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 $Z_0(\omega) = R$, 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 $Z_{n+1}(\omega) = f(Z_n(\omega))$, with the recursive function $f(z) = (i\omega Lz/i\omega L + z) + (z/1+i\omega Cz)$. 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 $Z_\infty (= R)$. 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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2007
- DOI:
- 10.48550/arXiv.0706.4384
- arXiv:
- arXiv:0706.4384
- Bibcode:
- 2007arXiv0706.4384K
- Keywords:
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- Condensed Matter - Disordered Systems and Neural Networks;
- Condensed Matter - Mesoscopic Systems and Quantum Hall Effect
- E-Print:
- 6 pages, 4 figures