IEEE Circuits and Systems Magazine - Q2 2018 - 37
in Table 1. The dual case of the
voltage-controlled elements is
omitted for the sake of simplicity.
It is obvious that the nonlinearity
of the characteristics is caused
by the dependence of the resistance on the current, whereas the
pinched hysteresis loop is due
to the resistance being governed
by the state variable, which is a
result of the inertial integrating
processes. The differential equation of ideal generic memristor
dx/dt = f (x) i can be arranged to
the form dx/f (x) = idt and inte grated: F (x) = # dx/f (x) = q. If an
inversion x = F -1 (q) ex ists for
the function F (x) = q, then the
behavior of ideal generic memristor can be modeled via ideal
memristor which complies with
the formula v = R M (F -1 (q)) i. For
f " 3, the gains of the integrating
cells tend to zero, and the memristance is not modulated via the
state. The hysteresis disappears:
The memristance of the extended
memristor is then modified only
by a current, and its v - i characteristic tends to the characteristic
of a nonlinear resistor. The memristance of the other memristors
is fixed, with the corresponding
straight-line v - i characteristic of
a linear resistor.
L. Chua revealed both the identification signs of the memristors,
the so-called fingerprints, and the
conditions the memristive system
must fulfill in order to be useful for
computer industry (particularly its
nonvolatility). As a subsequent important step, he put the memristor
into the existing order in the circuit
theory, the latter being developed
into the concept of Higher-Order
Elements (HOE) [7]. In the 1980s,
he showed that the memristor was
only the tip of the iceberg, and he
passed a tool of universal modeling of "anything in Electrical Engineering" on to next generations
of investigators.
SECOND quartEr 2018
Table 1.
The evolution of current-controlled resistive elements from linear
resistor to extended memristor, their circuit equations, diagrams, and v-i
characteristics.
Element
Model
Linear
resistor
v = Ri
i
v
R
v
Nonlinear
resistor
v = v (i ) = R (i ) i
i
i
v = R M (q) i
dq/dt = i
q
Ideal
generic
memristor
i
i
f→∞
v
f
i
v
i
0
v = R M (x ) i
dx /dt = f (x ) i
or
0
v
RM
v
i
f→∞
i
i
0
v
R
v
Ideal
memristor
v-i Characteristic
Diagram
x
0
RM
i
or
v = R M (x ) i
x = F -1 (q)
q = # idt
Generic
memristor
i
i
q
v
x
RM
v = R M (x) i
dx/dt = f (x, i )
i
Extended
memristor
x
i
i
v
0
RM
v = R M (x, i ) i
R M (x, 0) ! 3
d x/dt = f (x, i )
f→∞
v
f
i
v
F -1
i
f→∞
v
f
x
RM
i
0
i
IEEE CIrCuItS aND SyStEmS magazINE
37
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