290
7 Pulse Programming of Memristor Circuits
(x(t 0 ), y(t 0 )) = (0, 0).
(7.21)
Proof It suffices to rewrite the SEs (7.16) with the introduced notations.
7.4.2 State Equations in the Voltage-Current Domain
By time-differentiation of (7.20), the following system of second-order ODEs is
obtained (being ˙
x(t) = ˙
x M (t)):
M x ¨
x(t)
M y ¨
y(t)
= −
H 11 H 12
H 21 H 22
˙
x(t)
˙
y(t)
−
J F (x M (t))˙ x(t)
0
−
˙
u x (t)
˙
u y (t)
(7.22)
where J F (·) ∈ R n x ×n x is the Jacobian of F(·) (i.e., J F (·) is defined by memristances
and memconductances of the n x = n M memristors).
Theorem 7.2 If the memristor circuit N satisfies (A1)–(A3), then it has the SE
representation in the (v, i)-domain for t ≥ t 0
˙
x M (t) =
˙
ϕ M γ M
(t)
˙
q M λ M (t)
=
v C γ M (t)
i L λ M (t)
(7.23a)
M x
˙
v C γ M (t)
d
dt i L λ M (t)
= −(H 11 + J F (x M (t)))
v C γ M (t)
i L λ M (t)
− H 12
⎛
⎜
⎜
⎜
⎝
v C γ G (t)
i L λ R (t)
v C γ C (t)
i L λ L (t)
⎞
⎟
⎟
⎟
⎠
− ˙
u x (t)
(7.23b)
M y
⎛
⎜
⎜
⎜
⎝
˙
v C γ G (t)
d
dt i L λ R (t)
˙
v C γ C (t)
d
dt i L λ L (t)
⎞
⎟
⎟
⎟
⎠
= −H 21
v C γ M (t)
i L λ M (t)
− H 22
⎛
⎜
⎜
⎜
⎝
v C γ G (t)
i L λ R (t)
v C γ C (t)
i L λ L (t)
⎞
⎟
⎟
⎟
⎠
− ˙
u y (t).
(7.23c)