66
2 Fundamental Properties of Mem-Elements
Remark 2.6 It is important to note that the flux ϕ has to be intended as the integral
of the voltage applied to the memristor. However, it does not correspond to the flux
of a magnetic field as that known from physics. 9 Indeed, no flux of a magnetic field
appears to be involved in the physics of the HP memristor [28]. To make clear the
difference between magnetic flux and flux as integral of the voltage, the latter is
named voltage momentum in [24]. By duality the integral of the current results to be
the current momentum, a.k.a. charge [24].
Remark 2.7 The HP memristor is actually an ideal generic memristor and its
sibling correspond to an ideal memristor that can be recovered via the differentiable
globally one-to-one function given in (2.36).
Remark 2.8 (HP Memristor with Window Functions) The length w of the doped
part of an HP memristor is physically constrained in the interval [0, D], but the
equation (2.35) describing the dynamics of w does not guarantee that this constraint
is automatically satisfied. To solve this problem the state equation (2.35) can be
modified as follows [29, 30]:
dw(t)
dt
= μ V
R on
D
i(t)ζ(w(t))
where ζ(w) > 0, w ∈ (0, D), and ζ(0) = ζ(D) = 0. The function ζ(w), a.k.a.
window function, takes into account nonlinear drift-diffusion of ions and vacancies
and it also prevents w to escape from [0, D], being zero at the boundaries, i.e.,
when w can stick at 0 or D. A Boundary Condition Memristor (BCM) model has
been introduced to specify complex phenomena at the boundaries [31, 32].
Example 2.22 An example of window function ζ(w) is [29]
ζ(w) = 1 −
2w
D
− 1
2p
where p is a positive integer. Appendix 2 reports further details on window functions
that are used to develop mathematical models of memristor devices.
2.4.3 Generic Memristor
A memristor is called a generic memristor if it’s defined by the following statedependent Ohm’s law:
9 It is worth to observe that there exists no magnetic field in a synthetic inductor obtained via a
gyrator [7], thus in such a case the flux is just the integral of the voltage!
2 Fundamental Properties of Mem-Elements
Remark 2.6 It is important to note that the flux ϕ has to be intended as the integral
of the voltage applied to the memristor. However, it does not correspond to the flux
of a magnetic field as that known from physics. 9 Indeed, no flux of a magnetic field
appears to be involved in the physics of the HP memristor [28]. To make clear the
difference between magnetic flux and flux as integral of the voltage, the latter is
named voltage momentum in [24]. By duality the integral of the current results to be
the current momentum, a.k.a. charge [24].
Remark 2.7 The HP memristor is actually an ideal generic memristor and its
sibling correspond to an ideal memristor that can be recovered via the differentiable
globally one-to-one function given in (2.36).
Remark 2.8 (HP Memristor with Window Functions) The length w of the doped
part of an HP memristor is physically constrained in the interval [0, D], but the
equation (2.35) describing the dynamics of w does not guarantee that this constraint
is automatically satisfied. To solve this problem the state equation (2.35) can be
modified as follows [29, 30]:
dw(t)
dt
= μ V
R on
D
i(t)ζ(w(t))
where ζ(w) > 0, w ∈ (0, D), and ζ(0) = ζ(D) = 0. The function ζ(w), a.k.a.
window function, takes into account nonlinear drift-diffusion of ions and vacancies
and it also prevents w to escape from [0, D], being zero at the boundaries, i.e.,
when w can stick at 0 or D. A Boundary Condition Memristor (BCM) model has
been introduced to specify complex phenomena at the boundaries [31, 32].
Example 2.22 An example of window function ζ(w) is [29]
ζ(w) = 1 −
2w
D
− 1
2p
where p is a positive integer. Appendix 2 reports further details on window functions
that are used to develop mathematical models of memristor devices.
2.4.3 Generic Memristor
A memristor is called a generic memristor if it’s defined by the following statedependent Ohm’s law:
9 It is worth to observe that there exists no magnetic field in a synthetic inductor obtained via a
gyrator [7], thus in such a case the flux is just the integral of the voltage!