50
2 Fundamental Properties of Mem-Elements
As shown in the previous section, memristive devices and systems exhibit the
zero-crossing property by definition. The concept of passivity for a memristive
system is more intricate with respect to the passivity of an ideal memristor because
we cannot exploit the CR in the flux-charge domain to evaluate the memristance
ˆ
ϕ (q) or the memductance ˆ
q (ϕ) (see the proof of Property 2.1). Hence, the passivity
of memristive devices and systems defined in terms of DAEs results to be an
inherent input–output property which may be roughly specified in relation to energy
dissipation and transformation. 6 In the following, the passivity of memristor devices
and systems is intended in the common meaning of electrical circuit theory, that is,
the absence of any source of energy (e.g., a battery) inside the device package [7].
As a consequence, one cannot pull more energy out of a passive (dynamic) circuit
element than what was fed into it. Clearly, a sufficient condition for passivity of
a current-controlled (resp., voltage-controlled) memristive device is that we have
R(x, i) ≥ 0 (resp., G(x, i) ≥ 0) for any i (resp., for any v) and any x ∈ R n .
Indeed, under such condition, the instantaneous power entering the one-port is
always nonnegative, i.e., p(t) = R(x, i)i 2 ≥ 0 (resp., G(x, v)v 2 ≥ 0) [1]. Under
such condition, energy discharge is not possible, namely, it is not possible to extract
energy from the device by connecting it to a resistor (cf. Sect. 2.1.2).
A fundamental property of memristive devices, differentiating them from ideal
memristors, is that they may be either nonvolatile or volatile. In the latter case,
a memristive device has a unique state and corresponding memductance (memristance) when the input voltage (or current) is turned off. In the next section, we will
see that the concept of passivity is important to figure out the use of a memristive
device as a volatile or a nonvolatile memory element.
It is extremely important to observe that a volatile memory cannot be described
by an ideal memristor because an ideal memristor always features a continuum of
nonvolatile memory states and at least two different nonvolatile memductances. The
volatility feature is then peculiar to some classes of memristor devices and systems.
Remark 2.3 Let us consider a flux-controlled ideal memristor q = ˆ
q(ϕ) and
suppose it can store only one memductance value, for any flux ϕ, when power
is turned off. In other words, the slope ˆ
q (ϕ) of the CR q = ˆ
q(ϕ), i.e., the
memductance, is a constant G at any point ϕ. Then, the ideal memristor is linear
(q = Gϕ) and it acts as a linear resistor. In conclusion, a volatile ideal memristor
cannot be distinguished from a linear resistor.
6 The field of passivity for nonlinear dynamical systems constitutes an active research direction.
Such study is beyond the scope of this book and the reader is invited to refer to classic works
[13–15].
2 Fundamental Properties of Mem-Elements
As shown in the previous section, memristive devices and systems exhibit the
zero-crossing property by definition. The concept of passivity for a memristive
system is more intricate with respect to the passivity of an ideal memristor because
we cannot exploit the CR in the flux-charge domain to evaluate the memristance
ˆ
ϕ (q) or the memductance ˆ
q (ϕ) (see the proof of Property 2.1). Hence, the passivity
of memristive devices and systems defined in terms of DAEs results to be an
inherent input–output property which may be roughly specified in relation to energy
dissipation and transformation. 6 In the following, the passivity of memristor devices
and systems is intended in the common meaning of electrical circuit theory, that is,
the absence of any source of energy (e.g., a battery) inside the device package [7].
As a consequence, one cannot pull more energy out of a passive (dynamic) circuit
element than what was fed into it. Clearly, a sufficient condition for passivity of
a current-controlled (resp., voltage-controlled) memristive device is that we have
R(x, i) ≥ 0 (resp., G(x, i) ≥ 0) for any i (resp., for any v) and any x ∈ R n .
Indeed, under such condition, the instantaneous power entering the one-port is
always nonnegative, i.e., p(t) = R(x, i)i 2 ≥ 0 (resp., G(x, v)v 2 ≥ 0) [1]. Under
such condition, energy discharge is not possible, namely, it is not possible to extract
energy from the device by connecting it to a resistor (cf. Sect. 2.1.2).
A fundamental property of memristive devices, differentiating them from ideal
memristors, is that they may be either nonvolatile or volatile. In the latter case,
a memristive device has a unique state and corresponding memductance (memristance) when the input voltage (or current) is turned off. In the next section, we will
see that the concept of passivity is important to figure out the use of a memristive
device as a volatile or a nonvolatile memory element.
It is extremely important to observe that a volatile memory cannot be described
by an ideal memristor because an ideal memristor always features a continuum of
nonvolatile memory states and at least two different nonvolatile memductances. The
volatility feature is then peculiar to some classes of memristor devices and systems.
Remark 2.3 Let us consider a flux-controlled ideal memristor q = ˆ
q(ϕ) and
suppose it can store only one memductance value, for any flux ϕ, when power
is turned off. In other words, the slope ˆ
q (ϕ) of the CR q = ˆ
q(ϕ), i.e., the
memductance, is a constant G at any point ϕ. Then, the ideal memristor is linear
(q = Gϕ) and it acts as a linear resistor. In conclusion, a volatile ideal memristor
cannot be distinguished from a linear resistor.
6 The field of passivity for nonlinear dynamical systems constitutes an active research direction.
Such study is beyond the scope of this book and the reader is invited to refer to classic works
[13–15].