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The results of exact time domain analysis of the novel switched-mode tuned Class-FE power amplifiers with a parallel quarterwave transmission line are presented with analytical derivation of the optimum load network parameters. Effects of the quarterwave transmission line and collector capacitance on the current and voltage waveforms are discussed. A Class-FE power amplifier is based on the combination of a class-F operating mode with Class-E switching conditions, thus extending the Class-F operation to higher frequencies for driving signals with duty cycles D < 0.5. The ideal collector voltage and current waveforms demonstrate a possibility of 100% efficiency without overlapping between each other. A possibility of alternative load network implementation including a series quarterwave transmission line or lumped-element approximation is also considered. The switched-mode Class-FE power amplifiers with a quarterwave transmission line offer a new challenge for RF and microwave power amplification providing high-efficiency operation conditions.