Impedance Matching
Tune the transformer to kill the reflection
A line with characteristic impedance Z₀ = 50Ω is connected to a load Z_L = 100Ω, so left as it is, it reflects. Insert a 1/4-wavelength transformer between them and change its impedance Z₁ with the slider; at the right value, the impedance seen at the input, Z_in = Z₁²/Z_L, comes to 50Ω, the standing wave on the main line disappears, and the load turns green. The slider moves in whole-number steps, so the reflection nearly vanishes at the step closest to that value. How that right Z₁ is set is the question of this lesson.
Why match?
When a load is out of step with the characteristic impedance, part of the power sent is reflected back to the transmitter. The power delivered to the load drops by that much, and the returning power heats the transmitter or distorts the signal. Circuit theory also has the maximum power transfer theorem, which says a load receives the most power when it equals the complex conjugate of the source's internal impedance; at that point, though, the efficiency is only half, so power grids, where efficiency matters, are not matched this way. At high frequencies, moving small signals over without loss is what counts, so the last step of any high-frequency design joining antennas, amplifiers, and cables is matching: adjusting impedances so the load looks like Z₀.
The λ/4 transformer
The neatest matching tool is a single piece of line 1/4 of a wavelength long. A transmission line exactly λ/4 long has the property of making the impedance at its end appear inverted, so the impedance seen at the input becomes Z_in = Z₁²/Z_L. Here Z₁ is the characteristic impedance of that piece. Solving Z₁²/Z_L = Z₀ so that the input is Z₀ gives Z₁ = √(Z₀ Z_L), the geometric mean of Z₀ and Z_L. To match a 50Ω line to a 100Ω load, insert a piece of √(50·100) ≈ 70.7Ω; the two reflections from the two ends of the piece are then half a wavelength apart in round trip and cancel each other. It works neatly only for real loads, though, and is exact only at the one frequency for which the length is right.
Stub matching
When the load is complex, with some reactance mixed in, a transformer alone is not enough, and a stub is used instead. A stub is a short piece of line attached like a branch at one point on the main line, with its end left open or shorted. The impedance reflected back from that end is a pure reactance, so by choosing the length you can make whatever reactance you need. First find the spot on the main line where the real part of the admittance equals the reciprocal of Z₀, then attach in parallel a stub long enough to cancel the remaining susceptance there, and the load's complex impedance is pulled to Z₀. Matching with nothing but pieces of line, without adding components, is an advantage at high frequencies.
What matching achieves
Once matched, the reflection coefficient seen at the input is Γ = 0 and the standing wave ratio is SWR = 1. Only a traveling wave runs on the main line, with no standing wave, and all the power sent reaches the load. In the language of the Smith chart (EM-27), matching means moving the load's point to the center of the chart. A λ/4 transformer swings the load's point on the real axis along half of a circle that meets the real axis twice and sets it down at the center, while a stub moves the point along a reactance arc to land on the center. Whichever path is taken, the destination is the same: the one point with no reflection, the matched point.
Back to the first screen
On the first screen, as you moved the transformer impedance Z₁, the standing wave on the main line disappeared at the right value and the wave flowed smoothly. That value is Z₁ = √(Z₀ Z_L) ≈ 70.7Ω, the geometric mean of 50Ω and 100Ω. Then the impedance seen at the input, Z_in = Z₁²/Z_L, comes to 50Ω and the reflection coefficient becomes 0. Because the slider moves in whole-number steps, in practice the reflection nearly vanishes at the closest step. Matching means making a mismatched load look like Z₀, and a λ/4 transformer does it with a single piece of line. With 0 reflection, the power sent reaches the load in full.
Starting from the tiny elements of coordinate systems (EM-01), the track passed through electric and magnetic fields, Gauss's law and Ampère's law, Faraday's law and the displacement current, reached Maxwell's equations, and watched light emerge from them. Light splits into reflection and refraction at a boundary, becomes a signal when trapped in a transmission line, becomes a point on the Smith chart, and in the end is delivered in full through matching. From a single point charge to a matched antenna feed line, one path through electromagnetism has been walked to its end.