A hybrid coil (or bridge transformer, or sometimes hybrid) is a transformer that has three windings, and which is designed to be configured as a circuit having four ports that are conjugate in pairs.
A signal arriving at one port is divided equally between the two adjacent ports but does not appear at the opposite port. In the schematic diagram, the signal into W splits between X and Z, and no signal passes to Y. Similarly, signals into X split to W and Y with none to Z, etc.
Correct operation requires matched characteristic impedance at all four ports. Hybrids are a class of directional coupler in which the input port power is split equally between the two output ports. Forms of hybrid other than transfomer coils are possible; any format of directional coupler can be designed to be a hybrid. These formats include transmission lines and waveguides.
The primary use of a voiceband hybrid coil is to convert between 2-wire and 4-wire operation in sequential sections of a communications circuit, for example in a four-wire terminating set. Such conversion was necessary when repeaters were introduced in a 2-wire circuit, a frequent practice at early 20th century telephony. Without hybrids, the output of one amplifier feeds directly into the input of the other, resulting in a howling situation (upper diagram). By using hybrids, the outputs and inputs are isolated, resulting in correct 2-wire repeater operation. Late in the century, this practice became rare but hybrids continued in use in line cards.
Hybrids are realized using transformers. Two versions of transformer hybrids were used, the single transformer version providing unbalanced outputs with one end grounded, and the double transformer version providing balanced ports.