5.8 Formulation of Memristor Circuits Equations
203
The state variables in the (v, i)-domain are the n C capacitor voltages v C , the n L
inductor currents i L and the n M fluxes of flux-controlled memristors ϕ M . 9 First note
that the CRs of circuit elements in the (v, i)-domain can be written in vector form
as
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
C
d v C (t)
dt
= i C (t)
L
d i L (t)
dt
= v L (t)
v C (t 0 ) = v C 0
i L (t 0 ) = i L 0
(5.41)
for capacitors and inductors and
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
i M (t) = G(ϕ M (t))v M (t)
d ϕ M (t)
dt
= v M (t)
ϕ M (t 0 ) = ϕ M 0
(5.42)
for flux-controlled memristors, where
G(ϕ M (t)) = diag(G 1 (ϕ M 1 (t)), G 2 (ϕ M 2 (t)), . . . , G n M (ϕ M n M (t))).
Any circuit in LM can be represented as shown in Fig. 5.23, where all capacitors
and inductors are connected to a nonlinear (n C + n L )-port including ideal resistors,
memristors, and ideal independent voltage and current sources. Note that, for any
given value of the flux ϕ M k , each memristor q M k = f k (ϕ M k ) can be replaced by a
linear conductance G k (ϕ M k ) = f
k (ϕ M k ). For each fixed ϕ M the (n C + n L )-port is
thus a linear adynamic network.
By following a procedure analogous to that in Chap. 3, suppose to replace each
capacitor by a voltage source and each inductor by a current source. If a unique
solvability assumption is satisfied for any ϕ M by the linear adynamic circuit thus
obtained, then we can write
⎧
⎨
⎩
−i C (t) = H a (v C (t), i L (t), ϕ M (t), e(t), a(t))
−v L (t) = H b (v C (t), i L (t), ϕ M (t), e(t), a(t)).
(5.43)
Solving for v M the linear adynamic network we also obtain
v M (t) = H c (v C (t), i L (t), ϕ M (t), e(t), a(t)).
(5.44)
9 There are also the memristor charges q M for charge-controlled memristors.