5.8 Formulation of Memristor Circuits Equations
213
Remark 5.12 In Example 5.18 we have been able to find the SEs in the (v, i)domain explicitly and to show the existence in implicit form of the SEs in the
(ϕ, q)-domain, under suitable assumptions on f . Clearly, it would be difficult to use
the SEs in the (ϕ, q)-domain for further dynamic analysis when they are not known
in explicit form. Then, we are led to look for additional conditions on a circuit in
LM so that it admits of an SE representation and moreover such representation is
explicitly known and is sufficiently simple. This problem will be overcome for a
relevant subclass of circuits in LM in Chap. 7. For that class we will also show that
we can pass from the SEs in the (ϕ, q)-domain to those in the (v, i)-domain, and
conversely, simply by differentiation or integration in time.
We conclude this section with an example concerning a circuit not in the class
LM containing an operational amplifier and by completing the analysis in the
(ϕ, q)-domain of an example at the beginning of the chapter.
Example 5.19 (Memristor Circuit with Operational Amplifier) Consider a circuit
with an ideal operational amplifier operating in the linear region, a flux-controlled
memristor q M = f (ϕ M ), and two capacitors as shown in Fig. 5.27. The corresponding circuit in the (ϕ, q)-domain for finding the hybrid representation, where each
capacitor has been replaced by a flux-source, is shown in Fig. 5.27b.
KqL at cut-set C and node A , respectively, easily yield the following hybrid
representation:
q 1 (t; t 0 ) =
ϕ 1 (t; t 0 ) − ϕ e (t; t 0 )
R 1
+
ϕ 1 (t; t 0 )
R 2
+f (ϕ 1 (t; t 0 ) − ϕ 2 (t; t 0 ) + ϕ M 0 ) − f (ϕ M 0 )
q 2 (t; t 0 ) =
ϕ 1 (t; t 0 )
R 2
.
Then, the SEs in the (ϕ, q)-domain are given by the second-order system
C 1
dϕ C 1 (t; t 0 )
dt
= −ϕ C 1 (t; t 0 )(
1
R 1
+
1
R 2
)
−f
ϕ C 1 (t; t 0 ) − ϕ C 2 (t; t 0 ) + ϕ M 0
+f (ϕ M 0 ) +
ϕ e (t; t 0 )
R 1
+ C 1 v C 10
C 2
dϕ C 2 (t; t 0 )
dt
= −
ϕ C 1 (t; t 0 )
R 2
+ C 2 v C 20 .
By differentiation we may obtain the fourth-order SEs describing the circuit in the
(v, i)-domain (details are omitted).
Example 5.20 (Example 5.4 Continued) Consider again the circuit C 1 − R − C 2
in Example 5.4 for t ≥ 0, which we want to analyze here in the (ϕ, q)-domain by
213
Remark 5.12 In Example 5.18 we have been able to find the SEs in the (v, i)domain explicitly and to show the existence in implicit form of the SEs in the
(ϕ, q)-domain, under suitable assumptions on f . Clearly, it would be difficult to use
the SEs in the (ϕ, q)-domain for further dynamic analysis when they are not known
in explicit form. Then, we are led to look for additional conditions on a circuit in
LM so that it admits of an SE representation and moreover such representation is
explicitly known and is sufficiently simple. This problem will be overcome for a
relevant subclass of circuits in LM in Chap. 7. For that class we will also show that
we can pass from the SEs in the (ϕ, q)-domain to those in the (v, i)-domain, and
conversely, simply by differentiation or integration in time.
We conclude this section with an example concerning a circuit not in the class
LM containing an operational amplifier and by completing the analysis in the
(ϕ, q)-domain of an example at the beginning of the chapter.
Example 5.19 (Memristor Circuit with Operational Amplifier) Consider a circuit
with an ideal operational amplifier operating in the linear region, a flux-controlled
memristor q M = f (ϕ M ), and two capacitors as shown in Fig. 5.27. The corresponding circuit in the (ϕ, q)-domain for finding the hybrid representation, where each
capacitor has been replaced by a flux-source, is shown in Fig. 5.27b.
KqL at cut-set C and node A , respectively, easily yield the following hybrid
representation:
q 1 (t; t 0 ) =
ϕ 1 (t; t 0 ) − ϕ e (t; t 0 )
R 1
+
ϕ 1 (t; t 0 )
R 2
+f (ϕ 1 (t; t 0 ) − ϕ 2 (t; t 0 ) + ϕ M 0 ) − f (ϕ M 0 )
q 2 (t; t 0 ) =
ϕ 1 (t; t 0 )
R 2
.
Then, the SEs in the (ϕ, q)-domain are given by the second-order system
C 1
dϕ C 1 (t; t 0 )
dt
= −ϕ C 1 (t; t 0 )(
1
R 1
+
1
R 2
)
−f
ϕ C 1 (t; t 0 ) − ϕ C 2 (t; t 0 ) + ϕ M 0
+f (ϕ M 0 ) +
ϕ e (t; t 0 )
R 1
+ C 1 v C 10
C 2
dϕ C 2 (t; t 0 )
dt
= −
ϕ C 1 (t; t 0 )
R 2
+ C 2 v C 20 .
By differentiation we may obtain the fourth-order SEs describing the circuit in the
(v, i)-domain (details are omitted).
Example 5.20 (Example 5.4 Continued) Consider again the circuit C 1 − R − C 2
in Example 5.4 for t ≥ 0, which we want to analyze here in the (ϕ, q)-domain by