IEEE Circuits and Systems Magazine - Q2 2018 - 54

30

0.35
ϕm ,0 = 6 Vs

25
20

im /(A)

10
5
ϕm ,0 = -3 Vs

0

-10

0

10

20

t /s
(a)

30

ϕm ,0 = 3 Vs
ϕm ,0 = 3 Vs

0.2

ϕm ,0 = -3 Vs

0.15

ϕm ,0 = -6 Vs

0.1
0.05

ϕm ,0 = -6 Vs

-5

ϕm ,0 = 6 Vs

0.25

ϕm ,0 = 3 Vs

15
ϕm /(Vs)

0.3

ϕm ,0 = 3 Vs

40

50

0

0

10

20

t /s
(b)

30

40

50

Figure 3. non-fading memory in a dc-driven ideal flux-controlled memristor. memory state (a) and device current (b) over time
under dc excitation-the voltage Vm applied across the memristor has value 0.4 V-for a number of distinct initial conditions.

dynamic behaviour. Fig. 5, referring to the same AC
simulation as Fig. 4, illustrates the hysteresis loops observed after plotting the base-10 logarithm of the device
current modulus over the device voltage for all initial
states in the aforementioned set S.
All in all, this device does not have fading memory. In
fact, neither under DC excitation nor under AC periodic
forcing, history erase effects may ever appear in elements from the classes of ideal and ideal generic memristors. The absence of DC fading memory in elements
from these classes is related to a well-known result from
memristor theory ([40], [33]), whereby none of these
two-terminal circuit-theoretic components may ever at-

25
20

ϕm ,0 = 6 Vs

ϕm ,0 = 0 Vs

tain a steady-state behaviour, explaining the reason why
they are known as DC nullators.
Remark 1: In case the memductance (memristance)
of a voltage (current)-controlled ideal or ideal-generic
memristor were fixed to a lower (upper) limit for state
values below a negative (positive) threshold, then,
despite the memristor state solutions would keep
distinct for distinct initial conditions, the device currents would asymptotically lead to a unique steadystate behaviour under unipolar DC excitation. This is
expected since a memristor of this kind would behave
as a simple linear resistor as the state exceeds either
threshold [55].

0

ϕm ,0 = 3 Vs

-1

5

-5

-5

ϕm ,0 = -3 Vs
0

200

400

ϕm ,0 = 6 Vs
600

800

1,000

t /s
Figure 4. Evidence for absence of fading memory in an ideal
flux-controlled memristor: initial condition-dependent memory state over time under ac periodic voltage excitation (see
text for details).
54

-3
-4

0

-10

ϕm ,0 = 3 Vs

-2

10

im /A

ϕm /(Vs)

15

ϕm ,0 = 6 Vs ϕ
m ,0 = 0 Vs

IEEE cIrcuIts and systEms magazInE

ϕm ,0 = -6 Vs

ϕm ,0 = -3 Vs

-6
-0.4 -0.3 -0.2 -0.1

0
0.1
vm /V

0.2

0.3

0.4

Figure 5. sea-gull alike loci [54] of log 10 i m versus v m under
sine wave voltage excitation of an ideal flux-controlled memristor for a number of distinct initial states. this plot refers to
the same simulation as Fig. 4.

sEcOnd quartEr 2018



Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q2 2018

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