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Memristor Circuits: Flux-Charge Analysis Method / Corinto, Fernando; Forti, Mauro. - In: IEEE TRANSACTIONS ON
CIRCUITS AND SYSTEMS. I, REGULAR PAPERS. - ISSN 1549-8328. - STAMPA. - 63:11(2016), pp. 1997-2009.
[10.1109/TCSI.2016.2590948]
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Memristor Circuits: Flux-Charge Analysis Method
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS 1
Memristor Circuits: Flux—Charge Analysis Method
Fernando Corinto, Senior Member, IEEE, and Mauro Forti
Abstract—Memristor-based circuits are widely exploited to re-
alize analog and/or digital systems for a broad scope of applica-
tions (e.g., amplifiers, filters, oscillators, logic gates, and memristor
as synapses). A systematic methodology is necessary to understand
complex nonlinear phenomena emerging in memristor circuits.
The manuscript introduces a comprehensive analysis method of
memristor circuits in the flux-charge (ϕ, q)-domain. The pro-
posed method relies on Kirchhoff Flux and Charge Laws and
constitutive relations of circuit elements in terms of incremental
flux and charge. The main advantages (over the approaches in
the voltage-current (v, i)-domain) of the formulation of circuit
equations in the (ϕ, q)-domain are: a) a simplified analysis of
nonlinear dynamics and bifurcations by means of a smaller set
of ODEs; b) a clear understanding of the influence of initial con-
ditions. The straightforward application of the proposed method
provides a full portrait of the nonlinear dynamics of the simplest
memristor circuit made of one memristor connected to a capacitor.
In addition, the concept of invariant manifolds permits to clarify
how initial conditions give rise to bifurcations without parameters.
Index Terms—Bifurcations without parameters, circuit analy-
sis, circuit theory, memristor, nonlinear dynamics.
I. INTRODUCTION AND MOTIVATION
M
EMRISTOR was introduced in 1971 by Prof. L. O. Chua
[1] as a theoretical two-terminal circuit element model-
ing electrical devices in terms of the integral of the current i(t)
and the integral of the voltage v(t). Namely, a charge-controlled
memristor is described by
ϕ(t)=h (q(t)) (1)
where, following the nomenclature introduced in [2], the
“voltage momentum” (aka “flux”) and “current momentum”
(aka “charge”) are:
ϕ(t)=
t
−∞
v(τ)dτ (2)
q(t)=
t
−∞
i(τ)dτ. (3)
Manuscript received April 30, 2016; revised July 3, 2016; accepted July 4,
2016. The support from European Cooperation in Science and Technology
“COST Action IC1401” is acknowledged. This paper was recommended by
Associate Editor E. Blokhina.
F. Corinto is with the Department of Electronics and Telecommunications,
Politecnico di Torino, Torino 10129, Italy (e-mail: fernando.corinto@polito.it).
M. Forti is with the Department of Information Engineering and Mathemat-
ics, University of Siena, 53100 Siena, Italy (e-mail: forti@diism.unisi.it).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCSI.2016.2590948
An equivalent form (see [2, Th. 1]) of (1) includes the Ohm’s
Law and an Ordinary Differential Equation (ODE)
v(t)=R(q(t))i(t)
dq(t)
dt
= i(t) (4)
where R(q)=dh(q)/dq is the memristance.
The duality principle allows one to introduce the flux-
controlled memristor
q(t)=f (ϕ(t)) (5)
that is
i(t)=G (ϕ(t)) v(t)
dϕ(t)
dt
= v(t) (6)
where G(ϕ)=df (ϕ)/dϕ is the memductance.
A complete classification of memristor devices in terms of
the pairs of electrical variables (v(t),i(t)) and (ϕ(t),q(t)) is
provided in [2]. One main result there reported is the unified
description of memristor devices (i.e., without any regard to the
technological realization), under the formalism based on volt-
age and current momenta. This approach makes easier to show
that the memristor’s pinched loop in the (v(t),i(t)) description
represents just the specific response to a given external input
(refer to [2, Figs. 2 and 4]).
In 2008 researchers at Hewlett-Packard recognized memris-
tor features in single-devices based on a Pt/TiO
2
/P t struc-
ture [3]. Such achievement has promoted research activities
in memristor-based circuits and systems intended for a broad
scope of applications. Recently, memristor-based circuits are
widely exploited to realize analog and/or digital systems (e.g.,
amplifiers, filters, oscillators, logic gates, and memristor as
synapses) [4]–[6]. Such applications are based on the following
chief properties of memristor devices:
a) the fine-resolution programming of the memristance,
tuned by the input amplitude, pulsewidth and frequency,
in memristor acting as a non-volatile memory;
b) the inherent nonlinear dynamic behavior in memristor
acting as a volatile memory.
For demonstration of the first feature, memristors are subject
to low voltages during their operation as analog circuit elements
and high voltages to program their memristance, i.e., mem-
ristors are exploited as pulse-programmable resistances. The
programmability of memristance is also achieved with hybrid
CMOS/memristor circuits due to the flexibility, reliability and
high functionality of CMOS subsystems. Recently, memristor-
based synapses are also realized combining a Resistive RAM
memristor with a selector device (e.g., the building block is a
crossbar array made of cells with 1-transistor/1-memristor or
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2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS
Fig. 1. The simplest memristor-based circuit. (a) M −C circuit for any t ≥ t
0
.
