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In class-E switching-mode power amplifiers, the switch-current waveform includes a step change ("jump"), approximated by a ramp of <15% of the period. At a transistor's highest useful frequency, the large input drive required for fast-enough switching yields marginal power gain. Objective: a high-efficiency power amplifier with jumpless current and voltage waveforms. Previously, that was proven impossible for amplifiers using only linear passive components and an ideal switch. We present the theory of a new topology that does achieve the objective: a class-E amplifier with nonlinear passive or active components in the load network. A "biharmonic" version was simulated, built, and tested. It comprises a main stage switching at the output frequency f1, drawing DC power of approximately 3/4(POUT RF/drain efficiency), and an auxiliary amplifier switching at 2f1, injecting 2f1-current into the circuit node at the main-stage transistor's output port to shape jumpless voltage and current waveforms. That switching (nonlinear) output port converts 2f1 power from the auxiliary amplifier to approximately 1/4 of the f1 power at the load. Computer simulation, and measurement on a scaled-frequency 3.5-MHz prototype, show that switching losses practically disappear when the main-stage switch is operated in the jumpless regime.