Folding infinity into a circle
Almost every engineering student is a little scared of this thing. One look and you
can see why — it's a dartboard somebody left out in the rain. Circles inside circles.
Numbers crawling around the edge. Not one straight line in sight.
Here is what nobody tells you: it is not hard, and it was never meant to be clever.
A man drew it in 1939 because he was bored. He had to do the same slow arithmetic
every single day, and he would rather draw than calculate. Computers have been able
to do that arithmetic instantly since about 1975 — and every radio instrument built
this year still puts his circles on the screen anyway.
This is the story of why. You do not need an RF background. Every figure below
is alive: drag it, break it, watch what happens.
01 Two men, a field, and a lot of shouting
Deal Beach, New Jersey. 1929. A field full of wire, and two engineers who cannot see each other.
One of them is standing at the base of a large transmitting antenna — the kind used to carry telephone calls across the Atlantic and out to ships at sea. His job is to move a little piece of metal, a tuning stub, a few centimetres at a time. He has no idea whether he is helping.
The other one is far away at a measuring set, reading numbers off a dial. He shouts them back. Somebody then has to turn those numbers into something meaningful with a slide rule, decide which direction the stub should move next, and start over. If the guess was wrong, the whole loop runs again. This is how the afternoon goes.
The younger man is Phillip Hagar Smith, one year out of Tufts. He is not new to this frustration. He had been a radio amateur as a teenager, fighting exactly the same physics in his own backyard with equipment he could afford, which is to say almost none. Bell Labs hired him and handed him the professional-sized version of his childhood problem.
And here is the thing that should annoy you on his behalf: the physics was not a mystery. Oliver Heaviside had written the exact equations down in the 1880s. Nobody was confused. The answer was known and it was simply, brutally slow to compute — every time, all day, by hand.
To understand why two grown men were stuck in a field shouting numbers at each other, you need one fact about wires that first-year circuit theory never mentions.
02 The day a cable stops being a cable
Plug a lamp into the wall with a two-metre cable. Nothing interesting happens. The voltage at the plug is the voltage at the lamp. The cable is furniture.
That is true — but only because of a coincidence. Mains electricity in most of the world is 50 Hz, and one cycle of a 50 Hz wave is 6000 kilometres long. Next to that, your two-metre cable is a rounding error. So it behaves like a plain connection.
Hold on — what is a “wavelength”?
How far the signal travels during one full up-and-down cycle. Electrical signals move at roughly 300,000 km per second, so you get the wavelength by dividing that speed by the frequency. 50 Hz gives 6000 km. 3 GHz gives 10 cm.
This single number decides everything that follows. It is not the length of your cable that matters. It is the length of your cable compared to the wavelength.
Now push 3 GHz through the same cable — microwave territory, just past the top of the 2.4 GHz Wi-Fi band. The wavelength collapses to 10 centimetres. Suddenly a 2.5 cm scrap of copper is not a detail; it is a quarter of a wavelength. And a quarter wavelength happens to be the exact length that turns a dead short into an open circuit. Your wire is not a connection any more. It is a component, and you did not choose its value.
Why? Because the signal needs real time to reach the end of the cable. If whatever is sitting there does not swallow it completely, part of it comes back. Now two waves share one piece of copper: one heading out, one heading home. They add up. And what they add up to depends on where you are standing.
There is a rule of thumb for when you can stop worrying. If the wire is shorter than about one tenth of a wavelength, every point on it sits at nearly the same voltage and the circuit theory you already know works fine. Longer than that, and two points on the same wire are at different voltages at the same instant. At that moment the wire earns a new name: a transmission line.
Do not take my word for it. Below is that same two-metre cable, with the voltage drawn as it really behaves — swinging up and down. Turn the frequency up and watch it stop being a wire.
That is what Smith was fighting in the field. Not a hard idea — a moving target. Every centimetre his colleague walked, the answer changed.
03 One number for the whole load
Now the good news, and it is the key to the entire rest of this article.
Whatever you bolt onto the end of that cable — an antenna, an amplifier, a chip, a resistor, or literally nothing — the cable only ever learns two things about it: how much of the wave comes back, and how late it arrives. Two numbers. Not a schematic. Not a part number. Two numbers.
