The field coil was a form of technology, designed to transmit subspace emissions, which functioned as key components in starship and space station operations, including the sensors, warp drive and communications systems. (TNG: "Force of Nature")
After entering V'Ger, Spock discovered what he described as "a kind of plasma-energy conduit – possibly a field coil for a gigantic imaging system," where he discovered a series of images depicting objects encountered by V'Ger. (Star Trek: The Motion Picture)
Twenty-three field coils, working in concert from the surface of Bilana III, were the source behind the creation of a soliton wave, the first step towards "warp without warp drive." (TNG: "New Ground")
A verteron pulse is capable of overloading the field coils aboard a starship. This method was used by the Hekarans to disable several ships that traversed the Hekaras Corridor in mid-2370. (TNG: "Force of Nature")
In order to keep a transporter signal lock on an individual, and counteract the effects of a subspace transporter system, it is necessary to tie a ship's subspace field coils into the transporter system. This tie in can be used for implementing a subspace transport, by means of using the transporter system to shunt something through subspace. (TNG: "Bloodlines")
Clogged plasma injectors can cause both the inertial dampers and field coils to go off line. (VOY: "Parturition")
Field coils were utilized in Cardassian space stations, such as Terok Nor and Empok Nor. (DS9: "Destiny", "Empok Nor") While stranded on Empok Nor in 2373, Chief Miles O'Brien suggested the modification of the field coils to emit a series of covariant pulses through the deflector grid, and "use the station like like an old-fashioned telegraph and tap out an SOS," but in order for the signal to be strong enough, it was necessary to boost power to the station's induction grid. (DS9: "Empok Nor")