IEEE Circuits and Systems Magazine - Q2 2018 - 58

Table I.
Values for the parameters in equations (20) and (21)
according to Strachan's memristor model [28].
A /s -1
10 -10

v off / V
1.3 · 10 -2

B /s -1
1 ·10 -4

v on / V
4.5 · 10 -1

v p / (AV)
4 · 10 -5

G m /S
2.5 · 10 -2

b/ (A -1 V -1)
500
x on
6 · 10 -2

a /S
7.2 · 10 -6

b/ V -1/ 2
4.7

Vm = 0.5 V
Vm = 0.45 V
Vm = 0.4 V
Vm = 0.3 V
Vm = 0.2 V Vm = 0.01 V
Vm = 0.1 V Vm = 0.001 V

105

.
x/s

x off
4 · 10 -1

100
10-5
10-10
0

-10-10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
(a)

1

Vm = -0.001 VV = -0.01 VVm = -0.1 V
Vm = -0.2 V
m

.
x/s

-10-5
-100
-105
-1010

Vm = -0.3 V
Vm = -0.4 V Vm = -0.45 V Vm = -0.5 V
0

0.2

0.4

0.6

0.8

1

x
(b)
Figure 11. drm of the strachan memristor model under a number of positive (a) and negative (b) dc values of the memristor voltage with modulus lying within the closed set defined
as Vm ! " 0.001, 0.01, 0.1, 0.2, 0.3, 0.4, 0.45, 0.5 , .

Plots (a) and (b) in Fig. 11 show the DRM of Strachan's
memristor model for a number of positive and negative
values of the memristor voltage, respectively. As in the
case of Pershin's memristor model, any constant positive
(negative) DC voltage applied continuously across the
memristor inevitably drives the device towards its highest
(lowest) conduction state, keeping it there afterwards11.

However, here, under positive (negative) DC inputs, the
speed in the state evolution towards the upper unitary
(lower null) bound may vary over a large number of
decades within the state existence domain [0, 1] in dependence on the stimulus strength. This is one of the
characteristic features of the switching kinetics of realworld memristor devices. Moreover, unlike Pershin's
model, the on- and off-switching kinetics are clearly
asymmetric, yet another signature of physical resistance
switching memories.
The fading memory capability of this nano-device
under both DC and AC excitations was first discovered
by Ascoli and Tetzlaff in circuit- and system-theoretic
investigations of the Strachan's model [42] and later experimentally demonstrated at HP Labs. For details we invite the interested readers to consult the reference [33].
Here we wish to provide some further insight into the
emergence of memory loss in the tantalum oxide nanodevice from HP Labs. The complex mathematical form
of the state evolution and memductance (memristance)
functions in the DAE set of a physical voltage (current)controlled memristor device results in great difficulties
encountered in the attempt to derive the analytical state
solution under a general input. Thus researchers, willing
to analyse the peculiar dynamics of memristor devices
for their later exploitation in novel electronic circuits,
need to rely on numerical simulations of the underlying
models. However, in cooperation with Sirakoulis and
Ntinas [27], we were recently able to derive the following closed-form solution for the memory state after the
analytical integration of Strachan's DAE set under any
constant positive DC input voltage Vm (consult the reference [27] for details):
erfi -1 c 2 h (Vm) (t - t 0) + erfi ^a 1 x 0 - b 1 (Vm) hm
r

x (t) =

a1
b 1 (Vm)
+
,
a1

(22)

T

where x 0 = x (t 0) is the state initial condition at time t = t 0,
erfi -1 ( t) = - i erfi (i ( t)) stands for the inverse imaginary error function [58], plotted in Fig. 12(a) against its
argument, i = - 1 denotes the imaginary unit, erfi ($)
represents the imaginary error function [58], i.e.
T
erfi (p) = 2

r

#0

p

u2

e p dpu,

(23)

while a 1, b 1 (v m), and h (v m) are defined as
11

When, under the continuous application of a positive (negative) DC
voltage, the state attains the upper (lower) bound in its existence domain, the rate of change of the memory state drops suddenly from its
positive value-white-filled circle with Vm -dependent contour colour-
to 0 s - 1-note that the null value corresponds to - 3 on the logarithmically-scaled vertical axis in plot (a) ((b)) of Fig. 11-keeping there at
all times afterwards.

58

IEEE cIrcuIts and systEms magazInE

T

a1 =
T

b 1 (v m) =

1 ,
x on

(24)

x on v 2m ^
G m - a exp ^b v m hh,
2v p

(25)

sEcOnd quartEr 2018



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