2.4 Genealogy of Memristor Devices
73
It is easily seen that such two-terminal device, defined by equations similar to (2.44)
and (2.45) without the constrain (b) on the boundedness of R(x, i), would be
classified as an extended memristor.
To check the zero-crossing property and then pinched hysteresis loops, let us
apply a current source i = sin(t) and assume the initial state x(0) = −1. This
yields
x(t) = x(0) +
t
0
sin(τ )dτ = − cos(t).
Substituting for x in (2.51), we obtain
v(t) =
x(t)
i(t)
i(t).
The corresponding Lissajoux figure in the (v, i) plane is the unit circle shown in
Fig. 2.28a, which is not pinched at the origin (v, i) = (0, 0)!
This is a shocking result suggesting that one should accept the notion that not all
memristors exhibit the pinched hysteresis loop fingerprint.
Resolving this anomaly leads to a logical inconsistency which we will now
demonstrate. Following the systematic approach shown in the Examples 2.4 and 2.5,
it is easily understood that the problem is the plot of the Lissajoux figure in the
wrong plane. Considering the plot of v(t) as function x(t), i.e., the Lissajoux
figure in the v − x plane, a single-valued curve without any hysteresis is obtained
(Fig. 2.28b). Such curve represents the correct characteristic of the considered twoterminal device and, from the previous equations, it is easily derived that
v(t) = x(t) =
t
−∞
i(τ )dτ
.
= q(t)
which is just the definition of a 1 Farad capacitor! It would certainly be nonsense
to refer to a capacitor as (extended) memristor!
Remark 2.9 Although an extensive discussion of zero-crossing property and
pinched hysteresis loops is reported in Sect. 2.1.3, the Examples 2.4, 2.5, and 2.27
allow us to summarize the following chief statements:
• the pinched hysteresis loop is the hallmark of all memristors, and it can be
used for an experimental definition of memristors, whether they are ideal or not.
The pinched hysteresis loop signature of memristors must hold for all periodic
voltage or current inputs with zero time-average. It follows that the measurement
of a non-pinched hysteresis loop for even just one such testing signal would
disqualify a device from being classified as a memristor
73
It is easily seen that such two-terminal device, defined by equations similar to (2.44)
and (2.45) without the constrain (b) on the boundedness of R(x, i), would be
classified as an extended memristor.
To check the zero-crossing property and then pinched hysteresis loops, let us
apply a current source i = sin(t) and assume the initial state x(0) = −1. This
yields
x(t) = x(0) +
t
0
sin(τ )dτ = − cos(t).
Substituting for x in (2.51), we obtain
v(t) =
x(t)
i(t)
i(t).
The corresponding Lissajoux figure in the (v, i) plane is the unit circle shown in
Fig. 2.28a, which is not pinched at the origin (v, i) = (0, 0)!
This is a shocking result suggesting that one should accept the notion that not all
memristors exhibit the pinched hysteresis loop fingerprint.
Resolving this anomaly leads to a logical inconsistency which we will now
demonstrate. Following the systematic approach shown in the Examples 2.4 and 2.5,
it is easily understood that the problem is the plot of the Lissajoux figure in the
wrong plane. Considering the plot of v(t) as function x(t), i.e., the Lissajoux
figure in the v − x plane, a single-valued curve without any hysteresis is obtained
(Fig. 2.28b). Such curve represents the correct characteristic of the considered twoterminal device and, from the previous equations, it is easily derived that
v(t) = x(t) =
t
−∞
i(τ )dτ
.
= q(t)
which is just the definition of a 1 Farad capacitor! It would certainly be nonsense
to refer to a capacitor as (extended) memristor!
Remark 2.9 Although an extensive discussion of zero-crossing property and
pinched hysteresis loops is reported in Sect. 2.1.3, the Examples 2.4, 2.5, and 2.27
allow us to summarize the following chief statements:
• the pinched hysteresis loop is the hallmark of all memristors, and it can be
used for an experimental definition of memristors, whether they are ideal or not.
The pinched hysteresis loop signature of memristors must hold for all periodic
voltage or current inputs with zero time-average. It follows that the measurement
of a non-pinched hysteresis loop for even just one such testing signal would
disqualify a device from being classified as a memristor