And two numbers travelling together is exactly what a complex number is. We call this one the reflection coefficient, and we write it Γ (gamma). The size of Γ is how much bounced back. The angle of Γ is how late.
Quick refresher — impedance, and why it needs two numbers
Resistance is how hard a component pushes back against current. Impedance is the same idea for AC signals, but with a second half to it.
A resistor pushes back and keeps voltage and current marching in step. A capacitor or an inductor also shifts their timing relative to each other. One number cannot describe both effects, so impedance uses two: R, the honest push-back, and X, the timing shift. We stack them into one symbol and write Z = R + jX. The j is just a bookkeeping mark meaning “this part is the timing one”.
Positive X means inductive — coils, and anything that behaves like one. Negative X means capacitive. That is the whole vocabulary you need here.
Before the formula, a picture. Tie a rope to a wall and shake the free end at just the right speed. Something strange shows up. Some points on the rope barely move at all. Others swing hard between two limits. The rope looks frozen into one shape that only breathes in and out.
Nothing is actually standing still. A wave is racing along that rope the entire time. You are seeing two waves at once — the one you sent, and the one the wall sent back. At some points they cancel. At others they pile up. That is a standing wave, and it is the same pattern you just watched appear in the copper in Fig 1.
The ratio between the biggest point and the smallest point tells you how badly the far end failed to absorb your signal. That ratio has a name you have probably seen on an antenna datasheet: SWR.
Two things about this formula matter more than the formula itself.
First: if the load happens to equal Z0, the top of the fraction becomes zero and nothing comes back at all. This is called a match, and an embarrassingly large part of RF engineering is just trying to reach it.
What is Z₀, and why is it always 50 Ω?
Z0 is the characteristic impedance of the cable. It is not a resistor hidden inside it — you cannot find it with a multimeter. It is the voltage-to-current ratio a wave naturally settles into while it travels down that particular cable, decided purely by geometry: conductor thickness, spacing, and whatever plastic sits between them.
The famous 50 Ω is a compromise from the 1930s. Coaxial cable of a given size handles the most power at around 30 Ω, and has the lowest loss at around 77 Ω. Somebody split the difference, the industry agreed, and here we all still are.
Second, and this is the one to remember: a passive load cannot send back more energy than it received. Physics does not allow it. So |Γ| ≤ 1, always. Every possible passive load in the universe — every antenna ever built, every impedance you will ever measure on a bench — lives inside a circle of radius 1.
Hold on to that sentence. It is the whole idea.
|Γ|, SWR, return loss — three costumes, one fact
|Γ| is the fraction of the wave amplitude that comes back. 0 is perfect, 1 is total rejection.
SWR is the ratio between the largest and smallest voltage found along the line. 1 : 1 is perfect, 2 : 1 is a perfectly respectable antenna, ∞ : 1 means everything bounced.
Return loss is the same thing in decibels, and here bigger is better: 20 dB means only 1 % of the power came back.
They are three ways of saying one thing. Datasheets use all three, sometimes on the same page, which is exactly as helpful as it sounds.
04 The arithmetic that ate the afternoon
Back to the field. The question Smith actually needed answered, over and over, is a very reasonable one: my antenna is at the far end of a cable of length ℓ. What does it look like from where I am standing?
Heaviside had already answered it, forty years earlier. Here is the answer:
You do not have to like this formula. Nobody does. On a laptop it is one line of code and you will never think about it again.
In 1930 it is one complex multiplication, one complex division and one tangent — with a slide rule and a book of logarithm tables. Several minutes of careful, boring work. And one mistyped digit gives you a confidently wrong answer with no warning whatsoever.
Now do it again for the next frequency. And the next. And for the twenty cable lengths you are choosing between. And for every stub position your colleague is about to try, all afternoon, while he waits in the cold.
A stubborn engineer who genuinely loved radio had to grind this arithmetic hundreds of times a week, standing in a field. He said as much himself, years later: he simply preferred drawing to calculating. So he tried to draw his way out.
05 You cannot draw infinity — so fold it
His first instinct was the obvious one: plot the impedance. Impedance is a complex number, Z = R + jX, and for anything passive R can never go negative. So every impedance in existence lives in the right half of the complex plane.
