374
10 Extended Memristor Devices
of Chap. 2. A complete classification of memristor devices has been introduced in
Sect. 2.4 of the same chapter (see also the articles [8, 9]) to provide a systematic tool
to circuit designers. According to this classification, three main classes of memristor
device models can be identified: extended memristors, generic memristors and ideal
memristors.
An examination of the literature shows an evident gap between the experimental
characterization of memristor devices and their systematic description and modeling
in terms of fundamental circuit elements introduced in Chap. 1. In particular, the
following question naturally arises:
Which is the set of extended memristors that can be described by a composite circuit made
of the interconnection of linear/nonlinear circuit elements and ideal memristors?
This chapter provides a study on composite circuits obtained by interconnection
of nonlinear circuit elements and ideal memristors for describing and modeling
extended memristors. The results rely upon the concept of two-terminal algebraic
(α, β)-elements introduced in Chap. 1, i.e., elements whose constitutive relation
is an algebraic link between v (α) (t) (the α-derivative of v(t) for positive α, or
the α-integral of v(t) for negative α, see Chap. 1 for the exact definition), and
i (β) (t), where α, β are integers. Accordingly, it is possible to classify the four basic
algebraic circuit elements, i.e., (possibly) nonlinear resistors, inductors, capacitors
and memristors, and also higher-order algebraic circuit elements, by arranging them
into a periodic table. For example, a nonlinear resistor corresponds to an (α, β) =
(0, 0) element, whereas an ideal memristor corresponds to an (α, β) = (−1, −1)
element.
In the chapter, it is shown that we can identify a significant class of extended
memristors (i.e., memristor devices that cannot be reduced to just ideal memristors)
that admits of the description in terms of a combination of algebraic circuit elements.
To this end, it is preliminary observed that: (a) an ideal memristor with a linear
relationship between charge and flux is equivalent to an ideal resistor (Chap. 1);
(b) due to the Element Closure Property (Chap. 1), an arbitrary interconnection
of (α, β)-elements is equivalent to a (α, β) element. Then, in order to implement
an extended memristor, it is necessary to consider the interconnection of (α, β)
elements with at least two different pairs of (α, β). One of the main results in
the chapter is that an extended memristor can be realized by a combination of
ideal memristors ((α, β) = (−1, −1) elements) and nonlinear resistors ((α, β) =
(0, 0) elements). For brevity, we focus on the simplest class D ext of extended
memristors with one ideal memristor and one nonlinear (possibly locally active)
resistor. By massaging the characteristics of the constitutive elements, such class
D ext of extended memristors is shown to be able to approximate rectifying effects
and asymmetric pinched hysteresis loops of real nonvolatile switching memristor
devices. Finally, it is shown that if the nonlinear resistors are described in terms
of piecewise linear characteristics, then FCAM can still be employed to effectively
analyze the dynamics of nonlinear circuits with extended memristors in D ext .
10 Extended Memristor Devices
of Chap. 2. A complete classification of memristor devices has been introduced in
Sect. 2.4 of the same chapter (see also the articles [8, 9]) to provide a systematic tool
to circuit designers. According to this classification, three main classes of memristor
device models can be identified: extended memristors, generic memristors and ideal
memristors.
An examination of the literature shows an evident gap between the experimental
characterization of memristor devices and their systematic description and modeling
in terms of fundamental circuit elements introduced in Chap. 1. In particular, the
following question naturally arises:
Which is the set of extended memristors that can be described by a composite circuit made
of the interconnection of linear/nonlinear circuit elements and ideal memristors?
This chapter provides a study on composite circuits obtained by interconnection
of nonlinear circuit elements and ideal memristors for describing and modeling
extended memristors. The results rely upon the concept of two-terminal algebraic
(α, β)-elements introduced in Chap. 1, i.e., elements whose constitutive relation
is an algebraic link between v (α) (t) (the α-derivative of v(t) for positive α, or
the α-integral of v(t) for negative α, see Chap. 1 for the exact definition), and
i (β) (t), where α, β are integers. Accordingly, it is possible to classify the four basic
algebraic circuit elements, i.e., (possibly) nonlinear resistors, inductors, capacitors
and memristors, and also higher-order algebraic circuit elements, by arranging them
into a periodic table. For example, a nonlinear resistor corresponds to an (α, β) =
(0, 0) element, whereas an ideal memristor corresponds to an (α, β) = (−1, −1)
element.
In the chapter, it is shown that we can identify a significant class of extended
memristors (i.e., memristor devices that cannot be reduced to just ideal memristors)
that admits of the description in terms of a combination of algebraic circuit elements.
To this end, it is preliminary observed that: (a) an ideal memristor with a linear
relationship between charge and flux is equivalent to an ideal resistor (Chap. 1);
(b) due to the Element Closure Property (Chap. 1), an arbitrary interconnection
of (α, β)-elements is equivalent to a (α, β) element. Then, in order to implement
an extended memristor, it is necessary to consider the interconnection of (α, β)
elements with at least two different pairs of (α, β). One of the main results in
the chapter is that an extended memristor can be realized by a combination of
ideal memristors ((α, β) = (−1, −1) elements) and nonlinear resistors ((α, β) =
(0, 0) elements). For brevity, we focus on the simplest class D ext of extended
memristors with one ideal memristor and one nonlinear (possibly locally active)
resistor. By massaging the characteristics of the constitutive elements, such class
D ext of extended memristors is shown to be able to approximate rectifying effects
and asymmetric pinched hysteresis loops of real nonvolatile switching memristor
devices. Finally, it is shown that if the nonlinear resistors are described in terms
of piecewise linear characteristics, then FCAM can still be employed to effectively
analyze the dynamics of nonlinear circuits with extended memristors in D ext .