7.7 Examples on Memristor Circuit Programming
301
M(k) are obtained from (7.29) and (7.30) as
K(ϕ M , v C 1 , v C 2 , i L ) =
1
r + R
ϕ M + f (ϕ M )
+ C 1 v C 1 +
rC 2
r + R
v C 2 +
L
r + R
i L
(7.47a)
M(k) = {w ∈ R
4
: K(ϕ M , v C 1 , v C 2 , i L ) = k}
(7.47b)
where k ∈ R. Note that M(k) coincides with the expression obtained in Sect. 6.3 of
Chap. 6. The four-dimensional state space (ϕ M , v C 1 , v C 2 , i L ) in the (v, i)-domain is
completely spanned by the ∞ 1 three-dimensional manifolds M(k) by varying k in
R.
Since H 12 H
−1
22 B 21 = B 11 = −1/R and H 12 H
−1
22 B 22 = 0, a simple calculation
based on Property 7.2 and (7.34) yields
˙
k(t; t 0 , w 0 ) = a(t).
Then, the solution of the SEs (7.23) in the (v, i)-domain, with H and B in (7.45),
and ICs w 0 = (ϕ M (t 0 ), v C 1 (t 0 ), v C 2 (t 0 ), i L (t 0 )) T
• evolves on the invariant manifold M(k 0 ) for any t ≥ t 0 , being k 0 = K(w 0 ),
when q a (t; t 0 ) = 0 for any t ≥ t 0 (cf. Property 7.3)
• evolves on the invariant manifold M(k 1 ) for t ≥ T 1 , being k 1 = k 0 + ¯
aΔ, when
q a (t; t 0 ) is a constant momentum source for t ≥ T 1 as in (7.6.2) (e.g., q a (t; t 0 )
in Fig. 7.8)
• explores the manifolds M(k 0 + q a (t; t 0 )) when q a (t; t 0 ) is a time-varying
momentum source.
The same analysis makes clear that the external source ϕ e (t; t 0 ) does not
influence the switching of solutions between different manifolds.
The nonlinear dynamic behavior of MCC in Fig. 7.7 has been simulated for
t ≥ t 0 = 0 by using the SEs in the (ϕ, q)-domain (7.46). Note that M x = C 1 ,
M y = diag(C 2 , L) and introduce the dimensionless variables z 1 (τ ) = X(τ ),
z 2 (τ ) = Y 1 (τ ), z 3 (τ ) = RY 2 (τ ) via (7.26) (see also Sect. 6.3 in Chap. 6). Then,
the following normalized SEs are derived:
d z 1 (τ )
d τ
= α [−z 1 (τ ) + z 2 (τ ) − n(z 1 (τ )) − w 1 (τ ) + k 0 ]
(7.48a)
d z 2 (τ )
d τ
= z 1 (τ ) − z 2 (τ ) + z 3 (τ ) − w 2 (τ )
(7.48b)
d z 3 (τ )
d τ
= −βz 2 (τ ) − w 3 (τ )
(7.48c)
z 1 (0) = ϕ M (0)
301
M(k) are obtained from (7.29) and (7.30) as
K(ϕ M , v C 1 , v C 2 , i L ) =
1
r + R
ϕ M + f (ϕ M )
+ C 1 v C 1 +
rC 2
r + R
v C 2 +
L
r + R
i L
(7.47a)
M(k) = {w ∈ R
4
: K(ϕ M , v C 1 , v C 2 , i L ) = k}
(7.47b)
where k ∈ R. Note that M(k) coincides with the expression obtained in Sect. 6.3 of
Chap. 6. The four-dimensional state space (ϕ M , v C 1 , v C 2 , i L ) in the (v, i)-domain is
completely spanned by the ∞ 1 three-dimensional manifolds M(k) by varying k in
R.
Since H 12 H
−1
22 B 21 = B 11 = −1/R and H 12 H
−1
22 B 22 = 0, a simple calculation
based on Property 7.2 and (7.34) yields
˙
k(t; t 0 , w 0 ) = a(t).
Then, the solution of the SEs (7.23) in the (v, i)-domain, with H and B in (7.45),
and ICs w 0 = (ϕ M (t 0 ), v C 1 (t 0 ), v C 2 (t 0 ), i L (t 0 )) T
• evolves on the invariant manifold M(k 0 ) for any t ≥ t 0 , being k 0 = K(w 0 ),
when q a (t; t 0 ) = 0 for any t ≥ t 0 (cf. Property 7.3)
• evolves on the invariant manifold M(k 1 ) for t ≥ T 1 , being k 1 = k 0 + ¯
aΔ, when
q a (t; t 0 ) is a constant momentum source for t ≥ T 1 as in (7.6.2) (e.g., q a (t; t 0 )
in Fig. 7.8)
• explores the manifolds M(k 0 + q a (t; t 0 )) when q a (t; t 0 ) is a time-varying
momentum source.
The same analysis makes clear that the external source ϕ e (t; t 0 ) does not
influence the switching of solutions between different manifolds.
The nonlinear dynamic behavior of MCC in Fig. 7.7 has been simulated for
t ≥ t 0 = 0 by using the SEs in the (ϕ, q)-domain (7.46). Note that M x = C 1 ,
M y = diag(C 2 , L) and introduce the dimensionless variables z 1 (τ ) = X(τ ),
z 2 (τ ) = Y 1 (τ ), z 3 (τ ) = RY 2 (τ ) via (7.26) (see also Sect. 6.3 in Chap. 6). Then,
the following normalized SEs are derived:
d z 1 (τ )
d τ
= α [−z 1 (τ ) + z 2 (τ ) − n(z 1 (τ )) − w 1 (τ ) + k 0 ]
(7.48a)
d z 2 (τ )
d τ
= z 1 (τ ) − z 2 (τ ) + z 3 (τ ) − w 2 (τ )
(7.48b)
d z 3 (τ )
d τ
= −βz 2 (τ ) − w 3 (τ )
(7.48c)
z 1 (0) = ϕ M (0)