Apollonius unilateral transducer constant power gain circles on 3D Smith charts

Unilateral transducer constant power gain circles play an essential role in the design of radio-frequency amplifiers and active modulators, as they help to determine optimal impedance matching conditions to meet gain and stability specifications. It is shown that these gain circles are a subfamily of Apollonius circles. For better visualisation, unilateral transducer constant gain power circles have been plotted for the first time on the three-dimensional (3D) Smith chart. To this end, a natural relationship from an inversive geometry was required, in order to relate the gain circles with cutting planes for the 3D Smith chart.