182
5 Flux-Charge Analysis Method of Memristor Circuits
Consider a circuit N with a fixed topology for t ≥ t 0 , where t 0 is a finite initial
instant.
Definition 5.1 (Kirchhoff (Incremental) Charge Law (KqL)) The algebraic sum
of all incremental charges entering (leaving) any cut-set of N is zero at all times
t ≥ t 0 .
Definition 5.2 (Kirchhoff (Incremental) Flux Law (KϕL)) The algebraic sum of
all incremental fluxes around any loop of N is zero at all times t ≥ t 0 .
5.5.2 Constitutive Relations in the Flux-Charge Domain
The discussion in Sect. 5.5.1 shows that it is convenient to write the KϕL and
KqL equations in the (ϕ, q)-domain by using the incremental charge q k (t; t 0 ) and
incremental flux ϕ k (t; t 0 ) of each two-terminal element in LM. Then, we are led
in what follows to describe each two-terminal element by using its incremental
charge q k (t; t 0 ) and incremental flux ϕ k (t; t 0 ) as port variables, i.e., to express
the CR in terms of q k (t; t 0 ) and ϕ k (t; t 0 ). Once each circuit element is described
in the (ϕ, q)-domain by incremental flux and charge at its terminals, any circuit
in LM is represented in the (ϕ, q)-domain by connecting circuit elements using
such terminals. As a consequence, KϕL and KqL result to be independent of initial
conditions and provide the “usual” circuit topology constraints.
Next we discuss in detail how to obtain CRs and equivalent circuits in the (ϕ, q)domain for any two-terminal element in the class LM (we drop the index k to
simplify the notation).
5.5.2.1 Ideal Capacitor
Let us consider an ideal capacitor C described by
q C (t) = Cv C (t) = C
dϕ C (t)
dt
.
Let q C (t 0 ) = q C 0 = Cv C (t 0 ) be the initial charge at t 0 . We obtain for t ≥ t 0
q C (t; t 0 ) = q C (t) − q C (t 0 ) = C
dϕ C (t)
dt
− Cv C (t 0 )
that is
q C (t; t 0 ) = C
dϕ C (t)
dt
− q C 0 .
(5.23)
5 Flux-Charge Analysis Method of Memristor Circuits
Consider a circuit N with a fixed topology for t ≥ t 0 , where t 0 is a finite initial
instant.
Definition 5.1 (Kirchhoff (Incremental) Charge Law (KqL)) The algebraic sum
of all incremental charges entering (leaving) any cut-set of N is zero at all times
t ≥ t 0 .
Definition 5.2 (Kirchhoff (Incremental) Flux Law (KϕL)) The algebraic sum of
all incremental fluxes around any loop of N is zero at all times t ≥ t 0 .
5.5.2 Constitutive Relations in the Flux-Charge Domain
The discussion in Sect. 5.5.1 shows that it is convenient to write the KϕL and
KqL equations in the (ϕ, q)-domain by using the incremental charge q k (t; t 0 ) and
incremental flux ϕ k (t; t 0 ) of each two-terminal element in LM. Then, we are led
in what follows to describe each two-terminal element by using its incremental
charge q k (t; t 0 ) and incremental flux ϕ k (t; t 0 ) as port variables, i.e., to express
the CR in terms of q k (t; t 0 ) and ϕ k (t; t 0 ). Once each circuit element is described
in the (ϕ, q)-domain by incremental flux and charge at its terminals, any circuit
in LM is represented in the (ϕ, q)-domain by connecting circuit elements using
such terminals. As a consequence, KϕL and KqL result to be independent of initial
conditions and provide the “usual” circuit topology constraints.
Next we discuss in detail how to obtain CRs and equivalent circuits in the (ϕ, q)domain for any two-terminal element in the class LM (we drop the index k to
simplify the notation).
5.5.2.1 Ideal Capacitor
Let us consider an ideal capacitor C described by
q C (t) = Cv C (t) = C
dϕ C (t)
dt
.
Let q C (t 0 ) = q C 0 = Cv C (t 0 ) be the initial charge at t 0 . We obtain for t ≥ t 0
q C (t; t 0 ) = q C (t) − q C (t 0 ) = C
dϕ C (t)
dt
− Cv C (t 0 )
that is
q C (t; t 0 ) = C
dϕ C (t)
dt
− q C 0 .
(5.23)