Phillip H. Smith was 23 years old and fresh out of electrical engineering school when Bell Labs handed him a problem that would consume a decade of his life: stop a radio signal from bouncing back on itself before it ever reached an antenna. The challenge sounds dry. The solution he eventually drew on paper became so counterintuitive that generations of engineering undergraduates have described their first encounter with it the same way. ‘Holy. It’s like terrifying. It looks almost like a wormhole in some sci-fi movie.’ On many printed versions, the label at the top simply reads ‘black magic.’ But this chart, the Smith chart, is also embedded in nearly every piece of radio frequency measurement hardware on the planet, and the team at Veritasium recently traveled to Imperial College London’s screened radio frequency anechoic chamber to find out whether it actually works in a live experiment, cutting a real cable and watching real power readings move.
Why a bouncing signal nearly killed transatlantic radio
In 1928, Smith joined a Bell Labs team trying to beam radio signals from New Jersey to receiving stations in England and Argentina. The team’s solution was a directional array of more than 20 smaller antennas linked by over 2 km of transmission line, which could focus radiated power in a narrow beam roughly 10 degrees across, delivering 400 times the strength of a single antenna in that direction. But when Smith sent a signal down the line, a large portion of the power bounced back before reaching the antennas at all.
The culprit was a mismatch between the transmission line’s built-in impedance and the antenna’s impedance. Impedance, the AC equivalent of resistance, has two components: a magnitude that sets the relative size of voltage and current waves, and a phase angle that describes how far one leads the other. Capacitors and inductors each shift that phase by 90 degrees in opposite directions. A resistor changes the magnitude but does nothing to the phase. So simply bolting a resistor across the mismatch, as the Veritasium team demonstrated live in the chamber, failed completely. ‘It hasn’t helped us at all,’ one of the researchers noted, watching the power reading stall at half. A resistor also wastes power as heat, which is precisely the opposite of the goal.
The deeper insight was geometric. On the ordinary complex plane, impedance stretches off toward infinity. A short circuit sits at zero; an open circuit sits at infinity. No single printed chart can hold both. Smith realized that if he switched from plotting impedance directly to plotting the reflection coefficient, the ratio of the reflected wave to the forward wave, infinity disappeared. The reflected wave can never be larger than the forward wave, so the reflection coefficient is always between zero and one. Applying a conformal transformation, a type of map that curves straight lines into circles while preserving local angles, turned the entire infinite impedance plane into a compact, finite disk. Constant-resistance lines became one family of shrinking circles. Constant-reactance lines became another family curving perpendicular to the first. Any combination of resistance and reactance now lived at the single point where two circles crossed.
28 mm of copper tape and a trimmed stub that stopped every reflection
Inside the anechoic chamber, the team measured their antenna array at 36 ohms of resistance and 74 ohms of reactance. Normalized to the 50-ohm characteristic impedance of the line, that placed them at 0.7 plus 1.5J on the chart, with a reflection coefficient magnitude of 0.68. More than half the power was gone before it left the room.
The fix required two physical steps. First, locating the point on the existing transmission line where the resistance component already equaled one, then moving to that point by adding 28 mm of copper tape. That rotation on the chart took 12 degrees. Second, canceling the remaining capacitive reactance of roughly negative 1.8 by trimming a short-circuit stub, a dangling side branch of the same coaxial cable connected to nothing at its far end, down to the precise length where its reflected wave pushed back with exactly the right timing. As the cable was cut shorter and shorter in real time, the power reading climbed. ‘This is crazy. We’re getting way more power transfer, even though we’ve cut the line. How does that make any sense?’ And then, with the stub trimmed to length: ‘We stopped every reflection on this line with just a break in the circuit. A dangling piece of cable connected to nothing trimmed to the right length. And because the stub is just more of the same cable, the fix is made of the thing that it’s fixing.’
Smith finished his chart in 1937. Electronics magazine rejected it for two years before finally publishing it. Adoption was slow until World War II placed radar engineers at MIT’s Radiation Laboratory under impossible time pressure building microwave systems to detect German U-boats in complete darkness. Those engineers carried the chart into postwar universities, textbooks, and the companies they built. Two independent contemporaries, Tosaku Mizuhashi in Japan and Amiel Volpert in the Soviet Union, arrived at nearly identical representations the same year. Three groups, one solution.
Today the chart is as much a navigational tool as a calculation device. Dr. Stepan Lucyszyn of Imperial College London described the shift plainly: ‘Historically we would get a Smith chart and use rulers and protractors and compasses. These days we use it more like a back-of-the-envelope calculation. It’s like a map. If somebody asks how do you get to the nearest tube station, you say, here’s an envelope, you go down the street here, turn left there. In the same way, we use it to navigate from where we are to where we want to be.’
The dangling stub, still connected to nothing
At the end of the session in the chamber, the coaxial stub hung from the cable junction, its far end open to air, its inner and outer conductors unconnected, doing its work purely through the timing of a bouncing wave inside a short length of wire.
Smith’s chart eventually found its way into software, into every RF measurement instrument, and into the engineering curriculum on every continent, a reminder that the chart Phillip Smith drew in New Jersey in the late 1930s is still, a century later, the fastest way to explain which direction to go.
