This paper studies the throughput maximization problem for a three-node relay channel with non-ideal circuit power. In particular, the relay operates in a half-duplex manner, and the decode-and-forward (DF) relaying scheme is adopted. Considering the extra power consumption by the circuits, the optimal power allocation to maximize the throughput of the considered system over an infinite time horizon is investigated. First, two special scenarios, i.e., the direct link transmission (only use the direct link to transmit) and the relay assisted transmission (the source and the relay transmit with equal probability), are studied, and the corresponding optimal power allocations are obtained. By transforming two non-convex problems into quasiconcave ones, the closed-form solutions show that the source and the relay transmit with certain probability, which is determined by the average power budgets, circuit power consumptions, and channel gains. Next, based on the above results, the optimal power allocation for both the cases with and without direct link is derived, which is shown to be a mixed transmission scheme between the direct link transmission and the relay assisted transmission.