Resistance without resistors: An anomaly

The elementary 2-terminal network consisting of a resistively ($R-$) shunted inductance ($L$) in series with a capacitatively ($C-$) shunted resistance ($R$) with $R=\sqrt{L\mathrm{/}C}$, is known for its non-dispersive dissipative response, $i\mathrm{.}e\mathrm{.},$ with the input impedance $Z_0(\omega )=R$, independent of the frequency ($\omega$). 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 $Z_{n+1}(\omega) = f(Z_n(\omega))$,withtherecursivefunction$f(z) = (i\omega Lz/i\omega L + z)

• (z/1+i\omega 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 $Z_\infty (= R)$\mathrm{.}This,therefore,leadstorealizinginthelimit$n\to\infty$ 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.