IEEE Circuits and Systems Magazine - Q2 2018 - 56

20

x0 = 20 Ω
x0 = 15 Ω

0.6

x0 = 11 Ω
x0 = 7 Ω
x0 = 3 Ω

10
5
3
0

0.5

1 1.5
t /s
(a)

im /A

x/Ω

15

0

0.8
x0 = 3 Ω

0.4

x0 = 7 Ω
x0 = 11 Ω
x0 = 15 Ω
x0 = 20 Ω

0.2
0

2

0

0.5

1
t /s
(b)

1.5

2

vm /V

Figure 7. Fading memory in Pershin's memristor model under dc excitation. unique asymptotic behaviour for memory
state (a) and device current (b). Here Vm = 2 V.

2
0
-2
0

1

2

3

4

5

t /s
(a)
x0 = 20 Ω

x/Ω

20
10

0 x0 = 17.5 Ω x0 = 15 Ω x = 12.5 Ω x0 = 10 Ω
0
0

1

2

3

4

5

t /s
(b)
zim z/A

0.5
0.25
x = 20 Ω
0 0
-3
-2

x0 = 15 Ω x0 = 10 Ω
x0 = 12.5 Ω
x0 = 17.5 Ω
-1

0
vm /V
(c)

1

2

3

Figure 8. absence of fading memory in Pershin's model under periodic zero-mean ac excitation and state confinement within
its existence domain. triangular voltage applied across the
memristor modelled by the daE set (18)-(19) (a), and resulting state response over time (b), and hysteresis current-voltage loops (c) for a number of initial conditions, specifically for
x 0 ! " 10, 12.5, 15, 17.5, 20 , X.

memory state with constant rate into the on (off) value
x on ^ x offh . Keeping unaltered the DC voltage stimulus
across the device, after the time instant at which the
state attains the lower (upper) bound, xo would suddenly drop to 0 X/s, see the black-filled circle on the
point (x, xo ) = (x on, 0) ((x, xo ) = (x off, 0)) of the x-xo plane of
Fig. 6, and there would be no further change in x. In other words, after the aforementioned time instant x on ^ x offh
would be interpreted as a boundary condition-induced
equilibrium for the state equation (18) irrespective of
56

IEEE cIrcuIts and systEms magazInE

the positive (negative) value for Vm. It is thus straightforward to realize that under DC voltage excitation, both
the device state and its current will progressively experience memory loss, and, at steady state, will assume
the expressions x = xr and i m = ^1 xr h Vm, respectively,
where xr = x on step (Vm) + x off step (- Vm). Thus, due to the
saturation in the device state, the Pershin's memristor
experiences fading memory under any DC stimulation.
This is clearly demonstrated in plots (a) and (b) of Fig. 7,
respectively showing the time waveforms of memristor
state and current in response to a DC voltage Vm of value
2 V applied directly across the device for all initial conT
ditions within the set x 0 = x (0) ! " 3, 7, 11, 15, 20 , X.
Let us now focus on the device AC stimulation. The
ordinate of each straight line of the xo -x loci in Fig. 6
has fixed modulus for a given value of Vm . In other
words the on- and off-switching processes, respectively
emerging under positive and negative values for the
voltage stimulus, are symmetric. This feature, mathematically characterisable as g (x, v m) = - g (x, - v m), is
not observable in typical real-world resistance switching memories, and bears an important consequence on
the dynamic response of the memristor to periodic AC
excitations. Provided the AC input does never drive the
memory state into either of its two bounds10, the information embedded in the initial condition has a crucial
impact on the device dynamics, which do not experience the transitory memory loss phenomenon typical of real-world memristors with asymmetric on- and
off-switching kinetics. This is demonstrated in Fig. 8,
showing the simulation results obtained by applying a
triangular voltage waveform of amplitude vt m = 2.75 V
and period T = 1 s, shown in plot (a), directly across the
memristor for distinct initial conditions within the set
x 0 ! " 10, 12.5, 15, 17.5, 20 , X. As may be inferred by inspecting plot (b), the resulting time waveforms of the
memory state, undergoing no transitory memory loss,
keep distinct at all times. Plot (c) shows the different
hysteresis loops observed in the v m -i m plane for each of
the aforementioned initial conditions.
For the same parameter setting and input waveform,
choosing initial conditions closer to the lower bound,
specifically within the set x 0 ! " 3, 4, 5, 6, 7, 8, 9 , X, all
the state solutions hit the lower boundary within the
first input half cycle, and thereafter exhibit a unique
time behaviour, as shown in Fig. 9. Importantly, the AC
history erase effect emerging in the Pershin's model for the given set of initial conditions, which may
be interpreted as a local form of fading memory [47],
10

In Fig. 8 one of the initial conditions x0 -see plot (b)-is set to xoff, but
the input of plot (a), positive over the first half cycle, drives the memristor state away from the upper bound as soon as it is applied across the
device, and x returns to xoff only at the end of the input period.
sEcOnd quartEr 2018

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