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 alone it reflects. Insert a quarter-wave transformer between them and adjust its impedance Z₁. When Z₁ is just right, the impedance seen at the input Z_in = Z₁²/Z_L becomes exactly 50Ω, the standing wave on the main line vanishes, and it goes smooth. Find what that right Z₁ is.
Why match?
When a load is mismatched to the characteristic impedance, part of the power sent is reflected back to the transmitter. That much less power reaches the load, and the returning power heats the transmitter or distorts the signal. Moreover, that maximum power is transferred only when source and load are matched is a basic result of circuit theory. So the last step of every high-frequency design that joins antennas, amplifiers and cables is to fit the impedance so the load looks like Z₀ — matching.
The quarter-wave transformer
The cleanest matching tool is a single piece of line a quarter-wavelength long. A transmission line exactly λ/4 long has the striking property of inverting the impedance at its end: the impedance seen at the input becomes Z_in = Z₁²/Z_L, where Z₁ is the characteristic impedance of that piece. Solving Z₁²/Z_L = Z₀ for the input to be 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. It works for real loads and is exact at only one frequency.
Stub matching
When the load is complex, with reactance mixed in, a transformer alone is not enough. Then we use a stub — a short piece of line attached like a branch at a point on the main line, with its end either open or shorted. The impedance reflected from that end is pure reactance, so tuning its length cancels exactly the reactance the load creates. Placing one or two stubs of the right length at the right position pulls the load's complex impedance to Z₀. Matching with only pieces of line, adding no components, is its advantage at high frequencies.
What matching achieves
When matching is done, the reflection coefficient seen at the input is Γ = 0 and the standing-wave ratio is SWR = 1. The main line carries only a traveling wave, with no standing wave, and the power sent reaches the load in full. In the language of the Smith chart (EM-27), matching is moving the load's point to the very center of the chart. The λ/4 transformer pulls the point straight to the center along the real axis; a stub moves the point along a reactance arc to land on the center. Whichever path you take, the destination is the same: the one point without reflection, the matched point.
Back to the first screen
On the first screen, as you moved the transformer’s impedance Z₁, at the right value the standing wave on the main line vanished and the wave flowed smoothly. That value was Z₁ = √(Z₀ Z_L) ≈ 70.7Ω, the geometric mean of 50Ω and 100Ω. There the impedance seen at the input Z_in = Z₁²/Z_L became exactly 50Ω and the reflection coefficient went to zero. Matching is making a mismatched load look like Z₀, and the λ/4 transformer does it with a single piece of line. With zero reflection, the power sent reaches the load intact.
Beginning with the differential element of a coordinate system (EM-01), we passed through the electric and magnetic fields, Gauss and Ampère, Faraday and the displacement current, arrived at Maxwell's equations, and watched light spring out of them. Light reflects and refracts at boundaries, becomes a signal confined to a transmission line, becomes a point on the Smith chart, and at last is delivered intact by matching. From a single point charge to a matched antenna feed, you have walked one full path through electromagnetism.