5.8 Formulation of Memristor Circuits Equations
205
reduction equal to the number n M of memristors. Such a reduction is expected to
yield advantages in the dynamic analysis of memristor circuits in the (ϕ, q)-domain.
Remark 5.10 (Smoothing Effect) The SEs (5.36) in the (ϕ, q)-domain are defined
by a vector field containing the nonlinearities f M k (·) of memristors, while the
SEs (5.45) in the (v, i)-domain are defined by a vector field with terms G k (·) =
f
M k
(·). This means that the SEs in the (ϕ, q)-domain are defined by a smoother
vector field with respect to those in the (v, i)-domain. This represents a potential
advantage in view of the numerical simulations. Another relevant consequence
concerns mathematical issues related to the uniqueness of solutions for the SEs.
The latter aspect will be further discussed with specific examples.
Remark 5.11 In abstract mathematical form, the correspondence between the solution of the SEs (5.36)
Ψ ϕ,q (t, ϕ C (t 0 , t 0 ) = 0, q L (t 0 ; t 0 ) = 0) = (ϕ C (t; t 0 ), q L (t; t 0 ))
and the solution of the SEs (5.45)
Ψ v,i (t, v C 0 , i L 0 , ϕ M 0 ) = (v C (t), i L (t), ϕ M (t))
is expressed by the following relations:
d
dt
ϕ C (t; t 0 )
= v C (t)
d
dt
q L (t; t 0 )
= i L (t)
ϕ M (t) = ϕ M 0 +
t
t 0
H c (v C (τ ), i L (τ ), ϕ M (τ ), e(τ ), a(τ ))dτ.
Also an inverse relation can be obtained. These relationships between solutions will
be further illustrated by means of specific examples.
5.8.5 Examples
We have seen in the previous section how to write the DAEs and SEs of a circuit in
LM in the (ϕ, q)-domain and in the traditional (v, i)-domain. We provide here
some examples to illustrate how to apply these techniques. The examples also
discuss if it is possible to pass from the dynamic equations in a given domain to
the corresponding dynamic equations in the other domain via differentiation or
integration in time. In addition, the examples highlight some shortcomings in the
techniques discussed so far for writing the SEs in the (ϕ, q)-domain, that will be
solved later in the book in Chap. 7.
205
reduction equal to the number n M of memristors. Such a reduction is expected to
yield advantages in the dynamic analysis of memristor circuits in the (ϕ, q)-domain.
Remark 5.10 (Smoothing Effect) The SEs (5.36) in the (ϕ, q)-domain are defined
by a vector field containing the nonlinearities f M k (·) of memristors, while the
SEs (5.45) in the (v, i)-domain are defined by a vector field with terms G k (·) =
f
M k
(·). This means that the SEs in the (ϕ, q)-domain are defined by a smoother
vector field with respect to those in the (v, i)-domain. This represents a potential
advantage in view of the numerical simulations. Another relevant consequence
concerns mathematical issues related to the uniqueness of solutions for the SEs.
The latter aspect will be further discussed with specific examples.
Remark 5.11 In abstract mathematical form, the correspondence between the solution of the SEs (5.36)
Ψ ϕ,q (t, ϕ C (t 0 , t 0 ) = 0, q L (t 0 ; t 0 ) = 0) = (ϕ C (t; t 0 ), q L (t; t 0 ))
and the solution of the SEs (5.45)
Ψ v,i (t, v C 0 , i L 0 , ϕ M 0 ) = (v C (t), i L (t), ϕ M (t))
is expressed by the following relations:
d
dt
ϕ C (t; t 0 )
= v C (t)
d
dt
q L (t; t 0 )
= i L (t)
ϕ M (t) = ϕ M 0 +
t
t 0
H c (v C (τ ), i L (τ ), ϕ M (τ ), e(τ ), a(τ ))dτ.
Also an inverse relation can be obtained. These relationships between solutions will
be further illustrated by means of specific examples.
5.8.5 Examples
We have seen in the previous section how to write the DAEs and SEs of a circuit in
LM in the (ϕ, q)-domain and in the traditional (v, i)-domain. We provide here
some examples to illustrate how to apply these techniques. The examples also
discuss if it is possible to pass from the dynamic equations in a given domain to
the corresponding dynamic equations in the other domain via differentiation or
integration in time. In addition, the examples highlight some shortcomings in the
techniques discussed so far for writing the SEs in the (ϕ, q)-domain, that will be
solved later in the book in Chap. 7.