5.5 Flux-Charge Analysis Method (FCAM) for Class LM of Memristor Circuits
179
• The SE in the (ϕ, q)-domain keeps track of the initial conditions for the state
variables in the (v, i)-domain. Such initial conditions in fact appear as constant
inputs in the r.h.s. of the SE in the (ϕ, q)-domain.
• The natural choice of the state variables in the (ϕ, q)-domain is given by
the incremental flux of capacitors and incremental charge of inductors. The
memristor has no associated state variable in the (ϕ, q)-domain since, as it will
be discussed in more detail in Sect. 5.5.2, it is described by a nonlinear algebraic
relation between charge and flux in the (ϕ, q)-domain.
• For the analysis in the (ϕ, q)-domain, special care is needed to use Kirchhoff
laws in terms of charge and flux. The most effective form of Kirchhoff laws in
the (ϕ, q)-domain is that based on incremental charge and incremental flux of
two-terminal elements.
5.5 Flux-Charge Analysis Method (FCAM) for Class LM of
Memristor Circuits
We consider the class LM of memristor circuits containing at least one ideal fluxcontrolled memristor (or one ideal charge-controlled memristor) in addition to ideal
(linear) resistors, capacitors, inductors, and independent voltage or current sources.
All elements in LM, possibly with the exception of the independent sources, are
assumed to be time-invariant. The extension to the case where all elements may be
time-varying is straightforward and is briefly discussed in Sect. 5.6.2.
Consider a circuit N ∈ LM and suppose it has a fixed topology for t ≥ t 0 , where
t 0 is a finite initial instant. Our goal is to show that we can develop a method, named
flux-charge analysis method (FCAM), enabling to analyze the circuit dynamics for
t ≥ t 0 in the (ϕ, q)-domain. To this end we will address the following main issues:
• how to write Kirchhoff laws in the (ϕ, q)-domain;
• how to write the CR of each element in the class LM in the (ϕ, q)-domain.
We then address the issue of writing in a systematic way the DAEs and SEs
describing the circuit dynamics in the (ϕ, q)-domain.
5.5.1 Incremental Form of Kirchhoff Laws in the Flux-Charge
Domain
Let b be the number of two-terminal elements and n the number of nodes of a circuit
N ∈ LM. Then, KCL permits to write n − 1 fundamental cut-set equations in the
form (Chap. 3)
Ai(t) = 0
179
• The SE in the (ϕ, q)-domain keeps track of the initial conditions for the state
variables in the (v, i)-domain. Such initial conditions in fact appear as constant
inputs in the r.h.s. of the SE in the (ϕ, q)-domain.
• The natural choice of the state variables in the (ϕ, q)-domain is given by
the incremental flux of capacitors and incremental charge of inductors. The
memristor has no associated state variable in the (ϕ, q)-domain since, as it will
be discussed in more detail in Sect. 5.5.2, it is described by a nonlinear algebraic
relation between charge and flux in the (ϕ, q)-domain.
• For the analysis in the (ϕ, q)-domain, special care is needed to use Kirchhoff
laws in terms of charge and flux. The most effective form of Kirchhoff laws in
the (ϕ, q)-domain is that based on incremental charge and incremental flux of
two-terminal elements.
5.5 Flux-Charge Analysis Method (FCAM) for Class LM of
Memristor Circuits
We consider the class LM of memristor circuits containing at least one ideal fluxcontrolled memristor (or one ideal charge-controlled memristor) in addition to ideal
(linear) resistors, capacitors, inductors, and independent voltage or current sources.
All elements in LM, possibly with the exception of the independent sources, are
assumed to be time-invariant. The extension to the case where all elements may be
time-varying is straightforward and is briefly discussed in Sect. 5.6.2.
Consider a circuit N ∈ LM and suppose it has a fixed topology for t ≥ t 0 , where
t 0 is a finite initial instant. Our goal is to show that we can develop a method, named
flux-charge analysis method (FCAM), enabling to analyze the circuit dynamics for
t ≥ t 0 in the (ϕ, q)-domain. To this end we will address the following main issues:
• how to write Kirchhoff laws in the (ϕ, q)-domain;
• how to write the CR of each element in the class LM in the (ϕ, q)-domain.
We then address the issue of writing in a systematic way the DAEs and SEs
describing the circuit dynamics in the (ϕ, q)-domain.
5.5.1 Incremental Form of Kirchhoff Laws in the Flux-Charge
Domain
Let b be the number of two-terminal elements and n the number of nodes of a circuit
N ∈ LM. Then, KCL permits to write n − 1 fundamental cut-set equations in the
form (Chap. 3)
Ai(t) = 0