be the voltage and current that is incident on the boundary from the source side. and be the voltage and current that is transmitted to the load. and be the voltage and current that is reflected back toward the source. On the line side of the boundary and and on the load side where , , , , , and are phasors. At a boundary, voltage and current must be continuous, therefore All these conditions are satisfied by where the reflection coefficient going from the transmission line to the load. Source-end conditions At the source end of the transmission line, there may be waves incident both from the source and from the line; a reflection coefficient for each direction may be computed with , where Zs is the source impedance. The source of waves incident from the line are the reflections from the load end. If the source impedance matches the line, reflections from the load end will be absorbed at the source end. If the transmission line is not matched at both ends reflections from the load will be re-reflected at the source and re-re-reflected at the load end ad infinitum, losing energy on each transit of the transmission line. This can cause a resonance condition and strongly frequency-dependent behavior. In a narrow-band system this can be desirable for matching, but is generally undesirable in a wide-band system. Source-end impedance where is the one-way transfer function (from either end to the other) when the transmission line is exactly matched at source and load. accounts for everything that happens to the signal in transit (including delay, attenuation and dispersion). If there is a perfect match at the load, and Transfer function where is the open circuit (or unloaded) output voltage from the source. Note that if there is a perfect match at both ends and and then . Electrical examples Telephone systems Telephone systems also use matched impedances to minimise echo on long-distance lines. This is related to transmission-line theory. Matching also enables the telephone hybrid coil (2- to 4-wire conversion) to operate correctly. As the signals are sent and received on the same two-wire circuit to the central office (or exchange), cancellation is necessary at the telephone earpiece so excessive sidetone is not heard. All devices used in telephone signal paths are generally dependent on matched cable, source and load impedances. In the local loop, the impedance chosen is