Convergecast, in which packets originating from multiple sources are reported to a single sink, is a fundamental primitive for data collection in wireless sensor networks. This paper investigates the time-optimal link-scheduling problem for time-division multiple-access (TDMA)-based convergecast, aiming to minimize the amount of time required to complete convergecast. We observe that packet copying between the microcontroller and the radio transceiver in existing sensor platforms has a big impact on the packet forwarding delay, and we propose a novel model for convergecast in which packet copying is separated from packet transmission and reception. We establish tight lower bounds on the number of time slots required for convergecast in networks with line and tree routing topologies, and we present both centralized and distributed algorithms for constructing the time-optimal convergecast schedules. We evaluate our scheme in both simulations and experiments on hardware. The results show that our scheme can achieve a system throughput (defined as the number of data bits received by the sink per second) of 202.8 kb/s, which is 86.31% of the theoretical bound. In comparison with the traditional TDMA-based convergecast schemes, our scheme can achieve up to a 86.22% improvement on system throughput.