164
5 Flux-Charge Analysis Method of Memristor Circuits
different dynamics and attractors for the same set of circuit parameters and
the existence of special bifurcations phenomena named bifurcations without
parameters (Chap. 6).
• FCAM permits to obtain a reduction of order when dynamic circuit equations of
a memristor circuit in LM are derived in the (ϕ, q)-domain with respect to their
derivation in the (v, i)-domain. From a mathematical viewpoint, the reduction
of order enables a simpler dynamical analysis by exploiting standard theory of
low-order dynamical systems.
• FCAM in the (ϕ, q)-domain paves the way to programme different dynamics
and attractors in a systematic way by means of pulse voltage or current sources
(Chap. 7). Memristor programming is a crucial issue in several applications
including neuromorphic systems.
It is also worth to note that all the strong points reported above result from the
application of the FCAM to memristor circuits in LM in the (ϕ, q)-domain, because
it is more intricate to get a thorough understanding of nonlinear dynamics by means
of traditional circuit methodologies in the (v, i)-domain.
FCAM has been originally developed mainly in a series of works [1–3]. This
chapter is based on [1], but the treatment, presented in a didactic form, is also
extended along several directions, including multiterminal and multiport elements,
time-varying elements, etc. Finally, several examples are presented to facilitate the
acquisition of basic theoretic aspects of FCAM.
5.1 The Importance of Choosing the Correct Pair of
Variables
One of the chief concepts presented in Sect. 2.1.3 of Chap. 2 is that pinched
hysteresis loops displayed in the v −i plane by a memristor in response to any (zeromean) periodic signals don’t permit to derive a CR, thus pinched hysteresis loops are
not models! Changing the amplitude and frequency of the excitation would give rise
to a complicated scenario where the whole v − i plane is filled with differently
shaped hysteresis loops. The hysteresis loops are only the response to specific
excitations but they don’t have predicting ability since they do not permit to obtain
the response to other types of excitations. The only CR of a memristor is instead
obtained when using the correct pair of electric variables, i.e., flux and charge. The
CR is indeed given by a nonlinear characteristic relating flux and charge. For any
possible signal applied to the memristor, charge and flux lie on such characteristic,
thus implying that the CR in the flux-charge domain has predicting ability.
Hence, the selection of the correct pair of electric variables is of paramount
importance for characterizing and modeling any two-terminal circuit element. Flux
and charge are the proper variable for memristor devices. Next, we discuss three
additional examples, the first one in the electrical domain and the others in the
mechanical field.
5 Flux-Charge Analysis Method of Memristor Circuits
different dynamics and attractors for the same set of circuit parameters and
the existence of special bifurcations phenomena named bifurcations without
parameters (Chap. 6).
• FCAM permits to obtain a reduction of order when dynamic circuit equations of
a memristor circuit in LM are derived in the (ϕ, q)-domain with respect to their
derivation in the (v, i)-domain. From a mathematical viewpoint, the reduction
of order enables a simpler dynamical analysis by exploiting standard theory of
low-order dynamical systems.
• FCAM in the (ϕ, q)-domain paves the way to programme different dynamics
and attractors in a systematic way by means of pulse voltage or current sources
(Chap. 7). Memristor programming is a crucial issue in several applications
including neuromorphic systems.
It is also worth to note that all the strong points reported above result from the
application of the FCAM to memristor circuits in LM in the (ϕ, q)-domain, because
it is more intricate to get a thorough understanding of nonlinear dynamics by means
of traditional circuit methodologies in the (v, i)-domain.
FCAM has been originally developed mainly in a series of works [1–3]. This
chapter is based on [1], but the treatment, presented in a didactic form, is also
extended along several directions, including multiterminal and multiport elements,
time-varying elements, etc. Finally, several examples are presented to facilitate the
acquisition of basic theoretic aspects of FCAM.
5.1 The Importance of Choosing the Correct Pair of
Variables
One of the chief concepts presented in Sect. 2.1.3 of Chap. 2 is that pinched
hysteresis loops displayed in the v −i plane by a memristor in response to any (zeromean) periodic signals don’t permit to derive a CR, thus pinched hysteresis loops are
not models! Changing the amplitude and frequency of the excitation would give rise
to a complicated scenario where the whole v − i plane is filled with differently
shaped hysteresis loops. The hysteresis loops are only the response to specific
excitations but they don’t have predicting ability since they do not permit to obtain
the response to other types of excitations. The only CR of a memristor is instead
obtained when using the correct pair of electric variables, i.e., flux and charge. The
CR is indeed given by a nonlinear characteristic relating flux and charge. For any
possible signal applied to the memristor, charge and flux lie on such characteristic,
thus implying that the CR in the flux-charge domain has predicting ability.
Hence, the selection of the correct pair of electric variables is of paramount
importance for characterizing and modeling any two-terminal circuit element. Flux
and charge are the proper variable for memristor devices. Next, we discuss three
additional examples, the first one in the electrical domain and the others in the
mechanical field.