3D Smith Charts
July 16, 2012 9:08 PM   Subscribe

The 3D Smith chart differs from previous attempts to generalize the planar 2D Smith chart in a fundamental way: the way in which infinity is treated.
posted by Confess, Fletch (28 comments total) 15 users marked this as a favorite

As someone who's not an electrical engineer, I'd love to have some background on what Smith charts are, how they're used, why they're important, and why this work is interesting and novel. The combination of the linked article and the Wikipedia article for Smith chart are leaving me at a little bit of a loss.
posted by Serf at 9:24 PM on July 16, 2012 [2 favorites]

I just went to talk where the speaker used blackholes to make thermodynamics into something solvable with field theories, so can you also explain Smith charts to me using blackholes? It would really unify all the things I don't understand.
posted by Chekhovian at 9:27 PM on July 16, 2012

(fine, I'll say it for everyone here for whom complex transform functions and Riemann spheres are totally meaningless...)

posted by hanoixan at 9:28 PM on July 16, 2012

hanoixan: "What."

Well heavens knows we're miserable now.
posted by boo_radley at 9:30 PM on July 16, 2012 [3 favorites]

boo_radley, don't be sad. I just want to understand.
posted by hanoixan at 9:34 PM on July 16, 2012

posted by idiopath at 9:38 PM on July 16, 2012 [1 favorite]

(fine, I'll say it for everyone here for whom complex transform functions and Riemann spheres are totally meaningless...)
I get that there are alternative geometries to Newtonian, but would be helpful to know whether this will help me trade my personal pork bellies account to the point where I stand a reasonable chance of bankrupting Goldman Sachs.
posted by b1tr0t at 9:44 PM on July 16, 2012 [3 favorites]

This is pretty cool. Figure 2 really drives home the improvement in data analysis this model seems to provide.
posted by It's Raining Florence Henderson at 9:51 PM on July 16, 2012

I can still get remarkably peeved when reminded about an exhibition a vaguely friend of a friend of a friend artist held at the State Gallery a couple of years ago. Not so much because I think sticking a bunch of simple UJT LED flashers in a glass box and calling it "art" is stupid, but because sticking a bunch of Smith charts that pre-school kids have scribbled on in a glass-topped drawer and calling it "art" somehow offends my now-usually-latent technician's sensibilities.

Which is my roundabout way of saying I like this post ;-)
posted by Pinback at 9:51 PM on July 16, 2012 [1 favorite]

I don't know why, but I expected Agent Smith.
posted by Malice at 10:00 PM on July 16, 2012

Hrrm, I actually studied a lot of electromagnetics and microwave engineering in college. Even built a program that graphed Smith charts in the 80s. How to explain simply...

Radio Frequencies (RF) at the wavelength that you would look at are at the human scale, millimeter, meters, etc. If you can imagine an RF wave like a up and down sine curve traveling through space, the distance between 'humps' would be something you could measure with a ruler. For example, there are small holes in the 'window' of your microwave oven. The RF waves used in the microwave are so long, that can't fit through the small holes. Physicality of the environment has a huge effect on the RF waves.

RF at these wavelengths interacts with the physicality of whatever it's transmitted through, air, waveguide, cables, etc. The physical properties of the transmission medium affect properties like capacitance, impedance, and inductance. The math gets complicated fast and is difficult to visualize in your head.

A Smith chart allows an engineer to visualize a circuit, or partial circuit, to determine if it does what they want, or what changes to the circuit are necessary to make it do what they want.

In the days before widespread use of circuit design software, these kinds of diagrams were critical to properly designing things like radars, radio towers and receivers, televisions, and basically all forms of wireless communication and transmission.

Hope that makes sense.
posted by Argyle at 10:01 PM on July 16, 2012 [14 favorites]

Are they relevant at audio frequencies in analog circuits on PCBs or silicon?
posted by b1tr0t at 10:19 PM on July 16, 2012

Thanks, Argyle! Any chance you could take a shot at explaining the problem that this article is trying to solve?
posted by Serf at 10:22 PM on July 16, 2012

A Smith chart about infinity? Is that like starting something you couldn't finish? And where is the Light That Never Goes Out?
posted by Ratio at 10:35 PM on July 16, 2012 [2 favorites]

I remember when the Smith chart was trying to check into the motel and they said "Smith? Really? Your name is Smith?"

"No, I'm sorry, I meant to say Jones polynomial.
posted by twoleftfeet at 10:51 PM on July 16, 2012 [1 favorite]

Translation: A circuit analyst took an undergraduate complex analysis course.

Really, there's nothing new here. Stereographic projection has been around since the freakin' pyramids.

