5.8 Formulation of Memristor Circuits Equations
209
where v C is the capacitor voltage.
Let
ϕ M
.
= h(x) =
x
0
1
w(ρ)
dρ
for any x ∈ (−1, 1). Since w(x) > 0 for x ∈ (−1, 1), function h(·) is invertible in
the same interval and we have x = h −1 (ϕ M ) and
h
(x) =
1
w(x)
.
Substituting x with ϕ M we obtain the SE
C
dv C
dt
= − ˆ
G(ϕ M )v C
dϕ M
dt
= v C
where we have let
ˆ
G(ϕ M ) = G(h
−1 (ϕ M )).
Such an SE coincides with that obtained for the M − C circuit studied in
Example 5.15. Then, we can use the same procedure as in that example for studying
the dynamics of the more general circuit with an ideal generic memristor here
considered.
Example 5.17 Consider a circuit with a flux-controlled memristor q M 1 = f (ϕ M 1 )
and a charge controlled memristor ϕ M 2 = h(q M 2 ) as in Fig. 5.25a. We wish to
obtain the SE representation in the (ϕ, q)- and (v, i)-domain using the procedure
in Sect. 5.8.2. Consider the corresponding circuit in the (ϕ, q)-domain, extract L
and C, connect them to a two-port network containing the adynamic elements in
the (ϕ, q)-domain, and replace C and L by a flux source and a charge source,
respectively, as shown in Fig. 5.25b. It can be checked that the hypotheses in
Remark 5.8 for the existence of the hybrid representation of the two-port network
are satisfied. This representation is easily obtained as
q 1 (t; t 0 ) = f (ϕ 1 (t; t 0 ) + ϕ M 1 0 ) − f (ϕ M 1 0 ) − q a (t; t 0 ) − q 2 (t; t 0 )
ϕ 2 (t; t 0 ) = h(q 2 (t; t 0 ) + q M 2 0 ) − h(q M 2 0 ) + ϕ 1 (t; t 0 )
where the first equation derives from KqL applied to cut-set C in Fig. 5.25b and the
second equation from KϕL at the loop given by the source q 2 (t; t 0 ), the chargecontrolled memristor and the source ϕ 1 (t; t 0 ).
The SEs in the (ϕ, q)-domain are given by the second-order system
209
where v C is the capacitor voltage.
Let
ϕ M
.
= h(x) =
x
0
1
w(ρ)
dρ
for any x ∈ (−1, 1). Since w(x) > 0 for x ∈ (−1, 1), function h(·) is invertible in
the same interval and we have x = h −1 (ϕ M ) and
h
(x) =
1
w(x)
.
Substituting x with ϕ M we obtain the SE
C
dv C
dt
= − ˆ
G(ϕ M )v C
dϕ M
dt
= v C
where we have let
ˆ
G(ϕ M ) = G(h
−1 (ϕ M )).
Such an SE coincides with that obtained for the M − C circuit studied in
Example 5.15. Then, we can use the same procedure as in that example for studying
the dynamics of the more general circuit with an ideal generic memristor here
considered.
Example 5.17 Consider a circuit with a flux-controlled memristor q M 1 = f (ϕ M 1 )
and a charge controlled memristor ϕ M 2 = h(q M 2 ) as in Fig. 5.25a. We wish to
obtain the SE representation in the (ϕ, q)- and (v, i)-domain using the procedure
in Sect. 5.8.2. Consider the corresponding circuit in the (ϕ, q)-domain, extract L
and C, connect them to a two-port network containing the adynamic elements in
the (ϕ, q)-domain, and replace C and L by a flux source and a charge source,
respectively, as shown in Fig. 5.25b. It can be checked that the hypotheses in
Remark 5.8 for the existence of the hybrid representation of the two-port network
are satisfied. This representation is easily obtained as
q 1 (t; t 0 ) = f (ϕ 1 (t; t 0 ) + ϕ M 1 0 ) − f (ϕ M 1 0 ) − q a (t; t 0 ) − q 2 (t; t 0 )
ϕ 2 (t; t 0 ) = h(q 2 (t; t 0 ) + q M 2 0 ) − h(q M 2 0 ) + ϕ 1 (t; t 0 )
where the first equation derives from KqL applied to cut-set C in Fig. 5.25b and the
second equation from KϕL at the loop given by the source q 2 (t; t 0 ), the chargecontrolled memristor and the source ϕ 1 (t; t 0 ).
The SEs in the (ϕ, q)-domain are given by the second-order system