Preface
xvii
The first part is a summary of nonlinear circuit theory pillars, so that readers
are gently introduced to concepts underlying the FCAM via a self-contained book.
Although expert researchers might pass over this part, the holistic approach used
reveals the generality of the FCAM and poses the basis for its extension to nonlinear
circuits with mem-elements. The FCAM is the core of the second part, whereas
the third part is devoted to FCAM applications. A brief summary of the chapters
included in each part is reported in the following.
Foundation of Nonlinear Circuit Theory
Chapter 1 provides the reader with an axiomatic definition of the basic nonlinear
circuit elements that could be used to model a wide variety of nonlinear devices.
A black-box approach is used, independent of the internal composition, material,
geometry, and architecture of each device. The axiomatic approach leads naturally
to the definition of the fourth basic circuit element, i.e., the memristor, and also
higher-order circuit elements, such as the memcapacitor and meminductor.
Chapter 2 deals in more depth with the behavior of a memristor by examining
some of its main properties and signatures. A brief account of the HP memristor
is provided, as well as a discussion on extended memristors and a classification of
memristive devices. The technological realization of real memristive devices as well
as materials and physics phenomena underlying the memristor behavior are briefly
discussed.
Chapters 3 and 4 are devoted to synthetically give the needed theoretic background on nonlinear RLC circuits (without memristors). Especially, Chap. 3 deals
with a synthetic description of the main methods to analyze nonlinear RLC
circuits, i.e., tableau, nodal, and mesh analysis. Special emphasis is then paid to
give conditions for the existence of the state equation (SE) representation and the
techniques to write the SEs of nonlinear RLC circuits.
Chapter 4 briefly discusses some main dynamical phenomena that can be
observed in autonomous nonlinear RLC circuits of first, second, and third order,
including convergence of solutions, oscillations, and complex dynamic behavior.
The chapter ends with some basic considerations on local bifurcations of equilibrium points and global bifurcations of limit cycles in autonomous RLC nonlinear
circuits depending on parameters.
Flux-Charge Analysis Method (FCAM)
Chapters 5–7 are at the core of the book, and they are devoted to develop FCAM for
the dynamic analysis of nonlinear circuits containing memristors.
In particular, Chap. 5 starts with a critical review of Kirchhoff Laws in the (ϕ, q)domain, and, on this basis, it then proceeds to develop the new method of analysis in
the (ϕ, q)-domain (FCAM) for wide classes of memristor circuits. The chapter also
establishes an analogy between RLC circuits in the (v, i)-domain and memristor
circuits in the (ϕ, q)-domain and, via this analogy, gives a general formulation of
memristor circuit equations.
Chapter 6 discusses the applications of FCAM to some basic low-order
autonomous memristor circuits highlighting the main advantages of FCAM
related to the reduction of order, and smoother dynamics, in the (ϕ, q)-domain.
xvii
The first part is a summary of nonlinear circuit theory pillars, so that readers
are gently introduced to concepts underlying the FCAM via a self-contained book.
Although expert researchers might pass over this part, the holistic approach used
reveals the generality of the FCAM and poses the basis for its extension to nonlinear
circuits with mem-elements. The FCAM is the core of the second part, whereas
the third part is devoted to FCAM applications. A brief summary of the chapters
included in each part is reported in the following.
Foundation of Nonlinear Circuit Theory
Chapter 1 provides the reader with an axiomatic definition of the basic nonlinear
circuit elements that could be used to model a wide variety of nonlinear devices.
A black-box approach is used, independent of the internal composition, material,
geometry, and architecture of each device. The axiomatic approach leads naturally
to the definition of the fourth basic circuit element, i.e., the memristor, and also
higher-order circuit elements, such as the memcapacitor and meminductor.
Chapter 2 deals in more depth with the behavior of a memristor by examining
some of its main properties and signatures. A brief account of the HP memristor
is provided, as well as a discussion on extended memristors and a classification of
memristive devices. The technological realization of real memristive devices as well
as materials and physics phenomena underlying the memristor behavior are briefly
discussed.
Chapters 3 and 4 are devoted to synthetically give the needed theoretic background on nonlinear RLC circuits (without memristors). Especially, Chap. 3 deals
with a synthetic description of the main methods to analyze nonlinear RLC
circuits, i.e., tableau, nodal, and mesh analysis. Special emphasis is then paid to
give conditions for the existence of the state equation (SE) representation and the
techniques to write the SEs of nonlinear RLC circuits.
Chapter 4 briefly discusses some main dynamical phenomena that can be
observed in autonomous nonlinear RLC circuits of first, second, and third order,
including convergence of solutions, oscillations, and complex dynamic behavior.
The chapter ends with some basic considerations on local bifurcations of equilibrium points and global bifurcations of limit cycles in autonomous RLC nonlinear
circuits depending on parameters.
Flux-Charge Analysis Method (FCAM)
Chapters 5–7 are at the core of the book, and they are devoted to develop FCAM for
the dynamic analysis of nonlinear circuits containing memristors.
In particular, Chap. 5 starts with a critical review of Kirchhoff Laws in the (ϕ, q)domain, and, on this basis, it then proceeds to develop the new method of analysis in
the (ϕ, q)-domain (FCAM) for wide classes of memristor circuits. The chapter also
establishes an analogy between RLC circuits in the (v, i)-domain and memristor
circuits in the (ϕ, q)-domain and, via this analogy, gives a general formulation of
memristor circuit equations.
Chapter 6 discusses the applications of FCAM to some basic low-order
autonomous memristor circuits highlighting the main advantages of FCAM
related to the reduction of order, and smoother dynamics, in the (ϕ, q)-domain.