2.4 Genealogy of Memristor Devices
61
• voltage-controlled ideal generic memristor:
i = G(x)v
(2.27)
dx
dt
= g(x)v
(2.28)
where x = (x 1 , x 2 , . . . , x n ) T ∈ R n is the vector of state variables. If n = 1 and
f (x) = 1 (g(x) = 1), then x = q (x = ϕ), and so an ideal generic memristor
reduces to an ideal memristor, namely:
• charge-controlled ideal memristor:
v = R(q)i = ˆ
ϕ
(q)i
(2.29)
dq
dt
= i
(2.30)
• flux-controlled ideal memristor:
i = G(ϕ)v = ˆ
q
(ϕ)v
(2.31)
dϕ
dt
= v.
(2.32)
It is noted that, in Chap. 1, we used the notation M(q) (resp., W (ϕ)) for the
memristance (resp., memductance) of an ideal memristor.
Given an ideal memristor, it is possible to create an infinite number of ideal
generic memristor siblings by choosing a one-to-one differentiable function [4]. On
the other hand, the one-to-one correspondence can be also used to recover an ideal
memristor from its ideal generic memristor siblings.
The next examples illustrate such procedure for charge-controlled memristors
having a single scalar state variable x, but a similar approach can be used, mutatis
mutandis, for flux-controlled memristors.
Example 2.20 (Create an Ideal Generic Memristor Sibling from an Ideal Memristor) Let us consider a charge-controlled ideal memristor with CR
ϕ = ˆ
ϕ(q) = q +
1
3
q
3
and its corresponding state-dependent Ohm’s law in the (v, i)-domain
v = ˆ
ϕ
(q)i = (1 + q
2 )i
(2.33)
˙
q = i.
(2.34)
By introducing a cubic function q = x
1
3 , i.e., a differentiable one-to-one mapping
q = h(x) between the charge q and the internal scalar state variable x, then the CR
becomes