Resistance without resistance: An anomaly

Abstract

The elementary two-terminal network consisting of a resistively (R) shunted inductance (L) in series with a capacitatively (C) shunted resistance (R) with R= L/C, is known for its non-dispersive dissipative response, i.e. with the input impedance Z0(w) = R, independent of the frequency (w). In this communication, we examine the properties of a novel equivalent network derived iteratively from this two-terminal network by replacing everywhere the elemental resistive part R with the whole two-terminal network. This replacement suggests a recursion Zn+1(w) = f(Zn(w)), with the recursive function f (z) = (iw Lz/iw L + z) + (z/1 + iw Cz). This recursive map has two fixed points – an unstable fixed point Z*u = 0, and a stable fixed point Z*s = R. Thus, resistances at the boundary terminating the infinitely iterated network can now be made arbitrarily small without changing the input impedance Z¥ (= R). This, therefore, leads to realizing in the limit n ® ¥, an effectively dissipative network comprising essentially the non-dissipative reactive elements (L and C) only. Hence the oxymoron–resistance without resistance! This is best viewed as a classical anomaly akin to the one encountered in turbulence. Possible application as a formal decoherence device – the fake channel – is briefly discussed for its quantum analogue.