Chapter 5
Flux-Charge Analysis Method of
Memristor Circuits
Let us consider a relevant class of nonlinear networks, denoted by LM, containing
at least one ideal memristor in addition to ideal (linear) resistors, inductors,
capacitors, and independent voltage or currents sources. Thus, LM describes
nonlinear dynamic networks including ideal memristors.
This chapter provides an effective systematic methodology to write the dynamic
equations of any circuit in LM with the aim of exploring qualitative and quantitative
properties of nonlinear dynamic behavior. Memristor networks in the class LM
could be analyzed in the traditional voltage-current (v, i)-domain, but the CR of a
memristor is given in terms of flux and charge variables (see Chap. 2 and the next
Sect. 5.1 for a brief recap). Hence, due to the presence of the memristor, it is logical
to seek an answer to the following question:
Which is the domain, if any, where qualitative and quantitative nonlinear dynamic properties
of a memristor circuit in LM are described and can be studied in the most effective way?
Hereinafter, the ultimate goal of the book is to demonstrate that the flux-charge
(ϕ, q)-domain is the most effective framework for memristor circuits in the class
LM. The chief pillar of the systematic method for the analysis of memristor circuits
in LM is the writing of Kirchhoff laws and CRs of circuit elements in the (ϕ, q)domain, i.e., using flux ϕ = v (−1) and charge q = i (−1) as port variables of each
two-terminal element. This new technique is named Flux-Charge Analysis Method
(or FCAM, for short).
The application of FCAM to the class LM makes clear that the term “most
effective way” results into multiple practical and theoretical aspects. A concise
summary of the main advantages of FCAM with respect to traditional methods in
the (v, i)-domain is reported below:
• the analysis in the (ϕ, q)-domain via FCAM allows one to unfold the nonlinear
dynamics of memristor circuits by unveiling some new and peculiar features
that are not typically observable in standard RLC circuits. These include
the existence invariants of motion and invariant manifolds, the coexistence of
© Springer Nature Switzerland AG 2021
F. Corinto et al., Nonlinear Circuits and Systems with Memristors,
https://doi.org/10.1007/978-3-030-55651-8_5
163
Flux-Charge Analysis Method of
Memristor Circuits
Let us consider a relevant class of nonlinear networks, denoted by LM, containing
at least one ideal memristor in addition to ideal (linear) resistors, inductors,
capacitors, and independent voltage or currents sources. Thus, LM describes
nonlinear dynamic networks including ideal memristors.
This chapter provides an effective systematic methodology to write the dynamic
equations of any circuit in LM with the aim of exploring qualitative and quantitative
properties of nonlinear dynamic behavior. Memristor networks in the class LM
could be analyzed in the traditional voltage-current (v, i)-domain, but the CR of a
memristor is given in terms of flux and charge variables (see Chap. 2 and the next
Sect. 5.1 for a brief recap). Hence, due to the presence of the memristor, it is logical
to seek an answer to the following question:
Which is the domain, if any, where qualitative and quantitative nonlinear dynamic properties
of a memristor circuit in LM are described and can be studied in the most effective way?
Hereinafter, the ultimate goal of the book is to demonstrate that the flux-charge
(ϕ, q)-domain is the most effective framework for memristor circuits in the class
LM. The chief pillar of the systematic method for the analysis of memristor circuits
in LM is the writing of Kirchhoff laws and CRs of circuit elements in the (ϕ, q)domain, i.e., using flux ϕ = v (−1) and charge q = i (−1) as port variables of each
two-terminal element. This new technique is named Flux-Charge Analysis Method
(or FCAM, for short).
The application of FCAM to the class LM makes clear that the term “most
effective way” results into multiple practical and theoretical aspects. A concise
summary of the main advantages of FCAM with respect to traditional methods in
the (v, i)-domain is reported below:
• the analysis in the (ϕ, q)-domain via FCAM allows one to unfold the nonlinear
dynamics of memristor circuits by unveiling some new and peculiar features
that are not typically observable in standard RLC circuits. These include
the existence invariants of motion and invariant manifolds, the coexistence of
© Springer Nature Switzerland AG 2021
F. Corinto et al., Nonlinear Circuits and Systems with Memristors,
https://doi.org/10.1007/978-3-030-55651-8_5
163