2.1 Ideal Memristor: Basic Properties and Signatures
31
2.1.3 Zero Crossing Property and Pinched Hysteresis Loop
Property 2.3 An ideal charge-controlled memristor satisfies the following zerocrossing property
i(t) = 0 ⇒ v(t) = 0.
Similarly, for an ideal flux-controlled memristor we have
v(t) = 0 ⇒ i(t) = 0.
If a memristor is both charge-controlled and flux-controlled, then both the voltage
v(t) and the current i(t) must be zero at the same instants, thus the memristor
exhibits the identical zero-crossing property
i(t) = 0 ⇔ v(t) = 0.
Proof Follows straightforwardly from the state-dependent Ohm’s law (2.2) or (2.4).
The identical zero-crossing property holds for instance in the case of the HP
memristor discussed in Sect. 2.4.2.1. If an ideal memristor is active, in general the
identical zero-crossing property no longer holds (see Example 2.6).
The zero-crossing property for passive ideal memristors implies the following
zero phase-shift property. In any passive ideal memristor the phase shift 2 between
a periodic current waveform i(t) (resp., voltage waveform v(t)) and its associated
periodic voltage waveform v(t) (resp., current waveform i(t)) is zero. This property
implies that, unlike capacitors and inductors, it is impossible to store energy in a
passive memristor (cf. Sect. 2.1.2 for more details).
An additional consequence of the zero-crossing property is that the Lissajous
figure obtained combining the memristor voltage v(t) and current i(t) results to be
a parametric curve that is pinched in the origin, a.k.a. pinched hysteresis loop in
the (v, i)-plane. It is in fact a common practice to study the behavior of an ideal
flux-controlled memristor by applying a sinusoidal voltage source and measuring
the corresponding current, i.e., analyzing the pinched hysteresis loops in the (v, i)plane exhibited by the memristor.
The next examples illustrate the zero-crossing property and the corresponding
pinched hysteresis loops. All the numerical values in the examples are assumed to
be normalized with respect to a suitable set of physical quantities, i.e., parameters
are given in adimensional form.
2 The phase shift is considered as the lateral difference between two or more specific points (e.g.,
zeros, maxima, . . . ) on waveforms along a common axis.
31
2.1.3 Zero Crossing Property and Pinched Hysteresis Loop
Property 2.3 An ideal charge-controlled memristor satisfies the following zerocrossing property
i(t) = 0 ⇒ v(t) = 0.
Similarly, for an ideal flux-controlled memristor we have
v(t) = 0 ⇒ i(t) = 0.
If a memristor is both charge-controlled and flux-controlled, then both the voltage
v(t) and the current i(t) must be zero at the same instants, thus the memristor
exhibits the identical zero-crossing property
i(t) = 0 ⇔ v(t) = 0.
Proof Follows straightforwardly from the state-dependent Ohm’s law (2.2) or (2.4).
The identical zero-crossing property holds for instance in the case of the HP
memristor discussed in Sect. 2.4.2.1. If an ideal memristor is active, in general the
identical zero-crossing property no longer holds (see Example 2.6).
The zero-crossing property for passive ideal memristors implies the following
zero phase-shift property. In any passive ideal memristor the phase shift 2 between
a periodic current waveform i(t) (resp., voltage waveform v(t)) and its associated
periodic voltage waveform v(t) (resp., current waveform i(t)) is zero. This property
implies that, unlike capacitors and inductors, it is impossible to store energy in a
passive memristor (cf. Sect. 2.1.2 for more details).
An additional consequence of the zero-crossing property is that the Lissajous
figure obtained combining the memristor voltage v(t) and current i(t) results to be
a parametric curve that is pinched in the origin, a.k.a. pinched hysteresis loop in
the (v, i)-plane. It is in fact a common practice to study the behavior of an ideal
flux-controlled memristor by applying a sinusoidal voltage source and measuring
the corresponding current, i.e., analyzing the pinched hysteresis loops in the (v, i)plane exhibited by the memristor.
The next examples illustrate the zero-crossing property and the corresponding
pinched hysteresis loops. All the numerical values in the examples are assumed to
be normalized with respect to a suitable set of physical quantities, i.e., parameters
are given in adimensional form.
2 The phase shift is considered as the lateral difference between two or more specific points (e.g.,
zeros, maxima, . . . ) on waveforms along a common axis.