Charts/Science charts
Smith Chart
<SmithChart /> · 10 examples
Every possible impedance squeezed inside one circle, so matching an antenna becomes a walk instead of an algebra problem.
Overview
A radio engineer has a transmitter, a cable and an antenna, and the first law of the trade is that none of them agree. When the load on the end of a line does not match the line, part of the signal bounces back, power is wasted and amplifiers get warm in unhappy ways. The measure of this disagreement is a single complex number, the reflection coefficient, and the question is always the same: how do I make it go to zero?
Phillip Smith, an engineer at Bell Labs, answered that in 1939 with a chart that maps the entire half-plane of passive impedances onto the inside of a unit circle. Infinite resistance, a number that would normally run off the page, becomes a point on the rim. A perfect match sits in the exact middle. The whole thing was drawn by hand, used with a ruler and a compass, and is still taught in universities, partly because it works and partly because nothing else is as pretty.
A modern network analyser will happily print the numbers. People still ask for the Smith chart because the shape tells you what to do. Add a series inductor and your point slides along a circle. Add a shunt capacitor and it slides along another. Matching becomes a game of moving your dot to the centre, and you can see how far you have left to go.
Props
API
<SmithChart data={sweep} resistance={|s: &Sweep| s.r} reactance={|s: &Sweep| s.x} impedance={50.} />| Prop | Type | Default |
|---|---|---|
data | required | |
resistance, reactance | Fn(&T) -> f64, ohms | required |
impedance | f64: the reference impedance Z₀, ohms | 50. |
grid | Vec<f64>: normalised resistance circles and reactance arcs | [0.2, 0.5, 1, 2, 5] |
overlay | SmithGrid::Impedance, Admittance or Both: which grid to draw; the admittance grid is dashed | Impedance |
color, point_size | first series colour, 5. | |
legend, tooltip, class |
Each row becomes the reflection coefficient Γ = (z − 1) / (z + 1), with z = (R + jX) / Z₀, and the points are joined in order. The centre is a perfect match and the rim is total reflection. Hover a point to read Γ.
Run them
10 examples
Press Run to load an example into the live render at the top of the page. Open one to read its code.
How to read it
- 01Find the centre. A point there is a perfect match, and the closer your trace gets to it, the less power bounces back.
- 02The horizontal axis is the pure-resistance line. Above it, the load is inductive. Below it, capacitive.
- 03Follow the trace in the order of the rows, which is normally frequency. A tight loop near the middle is a good wideband match. A trace that runs out to the rim is a load that reflects almost everything.
- 04Hover a point to read its reflection coefficient and magnitude.
When to use it, and when not to
Reach for it when
- You have impedance versus frequency from a network analyser or a simulator and want to see how well a load matches its line.
- You are designing a matching network and want to see how each added component moves the point.
- You want to compare several sweeps (antennas, filters, amplifier inputs) on a shared, familiar background.
- You are teaching transmission lines. The chart is a better textbook than any table.
Skip it when
- Your audience does not do RF. Anyone else will see a circle of circles and quietly leave. Show them return loss in dB on a line chart.
- You need exact values. The chart is for the shape of the response, and a table or a magnitude-versus-frequency plot reads numbers better.
- Your load has negative resistance. The point leaves the unit circle, and the chart is not designed for it.
- You are plotting something that is not an impedance. Reusing the shape for decoration is allowed but will be noticed.
The math behind it
From impedance to reflection coefficient
Each row gives a resistance and a reactance in ohms, measured against a reference impedance (50 ohms unless you set impedance). Divide through by to get the normalised impedance . The point plotted is the reflection coefficient , which is a function of and always lands inside the unit circle for a passive load.
The grid is two families of circles
The chart does not distort anything by hand. It draws the images of the lines and under the same map. Both turn out to be circles. A line of constant resistance becomes a circle centred on the horizontal axis; a line of constant reactance becomes an arc of a circle whose centre is above or below the axis at the right-hand edge, and only the part inside the unit disc is drawn.
Admittance and mismatch
The default grid values are , and each is drawn for both signs of reactance. With overlay set to admittance, the same circles are reflected through the origin and drawn dashed; with both, you get the double grid used when you hop between series and shunt elements. A point's distance from the centre, , is the size of the reflection, and the standing-wave ratio that most people actually quote follows from it.
Real life usage
The antenna that preferred the wrong band
Tomás built weather balloons in a garage that had lost its battle with a pile of cardboard, and the next release needed a 433 MHz telemetry link that could reach a ground station 40 kilometres away. The radio worked on the bench. The antenna worked on the bench. Together, in the field, they delivered about a third of the range he had promised his two investors, both of whom were his sisters.
He tried everything a hobbyist tries. He moved the antenna. He moved the cable. He shortened the cable, then lengthened it, then shortened it again, and the range changed each time like a mood. A friend who does RF for a living looked at the setup and said, in the tone of a doctor looking at a cough, "Have you measured the match?"
He had not. He borrowed a cheap vector analyser and swept the antenna from 400 to 460 MHz, which gave him 201 rows of resistance and reactance, and a spreadsheet that gave him a headache.
He fed the columns into a zopra-charts <SmithChart> with impedance left at 50 ohms. The trace did not go near the centre. It looped on the left half of the chart, at a point about 0.6 of the way to the rim at 433 MHz, and swung into the middle only around 470 MHz, a band he was not licensed for and was not using.
The antenna was resonant, only at the wrong frequency. Cutting two millimetres from each element would have moved it, and he cut three, because garage engineering is an art. Ten minutes and a hacksaw later, the loop closed on the centre, and |Γ| at 433 MHz went from 0.6 to 0.08.
The next flight carried the same radio, the same cable and the same cardboard, and held the link for 41 kilometres. His sisters asked what had changed. He said, truthfully, "A point moved to the middle of a circle." They gave him a third round of funding on the strength of that sentence.
A hacksaw is the cheapest tuner ever sold. The Smith chart is the only thing that tells you which way to hold it.
Pitfalls
- Wrong reference impedance. The chart is normalised by
impedance. A 75 ohm system plotted at 50 ohms is a perfectly good chart of the wrong thing. - Plotting S11 as if it were Z. If your instrument gives you Γ directly, convert back to and first. This chart takes impedance and does the map itself.
- Order of rows. The points are joined in the order you give them. Sort by frequency before plotting, or the line will zigzag over the grid.
- Too many grid lines. Adding more values to
gridmakes the chart look rigorous and unreadable. Five circles per family is a good number.
From around the web