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, and the two faint lines mark the edges of how far the wave swings. Set the load impedance Z_L with the slider far from the characteristic impedance Z₀, and part of the wave comes back and overlaps the outgoing wave, making a standing wave whose outline bulges and pinches in turn. Match Z_L to Z₀ and the load turns green, the reflection drops to 0, and the outline straightens out. The condition under which reflection disappears is the question of this lesson.
Not a circuit but a wave
Treating the voltage as appearing across the whole circuit at once when you flip a switch only works when the wires are far shorter than the signal's wavelength. Once a line becomes comparable to the wavelength, like the high-frequency traces on a phone's circuit board or a long feed line to a broadcast antenna, voltage and current no longer move as one lump. A change at one end spreads along the line at a speed somewhat below that of light, so at the same instant the voltage differs from place to place along the line. A line that has to be treated this way is called a transmission line. A common rule of thumb is that once a line is longer than about a tenth of a wavelength, this effect has to be taken into account.
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 sets up a magnetic field around it, which gives the inductance, and voltage between the conductors sets up an electric field, which gives the capacitance. The line is like an endless ladder of tiny L's and C's, which is what lets it carry waves; with no loss, the wave speed is v = 1/√(LC) and the ratio of voltage to current is the characteristic impedance Z₀ = √(L/C). Z₀ is set by the conductors' thickness and spacing and the material between them, and does not depend on the line's length. Coaxial cables are usually 50Ω, and broadcast cables such as TV antenna leads are 75Ω: for an air-filled coaxial line, the first is a compromise between the value with the lowest loss and the value that handles the most power, and the second is chosen to lean toward low loss.
The reflection coefficient Γ
When the wave reaches the load at the end of the line, the ratio of voltage to current across the load has to be Z_L, set by the load. The incoming wave alone has a ratio of Z₀, so if Z_L equals Z₀, the wave is fully absorbed, as if the line went on forever. This is matching. If Z_L differs from Z₀, the incoming wave alone cannot meet that condition, so a returning wave appears as well, and its size relative to the incoming wave is the reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀). With an open load (Z_L = ∞), Γ = +1; with a short (Z_L = 0), Γ = −1; either way the wave bounces back whole. At an open end the voltage returns with the same sign, and at a shorted end with the sign flipped.
The standing wave
A wave going forward and a wave reflected back overlap to form a standing wave: a pattern in which places that always swing widely (antinodes) and places that barely move (nodes) alternate a quarter wavelength apart along the line. How bumpy this pattern is gets measured by the standing wave ratio SWR = (1 + |Γ|)/(1 − |Γ|); with perfect matching, Γ = 0 and SWR = 1, the pattern is smooth, and all the power sent goes into the load. For a resistive load, the SWR equals the larger of Z_L and Z₀ divided by the smaller. The larger the reflection, the larger the SWR, and the returning power heats the transmitter or distorts the signal, which is why high-frequency design always aims for matching.
Back to the first screen
On the first screen, the farther you set the load Z_L from the characteristic impedance Z₀, the more the outgoing and returning waves overlapped and the sharper the antinodes and nodes of the standing wave. The moment Z_L matched Z₀, the reflection coefficient Γ dropped to 0, 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 takes that ratio as it is, nothing reflects; if not, Γ = (Z_L − Z₀)/(Z_L + Z₀) of it comes back. Matching is the way to hand all the power sent over to the load.
Instead of working out the reflection coefficient Γ by hand for each load, there is a tool that spreads every impedance and reflection over a single disc so they can be read at a glance. The next lesson (EM-27) covers that Smith chart: a map on which complex impedances are drawn as circles and arcs, and moving along the line away from the load turns a point around a circle.