0.35 or 0.5? Two Constants, Two Different Questions

The confusion

Ask two engineers to convert a rise time into a bandwidth and you'll get two answers: 0.35 divided by the rise time, or 0.5 divided by the rise time. They differ by more than 40%, both are correct, and they answer different questions.

Worth getting straight, because using the wrong one either buys you an amplifier you didn't need or leaves you with an interconnect model that quietly stops being valid halfway up your spectrum.

0.35 is exact, not a rule of thumb

This one surprises people. The 0.35 isn't empirical and it isn't rounded from measurements. It falls out of the single-pole step response.

A first-order system reaches 10% at $t = -\tau\ln(0.9)$ and 90% at $t = -\tau\ln(0.1)$. Subtract:

And the 3 dB bandwidth of that same single pole is $f_{3dB} = 1/(2\pi\tau)$. Multiply:

So 0.35 is $\ln 9 / 2\pi$ to two figures. Nothing approximate about it, given the single-pole assumption.

What it answers: what analogue bandwidth do I need to reproduce this edge? That's your amplifier, your receiver front end, your oscilloscope.

0.5 is a design bound

Johnson and Graham's knee frequency answers something else entirely:

Above the knee, a digital edge carries so little energy that channel behaviour up there stops mattering. It's deliberately conservative, and it's the number you want when deciding how far up your interconnect model has to stay valid — how far to trust a via model, a connector S-parameter file, a trace loss estimate.

So: design the channel to the knee, design the electronics to the 3 dB figure. Different jobs, different constants.

The 20–80% trap

Datasheets quote rise time either 10–90% or 20–80%, and the two are not interchangeable. For a single pole:

Feed a 20–80% number into a 10–90% formula and you under-predict rise time by 37% and over-predict bandwidth by the same factor. That's not a small error and it's entirely silent.

Modern SerDes and DDR datasheets increasingly use 20–80%, because it avoids the noisy tails of the waveform where the measurement gets unreliable. Older logic families usually quote 10–90%. Check which one you're reading. If the datasheet doesn't say, that's worth an email to the FAE.

One caveat on 1.585: it's the single-pole value and the largest factor in circulation. A Gaussian edge gives about 1.52, a linear ramp 1.33. So for a real driver, 1.585 slightly over-states the rise time and therefore under-states the knee. Where margin is tight, design to the shorter estimate.

The mistake that causes EMC failures

Here's the one that actually costs board spins.

Rise time comes from the driver, not the clock. A 25 MHz oscillator with a modern CMOS output stage can have a 500 ps edge. That puts the knee at 1 GHz.

One gigahertz. On a 25 MHz clock.

So the interconnect has to behave to 1 GHz — controlled impedance, continuous reference plane, terminated if it's electrically long — even though the fundamental is 25 MHz and the design "feels" slow. This is the single most common misjudgement I see: slow clock, fast edges, channel modelled only to the fundamental, then a surprise at the EMC lab.

Emissions follow the edge rate. Always.

The cheapest fix in the business

If a signal doesn't need to be fast, slow it down. A series resistor at the driver, or a slew-rate-limited output if the part offers one. Slowing an edge from 500 ps to 2 ns drops the knee from 1 GHz to 250 MHz, and everything downstream gets easier — emissions, crosstalk, termination requirements.

It costs one resistor. Do it on resets, LEDs, configuration straps, anything where speed buys you nothing.

While we're here: measuring it

A couple of practical points that catch people out.

Rise times add in quadrature. Your scope has its own rise time, and what you measure is:

Measure a 500 ps edge on a scope with a 500 ps rise time and you read 707 ps — 41% high. Pick an instrument with at least 3× the signal's 3 dB bandwidth so its contribution stays under about 5%.

Design to the fast corner, not the typical. Datasheet typical rise times are typical. Lightly loaded fast-corner silicon can be two or three times quicker, and that's the corner that sets your emissions. Assume the fastest edge the driver can produce.

The rise time to bandwidth calculator gives both constants side by side, handles the 20–80% conversion, and tells you how many harmonics of your clock fall below the knee.