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The Fourth Element

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Abstract

This tutorial clarifies the axiomatic definition of \((v^{(\alpha )},i^{(\beta )})\) circuit elements via a look-up-table dubbed an A-pad, of admissible (vi) signals measured via Gedanken Probing Circuits. The \((v^{(\alpha )},i^{(\beta )})\) elements are ordered via a complexity metric. Under this metric, the memristor emerges naturally as the fourth element Tour (Nature 453:42–43, 2008 [1]), characterized by a state-dependent Ohm’s law. A logical generalization to memristive devices reveals a common fingerprint consisting of a dense continuum of pinched hysteresis loops whose area decreases with the frequency \(\omega \) and tends to a straight line as \(\omega \rightarrow \infty \), for all bipolar periodic signals and for all initial conditions. This common fingerprint suggests that the term memristor be used henceforth as a moniker for memristive devices.

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Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    Observe that the voltage v and the current i are defined axiomatically via two instruments called voltmeter and ammeter, without invoking any physical concepts such as electric field, magnetic field, charge, flux linkages, etc. One does not even have to know how a voltmeter, or an ammeter, works. They are just names assigned to the instruments.

  2. 2.

    In practice one can never know the precise signal i(t) over the infinite past. Rather we can only set up our measurements to begin at some initial time \(t=t_0\). Consequently, the initial condition \(q_0\) in Eq. (8) represents a summary of the past memory of q(t) measured at \(t=t_0\).

  3. 3.

    Hodgkin and Huxley were awarded the 1965 Nobel Prize in physiology for their derivation of the circuit shown in Fig. 4a, where the 2 memristors were drawn as time-varying resistors in Fig. 1 (page 501) of [17].

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Chua, L. (2019). The Fourth Element. In: Chua, L., Sirakoulis, G., Adamatzky, A. (eds) Handbook of Memristor Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-76375-0_1

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  • DOI: https://doi.org/10.1007/978-3-319-76375-0_1

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