IEEE Circuits and Systems Magazine - Q2 2018 - 55

150 V = 0 V
Vm = -3.5 V
V = -2.75 V
m
Vm = -1.75 V m
Vm = -3 V
100
x/(Ωs-1)

Conversely, the reason behind the non-fading memory in periodically-forced ideal or ideal generic memristors lies in another memristor circuit-theoretic notion
([40], [55], [56]), according to which under excitations
of this kind transient dynamics may never emerge in
any of these devices, which thus experience steadystate behaviour from the very beginning, i.e. from the
time instant at which the AC voltage stimulus is applied
across them.
Remark 2: In case the memductance (memristance)
of a voltage (current)-controlled ideal or ideal-generic
memristor were fixed to some value for state values
within a certain range, then, for a number of distinct
initial conditions within that range, the device excitation by means of a periodic voltage (current) signal with
sufficiently-small amplitude may prove insufficient to
drive the memory state out of that range [55]. Despite
the state waveforms originating from that set of distinct
initial conditions would differ one from another, the current (voltage) response would inevitably be unique.
Before investigating models of real world memristors, it is instructive to examine a very popular DAE
set, first introduced by Pershin et al. back in [57], which
falls into the class of generic voltage-controlled memristors [40], but does not mimic quantitatively the physical
mechanisms behind the complex behaviour of a particular real-world resistance switching memory.

.

50
0
-50
-100
-150

Vm = 1.75 VVm = 2.75 V Vm = 3 V Vm = 3.25 V
3 4

6

8

10

12
x /Ω

14

16

18

20

Figure 6. drm of the Pershin memristor model under a set
of positive and negative dc values for the input voltage Vm .
note the similarity between the state dynamic routes of the
ideal flux-controlled memristor shown in Fig. 2. Here, however, when, under the continuous application of a constant
positive (negative) dc voltage across the memristor, the
state attains the lower (upper) bound, the rate of change of
the memory state drops suddenly from its negative (positive) constant value-white-filled circle with Vm -dependent
contour colour-to 0 Ωs - 1 -black-filled circle with black
contour colour at x = x on ^ x = x off h -keeping there at all
times afterwards.

(19)

where the voltage-controlled generic memristor state
is defined as the inverse of the device memductance,
x on a nd x off denoting its lower and upper bound,
while m 1 and m 2 denote two positive real coefficients
^ m 2 2 m 1 h, Vt repre sents a threshold voltage, and
step ( t) represents the Heaviside function, set to 1 if t $ 0
and 0 otherwise.
Fig. 6 depicts the DRM of the Pershin model for the
following set of parameters, extracted from the reference9 [57]: Vt = 2.5 V, x on = 3 X, x off = 20 X, m 1 = 10 X.
V -1 $ s -1, and m 2 = 100 X. V -1 $ s -1 . With reference to
the Pershin model state equation (18), for each positive (negative) value of the memristor DC voltage Vm the
T
xo = ^dx dt h - x loci is a straight line with negative (positive) ordinate, as shown in Fig. 6. It is worthy to point out
that the POP, consisting of the finite length segment extending over the horizontal axis from x = x on to x = x off,
reveals the analogue non-volatile memory capability of
the memristor with state equation (18). At power off the
memristor may keep the information embedded into its
memory state indefinitely, since any value for x within
its existence domain [x on, x off] represents an equilibrium
T
xr = x xo = 0 for the state equation under Vm = 0 V. Looking
at the direction of the arrows drawn on the xo -x loci, and
taking into account that x is constrained to lie over the
closed range defined as x ! [x on, x off], any constant positive (negative) DC voltage would progressively drive the

It is instructive to point out, already at this point, that, in general, the
AC fading memory of a memristor is not a boundary condition effect.

9
The value for m1 was increased by 100 times as compared to the reference [57] in order to improve visualisation quality, but this has no
impact on the theoretical results of this section.

C. A Generic Memristor with Non-Fading
Memory at AC: The Pershin's Model
As will become apparent shortly, due to the boundary
conditions imposed on the memory state, the Pershin
mathematical model [57] is subject to input-induced fading memory under any DC stimulation. Most interestingly, since the non-ideal memristor described by Pershin's
DAE set features symmetric on- and off-switching kinetics, its AC excitation induces no history erase effect,
similarly as in the case of ideal or ideal generic memristors, in case the memory state keeps away from either of
its two bounds in its time evolution8. The Pershin model
is defined as
dx = g (x, v )
m
dt
= ` - m 2 · v m + m 2 - m 1 · ^ v m + Vt - v m - Vt hj
2
· ^step (v m) · step (x - x on) + step (- v m)
· step (x off - x) h,
(18)
i m = G (x) v m
= 1 · v m,
x
8

sEcOnd quartEr 2018

IEEE cIrcuIts and systEms magazInE

55



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