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6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
Fig. 6.18 The M–L–C
obtained from the M–C
circuit in Fig. 6.14 by
inserting a parasitic
inductance L in series with
the memristor
h(q M )
q M0
ϕ M0
L
ϕ L0
C
q C0
q L (t; t 0 )
q C (t; t 0 )
q M (t; t 0 )
ϕ M (t; t 0 )
ϕ C (t; t 0 )
ϕ L (t; t 0 )
Fig. 6.19 Equivalent circuit in the (ϕ, q)-domain of the M–L–C circuit in Fig. 6.18 for t ≥ t 0
q L (t; t 0 ) = −q C (t; t 0 )
(6.28b)
q L (t; t 0 ) = q M (t; t 0 )
(6.28c)
q C (t; t 0 ) = −q C 0 + C
d
dt
ϕ C (t; t 0 )
(6.28d)
ϕ L (t; t 0 ) = −ϕ L 0 + L
d
dt
q L (t; t 0 )
(6.28e)
ϕ M (t; t 0 ) = −h(q M 0 ) + h(q M (t; t 0 ) + q M 0 )
(6.28f)
for t ≥ t 0 . These constitute the set of DAEs describing the circuit dynamics in the
(ϕ, q)-domain.
It is an easy matter to verify that the circuit in Fig. 6.18 satisfies the conditions
given in Property 3.1 of Chap. 5 for the existence of the SE representation. By
substitution in the previous DAEs, we indeed obtain for t ≥ t 0 the second-order
SEs in the (ϕ, q)-domain in terms of the state variables ϕ C (t; t 0 ) and q L (t; t 0 )