The instantaneous luminosity of the LHC will be increased by up to a factor of three with respect to the original design value at Run-3 (starting 2021). The ATLAS Level-1 end-cap muon trigger in LHC Run-3 will identify muons by combining data from the Thin-Gap Chamber detector (TGC) and the New Small Wheel (NSW), which is a new detector and will be able to operate in a high background hit rate at Run-3, to suppress the Level-1 trigger rate. In order to handle data from both TGC and NSW, a new trigger processor board has been developed. The board has a modern FPGA to make use of Multi-Gigabit transceiver technology. The readout system for trigger data has also been designed with TCP/IP instead of a dedicated ASIC. This letter presents the electronics and its firmware of the ATLAS Level-1 end-cap muon trigger processor board for LHC Run-3.