198
5 Flux-Charge Analysis Method of Memristor Circuits
ϕ(t 0 ; t 0 ) = 0
gives the evolution of q(t; t 0 ) and ϕ(t; t 0 ) for t ≥ t 0 .
Remark 5.7 Although the DAEs (5.30)–(5.33) involve the incremental variables in
the (ϕ, q)-domain, their solution depends on q C 0 , ϕ L 0 and ϕ M 0 , 6 that is, the initial
conditions at t 0 for the state variables in the (v, i)-domain. Indeed, in the obtained
formulation, such initial conditions appear as constant inputs in the r.h.s. of (5.32)
and (5.33).
5.8.2 State Equations in the Flux-Charge Domain
Circuit analysis methods based on DAEs are fundamental in numerical simulation
(using, e.g., PSpice software [9]) of linear and nonlinear circuits [10]. On the other
hand, qualitative aspects of the nonlinear dynamics (e.g., the existence of no-finiteforward-escape-time solutions, eventual boundedness of solutions, local and global
asymptotic stability properties, bifurcation phenomena, etc.) are more effectively
analyzed by means of the SE formulation. Moreover, as pointed out in Chap. 3,
circuits that do not admit an SE description may be bad modeled from a physical
viewpoint due to the presence of impasse points.
The SE description in the (ϕ, q)-domain of a memristor circuit in LM, admitting
it exists, can be obtained by inspection of the equivalent circuit in the (ϕ, q)-domain
for simple low-order circuits. However, for more complex circuits, it is desirable to
develop a systematic procedure for writing the SEs and to give easily checkable
conditions guaranteeing that the SE representation exists.
In the following, we discuss a systematic approach for obtaining the SE
description of a circuit in LM in the (ϕ, q)-domain. The formal derivation of an
SE formulation of nonlinear RLC circuits in the (v, i)-domain, and the conditions
for the existence of such an SE formulation, have been provided in Sect. 3.3.2.3 of
Chap. 3. As previously observed, a memristor circuit in LM, when described in the
(ϕ, q)-domain, is analogous to a nonlinear RLC circuit in the (v, i)-domain. On
the basis of this observation, we are in a position to derive the SE description in
the (ϕ, q)-domain by exploiting the results in Chap. 3. It is worth to observe that
the formulation here proposed is based on using the vector of capacitor incremental
fluxes ϕ C (t; t 0 ), and the vector of incremental inductor charges q L (t; t 0 ), as state
variables in the (ϕ, q)-domain. This is consistent with the fact that while capacitors
and inductors are memory elements in the (ϕ, q)-domain, a memristor is instead
an adynamic element in the (ϕ, q)-domain, hence no state variable is needed for a
memristor in such domain.
6 The DAEs (5.30)–(5.33) depend also on q M k 0 if charge-controlled memristors are included in the
class LM.
5 Flux-Charge Analysis Method of Memristor Circuits
ϕ(t 0 ; t 0 ) = 0
gives the evolution of q(t; t 0 ) and ϕ(t; t 0 ) for t ≥ t 0 .
Remark 5.7 Although the DAEs (5.30)–(5.33) involve the incremental variables in
the (ϕ, q)-domain, their solution depends on q C 0 , ϕ L 0 and ϕ M 0 , 6 that is, the initial
conditions at t 0 for the state variables in the (v, i)-domain. Indeed, in the obtained
formulation, such initial conditions appear as constant inputs in the r.h.s. of (5.32)
and (5.33).
5.8.2 State Equations in the Flux-Charge Domain
Circuit analysis methods based on DAEs are fundamental in numerical simulation
(using, e.g., PSpice software [9]) of linear and nonlinear circuits [10]. On the other
hand, qualitative aspects of the nonlinear dynamics (e.g., the existence of no-finiteforward-escape-time solutions, eventual boundedness of solutions, local and global
asymptotic stability properties, bifurcation phenomena, etc.) are more effectively
analyzed by means of the SE formulation. Moreover, as pointed out in Chap. 3,
circuits that do not admit an SE description may be bad modeled from a physical
viewpoint due to the presence of impasse points.
The SE description in the (ϕ, q)-domain of a memristor circuit in LM, admitting
it exists, can be obtained by inspection of the equivalent circuit in the (ϕ, q)-domain
for simple low-order circuits. However, for more complex circuits, it is desirable to
develop a systematic procedure for writing the SEs and to give easily checkable
conditions guaranteeing that the SE representation exists.
In the following, we discuss a systematic approach for obtaining the SE
description of a circuit in LM in the (ϕ, q)-domain. The formal derivation of an
SE formulation of nonlinear RLC circuits in the (v, i)-domain, and the conditions
for the existence of such an SE formulation, have been provided in Sect. 3.3.2.3 of
Chap. 3. As previously observed, a memristor circuit in LM, when described in the
(ϕ, q)-domain, is analogous to a nonlinear RLC circuit in the (v, i)-domain. On
the basis of this observation, we are in a position to derive the SE description in
the (ϕ, q)-domain by exploiting the results in Chap. 3. It is worth to observe that
the formulation here proposed is based on using the vector of capacitor incremental
fluxes ϕ C (t; t 0 ), and the vector of incremental inductor charges q L (t; t 0 ), as state
variables in the (ϕ, q)-domain. This is consistent with the fact that while capacitors
and inductors are memory elements in the (ϕ, q)-domain, a memristor is instead
an adynamic element in the (ϕ, q)-domain, hence no state variable is needed for a
memristor in such domain.
6 The DAEs (5.30)–(5.33) depend also on q M k 0 if charge-controlled memristors are included in the
class LM.