Slowing down velocity and controlling time delay of optical pulses in fiber have many potential applications in optical communication systems, and a number of theoretical and experimental studies have been done. In this paper, the transmission spectrum and reflective spectrum of active fiber Bragg grating (FBG) couplers are studied, and the analytic expressions of dispersion and time delay are obtained. By changing the detuning and gain coefficient, different dispersions and time delays in active fiber Bragg grating coupler are obtained. The results show that different detunings and gain coefficients can result in various time delays, and thus tunable time delay could be achieved by changing signal frequency or gain coefficient.