A contribution to element reduction in transformerless driving-point impedance synthesis

A theorem formulating the necessary and sufficient conditions for the decomposition of a real positive function into a sum of real positive functions is presented. It is shown that every such decomposition depends on an arbitrary real positive function and on a positive number only. The appropriate choice of this function gives decompositions with desired properties. This theorem enables us to treat the known methods of transformerless synthesis, which do not use bridge networks, as special cases of a more general approach. Resistance extraction, Bott and Duffin´s method and Miyata´s method are shown as special cases. It is possible to compare these methods and derive some new procedures. The discussion and examples show that a reduction of the number of elements required in the synthesis procedure can be obtained. This reduction is due to the fact that the decomposition can be done without common poles between the two summands in some cases; in this way it is possible to avoid the exponential growth of the number of required elements which is found in other methods of transformerless drivingpoint impedance synthesis.