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This paper concerns with the master-slave exponential synchronization analysis for a class of general Lur'e systems with time delay. Different from the previous methods based on the differential inequality technique, a new approach is proposed to derive some new exponential synchronization criteria. The restriction that the control width has to be larger than the time delay is removed. This leads to a larger application scope for our method. Moreover, no transcendental equation is involved in the obtained result, which reduces the computational burden. Two examples are given to validate the theoretical results.