Which is a lovely idea until you try to print it. That region runs off to infinity in two directions. Any sheet of graph paper you make shows a small window and throws the rest away. Smith's first attempt, in 1931, had exactly this problem — it worked, but only for a narrow range of impedances, which is a polite way of saying it did not really work.
Then came the idea that fixed everything.
We already know Γ carries the same information as Z. And we know Γ can never escape a circle of radius 1. So: stop drawing impedance. Draw Γ instead — and print the impedance numbers on top of it. The infinite half-plane folds down into something the size of a coin.
Why is that beautiful rather than merely clever? Because of what happens to the grid lines. The map from Z to Γ is what mathematicians call a Möbius transformation, and those have one famous habit: they turn lines and circles into lines and circles, and never into anything else. Bend the paper however you like — a circle stays a circle.
So the plain square grid of “constant R” and “constant X” lines does not get destroyed by the fold. It survives, bent into circles. Watch it happen:
At 0 % you are looking at the impedance plane: plain square graph paper. Vertical lines are constant resistance. Horizontal lines are constant reactance. The edges are where the numbers run away to infinity.
At 100 % you are looking at a Smith chart. Same lines, same labels. Nothing was thrown away — the paper was only bent.
06 Learning to read the map
After the fold, the chart behaves like a map. And every map has landmarks you learn once and then never think about again.
- The centre is Z0 — 50 Ω, a perfect match, Γ = 0. This is where you are trying to get to.
- Far left is a short circuit. Far right is an open circuit. Both send everything back, but with opposite sign.
- The horizontal line through the middle holds every purely resistive load — anything with no timing shift at all.
- Above the line is inductive, below is capacitive. Nothing to memorise: on a chart, coils go up.
- Distance from the centre is |Γ| — how bad your match is. A circle drawn around the centre is a constant-SWR circle.
Why the chart says “1” where you expect 50 Ω
Engineers almost never write real ohms on a chart. They divide everything by Z0 first, so 50 Ω becomes 1, 25 Ω becomes 0.5, 100 Ω becomes 2. That is what the lowercase z means: normalised impedance.
The payoff is that one printed chart then works for every cable on Earth, whether it is 50 Ω, 75 Ω or something exotic. You just multiply by Z0 at the very end.
The two blue curves through the marker are its constant-resistance circle and its constant-reactance arc. These are the chart's grid lines — the ones that were straight before the fold. The dashed circle is constant SWR.
07 Walking down a cable is just turning a dial
Here is the payoff. This is the moment the chart stopped being a nice picture and became a tool.
On a cable with no loss, moving away from the load does not change how much comes back. It only delays it. In Γ language: the size stays exactly the same, and only the angle turns.
Which means walking along a cable is precisely one thing on this chart: turning around the centre at a fixed radius. Like a dial.
That miserable tangent formula from section 4 — the one that cost several minutes with a slide rule — becomes a compass. Put the point in the centre. Set the radius to your load. Turn. Half a wavelength of cable is one full circle.
08 Matching, as a route
Rotation is not the only legal move. There are two more, and together they turn matching from algebra into navigation.
Put a component in series with the load. It can only change the reactance, never the resistance — so the point slides along a constant-resistance circle.
Put a component in parallel. Now it can only change the susceptance — so the point slides along a constant-conductance circle, which is the same family of circles, mirrored.
“Susceptance”? “Conductance”? — admittance, in one minute
Admittance is impedance turned upside down: Y = 1 / Z. Conductance G is the flipped version of resistance, and susceptance B is the flipped version of reactance. Same physics, inverted bookkeeping.
Why bother? Because components wired in series add their impedances, and components wired in parallel add their admittances. Picking the right one turns an ugly division into a simple sum.
That is the entire reason the chart carries a second, mirrored grid — the blue one you can switch on in the figure below. Series moves ride the black grid, parallel moves ride the blue one.
Put both grids on the chart at once and every possible move becomes a road on a map. Matching an antenna is no longer algebra. It is navigation: get from the load to the centre using the roads you are allowed to drive on. Usually there are two routes, and you can see both of them before you calculate a single thing.
