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The feasibility of a new three-terminal linear power amplifier has been demonstrated both theoretically and experimentally from 0.5 to 3.0 GHz. The new amplifier is similar to an n-p-n bipolar transistor in configuration but develops extra power gain through avalanche multiplication and by the use of transit time in the collector. Major differences in the construction of the two devices are in their collector doping profiles and depletion layer widths. It is estimated that this new amplifier will be capable of several watts of power output at 10 GHz with useful gain, good linearity, and wide dynamic range. An acronym, CATT, which stands for controlled avalanche transit-time triode, is used to designate this new microwave semiconductor device. In this paper, the theory of the CATT is developed. It is found to be dc and RF stable. The necessary conditions on the ionization coefficients for signal amplification are investigated. The emitter-base dynamics of the CATT are shown to be quite different from a transistor due to hole feedback from the avalanche multiplication region. This phenomena results in a more uniform emitter current injection and better use of the emitter finger area than for transistors.