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2 Fundamental Properties of Mem-Elements
where x ∈ R. We aim to show that h(x, v) can be factorized in the form of a statedependent Ohm’s law (2.44) under the assumption that, for any x ∈ R
lim
v→0
h(x, v) = 0.
In other words, we want to write
i = h(x, v) = G(x, v)v
where
G(x, v) =
h(x, v)
v
is the memductance. Note that G(x, 0) = 0/0 is indeterminate, however, the
memductance G(x, 0) is bounded in a neighbor of (x, 0) for any x. By de l’Hôpital
rule we obtain
G(x, 0) = lim
v→0
h(x, v)
v
= lim
v→0
∂h(x,v)
∂v
∂(v)
∂v
= lim
v→0
∂h(x, v)
∂v
.
=
∂h(x, 0)
∂v
.
As an application, reconsider the Pt/TaO x /Ta memristor device in the Example 2.15, which is defined as
i = h(x, v) = α(1 − e
−βv ) + γ x sinh(vδ) = G(x, v)v
for any x ∈ [0, 1]. According to de l’Hôpital rule, we obtain
G(x, 0) =
∂h(x, 0)
∂v
= αβ + γ δx.
Example 2.27 (Extended Memristor is Not for Everything!) Consider a twoterminal circuit element defined by a DAE
v = R(x, i)i
(2.51)
where
R(x, i) =
x
i
and the scalar state variable x ∈ R satisfies the equation
dx
dt
= f (x, i) = i.