Synthesis Without Ideal Transformers

Why RLC Realizations of Certain Impedances Need Many More Energy Storage Elements Than Expected

“…The networks discovered in the 1950s [10]- [12], which simplify the famous Bott-Duffin networks [9], contain a surprisingly large number of energy storage elements, and have non-minimal state-space realizations with states as inductor currents and capacitor voltages. In this paper, we showed that these networks actually contain the least possible number of energy storage elements for realizing certain impedances (almost all biquadratic minimum functions).…”

“…Slight simplifications of the Bott Duffin networks were discovered by Reza, Pantell, Fialkow and Gerst in the 1950s [10]- [12] (the RPFG networks). In this section, we present some newly discovered alternatives to these networks.…”

“…Notably, an RLC network can contain more energy storage elements than the McMillan degree of its impedance, and possess a nonminimal state-space representation whose states correspond to the inductor currents and capacitor voltages. Indeed, this is the case for the famous Bott-Duffin networks [9], and their simplifications [10]- [12]. The purpose of this paper is to demonstrate the necessity of this apparent non-minimality in the RLC realization of certain PR functions.…”

“…Stage 1 (described above) is achieved by the Foster preamble, as discussed in [13,Section II]. The contribution of [10]- [12] was the discovery of the networks in Fig. 4a, which pertain to stage 2.…”

“…The famous Bott-Duffin networks [9], and their simplifications [10]- [12] (the RPFG networks), prove that any given PR function can be realized by an RLC network. However, the number of energy storage elements in these networks is considerably greater than the McMillan degree of their impedance.…”