Sign reflections in digital circuits

In a few of the earlier articles on this weblog, I talked about two necessary and hard-to-grasp phenomena that intrude with the design of high-speed digital circuits. The primary one are signal distortions contributed by PCB parasitics; the opposite is the peculiar nature of square waves that makes them way more unruly than sinusoidal waveforms.

As we speak, I’d wish to deal with one of many boss fights of digital circuit design: sign reflections. This peculiar phenomenon was first noticed on long-distance transmission traces, after which saved cropping up in radio design. It manifested as echoes of previously-transmitted indicators, seemingly “bouncing again” off impedance discontinuities. It occurred everytime you had a supply with excessive present capability driving a low-power load over a sufficiently future of wire. Extra confusingly, the identical situation occurred with wimpy sources driving power-hungry gadgets.

In fashionable occasions, reflections are a recurring headache for ultra-high-speed digital electronics — and though they seldom intrude with pastime initiatives, they’re nonetheless value studying about. Alas, a lot of the common sense explanations you discover on-line don’t make a complete lot of sense; for instance, the optical analogy utilized by Wikipedia will depart you scratching your head. So, let’s attempt one other strategy.

Digital indicators don’t propagate via wires instantaneously, however we all know they journey quick; to measure their pace, it’s attainable to conduct a easy experiment. Let’s take a 100 ft (30.5 m) run of coaxial cable, join one aspect it to a sign generator, and the opposite to a 47 Ω resistor (this roughly matches the present capability of the sign generator). Lastly, let’s carry the ends shut collectively and let’s hook up a pair of oscilloscope leads.

The fantastic thing about coaxial cables is that just about your entire electromagnetic subject is contained inside the construction of the cable, so the sign can’t take any shortcuts — even when the coax is coiled on the workshop ground. By measuring the delay between the sign getting into the cable (yellow hint) and exiting it (blue), we are able to determine how rapidly it traversed 100 ft of wire:

The measurement factors to a propagation delay of about 127 nanoseconds. This means sign pace of 0.24 m/ns — 240,000 km/s — or about 80% the pace of sunshine in vacuum.

This result’s, in essence, the elemental constraint on the trade of knowledge in that medium. And it’s fairly cool that we are able to make this measurement at residence!

Subsequent, let’s take away the 47 Ω resistor on the far finish of the cable and change it with 10 Ω. This resistor will admit way more present than the sign generator can present, so one would anticipate to see the ensuing voltage swing of the sq. wave to be drastically attenuated. However does this occur immediately, or is there a delay? Let’s discover out:

The yellow hint represents the specified waveform; the blue one is the precise voltage as measured on the sign generator’s output port. For the primary ~255 nanoseconds after each transition, the blue waveform seems pretty regular; however then, the rug is pulled and the voltage all of a sudden drops.

What’s happening? Merely put, it might violate causality if the sign generator may know what the resistor linked on the opposite finish goes to do earlier than the electromagnetic wave makes it there and again. During that roundtrip, pushing electrons into the coax cable essentially takes the identical quantity of effort whether or not it’s an open circuit or a brief. Solely after that, the truth of driving a low-impedance load all of a sudden units in.

If this sounds unconvincing, let’s swap from sq. waves to shorter pulses. With this transformation, the echo will present up after the very fact, exactly on schedule — and now pulling the voltage damaging:

The generator may attempt more durable to battle it, however that will solely make issues worse; the phantom electromotive drive it sees throughout its terminals is an echo of its personal actions, shifted in time.

On the threat of restating what may be apparent, the experiment we simply carried out concerned a high-impedance (low-current) sign generator driving a low-impedance (power-hungry) load. Coincidentally, that is the place most different non-mathematical explanations of sign reflections fall quick: in any case, when you have a trickle of water flowing into a big drainpipe, why does something bounce again in any respect? And what’s up with the inverted polarity?

For these causes and extra, deciphering the impact as a relativistic consequence of prior actions — reasonably than leaning on wonky optical or hydraulic analogies — makes extra sense.

