Download Smith Chart Pdf
Sandrine Willert
I'm not an EE graduate, but practical electronics has always beenmy hobby (albeit typically on the back burner) and I also happen towade into antenna and RF technology on my job every now and then.On a recent encounter with WiFi antennas and directional couplers,as I was simulating some stuff in Qucs, I came across the S-parametersand the Smith chart (once again). I got intrigued, because thisstuff seemd related to my "research into directional couplers".(Well... not all that useful, as I later concluded, but anyway.)
I felt I had to "start from someplace in the middle", as I didn'thave enough of a maths background to start "properly from the ground up".
I started out by spending some time reading primers on the S-parametersand the Smith chart(kudos to Microwaves101.com): what variables are on each axis, why the grid is curved, the distinction between the impedance vs. admittance version of the Smith chart. I was wondering what some typical circuits looked likein the Smith chart - and the interwebs provided surprisingly fewanswers. That's when I started to squint back at Qucs: "Oh wait... I can sketch some circuits in Qucs. And, Qucs contains some Smith charts in the graphing tabs. Now how do I go about this, does it actually work? I could then try fiddling with somecomponent values to see the effect of my changes..."
Essentially, you need a "power source" for each port of your circuit.This component looks like a voltage source with a series resistorin one package. Next, you need an "S-parameter simulation".This will automatically set up the matrix of N by N ports.Once you compute the simulation, in the graphing tab you can select the two versions of the Smith chart, or a "cartesian" Bode plot. You will be offered the S-matrix members for usein your charts/plots.
The following is an LC circuit, coupled by two different series resistors to two different power sources.Note the resulting shape on the Smith chart, the differencesand similarities between the impedance and admittance versionof the Smith chart, and the differences vs. similaritiesamong the various S-parameters.
I've taken screenshots of two, slightly different versionsof the schematic: the difference is in the L and C values,both versions resonate at 50 MHz, but each one with a different Q.Note how this affects the shape of the Bode plot and the"spacing of nodes" in the Smith chart :-)
I want to use a Smith chart in an upcoming article here, but it occurs tome that most readers probably aren't familiar with them. Smith charts don'tcome up very often in audio, being mostly an RF thing. So as with complex numbers, I'm postingthis separate article to introduce Smith charts, and then I can refer backto it when I use them later.
Smith charts are appealing because they look cool. However, they're alsoa bit scary and mysterious, because they're associated with the black magicof RF engineering. In my article on PCB design mistakes, I talked a bitabout the electromagnetic considerations involved in designing a circuitboard. At audio frequencies, we mostly expect wires to be wires andinsulators to be insulators. We want a signal in one PCB trace to stay inthat trace, not hop into a neighbouring trace by means of electromagneticradiation; and the extent of "electromagnetic considerations" relevant toaudio PCB design basically just comes down to making sure that leakagebetween traces isn't going to happen.
With the Smith chart, the general problem being solved is the problem ofvisualizing complex numbers, especially complex impedances for the purposesof matching them. This is a context where it's necessary to think aboutboth large and small values, including some that are theoretically zero orinfinite.
Plotting points in Cartesian coordinates as shown above doesn't work wellif some of your numbers are much larger than others. You might easily haveto deal with numbers like 1Ω and also numbers like 1kΩ in the same problem;a scale that shows one well will either put the other off the chart, or haveit merge into zero. Using log-log graph paper might help, but that hasproblems with zero and negative numbers. There's also an obvious problem ifyou want to plot a point representing infinite impedance.
Now here's a more specific statement of at least one problem the Smithchart is meant to solve: the Smith chart is an answer to the questionhow can we fit both zero and infinity onto the page, and have it allwork out nicely, when making a diagram of complex impedances that happen tohave nonnegative resistive component?
One way to define the Smith chart involves replacing a complex numberz with (z-1)/(z+1) and proving thingssymbolically about the result. That is perfectly correct, but it may not beeasy to understand, especially if complex numbers are unfamiliar. I preferto think of it in geometric and visual intuitive terms, as two stereographicprojections with a rotation in the middle. The resulting derivation isquite straightforward and I'm probably not the first to think of it, but Ihaven't been able to find this approach documented elsewhere on the Web.
