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Given large circuit sizes, high clock frequencies, and possibly extreme operating environments, Field Programmable Gate Arrays (FPGAs) are capable of heating beyond their designed thermal limits. As new circuits are developed for FPGAs and deployed remotely, engineers are challenged to determine in advance if the device will operate within recommended thermal ranges. The amount of power consumed by the circuit depends on how an algorithm is compiled into hardware, how the circuit is placed and routed, and the patterns of data that pass through the system. The amount of heat that can be dissipated depends on the thermal transfer characteristics of the package, the air flow that passes over the package, and the ambient temperature of the remote systems. Rather than designing a system to handle unreasonable worst-case situations, we have implemented a thermal management system that continuously monitors the temperature of the FPGA and reprograms the device if the temperate approaches the outer limits of safe operating conditions. Our system measures the junction temperature of a Xilinx Virtex FPGA using a built-in thermal diode. Using the temperature monitoring mechanism, we have studied the steady-state and transient conditions of multiple benchmark circuits implemented in an FPGA logic on the Field-programmable Port Extender (FPX) development platform. We observed properties of these benchmark circuits that enable us to predict power and thermal characteristics for real applications. We propose a Dynamic Thermal Management (DTM) strategy for FPGAs based on temperature feedback.