SI/PI Units, Conventions, and Reference Planes

Why conventions cause real failures

In low-speed board design, numbers are forgiving. If a supply rail is labeled 3.3 V or a clock is labeled 50 MHz, everyone on the team understands the circuit in roughly the same way. In signal and power integrity, almost every number you encounter — whether it is a rise time, jitter budget, insertion loss, or trace impedance — is fundamentally incomplete without its convention, bandwidth limit, or reference plane.

A unit mistake is not a rounding error. It is usually a factor of two, or 2π, or the difference between a standard deviation and a peak-to-peak width — and those do not show up as a slightly wrong answer. They show up as a design that passed review and fails on the bench.

Worse, most of them are silent. Nothing in a spreadsheet objects when a 20–80% rise time is fed into a formula expecting 10–90%. The number that comes out is plausible, and it is wrong by about 40%.

Rise time: two definitions, and no single factor between them

The 10–90% rise time is the standard signal-integrity convention. The 20–80% figure is common in high-speed serial-link specifications, because channel loss rounds off the corners of an edge, making the 0–10% and 90–100% tails noisy and difficult to measure cleanly on an oscilloscope.

The ratio between them depends on the shape of the edge, and the widely-repeated "about 0.6" is not right for any of the shapes people actually mean. Each of these has a closed form, so they are facts rather than fits:

So the spread across plausible shapes is 0.63 to 0.75 — nearly 20% — and using 0.6 is wrong by about 9% even for the Gaussian edge it is usually attributed to. Mixing the two conventions without converting inflates a 10–90% time by 1.52× for a Gaussian edge, not the "almost twice" a 0.6 factor suggests. Every knee frequency and critical length built on it then comes out wrong in the dangerous direction — too optimistic.

A concrete worked example: Suppose an ASIC driver produces an edge with a measured 20–80% rise time of 30 ps. Modeled as a Gaussian cumulative step (ratio 0.657), its equivalent 10–90% rise time is:

The habit worth having is not to memorise a better factor. It is to identify the shape before converting, and to say which one you assumed when you write the number down. If you cannot tell — which is common, because a real edge is none of these exactly — measure both percentages off the same waveform rather than converting one into the other.

ω and f, and the 2π that goes missing

Frequency f is in hertz — cycles per second. Angular frequency ω is in radians per second, and ω = 2πf. Impedance formulas are written in ω; instruments and specifications are written in f.

Rate, symbol rate, and the unit interval

These three are the same thing only for binary signalling, and PAM4 broke that assumption for good.

• Bit rate — bits per second. What the interface is sold as.
• Symbol rate — symbols per second, in baud. For NRZ it equals the bit rate; for PAM4 it is half.
• Unit interval (UI) — the duration of one symbol. UI = 1/symbol rate.
• Nyquist frequency — half the symbol rate. This is the frequency a channel budget is written at.

So PCIe Gen 6 at 64 GT/s has a 32 Gbaud symbol rate, a 31.25 ps UI, and a 16 GHz Nyquist — the same Nyquist as Gen 5 at half the data rate. Quoting "64 GT/s" and reaching for 32 GHz is the single most common error in this family.

RMS, peak, and peak-to-peak

Three ways to describe an amplitude, and they are not interchangeable — most importantly, they combine differently.

• RMS is not the same thing as σ, except when the mean is zero. In general RMS² = mean² + σ², so the two coincide only for a zero-mean quantity. For jitter about its own average that is fine and the shorthand is harmless; for a supply ripple sitting on a DC rail, or a skew measured against an absolute reference, it is a real error — the RMS of a 1 V rail with 10 mV of noise is 1.00005 V, not 10 mV. Check that the mean has been removed before using the words interchangeably.
• Quadrature addition needs more than "random". σ = √(σ₁² + σ₂²) is the independent case. The general form carries the covariance, σ² = σ₁² + σ₂² + 2·ρ·σ₁·σ₂, and ρ is not always zero: two jitter terms driven by the same supply, or two skew terms sharing a clock tree, are correlated by construction. With ρ = +1 they add linearly; with ρ = −1 they cancel. Root-sum-square is an assumption you are making, and it is worth saying so.
• Peak-to-peak is what you quote for anything bounded. Bounded contributions add directly, worst case to worst case.
• Peak is half of peak-to-peak for a symmetric signal, and the source of a great many factor-of-two errors. For an asymmetric one it is not half of anything — say which peak you mean.

Adding a peak-to-peak deterministic jitter to an RMS random jitter in quadrature understates the total. Adding two independent RMS contributions linearly overstates it. Neither error announces itself — and neither does assuming independence between two terms that share a cause.

Decibels, and what they are relative to

A decibel is always a ratio, and the suffix says what the reference is. Dropping the suffix drops the meaning.

