The transporter circuit was an important component of the transporter's function. On Constitution-class starships, the circuit was located with the abort control circuit on the lower level of the engineering deck. There were at least two of these circuits on a ship.
In 1968, a transporter beam from an unnamed planet a thousand light years from Earth was intercepted by the USS Enterprise. The beam locked the circuits "wide open", resulting in Gary Seven beaming into the Enterprise. Later that day, Lieutenant Uhura informed Captain James T. Kirk that the circuits detected Gary Seven preparing to beam down to Earth. Kirk ordered an override; however, he couldn't stop Gary Seven. (TOS: "Assignment: Earth")
In 2266, aboard the Enterprise, Chief Engineer Montgomery Scott reported to Captain James T. Kirk that there was a transporter breakdown. The transporter beamed up two opposites of an canine from Alfa 177. While the engineer was speaking with Kirk, an operations division crewmember was testing the circuits. The circuits were later burned out by a phaser blast, necessitating emergency repairs. Before the opposites of James T. Kirk were brought together, Lieutenant Commander readied the transporter room by checking the circuits. (TOS: "The Enemy Within")
In 2267, an ion storm with a violent magnitude occurring simultaneously in two parallel universes caused a power surge in the circuits of the Enterprise and the ISS Enterprise. This event created a "momentarily interdimensional contact" between the universes, with individuals in the act of beaming being transposed with their counterparts. After beaming aboard the Enterprise in the mirror universe, Kirk requested that Commander Spock have the circuits checked. Before Kirk's landing party could beam back to their universe, Spock had ordered the circuits, located in engineering, be de-activated. This gave the commander the time needed for getting to the transporter room. Once in the room, he ordered the circuits re-activated. (TOS: "Mirror, Mirror")