Transmission Lines and Characteristic Impedance
Match the load to kill the reflection
A voltage wave sent from the left runs along the transmission line to the load on the right. Set the load impedance Z_L far from the characteristic impedance Z₀ and part of the wave returns, overlapping the forward wave into a lumpy standing wave. Match Z_L to Z₀ and the reflection vanishes and the wave flows smoothly. Find when the reflection becomes zero.
Not a circuit but a wave
We learned that flipping a switch puts the same voltage across the whole circuit at once. But that holds only when the line is far shorter than the signal's wavelength. When the line length becomes comparable to the wavelength — the high-frequency traces on a phone board, or a long feeder to a broadcast antenna — voltage and current no longer move as one lump. A change at one end spreads along the line like a wave at nearly the speed of light. Then we call the line a transmission line.
Distributed L and C, and Z₀
The two conductors of a transmission line carry an inductance L and a capacitance C per unit length. Current in a conductor makes a magnetic field, giving the inductance; voltage between the conductors makes an electric field, giving the capacitance. Linked like a ladder, this L and C carry a wave. The wave’s speed is v = 1/√(LC), and the ratio of voltage to current is the characteristic impedance Z₀ = √(L/C). Z₀ is set by the thickness and spacing of the conductors and the material between them; coaxial cable is usually 50Ω, and broadcast cable 75Ω.
The reflection coefficient Γ
When the wave reaches the load at the end of the line, it is absorbed smoothly only if the ratio of voltage to current the load accepts matches Z₀. If the load impedance Z_L equals Z₀, the wave is fully absorbed, as though the line ran on forever. This is a match. But if Z_L differs from Z₀, part of the wave is reflected back to force the ratio to fit. The fraction returning is the reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀). An open load (Z_L = ∞) gives Γ = +1 and a short (Z_L = 0) gives Γ = −1; both bounce the whole wave back.
The standing wave
When the forward wave and the reflected wave overlap, a standing wave forms — a lumpy pattern where some places always swing wide (antinodes) and others barely move (nodes). The lumpiness is measured by the standing-wave ratio SWR = (1 + |Γ|)/(1 − |Γ|). With a perfect match, Γ = 0 and SWR = 1: no pattern, smooth, and all the power is delivered to the load. The larger the reflection, the larger the SWR, and the returning power heats the transmitter or distorts the signal. So the goal of high-frequency design is always a match.
Back to the first screen
On the first screen, the farther you set the load Z_L from the characteristic impedance Z₀, the sharper the antinodes and nodes of the standing wave, as forward and reflected waves overlapped. The moment you matched Z_L to Z₀, the reflection coefficient Γ went to zero, the pattern vanished, and the wave flowed smoothly. On a transmission line, voltage and current are waves, and their ratio is Z₀ = √(L/C). If the load accepts that ratio as is, there is no reflection; if it differs, a fraction Γ = (Z_L − Z₀)/(Z_L + Z₀) returns. Matching is the way to deliver all the power intact.
Instead of computing the reflection coefficient Γ by hand for every load, there is a tool that spreads all impedances and reflections on a single disc to read at a glance. Next (EM-27) looks at that Smith chart. You will learn the map where complex impedances are drawn as circles and arcs, and moving along the line makes the point circle around.