IEEE Circuits and Systems Magazine - Q2 2018 - 46

(a)

(b)

(c)

(d)

Figure 9. Example of testing the emulator of two-state memristor with the PWL flux-charge constitutive relation from [31].
testing by: (a), (b): sinusoidal voltage, (c), (d): audio signal;
(a) voltage (upper curve) and current waveform; (b), (c): v-i
PHLs, (d) flux-charge constitutive relation.

This can be considered the "native" fingerprint of the
memristor. The test for a given element can be done via
a driving signal of random character. One example is illustrated in Fig. 7.
Six best known fundamental elements from Chua's table, the resistor R, capacitor C, inductor L, memristor MR,
memcapacitor MC, and meminductor ML, form a one-port
driven by a voltage waveform of random character. The
voltages, currents, fluxes, charges and integrals of fluxes
and charges on all elements are sensed and continuously
plotted as two-dimensional graphs in coordinates which
are combinations of all the above variables. Since the fundamental elements preserve their constitutive relations,
the portraits in the corresponding coordinates will be unambiguous curves. The other curves seem to be chaotic.
Results of the experiment are displayed in Fig. 8.
Let us focus on the right-side column of Fig. 8, containing the characteristics of memelements. Take note of
the characteristics located "one cell right and up" from
the constitutive relations, which are highlighted by colored frames. Their coordinates are time derivatives of
the constitutive variables of the element, thus i and v
for the memristor, i and { for the meminductor, and q
and v for the memcapacitor. All these cases provide a
set of pinched hysteresis loops which conforms to the
explanation given in Fig. 4: The unambiguous constitutive relation of the element in the storeyed structure is
projected one floor below as a pinched hysteresis loop
[25]. A comparison of the left and the right side of Fig. 8
also shows what happens when the memory of the el46

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ement is lost: the memelement changes into a classical
element of unambiguous characteristic, i.e. without the
hysteresis property.
It follows from the principle of this method that
it is useful for a reliable identification of an arbitrary
(a, b ) element from Chua's table. For example, it can
be used for a conclusive verification of the correct operation of emulators of memristors. Any deflection of
the measured characteristic from the unambiguous
charge-flux constitutive relation may indicate a problem. The charge and flux waveforms can be easily obtained via analog integrators. One can utilize the fact
that the emulator itself usually performs the integration
of either voltage or current and thus the corresponding
waveform is then directly available. For a correctly designed emulator one can easily identify the conditions
when the ability to emulate the memristor is reduced,
particularly due to overdriving the active component in
the emulator or exceeding its bandwidth. Fig. 9 contains
oscillograms from tests of a memristor emulator with
PWL flux-charge constitutive relation [31]. The pinched
hysteresis loop under sinusoidal excitation (Fig. b) is
not a proof that the system behaves as ideal memristor,
but portraying an unambiguous constitutive relation for
the audio driving signal (Fig. d) is conclusive.
IV. Conclusion
With the benefit of hindsight, when L. Chua's lifework
can be perceived in its entirety, it turns out that from
the very beginning he approached the development of
a modern circuit theory systematically from the foundations up to a roof as though at that time already he had
an exact notion of the final form of the work. The milestones in this evolution are memristors, Chua's pseudoperiodical table, Chua's circuits, neural networks, chaos, generalized law of thermodynamics, and generalized
Wolfram's concept of a New Kind of Science.
It is shown in the paper on the example of the first two
milestones what a well-thought-out and interlocking unit
results just from the concepts of predictive modeling,
memristors, and other nonlinear elements from Chua's
table, and that many present-day "discoveries" are mere
rediscoveries of what was previously described by an exact mathematical language in L. Chua's classical works.
This also holds without exception for various memristor
fingerprints which only represent different points of view
on the manifestations of a beautiful natural principle - a
principle that governs the behavior of quite a lot of nonlinear systems of not only electrical nature.
Acknowledgment
This work was supported by the Czech Science Foundation under grant No 18-21608S.
SECOND quartEr 2018

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