Standing-wave ratio and impedance matching

As position changes, the relative phase between forward and backward waves winds twice as rapidly as the phase of either one alone. At constructive alignment their voltage magnitudes add; at destructive alignment they subtract. The voltage standing-wave ratio is the larger envelope magnitude divided by the smaller. A matched line gives a ratio of one, and an ideal open or short gives an unbounded ratio because the minimum reaches zero. This is an envelope ratio, not a ratio of instantaneous voltages at arbitrary times.

VSWR discards reflection phase. Many complex loads can share the same ratio while placing maxima at different distances from the load. To recover the impedance, one needs additional phase or position information and the known line properties. The simple constant-extrema formula assumes a uniform lossless line. On a lossy line the two wave envelopes have different spatial dependence, so maxima and minima measured far apart cannot automatically be interpreted with the same lossless expression.

Insert a lossless line section between a feed line and a positive real load. At the design frequency choose its electrical length to be one quarter wavelength. Taking the appropriate limit of the input-impedance formula shows that its input impedance equals its own characteristic impedance squared divided by the load. Choosing that section impedance as the geometric mean of feed and load resistances makes the transformed input exactly equal to the feed impedance.

The transformer section's wavelength is determined by its own propagation speed, which can differ from that of the feed. Its characteristic impedance must also be physically realizable. The elementary geometric-mean formula assumes a real load at the design frequency; a complex load generally requires first canceling or transforming its reactive part, or using a more general network. Merely inserting a quarter-wave section of the feed's existing impedance shifts the reflection phase without reducing its magnitude.

Reflectionless matching asks whether a terminating input equals a line's characteristic impedance. Maximum power from a Thevenin source instead requires load impedance equal to the complex conjugate of source impedance at the chosen frequency. The two conditions coincide in the common real-resistance case but differ for a complex source. Neither condition alone proves that an amplifier is stable or that a network dissipates little power. A matched attenuator can have excellent input matching while intentionally absorbing much of the signal.

A quarter-wave transformer loses its exact quarter-wave property away from its design frequency, and real loads may themselves disperse. Multi-section or reactive networks can trade complexity against bandwidth and sensitivity. General passive matching constraints depend on the load and allowed network class; a one-frequency construction is not a promise of arbitrary broadband matching. Specify acceptable reflection over a band, component losses, and power or voltage limits before calling a design successful. The appropriate performance measure is more informative than the smallest achievable reflection at one isolated point.