5.8 Formulation of Memristor Circuits Equations
197
circuits including either charge-controlled memristors or flux- and charge-controlled
memristors, as illustrated via examples.
5.8.1 Differential Algebraic Equations in the Flux-Charge
Domain
Following the classical tableau analysis approach (cf. Chap. 3), the set of b linear
algebraic equations due to the topological constraints can be obtained by putting
together the n−1 KqL-equations in (5.21) and the b−n+1 KϕL-equations in (5.22),
that is
⎧
⎨
⎩
Aq(t; t 0 ) = 0
Bϕ(t; t 0 ) = 0.
(5.30)
The collection of all CRs, one for each of the b two-terminal elements in a circuit
in LM, provides a set of algebraic and differential equations that can be written in
terms of the incremental charge and flux as follows:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ϕ R s (t; t 0 ) = R s q R s (t; t 0 ), (s = 1, . . . , n R )
ϕ w (t; t 0 ) = ϕ e w (t; t 0 ), ∀q w (t; t 0 ), (w = 1, . . . , n E )
q z (t; t 0 ) = q a z (t; t 0 ), ∀ϕ z (t; t 0 ), (z = 1, . . . , n A )
(5.31)
⎧
⎪ ⎨
⎪ ⎩
C j
dϕ C j (t;t 0 )
dt
= q C j (t; t 0 ) + q C j 0 , (j = 1, . . . , n C )
L m
dq Lm (t;t 0 )
dt
= ϕ L m (t; t 0 ) + ϕ L m 0 , (m = 1, . . . , n L )
(5.32)
q M k (t; t 0 ) = f M k (ϕ M k (t; t 0 ) + ϕ M k 0 ) − f M k (ϕ M k 0 ), (p = 1, . . . , n M )
(5.33)
where n R is the number of resistors, n E the number of independent voltage sources,
n A the number of independent current sources, n C the number of capacitors, n L the
number of inductors, and n M the number of flux-controlled memristors.
Equations (5.30)–(5.33) provide a system of 2b DAEs involving only the
incremental variables q(t; t 0 ) and ϕ(t; t 0 ). The solution of DAEs (5.30)–(5.33), with
the initial conditions
q(t 0 ; t 0 ) = 0
and
197
circuits including either charge-controlled memristors or flux- and charge-controlled
memristors, as illustrated via examples.
5.8.1 Differential Algebraic Equations in the Flux-Charge
Domain
Following the classical tableau analysis approach (cf. Chap. 3), the set of b linear
algebraic equations due to the topological constraints can be obtained by putting
together the n−1 KqL-equations in (5.21) and the b−n+1 KϕL-equations in (5.22),
that is
⎧
⎨
⎩
Aq(t; t 0 ) = 0
Bϕ(t; t 0 ) = 0.
(5.30)
The collection of all CRs, one for each of the b two-terminal elements in a circuit
in LM, provides a set of algebraic and differential equations that can be written in
terms of the incremental charge and flux as follows:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ϕ R s (t; t 0 ) = R s q R s (t; t 0 ), (s = 1, . . . , n R )
ϕ w (t; t 0 ) = ϕ e w (t; t 0 ), ∀q w (t; t 0 ), (w = 1, . . . , n E )
q z (t; t 0 ) = q a z (t; t 0 ), ∀ϕ z (t; t 0 ), (z = 1, . . . , n A )
(5.31)
⎧
⎪ ⎨
⎪ ⎩
C j
dϕ C j (t;t 0 )
dt
= q C j (t; t 0 ) + q C j 0 , (j = 1, . . . , n C )
L m
dq Lm (t;t 0 )
dt
= ϕ L m (t; t 0 ) + ϕ L m 0 , (m = 1, . . . , n L )
(5.32)
q M k (t; t 0 ) = f M k (ϕ M k (t; t 0 ) + ϕ M k 0 ) − f M k (ϕ M k 0 ), (p = 1, . . . , n M )
(5.33)
where n R is the number of resistors, n E the number of independent voltage sources,
n A the number of independent current sources, n C the number of capacitors, n L the
number of inductors, and n M the number of flux-controlled memristors.
Equations (5.30)–(5.33) provide a system of 2b DAEs involving only the
incremental variables q(t; t 0 ) and ϕ(t; t 0 ). The solution of DAEs (5.30)–(5.33), with
the initial conditions
q(t 0 ; t 0 ) = 0
and