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A general theory of the n-section quarter-wave transformer is presented. It is shown that optimum bandwidth with a minimum pass band tolerance is obtained when the power loss ratio is chosen to give Tchebycheff behavior in the pass band. A comparison is made of the Tchebycheff transformer and the maximally flat transformer, and shows that the former gives a large increase in bandwidth-e.g., up to 44 per cent for a 2-section transformer and up to 75 per cent for a 4-section transformer. Design formulas are given for the 2, 3 and 4-section transformers.