The Scariest Chart In Electrical Engineering
Veritasium
A Chart That Frightens Engineers 0:00
This chart, often called the Smith Chart, has terrified generations of undergraduate electrical engineering students, some even printing "black magic" on its versions. Despite the fear, millions of copies exist and it remains embedded in today's most advanced engineering software. The reason it endures is that it solves a genuinely paradoxical problem, one worth testing directly: cutting a transmission line can, strangely, increase power transfer rather than reduce it, a result that seems to defy common sense until you understand the physics behind it.
Smith's Original Problem At Bell Labs 1:04
In 1928, Phillip H. Smith joined Bell Labs just as the telephone industry was booming, with Americans placing over 65 million calls a day. Since no telephone cables yet crossed the Atlantic, long distance calls relied on radio waves. Smith worked on sending signals from New Jersey to receivers in England and Argentina. A single antenna wastes most of its power radiating in all directions, but combining more than 20 antennas into a directional array, linked by over 2 kilometers of transmission line, could focus the beam to about 10 degrees and boost power 400 times. Testing this array, Smith noticed part of his signal bouncing back down the line instead of reaching the antennas, a reflection problem he needed to solve.
Why Alternating Current Reflects 2:33
Unlike the steady direct current from a battery lighting a bulb, radio signals require alternating current, with oscillating electrons producing rising, falling, and reversing voltage and current, typically as sine waves. Demonstrating this with a slinky shows how waves have a wavelength and frequency, and their product gives wave speed, a value fixed for a given line. Because speed is fixed, frequency and wavelength are inversely related. When wavelength is long compared to the line, reflections barely disturb the signal, but when wavelength shrinks below the line's length, reflected and forward waves interfere to create a standing wave pattern, where voltage peaks can reach twice the input voltage and burn out equipment. At household frequencies of 50 or 60 Hz, wavelengths span thousands of kilometers, so this isn't an issue, but Smith's radio frequencies in the megahertz range produced wavelengths around 30 meters against a 2 kilometer line, making reflections severe.
Recreating The Experiment 6:32
At Imperial College London's anechoic chamber, a scaled recreation of Smith's setup showed exactly this power loss. Expecting a signal around -55 dB, the measured result was -59 dB, meaning more than half the power was lost to reflection. Modeling the system simply as a source, a transmission line, and an antenna, represented by slinkies with different mass per unit length, showed that reflections arise whenever properties mismatch at a boundary, quantified by the reflection coefficient. Matching the antenna's 12.5 ohms to the line's 50 ohms by adding a 37.5 ohm resistor didn't help at all, since resistors dissipate power as heat and are inherently lossy.
Capacitance And Inductance Shift Timing 10:35
Beyond resistance, real transmission lines and antennas have capacitance, like the charge-storing plates in a disposable camera flash, and inductance, from magnetic fields generated by current flow. Both cause voltage and current waves to fall out of sync: a capacitor makes voltage lag current by 90 degrees, while an inductor makes voltage lead current by 90 degrees. This means matching resistance alone isn't enough, because it says nothing about timing, or phase. A true impedance match requires aligning both magnitude and phase together.
Impedance As A Complex Number 13:33
Electrical engineers represent this combined magnitude and phase using complex numbers, with resistance plotted along the horizontal axis and reactance, the combined effect of capacitance and inductance, plotted vertically, using J instead of I for the imaginary unit since I denotes current. This two-dimensional quantity is called impedance, written as Z, and defined as voltage divided by current, essentially Ohm's law extended to AC circuits. A transmission line has its own fixed characteristic impedance, often 50 ohms in radio frequency systems, and matching a component's impedance to this value eliminates reflections entirely.
Matching Without Resistors 15:33
Since resistors waste power as heat, engineers needed another way to match resistance. The key insight is that impedance along a transmission line isn't constant, because forward and reflected waves combine differently at different points, meaning voltage and current interact in shifting ways along the line. This suggests there's likely a specific point along the line where the resistance already matches, leaving only a leftover reactive part that can be canceled using a lossless capacitor or inductor, avoiding energy loss altogether. Finding that exact point, however, required working through Oliver Heaviside's transmission line equations by hand, a slow process using slide rules that became urgent as global political tensions in the 1930s made reliable long-distance radio communication strategically vital, prompting parallel work by Soviet engineer Amiel Volpert and Japanese engineer Tosaku Mizuhashi.
