5.8 Formulation of Memristor Circuits Equations
201
• if needed, to obtain the SEs in normal form, it is enough to multiply both sides
of the first two equations (5.36) by C −1 and L −1 , respectively.
Remark 5.8 Criteria ensuring the existence of the SEs (5.36) and the hybrid
representation (5.35) of the adynamic (n C +n L )-port can be derived via the approach
presented in Chap. 3.
Taking into account the analogies in Sect. 5.7, from Property 3.1 in Sect. 3.3.2.3
of Chap. 3, it is not difficult to see that a set of sufficient conditions for the existence
of the SEs in the (ϕ, q)-domain is as follows:
1. There is no loop formed exclusively by capacitors, inductors, and/or independent
flux sources. Furthermore, there is no cut-set formed exclusively by capacitors,
inductors, and/or independent charge sources.
2. Each flux-controlled (but not charge-controlled) memristor is in parallel with a
capacitor. Each charge-controlled (but not flux-controlled) memristor is in series
with an inductor.
3. Each remaining memristor is strongly locally passive, or else it is either in
parallel with a capacitor or in series with an inductor.
A verification is left to the reader. Here we have considered also the possible
presence of charge-controlled memristors. A flux-controlled memristor q M =
f (ϕ M ) is said to be strongly passive if there exist constants 0 < γ < γ such
that γ ≤ (f (ϕ M,1 ) − f (ϕ M,2 ))/(ϕ M,1 − ϕ M,2 ) ≤ γ for any ϕ M,1 = ϕ M,2 . A
similar definition holds for a charge-controlled memristor.
5.8.3 Differential Algebraic Equations in the Voltage-Current
Domain
Given a circuit in the class LM, the common formulation of DAEs in the
(v, i)-domain can be readily derived either by differentiating the DAEs in the (ϕ, q)domain, or otherwise by using KCLs, KVLs, and CRs in terms of current and
voltage. The former approach is briefly discussed in this section, whereas the latter
is widely reported in the literature (see for instance [11]). It turns out that both
methods provide the same circuit equations.
Since
d q(t;t 0 )
dt
=
d q(t)
dt = i(t)
d ϕ(t;t 0 )
dt
=
d ϕ(t)
dt = v(t)
by differentiating (5.30)–(5.32), the “usual” KCLs and KVLs
201
• if needed, to obtain the SEs in normal form, it is enough to multiply both sides
of the first two equations (5.36) by C −1 and L −1 , respectively.
Remark 5.8 Criteria ensuring the existence of the SEs (5.36) and the hybrid
representation (5.35) of the adynamic (n C +n L )-port can be derived via the approach
presented in Chap. 3.
Taking into account the analogies in Sect. 5.7, from Property 3.1 in Sect. 3.3.2.3
of Chap. 3, it is not difficult to see that a set of sufficient conditions for the existence
of the SEs in the (ϕ, q)-domain is as follows:
1. There is no loop formed exclusively by capacitors, inductors, and/or independent
flux sources. Furthermore, there is no cut-set formed exclusively by capacitors,
inductors, and/or independent charge sources.
2. Each flux-controlled (but not charge-controlled) memristor is in parallel with a
capacitor. Each charge-controlled (but not flux-controlled) memristor is in series
with an inductor.
3. Each remaining memristor is strongly locally passive, or else it is either in
parallel with a capacitor or in series with an inductor.
A verification is left to the reader. Here we have considered also the possible
presence of charge-controlled memristors. A flux-controlled memristor q M =
f (ϕ M ) is said to be strongly passive if there exist constants 0 < γ < γ such
that γ ≤ (f (ϕ M,1 ) − f (ϕ M,2 ))/(ϕ M,1 − ϕ M,2 ) ≤ γ for any ϕ M,1 = ϕ M,2 . A
similar definition holds for a charge-controlled memristor.
5.8.3 Differential Algebraic Equations in the Voltage-Current
Domain
Given a circuit in the class LM, the common formulation of DAEs in the
(v, i)-domain can be readily derived either by differentiating the DAEs in the (ϕ, q)domain, or otherwise by using KCLs, KVLs, and CRs in terms of current and
voltage. The former approach is briefly discussed in this section, whereas the latter
is widely reported in the literature (see for instance [11]). It turns out that both
methods provide the same circuit equations.
Since
d q(t;t 0 )
dt
=
d q(t)
dt = i(t)
d ϕ(t;t 0 )
dt
=
d ϕ(t)
dt = v(t)
by differentiating (5.30)–(5.32), the “usual” KCLs and KVLs