Holy frijoles, that is a heap of buzzwords right there.
posted by erniepan at 11:05 PM on July 16, 2012 [1 favorite]

One thing Smith charts are good for is impedance matching cables to antennas. If a cable isn't the right length some energy from the transmitter will never make it to the antenna. One thing you can do to fix that is to add a stub. Quickly find yourself a length of coaxial cable, short it out at one end, add it to the circuit, problem solved. Play around with the Smith chart and it will tell you how many centimeters long that stub needs to be. For more details, see here. I say quickly because I've seen how fast old hands can get it done out in the mud.

Smith charts look like alchemy, and using them feels like alchemy. It's primal electrical engineering. Allow this unquestionably occult diagram to guide your knife and hot iron and you can command invisible forces to do your bidding.
posted by justsomebodythatyouusedtoknow at 11:31 PM on July 16, 2012 [9 favorites]

Thanks, Argyle! Any chance you could take a shot at explaining the problem that this article is trying to solve?

I can have a go, if you like.

The 2D chart is trying to represent what is, in fact a 3D phenomenon. It is analogous to representing the 3D surface of the earth on a 2D surface. It is a map.

In visualising the surface of the spherical earth on a piece of paper, we use a projection, for example the Mercator. In a projection you always have to compromise something. In an equatorial Mercator projection, the line around the equator yields a proportional representation of distance. The further away you get from the equator, the more that breaks down until, at the poles, there is no correspondence at all.

An equatorial Mercator projection is great for exploring the equatorial region. It is crap for exploring the poles.

In the 2D Smith projection, the "thing that is being compromised" is the behaviour of certain types of electronic circuits called "active circuits" under condition of open circuit. This is an important region for people like microwave designers and negative amplifier designers. Using 2D Smith projections, they are like polar explorers trying to use an equatorial Mercator projection - they can't "see" anything.

The choice of the equator in a Mercator projection is arbitrary. If you wanted to explore the poles, you could construct a different projection that preserved polar distances but compromised "east" and "west". You can do that, too, with a 2D Smith chart but then you can't "see" short circuits.

There are certain design situations where you need to "see" open and short circuit conditions simultaneously. The 3D Smith chart seeks to let you simultaneously "see" the north, south, east, and west poles so that you can solve certain problems graphically. It projects the 3D space from an arbitrary viewpoint together with geometry in the 3D space corresponding to things like amplifier stability regions. It becomes a trivial exercise to modify designs to ensure the relevant polar regions are covered.
posted by falcon at 11:58 PM on July 16, 2012 [5 favorites]

Even Riemann couldn't figure out what was happening until he realized that he was being stereographically projected.
posted by twoleftfeet at 12:16 AM on July 17, 2012

Another attempt to explain:

In AC circuits, the concept of resistance gets generalized to impedance; where resistance is a real number, impedance is a complex number. (The imaginary component has to do with the circuit's effect on phase, and the reason complex numbers are a reasonable way to represent this has to do with Euler's identity.)

The Smith chart is just a chart of the complex plane, so a point on the chart represents a particular possible circuit impedance. The various intersecting curves are "natural directions": adding or modifying various kinds of circuit elements will shift a circuit's impedance along those lines. So, a Smith chart lets you transform an algebraic problem ("given these building blocks, how do we form an equation that has the value X?" / "What happens if the circuit element represented by this term in the polynomial changes?") into a spatial problem ("you're at THIS point in a maze, how do you get to THAT point?" / "If you go a little farther along THIS segment of the path, how does that affect where you end up?"), which is much, much easier for humans to manipulate and understand.

The "natural directions", the lines on the Smith chart, are like the meridians and latitudes and rhumb lines on a map. They represent pathways a traveler could take. If a ship sails due South, it'll follow its meridian. That might or might not be a straight line on your map (depending on your map's projection). But if you're giving instructions to the captain, or trying to chart the ship's course from dead reckoning, then you need to be able to translate between the "natural" straight-line compass directions and their corresponding curves on your map. The Smith chart has an analogous function in circuit design.

The 2D chart is trying to represent what is, in fact a 3D phenomenon.

As I understand it, that's not really true. The (extended) complex impedance plane is a 2-dimensional manifold, just like the Cartesian plane is— it's just that its geometry isn't very plane-like, it's not Euclidean. Like a projection of a globe onto a flat map, parts of the complex plane are increasingly distorted, or even pushed off to infinity, like the north and south poles on a Mercator map. The traditional Smith chart only shows half the plane: the half that represents so-called passive electronic components. The 3D smith chart is basically an observation of the fact that you can also map the complex numbers to a sphere's surface, the Riemann sphere, and this lets you chart a much wider variety of circuits.

