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11 Nonlinear Dynamics of Circuits with Mem-Elements
11.4.1 Memristor M ϕ
A flux-controlled memristor M ϕ is defined by the algebraic CR q M (t) = f M (ϕ M (t))
in the (ϕ, q)-domain. CRs need to be expressed in terms of the incremental charge
and flux, i.e., the CR of M ϕ becomes
q M (t; t 0 ) = f M (ϕ M (t; t 0 ) + ϕ M 0 ) − f M (ϕ M 0 )
for t ≥ t 0 , where ϕ M 0 = ϕ M (t 0 ). Hence, O
(0,0)
M ϕ
= 0 and then there is no state
variable in view of the fact that a flux-controlled memristor acts as an algebraic
element in the (ϕ, q)-domain (it is the analogous of a nonlinear resistor in the (v, i)domain). It is worth noting that the CR depends upon ϕ M 0 . The corresponding
equivalent circuit in such domain is reported in Fig. 5.12 of Chap. 5. The timederivative of the previous CR gives the state dependent Ohm’s law describing M ϕ
in the (v, i)-domain, i.e.,
i M (t) = f
M (ϕ M (t))v M (t)
and
˙
ϕ M (t) = v M (t).
Hence, O
(0,0)
M ϕ
= 1 and the state variable is ϕ M (t). The IC is ϕ M 0 .
11.4.2 Memristor M q
Similar arguments hold for a charge-controlled memristor M q defined by the
algebraic CR ϕ M (t) = f M (q M (t)), or equivalently by
ϕ M (t; t 0 ) = f M (q M (t; t 0 ) + q M 0 ) − f M (q M 0 )
for t ≥ t 0 , where q M 0 = q M (t 0 ). Hence, O
(−1,−1)
M q
= 0 and there is no state variable
in the (ϕ, q)-domain. The equivalent circuit is in Fig. 5.13 of Chap. 5. In the (v, i)domain, M q is described by
v M (t) = f
M (q M (t))i(t)
and
˙
q M (t) = i(t)