Impedance synthesis without mutual coupling

Abstract

the Cartesian co-ordinates of A, , A2 and A3 being (xt , yj, (x2 , y2) and (x3 , y3) respectively, intersect at L. Let the lines H3M and H,M be the lines derived from the lines H3L and H^L by the procedure by which in Fig. 2 the line 12 was derived from the line 34. Let them intersect at the point M, ysr). Using basic methods of analytical geometry we obtain for ?/."and a similar formula for xM • These formulae are symmetrical in the co-ordinates of A i , A 2 and A3 , so that the line IL.M derived from the line H2L by the same procedure passes through M. In other words, if the points 0, 0' and 0" in Fig. 2 are the mid-points of the sides of the triangle ABC, the line 02 for the side AB and the corresponding lines 0'2 and 0"2 for the sides BC and CA intersect in one point 2.If (see Fig. 3) the ratiothat corresponds to the ratio <T-r0.,/lAB of Fig. 2 is computed, we obtain after some manipulating 27 = 4-kg ^7) cot ip2 -~ cot tan tp\ tan (p2 Equation (7) is identical with Eq. (2), as can easily be shown. Hence Eq. (2) represents the special case of Eq. (4), if the point 0 (Fig. 2) and the corresponding other two points 0' and 0" are the mid-points of the sides of the triangle ABC. Tensor conductivities can be dealt with also by a change of variables [3], This method leads to the same results.

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).…”

“…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.…”

“…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.…”

“…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.…”

Synthesis of positive-real functions with low-complexity series-parallel networks

“…The apparent non-minimality of the construction has subsequently intrigued many researchers and there were a number of important papers which followed up on this question, e.g. [2], [3], [4], [5]. Interest in the topic lost momentum in the early 1970s due to the growing importance of integrated circuits.…”