6.1 First-Order Memristor Circuits
227
d Q(t)
dt
= G(ϕ M (t))
d ϕ M (t)
dt
+ C
d v C (t)
dt
= 0.
(6.11)
It follows that Q(t) is constant in time. Hence, Q(t) keeps the initial value Q(t 0 ) =
Q 0 for all t ≥ t 0 .
Property 6.1 expresses the physical law of conservation of charge, i.e., the state
variables in the (v, i)-domain, v C (t) and ϕ M (t), evolve according to (6.7), but the
total charge Q(t) remains constant to the initial value Q 0 set by v C 0 and ϕ M 0 .
Remark 6.4 We remark that the conservation of charge in the M–C circuit directly
follows from KqL as well. Indeed, considering the equivalent circuit in the (ϕ, q)domain, KqL yields
q C (t; t 0 ) + q M (t; t 0 ) = q C (t) − q C 0 + q M (t) − q M 0
= (Cv C (t) + f (ϕ M (t))) − (Cv C 0 + f (ϕ M 0 ))
= Q(t) − Q(t 0 ) = 0
for any t ≥ t 0 .
Since Q(t) is a function of the state variables in the (ϕ, q)-domain, Property 6.1
means that the total charge is an invariant of motion for the SEs in the (v, i)-domain.
Define, for any Q 0 ∈ R, M(Q 0 ) as the subset of the phase-space in the (v, i)domain where Q 0 ∈ R has a fixed value, that is
M(Q 0 ) = {(v C , ϕ M )
T
∈ R
2
: f (ϕ M ) + Cv C = Q 0 }.
(6.12)
The following observations, whose verification is straightforward, make clear
some salient features of M(Q 0 ).
• The set M(Q 0 ) is a one-dimensional manifold (i.e., a curve) in the phase-space
R 2 of the SEs (6.7) in the (v, i)-domain
• there are ∞ 1 1D nonintersecting manifolds M(Q 0 ), spanning the whole phasespace R 2 of the SE (6.7) in the (v, i)-domain, obtained by varying Q 0 in R. Some
of these manifolds, for different values of Q 0 , are depicted in Fig. 6.7 in the case
where the memristor has a cubic characteristic q M = f (ϕ M ) = −ϕ M +
1
3 ϕ 3
M as
shown in Fig. 6.9
• since Q(t) is constant for any t ≥ t 0 , it follows that for any given Q 0 ∈ R the
manifold M(Q 0 ) is positively invariant for the evolution of (v C (t), ϕ M (t)), i.e.,
if (v C 0 , ϕ M 0 ) T ∈ M(Q 0 ) then the solution of (6.7) with such initial conditions
belongs to M(Q 0 ) for any t ≥ t 0
• the manifold where a solution evolves can be changed by varying Q 0 , i.e., by
varying the initial conditions at t 0 for the state variables in the (v, i)-domain
227
d Q(t)
dt
= G(ϕ M (t))
d ϕ M (t)
dt
+ C
d v C (t)
dt
= 0.
(6.11)
It follows that Q(t) is constant in time. Hence, Q(t) keeps the initial value Q(t 0 ) =
Q 0 for all t ≥ t 0 .
Property 6.1 expresses the physical law of conservation of charge, i.e., the state
variables in the (v, i)-domain, v C (t) and ϕ M (t), evolve according to (6.7), but the
total charge Q(t) remains constant to the initial value Q 0 set by v C 0 and ϕ M 0 .
Remark 6.4 We remark that the conservation of charge in the M–C circuit directly
follows from KqL as well. Indeed, considering the equivalent circuit in the (ϕ, q)domain, KqL yields
q C (t; t 0 ) + q M (t; t 0 ) = q C (t) − q C 0 + q M (t) − q M 0
= (Cv C (t) + f (ϕ M (t))) − (Cv C 0 + f (ϕ M 0 ))
= Q(t) − Q(t 0 ) = 0
for any t ≥ t 0 .
Since Q(t) is a function of the state variables in the (ϕ, q)-domain, Property 6.1
means that the total charge is an invariant of motion for the SEs in the (v, i)-domain.
Define, for any Q 0 ∈ R, M(Q 0 ) as the subset of the phase-space in the (v, i)domain where Q 0 ∈ R has a fixed value, that is
M(Q 0 ) = {(v C , ϕ M )
T
∈ R
2
: f (ϕ M ) + Cv C = Q 0 }.
(6.12)
The following observations, whose verification is straightforward, make clear
some salient features of M(Q 0 ).
• The set M(Q 0 ) is a one-dimensional manifold (i.e., a curve) in the phase-space
R 2 of the SEs (6.7) in the (v, i)-domain
• there are ∞ 1 1D nonintersecting manifolds M(Q 0 ), spanning the whole phasespace R 2 of the SE (6.7) in the (v, i)-domain, obtained by varying Q 0 in R. Some
of these manifolds, for different values of Q 0 , are depicted in Fig. 6.7 in the case
where the memristor has a cubic characteristic q M = f (ϕ M ) = −ϕ M +
1
3 ϕ 3
M as
shown in Fig. 6.9
• since Q(t) is constant for any t ≥ t 0 , it follows that for any given Q 0 ∈ R the
manifold M(Q 0 ) is positively invariant for the evolution of (v C (t), ϕ M (t)), i.e.,
if (v C 0 , ϕ M 0 ) T ∈ M(Q 0 ) then the solution of (6.7) with such initial conditions
belongs to M(Q 0 ) for any t ≥ t 0
• the manifold where a solution evolves can be changed by varying Q 0 , i.e., by
varying the initial conditions at t 0 for the state variables in the (v, i)-domain