformer or semi-floating differential amplifier in one port's path, which breaks the braid loop without breaking the signal path. Needed below roughly a megahertz, where the braid's own inductance has not yet made it a high impedance.
Measure a known short and a known milliohm standard. Without an isolator the two read almost the same at low frequency; with one, they separate. It is a two-minute check and it is the difference between a measurement and a number.
At direct current and up to the low kilohertz, the network-analyser methods are replaced by the oldest trick in low-resistance measurement: separate the wires that carry the current from the wires that sense the voltage.
A two-wire measurement puts the lead and contact resistance in series with the unknown and reports the sum. Since the leads are themselves several milliohms, the measurement is mostly of itself. A four-point Kelvin arrangement forces current through one pair and senses voltage with another, and because the sense pair carries essentially no current its own resistance drops out.
Kelvin sensing is exact and slow; shunt-through is fast and fails below the frequency where the ground loop bites. A complete impedance profile is therefore usually stitched from two measurements, and the overlap region is where they are checked against each other. A discontinuity at the join is the usual sign that the isolator is missing.
The same principle appears again in the board itself. A regulator that senses its output at its own pins regulates the voltage at its own pins; one that senses at the load regulates the voltage at the load, and the difference is the IR drop along the path — which for a hundred-amp rail at a milliohm is a hundred millivolts, or more than the entire ripple budget.
Deck 12 established that impedance is a property of a pair of points. That makes the choice of probe location part of the measurement's definition rather than a practical detail, and there are only a few defensible choices.
| Where | What it tells you | What it misses |
|---|---|---|
| At the load's own pads, or as close as the package allows | What the load actually experiences — the right answer | Usually inaccessible, and adding a probe changes it |
| At a dedicated test point near the load | A good proxy, if the spreading inductance between the two is small | The spreading inductance between the two, which slide 02 of deck 12 says is logarithmic and non-zero |
| At the regulator | The regulator's view, useful for loop stability | Almost everything above a megahertz |
| Between two arbitrary plane points | The cavity's behaviour | The die, the package, and therefore the Bandini Mountain entirely |
The last row is worth dwelling on. A board-level measurement cannot see the largest feature in the profile, because that feature is on the far side of the package inductance. A board that measures beautifully can still sit under a die that is seeing a tall peak, and no amount of care with the instrument will reveal it — which is why on-die voltage monitors exist.
The purpose of measuring is usually to check a simulation, and the same warning applies here as in deck 16: agreement on the quantity that is easy to plot does not imply agreement on the quantity that matters.
The broad shape: the capacitive fall, the resonances of each tier, the overall level. Getting these right mostly requires having the right parts in the model, which is bookkeeping.
The height of the anti-resonant peaks, which depends on damping and therefore on the equivalent series resistance of every branch, on board copper loss, and on the dielectric loss of the planes. Those are the numbers least likely to be in a datasheet and most likely to be guessed.
Since deck 11 showed that the peaks are what decide a design, a model that matches everywhere except the peaks has failed at the only place it was needed. The practical discipline is to correlate on peak height explicitly, and to treat a model that is too good at the peaks — predicting a lower impedance than measured — as the more dangerous error, since it is the direction that ships a board.
A frequency-domain correlation can be satisfied while the transient response is wrong, for the same reason deck 16 gives: the phase relationships across the band decide where the energy lands in time. Comparing a measured droop against a simulated one, as deck 12 computes it, is a stronger test than comparing two impedance curves.
| Method | Expression | Range and failure mode |
|---|---|---|
| One-port reflection | $Z=Z_0\frac{1+S_{11}}{1-S_{11}}$ | Down to about an ohm; limited by directivity, not by noise |
| Two-port shunt-through | $Z=\frac{Z_0}{2}\frac{S_{21}}{1-S_{21}}$ | Down to a milliohm and below; fails at low frequency on the braid loop |
| A milliohm shunt | $S_{21}\approx -88$ dB | A small signal in a large dynamic range, which is why it works |
| Ground-loop error | Parallel with the braid | Always reads low; cure with a common-mode isolator |
| Four-point Kelvin | Force and sense separately | DC to kilohertz; exact, slow |
| Where to probe | — | Part of the definition; a board measurement cannot see the Bandini peak |
| Correlation | — | Correlate on peak height, and check the transient |
Deck 13 of eleven in Signal Integrity & High-Speed Digital Design. Every figure on this page is computed by si_models/deck13 and embedded as data; nothing is typed in by hand.
Single-page HTML · KaTeX-rendered maths · no build step. Source on GitHub