Building The Chart From Scratch 19:34
With no shortcut available, Smith began constructing his chart by plotting impedance on the complex plane, but dividing all values by the line's characteristic impedance so that a perfect match always appeared as the value one, regardless of the actual impedance involved. The remaining problem was that impedance can range from zero, a short circuit, to infinite, an open circuit, meaning any complete chart would need to be infinite in size. Smith enlisted colleagues Ferrell and McRae, and together they turned to a property of complex numbers called a conformal map, a transformation that scales and rotates values while preserving shapes and angles locally, and which can fold infinite values into a finite space.
Circles Replacing Infinite Lines 22:31
Rather than plotting impedance directly, Smith worked with the reflection coefficient, the ratio of the reflected wave to the forward wave, whose magnitude never exceeds one on a lossless line, avoiding the infinity problem entirely. Applying the conformal transformation, lines of constant resistance on the impedance plane become circles on this new plane, shrinking and shifting toward the point one as resistance increases, with the r-equals-one circle marking the matched case. Lines of constant reactance become circles too, curving above the axis for inductance and below for capacitance, shrinking as their values grow, while a reactance of zero appears as a straight line, setting up the layout that becomes the finished chart.
How the Smith chart works 25:01
On the finished chart, one family of circles shows resistance and another shows reactance, so any point where two circles cross gives you an impedance. Because the chart is really a plot of the reflection coefficient, the distance of that point from the center tells you the reflection coefficient's magnitude, and rotating around the center walks you through every impedance you would measure along the line. This lets an infinite range of impedance values fit inside a single finite circle, which is the trick Phillip Smith needed to make the chart work.
Matching the measured 36 ohm load 26:03
Using the earlier lab measurement of 36 ohms resistance and 74 ohms reactance, normalizing by 50 ohms gives 0.7 plus 1.5j, which can be located where the corresponding resistance and reactance circles intersect. Since a matched line needs a normalized resistance of one and zero reactance, the goal is the exact center of the chart, and the arrow from center to the starting point has a length of 0.68, matching the reflection magnitude seen in the demo. Rotating around that fixed radius traces a circle, where every point represents a different position along the line, and a full 360 degree turn corresponds to half a wavelength of physical distance.
Adding length and an inductor 29:03
Rotating 12 degrees to reach a point of resistance one would only require 3.1 millimeters of extra line at 1085 MHz, which was too short to use practically, so the team moved to the next matching spot instead, adding 28 millimeters of copper tape line. That left a remaining reactance of about negative 1.8, which was canceled with a series inductor calculated at 13.2 nanohenries; after testing a smaller inductor and then the correctly sized one, the reflections disappeared and the team measured their highest power output yet.
Independent inventors and stub matching 31:31
Smith finished his chart in 1937, and almost simultaneously Mizuhashi in Japan and later Volpert in the Soviet Union in 1939 arrived at similar solutions independently. The chart also shows that an open or short circuit sits on the outer rim, reflecting the entire wave, and moving along a line from that point cycles through every possible reactance, meaning a length of line called a stub can replace an inductor or capacitor. For the 1.8 reactance needed, walking around the chart from the open circuit position to 302 degrees gave a stub length of 77 millimeters, and cutting a longer stub down to that length in the lab produced a clean match with no reflections.
Parallel stubs and frequency limits 35:01
A series stub works but requires cutting into the line, so in practice engineers solder stubs in parallel instead, which requires switching to admittance, the inverse of impedance, and a flipped admittance version of the Smith chart. Every chart only applies to a single frequency, so real signals that span a range of frequencies trace a curve on the chart rather than a single point, and the goal becomes bringing that whole curve as close to the center as possible.
Slow adoption and wartime use 36:04
Smith's chart took about two years of rejections before Electronics magazine finally published it, since engineers already had their own methods. World War II changed that, as MIT Radiation Laboratory scientists building microwave radar to detect German U-boats in the Battle of the Atlantic needed fast, reliable fixes and used the chart constantly, after which those engineers carried it into industry, universities, and textbooks, while the Japanese and Soviet versions stayed local, which is why it is now known simply as the Smith chart.
Why the chart still matters 37:34
Computers can now calculate impedance matches faster than any chart, but the Smith chart is still taught worldwide because it shows engineers what to try and why it works, giving an intuitive sense of direction the way a hand drawn map gives directions to a train station. It remains a standard visualization in radio frequency software and measurement tools, and it is compared to representations like the periodic table or Feynman diagrams, since progress often comes from new ways of representing a problem rather than new discoveries, and this chart helped enable modern radar, sensing, and communication networks.
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