Resistance without resistors: An anomaly

Abstract

The elementary 2-terminal network consisting of a resistively ($R-$) shunted inductance ($L$) in series with a capacitatively ($C-$) shunted resistance ($R$) with $R=\sqrt{L/C}$, is known for its non-dispersive dissipative response, $i.e.,$ with the input impedance $Z_0(\ddot{I}‰)=R$, independent of the frequency ($\ddot{I}‰$). In this communication we examine the properties of a novel equivalent network derived iteratively from this 2-terminal network by replacing everywhere the elemental resistive part $R$ with the whole 2-terminal network. This replacement suggests a recursion $Z_{n+1}(\ddot{I}‰)=f(Z_n(\ddot{I}‰))$, with the recursive function $f(z)=(i\ddot{I}‰Lz/i\ddot{I}‰L+z)+(z/1+i\ddot{I}‰Cz)$. The recursive map has two fixed points -- an unstable fixed point $Z_u^{\star }=0$, and a stable fixed point $Z_s^{\star }=R$. Thus, resistances at the boundary terminating the infinitely iterated network can now be made arbitrarily small without changing the input impedance $Z_{\infty }(=R)$. This, therefore, leads to realizing in the limit $n\to \infty$ an effectively dissipative network comprising essentially non-dissipative reactive elements ($L$ and $C$) only. Hence the oxymoron -- resistance without resistors! This is best viewed as a classical anomaly akin to the one encountered in turbulence. Possible application as a formal decoherence device -- the {\it fake channel} -- is briefly discussed for its quantum analogue.

6 pages, 4 figures