And the factor of 10 or 20 carries a hidden condition. A decibel is defined on power: 10·log₁₀(P₂/P₁). The voltage form with 20 in front of it is a consequence of P = V²/Z, and it is only equivalent to the power form when the two voltages are measured across the same impedance. In a 50 Ω instrument chain that holds everywhere and the distinction never bites. It stops holding the moment an impedance changes between the two points — across a transformer, at a probe tip loading a high-impedance node, at the input of an amplifier with a different input impedance from its source, or between the two sides of the impedance step that the whole of this site is otherwise about. A "−6 dB voltage divider" into a different impedance is not −6 dB of power, and which of the two a specification means is worth reading rather than assuming.

The 10-versus-20 distinction is the one that bites. S-parameters are voltage ratios, so S₂₁ in dB uses 20·log₁₀. Use 10 and a 14 dB channel becomes a 7 dB one.

Insertion loss and return loss, and their signs

Both are quoted as positive numbers in conversation and are negative quantities in a file, which causes endless confusion.

• S₂₁ is transmission. For a passive channel it is at or below 0 dB. "12 dB of loss" and "S₂₁ = −12 dB" are the same statement.
• S₁₁ is reflection. "20 dB of return loss" means S₁₁ = −20 dB, and more return loss is better — which is the opposite of how "loss" reads everywhere else.
• α, the attenuation constant, is a rate in dB per inch and is positive and rising with frequency. A plot in dB/inch going up and an S₂₁ plot going down are consistent; the giveaway is the "per inch".

Single-ended, differential, and common mode

A differential pair has two impedances, and four ways to normalise a measurement. The relationships are fixed and worth memorising:

In mixed-mode S-parameters the naming is output mode first: SCD21 is differential in, common out. That ordering is the single most common misreading, and SCD21 is the plot that predicts an emissions failure.

Reference planes: the one that invalidates everything

Every S-parameter, every loss figure and every eye measurement is relative to a place. Measured at the connector it includes the launch; de-embedded to the board it does not; after the receiver's equaliser it is a different picture again.

Two numbers measured to different planes are not comparable, and nothing in either number says so. When a simulation and a measurement disagree, this is a more common cause than the physics.

Go deeper — the factor-of-two table

Nearly every units error on this site's subject matter is one of these. Worth reading once and recognising later.

Confusion	Factor	Which way it hurts
20–80% vs 10–90% rise time	0.6	edge looks slower → knee too low → under-designed
peak vs peak-to-peak	2	either way, depending on which is assumed
± tolerance vs peak-to-peak window	2	a PDN twice as expensive, or half as good
10·log vs 20·log	2 (in dB)	a 14 dB channel read as 7 dB → wildly optimistic
ω vs f	6.28	every reactance wrong by 2π
bit rate vs symbol rate (PAM4)	2	Nyquist doubled → a channel judged impossible
TDR distance vs delay	2	a discontinuity located twice as far away as it is
MT/s vs clock MHz (DDR)	2	UI halved or doubled

The TDR one deserves a note because it is geometric rather than definitional. A TDR sees a reflection after a round trip, so a feature 2 inches away shows up at 4 inches' worth of delay. Instruments divide by two to report distance — but only if you told them the right propagation velocity, which depends on D_k. Get that wrong and every distance on the plot is wrong by the same ratio.

Worked conversion: Suppose a TDR trace on a low-loss laminate (D_k ≈ 3.3, propagation delay ≈ 150 ps/inch) shows an impedance bump at 1 ns after the launch. The one-way electrical delay is 1000 ps / 2 = 500 ps. The physical distance to the discontinuity is:

Distance = (1000 ps / 2) / (150 ps/inch) = 500 ps / 150 ps/inch ≈ 3.33 inches forgetting the round-trip factor of 2 puts you at 6.67 inches — past the connector you were looking for

A related habit worth building: quote delays in picoseconds in a report and inches only in conversation. Picoseconds are what was measured. Inches are an interpretation that carries an assumption about the material.

In the real world

The reason this page exists is that units errors are the most common preventable failure in the subject and the least interesting to find. They do not require insight to discover, only attention — and they cost exactly as much as a genuine physics mistake.

The habit that prevents most of them is writing the unit next to every number, everywhere, including in variable names. This site's code does it: trUI, lossDb, rjRms, trSamples. It looks pedantic right up until the afternoon it saves.

Related

• Signal Bandwidth: Why Edge Rate Matters — where rise time actually matters
• Jitter — RJ, DJ, DCD, PJ and TJ — why RMS and peak-to-peak cannot be added
• S-Parameters: Insertion Loss, Return Loss, and Mixed Mode — reference planes, port order, normalisation
• Differential signalling — odd, even, differential and common
• Time-Domain and Frequency-Domain Analysis — the transforms these units describe