IEEE Circuits and Systems Magazine - Q2 2018 - 38
The predictive model must, with a reasonable accuracy, mimic the behavior
of the original in various modes of its operation.
The concept of the so-called predictive modeling [8]
forms the basic idea of the HOEs, organized in Chua's
table in Fig. 1. The predictive model must, with a reasonable accuracy, mimic the behavior of the original in
various modes of its operation. For example, the linearized model of a BJT amplifier cannot predict the behavior of the original within its large-signal operation. All
HOEs from Chua's table have their predictive models,
and therefore they can serve as building blocks for the
construction of predictive models of more complex systems. The predictive model of the (a, b ) HOE, where a
and b are integers, the constitutive relation, is generally
a nonlinear algebraic relation between two constitutive
variables v (a) and i (b). The notation v (a) specifies the a
-order time integral (derivative) of voltage (integral for
negative and derivative for positive a ). A similar notation i (b) also holds for the integral/derivative of current.
The generalization of the well-known R-C-L-M square
schematic, the so-called storeyed structure in Fig. 2, provides a different view on Chua's table. It is obvious from
Fig. 2 that the memristor (the ideal memristor according to
current terminology) is a fundamental element, located
not on the ground floor of classical R, C, L elements but on
the higher floor of their memory versions.
Despite the common opinion, the elements from Chua's
table or an equivalent storeyed structure do not have the
potential to model an arbitrary system. Back in 1980, L.
Chua notices that "..it is generally impossible to model an
(n+1)-terminal or n-port device using only 2-terminal elements
as building-blocks" [7]. That it why he developed, in parallel
to two-terminal fundamental elements, the theory of multipoles or multiports [8]. Moreover, some two-terminal elements cannot be compounded of the models of ideal memristors and the other HOEs. To do this, we would also need,
for example, controlled sources, thus at least three-terminal
elements. Within this context, the extension of the original
definition of the ideal memristor to its other more complex
versions from Table 1 seems to be a pragmatic step, forming
a bridge between Chua's table of fundamental elements and
the models of much more complex existing systems.
β
0
0,2
1,2
2,2
-1
2
1
-2,1
-1,1
0,1
1,1
2,1
-2
1
0
-2,0
-1,0
0,0
1,0
2,0
-3
0
-1
-2,-1 -1,-1 0,-1
1,-1
2,-1
-4
-1
-2
-2,-2 -1,-2 0,-2
1,-2
2,-2
-5
1
2
-1
,-
2
ur
re
rg
e
C
ur
re
-2
0
1
2
3
α
ge
TI
(a)
-1
Vo
lta
-2
Fl
ux
-3
-3
F
α
Q
3
C
ha
lta
ge
ux
0
Vo
TI
-1
Fl
-2
F
-3
-3
TI
C
ha
r
C
Q
0
-1
0,
0
-1,2
0,
-1
-2,2
nt
2
TI
β -α : 1
3
-1
,0
1
-1
,-
1
2
-2
,-
1
3
nt
4
ge
3
β
β -α : 5
(b)
Figure 1. (a) Illustration of Chua's table of fundamental (a, b) elements. the integral of voltage (a = - 1) is the flux {; the integral
of flux, thus the second integral of voltage (a = - 2) is tIF (time-Integral of Flux) t. Similarly, the integral of current (b = - 1) is
the charge q; the integral of charge, thus the second integral of current (b = - 2) is tIq (time-Integral of Charge) v. (b) resistor, inductor, capacitor are defined by their constitutive relations between voltage and current, flux and current, and voltage and
charge. they are therefore (0,0), (-1,0), and (0,-1) elements. their memory versions, i.e. memristor, meminductor, and memcapacitor, have the coordinates (-1,-1), (-2,-1), and (-1,-2), such that the memristor is unambiguously defined via its flux-charge
constitutive relation.
38
IEEE CIrCuItS aND SyStEmS magazINE
SECOND quartEr 2018
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