Classical Network Synthesis Revisited

The talk consists of four parts. In the first part, the motivation is explained for revisiting certain questions in circuit theory, in particular, the synthesis theory of “one-port” (i.e. with a pair of external driving terminals) RLC networks and the questions of minimality associated with the Bott-Duffin procedure. The motivation relates to the synthesis of passive mechanical impedances and the need for a new ideal modelling element the “inerter” [6]. The development of the inerter from a mathematical concept and ideal modelling element through to its adoption as a standard component in Formula One racing and beyond is described [2]. In the second part, classical results from electrical circuit synthesis are reviewed including the procedures of Foster, Cauer, Brune, Darlington, Bott and Duffin. The reactance theorem of Foster [3] for lossless networks and the Bott-Duffin construction [1] for arbitrary positive-real functions are highlighted. In the third part, the concept of regular positive-real functions is described. A positivereal function Z(s) is defined to be regular if the smallest value of Re (Z(jω)) or Re ( Z−1(jω) ) occurs at ω = 0 or ω = ∞. It is shown how the concept can aid the classification of lowcomplexity networks [4]. The fourth part describes recent work and new results on network synthesis [5].