The Riemann-sphere-projected-to-2D chart seems kind of useful for analysis or visualization, but not so much for design. Though I do like the idea of some alternate-timeline electrical engineers using polished-and-engraved bocce-ball Smith spheres and wax pencils for circuit design, along with their slide rules and nomographs.
posted by hattifattener at 12:57 AM on July 17, 2012 [9 favorites]

Picked up Microwave Journal hoping to find good tips for reheating last night's dinner, cooking up frozen burritos and defrosting turkey; was disappointed.
posted by kcds at 3:11 AM on July 17, 2012 [2 favorites]

To me, the best use of a Smith chart is having one on your over-sized coffee cup. It chases off non-EE's and is a quiet statement of relative intellectual horsepower, useful in tense interdepartmental meetings when things may or may not go your way and you need to intimidate your opposition from say, the marketing or HR departments. Timely word drops of reactance, admittance, mohs, normalize and complex impedance and all of a sudden, silence in the room.

Some yoyo with an Excel spreadsheet gets in your face, and out comes your Smith chart. Problem solved.

(can't express my delight that Smith charts made it to a metafilter post. interesting new visualization. maybe i can get one silkscreened on my Klein stein. )
posted by FauxScot at 4:27 AM on July 17, 2012 [4 favorites]

As I understand it, that's not really true.

No it's not. The projection of the manifold is 3D. The space itself is not. There are only so many levels of abstraction a simplification can span, I guess.
posted by falcon at 6:29 AM on July 17, 2012

Serf: As someone who's not an electrical engineer, I'd love to have some background on what Smith charts are, how they're used, why they're important, and why this work is interesting and novel.
As someone who graduated cum laude in Electrical Engineering, ... I think FPPs should throw a bit more of a bone to the readers, and I'm getting tired of "Inscrutable Lede!" posts.
posted by IAmBroom at 10:13 AM on July 17, 2012 [1 favorite]

Isn't what makes a 2D Smith chart really useful the fact that you can plot actual values on it and make actual measurements to decide what to do? So while the 3D Smith chart is a great concept, don't you need an actual physical object to make it useful?

Something like a globe with a grid you can plot points on, and then a moveable transparent plastic hemisphere (with a "natural" grid ruled on it) that you could lay over it to read displacements relative to your point. Is the article advocating this, or does someone make one of these already?

(And since I only know math, I'm curious about what distinguishes the active and passive impedance hemispheres? I'd put a few bucks on it being the sign of the real part of the impedance, but that's just a guess.)
posted by benito.strauss at 11:02 AM on July 17, 2012

b1tr0t: "Are they relevant at audio frequencies in analog circuits on PCBs or silicon?"

They can be, though the parameters & effects you're designing around are much less critical / problematic at lower frequencies &/or with silicon.

Here's an example of somebody discovering the use of a Smith chart for visualising speaker impedance (I'm not sure that it's as novel as they think it is, since essentially the same thing was taught to me as an apprentice nearly 30 years ago using engineering texts from the 50's).

Audio circuits, particularly ones with long lines (e.g. old-style telco trunks) can be thought of as a transmission line of complex impedance, with more complex impedance terminations at either end. Same goes for old-school amplifiers with valve output stages & impedance matching transformers. As such, Smith charts can come in handy at the design stage, particularly when you're getting towards the edges of what you can squeeze out of them.

Typical modern analogue audio electronics? Not so much, although they still have application for designing modern wideband amplifiers (e.g. some people like to think they need Hi-Fi amplifiers that are essentially flat from DC to 100kHz). You still occasionally see Smith charts for high-end high-power audio transistors because, while their inherent impedance is complex but small, it can be important to understand when designing circuits that may push them towards the edge of their parameters.
posted by Pinback at 3:37 PM on July 17, 2012

What about analog signal processing - multipole filters, wave shapers, and that sort of thing?
posted by b1tr0t at 4:49 PM on July 17, 2012

And since I only know math, I'm curious about what distinguishes the active and passive impedance hemispheres? I'd put a few bucks on it being the sign of the real part of the impedance, but that's just a guess

It's a good guess though. Corresponding to the sign of the real part of the impedance, passive devices absorb/consume energy and active devices produce/inject energy. (They usually "produce" it by drawing it from a power supply, of course, but the existence of power supplies is generally outside the realm of what you're using a Smith chart for.)

I'm surprised that Smith charts are very useful for audio applications. Smith charts are great for narrowband circuits, but audio stuff is wideband; the impedances of various components will vary depending on the frequency. I guess you could plot the locus of the circuit/speaker(/cabinet?) impedance across a range of frequencies, kind of like a Nyquist plot?
posted by hattifattener at 9:19 PM on July 17, 2012 [1 favorite]

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