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(ω) = R, independent of the frequency (ω). 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 Z_n+1(ω) = ƒ(Z_n(ω)), with the recursive function ƒ(z) = (iωLz/iωL + z) + (z/1 + iωCz). 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.