For completeness, let’s observe what occurs if we take away the 4.7 Ω resistor on the finish of the road and change it with 1 kΩ. This enables solely a fraction of the generator’s most output present to movement via the load. In impact, we now have a relatively low-impedance supply driving a high-impedance system:

As soon as once more, for the primary ~255 nanoseconds, the generator is merrily pushing electrons onto the wire, full steam forward. Solely then, our acquainted phantom electromotive drive reveals up and says “truly, these had nowhere to go, so right here’s your vitality again.” Within the earlier experiment, the voltage all of a sudden sagged; right here, it shoots as much as nearly twice the anticipated worth.

If we repeat the experiment with quick pulses as an alternative of sq. waves, we are able to isolate the echo and see that in distinction to the sooner check, it has the identical polarity because the preliminary pulse:

The dialogue to this point thought of the implications of driving a mismatched system on the opposite finish of a wire, however we glanced over the query of how tough it’s to push electrons down the transmission line within the nanoseconds earlier than we are able to “see” the load.

Throughout that interval, the road is clearly not behaving as {an electrical} quick: in any case, within the first experiment, the generator efficiently created the specified voltage throughout its terminals, a minimum of for a short whereas. The coaxial cable can also be not an open circuit: within the second experiment, we evidently succeeded at transmitting vitality towards the load, which got here again later and triggered the voltage to shoot up.

The precise reply is a parameter referred to as “particular impedance.” Impedance is an overloaded time period, however the thought may be defined by modeling our coaxial cable as a collection of segments which are quick in proportion to the wavelength of our sign. On this mannequin, every of the chained segments is basically a small capacitor between the sign line and the return path, the phase’s capacitance ruled by the conductor’s geometry and the properties of the encircling dielectric:

At any given time, the propagating wavefront is charging only a small handful of those capacitors; those behind are already taken care of, and those in entrance are out of attain. It follows that the obvious impedance of the stays fixed at some stage in sign propagation, whatever the cable’s size.

Ideally, the precise impedance of the transmission line ought to be matched to the transmitter; that stated, a mismatch on the supply is much less dramatic than a mismatch on the vacation spot. If the precise impedance of the cable is simply too low, the transmitter will be unable to construct up the specified voltage straight away — not till the impedance of the load begins to maintain the present in examine. Conversely, if the precise impedance of the road is simply too excessive, the preliminary present via the load will likely be decrease than it could possibly be.

Most coaxial cables are designed to exhibit particular impedance of fifty Ω or 75 Ω; in the meantime, commonplace traces on a two-layer PCB often measure round 100 to 150 Ω. To be clear, regardless of using a well-known unit of measurement, this doesn’t imply that the conductor displays such resistance to regular currents; it’s merely a approach to cause about what occurs throughout sign propagation in a sufficiently lengthy transmission line.

That relies upon. When your sign’s journey time is brief in proportion to the its fee of change, the reflections from picoseconds or nanoseconds in the past are practically similar to what you’re pushing onto the wire proper now — so the phenomenon has no sensible penalties.

For sine waves, the conservative rule of thumb is that in case your wire or PCB hint is shorter than one tenth of the wavelength, you don’t want to fret about impedance matching. For indicators contained on a small printed circuit board, sine frequencies under 200-400 MHz ought to require no particular precautions on this entrance. That stated, on the higher finish, you continue to want to consider standard parasitics and RFI.

In fact, there may be that gotcha we talked about for sq. waves: they are often regarded as a sum of sine harmonics, and relying on edge rise occasions, there may be a good quantity of vitality current as excessive as 11 occasions the elemental sign frequency; in different phrases, a ten MHz sign may need some properties of a 110 MHz pure-sine wave. This doesn’t at all times spell hassle — digital indicators are supposed to stand up to a good quantity of distortion — however broadly talking, some warning is suggested at PCB information bus frequencies above 50-100 MHz.

When PCB reflections begin getting in the best way of digital signaling, the same old offender is a low-impedance supply driving a relatively high-impedance load (e.g., a MOSFET gate). The only treatment could also be including a “sink” resistor on the receiving finish, linked to the sign’s return path. That is often paired with a collection resistor on the driving aspect, each to restrict peak present and to a minimum of roughly match the precise impedance of the hint.

At gigahertz frequencies, the duty will get extra sophisticated: the impedance would possibly have to modeled with larger precision after which saved in examine with particular board stackups, unique substrate supplies, and the avoidance of vias. That stated, at that time, many different handy circuit design abstractions break down too.

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