Now the two halves of this article finally meet. Go high enough in frequency and components stop existing in any honest sense: at 10 GHz, a “1 nH inductor” is mostly the parasitics of its own legs. So you stop buying parts and start cutting copper. Section 2 said a piece of wire becomes a component. Section 7 said a length of line is a rotation. Put those two facts together and you get stub matching — the trick the Smith chart is most famous for:
- Run the main line away from the load and rotate until you land on the circle where the conductance is exactly 1. This fixes the real part and costs you nothing but distance. Call that distance d.
- At that point, hang a dead-end piece of the same line off the side — a stub. Cut it to the length whose susceptance exactly cancels whatever is left over. Done.
Two lengths of copper. No components at all. And you read both numbers straight off the chart.
What exactly is a “stub”?
A short piece of transmission line soldered onto the side of the main line and going nowhere — either left open at the far end, or shorted to ground.
Because a length of line has an impedance of its own (that was section 2), a dead-end stub behaves exactly like a capacitor or an inductor whose value you set by cutting it to length. At 10 GHz that is far more accurate, more repeatable and cheaper than any component you could buy — and it is why microwave boards look like someone drew a maze on them.
09 The man who would rather draw
Phillip Hagar Smith
1905 – 1987 · TUFTS COLLEGE 1928 · BELL TELEPHONE LABORATORIES
He stayed at Bell Labs for 42 years, nearly all of it on antennas and transmission lines. He collected 21 US patents and published more than 35 papers. One more detail that is too good to leave out: Eli Whitney, the inventor of the cotton gin, was one of his ancestors. Two American engineers a century apart, both of whom got tired of a slow job and built a shortcut around it. The chart carries his name because it is genuinely his — but it was never a product. It spread in the cheapest way imaginable: as pads of printed graph paper, handed around a laboratory. By the time he died in 1987, roughly nine million copies of that graph paper had been printed and sold.
Two things in that timeline deserve a second look.
The first: Smith's original 1931 chart was not very good. He drew it on rectangular coordinates, so it could only show a narrow range of impedances. To get the version that changed everything, he had to stop drawing the obvious thing (impedance) and draw the strange thing instead (reflection). That is usually where the good version is hiding.
The second: he was not alone. Tosaku Mizuhashi published a very similar chart in Japan in 1937. Amiel Volpert did the same in the Soviet Union in 1939. Three people, three countries, no contact between them, one answer. When that happens, it tells you the problem was real and the solution was simply waiting to be found.
10 Why computers could not kill it
Every other graphical calculator from that era is dead. Nomograms, slide rules, planimeters, the Mollier diagrams gathering dust in mechanical engineering departments — all of them lost their jobs to a machine that did the arithmetic faster and without mistakes.
The Smith chart lost that job too, somewhere around 1975. It did not care.
It did not care because the number was never the interesting part. Look again at what the chart actually shows you. A match is a place. Cable length is a rotation. A series inductor is a direction. Bandwidth is how long your curve stays near the middle. A tuning screw is a path.
The chart does not answer “what is Z?” A calculator does that better and always will. The chart answers “what should I do next?” — and nobody has automated that question yet.
So it is still here. Open any vector network analyser built this year and one of the display modes is a Smith chart. It is on antenna datasheets, in every RF simulator, and in the notebook of every engineer tuning a filter. It is 87 years old, it came out of a problem about telephone cables, and it is still the fastest way to see what a wave is doing.
Not bad for a man who just wanted to stop doing sums.
Now go break it yourself
The figures above are deliberately small — each one makes exactly one point. The full tool is next door: drag the reflection point with every value live, rotate along a lossy or lossless line, turn the admittance grid on and off, and work through seven guided lessons that end with L-network matching.
Sources & further reading
- P. H. Smith, “Transmission Line Calculator,” Electronics, vol. 12, no. 1, Jan. 1939.
- P. H. Smith, “An Improved Transmission Line Calculator,” Electronics, vol. 17, no. 1, Jan. 1944.
- P. H. Smith, Electronic Applications of the Smith Chart, McGraw-Hill, 1969.
- IEEE History Center, oral history interview with P. H. Smith, 1973.
- D. M. Pozar, Microwave Engineering, 4th ed., Wiley — chapter 2, for the transmission-line theory behind every figure here.