Now, here's a close-up of the same sphere, with the original impedanceplane removed. One thing visible is that the entire impedanceplane with positive and negative resistance and reactance from zero toinfinity, fits onto the surface of the finite-sized sphere. So we havealready achieved part of the goal by making the range from zero to infinitycomfortably visible; but it's still rather inconvenient to have a sphericalchart that won't fit on a flat page.
The warped coordinate system of the Smith chart makes it convenient toplot impedances, especially for solving RF design problems, because itcaptures both zero and infinity in the same view. As long as the resistancecomponent is nonnegative, everything stays on or within the boundary circlecorresponding to the reactance axis. So it's an appealing way to presentany data that consists of complex numbers with nonnegative real componentbut potentially a large dynamic range. And that's how I plan to use it inmy future article.
For RF design purposes, it turns out that the Smith chart transformationhas other desirable properties too, largely flowing from the fact thatstereographic projection preserves circles and lines, so not only the gridlines but also other circles and lines on the original impedance planetranslate to circles and lines on the Smith chart.
The ordinary Cartesiancoordinates measured on the Smith chart from the centre point (50Ω or otherstandardized impedance) end up corresponding to the "reflectioncoefficient," which is important in RF transmission-line problems. Important parameters like standing wave ratio and transmission line lengthcan be calculated by just measuring distance from the centre, orangle around the centre or around some other point.
Because of the geometric significance of locations on Smith charts, thesecharts have historically been used not only for presenting data but also asanalog computers of a sort, sometimes involving physical devices likerotating rulers and extra scales on the side. You could buy pads of Smithchart graph paper featuring the special multi-circular grid lines andprinted to precise scale, for working out your own impedance matchingproblems. That kind of computational usage is much less common now thatdigital computers are readily available; now the chart is mostly just usedfor presenting information measured or calculated in some other way.
The (z-1)/(z+1) transformation seems to have beenproposed at least three times independently in the 1930s, by TousakuMizuhashi in 1937; Amiel Volpert in 1939; and Philip H. Smith, also in1939. So it's debatable whether it's really right to call it a "Smithchart"; but that is what's normally done, at least in English-speakingplaces.
In fairness to Smith, what he developed was more than just the(z-1)/(z+1) transformation. He also originated a layout ofcircular scales around the main chart, linear scales arranged at the bottom(originally, on a rotating ruler, turning the whole thing into a kind ofcircular slide rule), and calculation techniques associated with theseelements.
When preparing this article, I thought it would be fun to include somedownloadable PDFs of blank Smith chart forms, and I had a look online. WhatI found was interesting, and I think it reveals a cultural insight into therole of these charts.
There are many downloadable Smith chart forms available on the Web, andmost of them are pretty bad. In particular, many downloadable Smith chartforms are pretty clearly bitmapped scans, of varying quality, of the blankforms that Smith's own company used to sell. Every time, the people doingthe scans have added their own branding to the image. There are also somefiles out there that have been redrawn (Wikipedia includes an SVG versionlike this). One popular version seems to have been redrawn by "Black MagicDesign" - and then heavily copied by a bunch of other people, some of whomadded their own branding too or did things like changing the overall fileformat.
Nonetheless, in every redrawn version of the Smith chart, the "RADIALLYSCALED PARAMETERS" scales are reproduced, even down to non-functionaldetails of graphic design like the way the labels attach to the scales alongslanted leader lines. And except for occasional typos, the spelling andphrasing of the abbreviations for the scale names is carefullypreserved in copies.
The blank chart form has gone through multiple generations of copyingwithout anybody removing the scales that are unlikely to ever be usedanymore, nor correcting or improving the design of still-relevant scales tomake them more usable. Efforts are made to exactly reproduce superficialfeatures like the inconsistent traditional punctuation of the abbreviations,and the uneven selection of which scale ticks get numbers; but when acopyist makes a mistake, it doesn't get corrected because the copyists andeventual readers aren't paying attention to the meaning, only the form.
I also had a look at the PGFplots Smith chartmodule, software for drawing Smith charts in the context of agraphics package I already use for other things. I wasn't too pleased withit. The grid it generates, when on the finest setting, is kind of ugly andhard to read, not using the well-organized fine-tick spacing of thetraditional Smith chart and not easily configurable to do so. It doesn'tuse the traditional pattern of on-chart labels for the ticks, and again,it cannot be easily configured to do so.
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