(b) M − C circuit including the networks L
a
and L
b
that set independent
initial conditions v
C
0
and ϕ
M
0
through the evolution of the electrical variables
v
C
(t) and ϕ
M
(t) for t<t
0
.
2-transistor/1-memristor) [7]. In such a case the compatibility
between memristor and CMOS technologies is an issue that is
still under investigation.
Nonlinear dynamic behavior of memristors is exploited in
oscillatory [8], [9] and chaotic circuits [10], [11]. A thorough
study is necessary to understand the rich complex nonlinear
phenomena emerging in memristor circuits. Nonlinear dynam-
ics of dual coupled memristors and of a thermally-activated
locally-active memristor (based on a microstructure consisting
of a bi-layer of Nb
2
O
5
and Nb
2
O
x
materials) has been inves-
tigated in [12] and [13]. Stability properties of attractors, local
and global bifurcations, and the role of the initial conditions
have been extensively investigated as well [14]. A systematic
description (mainly based on the network theory technique
referred to as the tableau method) is proposed in [15], but
this leads to large systems of nonlinear Differential Algebraic
Equations (DAEs), whose solution requires efficient numerical
simulation tools.
The aim of the manuscript is to present a novel systematic
methodology for the analysis of a large class of nonlinear
circuits containing memristors. The class, which is denoted
by LM, is constituted by ideal capacitors, ideal inductors,
ideal resistors, ideal independent voltage and current sources,
and memristors that are either flux-controlled and/or charge-
controlled.
1
Any practical application of a circuit in LM actually starts
at a finite time instant t
0
, i.e., −∞ <t
0
< +∞, representing,
e.g., the instant when switches are turned on or off. Given a
circuit in the class LM, with fixed topology for t ≥ t
0
, our
goal is to develop a method to analyze its dynamics for t ≥ t
0
.
An illustrative example allows us to highlight the key issues
addressed by the proposed methodology and its advantages over
current approaches (see in particular [8], [9], [15]).
Example: Consider the simplest memristor-based circuit in
the class LM composed of just one memristor M connected to
a capacitor C [see the M − C circuit in Fig. 1(a)]. To analyze
its dynamics for t ≥ t
0
we first need to set suitable initial
conditions at t
0
for the state variables v
C
(t) and ϕ
M
(t),i.e.,
v
C
(t
0
)=v
C
0
and ϕ
M
(t
0
)=ϕ
M
0
. To this end, let us consider
the circuit reported in Fig. 1(b), which is equivalent to the
M − C circuit for t ≥ t
0
(i.e., S
1
and S
3
are open whereas S
2
1
The (linear) resistors, capacitors, and inductors can be either passive or active.
is closed), but includes the evolution of the electrical variables
v
C
(t) and ϕ
M
(t) for t<t
0
as well (i.e., S
1
and S
3
are closed
while S
2
is open). The two-terminal circuit elements L
a
and
L
b
can be any linear networks made of resistors, capacitors,
inductors, voltage and current sources.
It is clear that we can set independent initial conditions v
C
0
and ϕ
M
0
for the state variables v
C
(t),ϕ
M
(t) by means of the
dynamics for t<t
0
of the circuits L
a
− M (with S
1
closed),
and L
b
− C (with S
3
closed), respectively. We may assume
that the elements M and C are not energized at −∞,i.e.,
v
C
(−∞)=0, ϕ
M
(−∞)=0.
Analysis by inspection of the M − C circuit permits to
derive that v
C
(t) and ϕ
M
(t) obey the following Initial Value
Problem (IVP) for a second-order ODE [obtained by (6)]:
C
dv
C
(t)
dt
= −G (ϕ
M
(t)) v
C
(t) (7)
dϕ
M
(t)
dt
= v
C
(t)
v
C
(t
0
)=v
C
0
ϕ
M
(t
0
)=ϕ
M
0
(8)
for any t ≥ t
0
,wherev
C
0
, ϕ
M
0
are independent initial condi-
tions for the state variables v
C
(t) and ϕ
M
(t) at t
0
.
Note that the r.h.s. of (7) can be written as
−G(ϕ
M
(t))v
C
(t)=−
df (ϕ
M
)
dϕ
M
dϕ
M
(t)
dt
=−
d
dt
f(ϕ
M
(t)) (9)
and, as a consequence, by integrating (7) over (t
0
,t),where
t ≥ t
0
, we obtain
C (v
C
(t) − v
C
(t
0
)) = −(f (ϕ
M
(t)) − f (ϕ
M
(t
0
))
= −(q
M
(t) − q
M
(t
0
)) . (10)
The chief result drawn by (10) is twofold.
First, the nonlinear dynamical behavior for t ≥ t
0
of the
M − C circuit is described by the following IVP for a first-
order ODE [derived from (10) and (8)]:
dϕ
M
(t)
dt
= −
f (ϕ
M
(t))
C
+
f(ϕ
M
0
)
C
+ v
C
0
ϕ
M
(t
0
)=ϕ
M
0
(11)
where the state variable is ϕ
M
(t) and the initial conditions v
C
0
and ϕ
M
0
appear as constant inputs in the r.h.s.
It turns out that th e IVP (7), (8) for a second-order ODE in
the voltage-current (v, i)-domain can be reduced to an IVP (11)
for a first-order ODE in the flux-charge (ϕ, q)-domain where
the r.h.s. depends on the initial conditions v
C
0
and ϕ
M
0
at
t
0
for the state variables in the (v, i)-domain.
2
This reduction
is crucial in analyzing nonlinear dynamics and bifurcations
in the M − C circuit. The key electrical variable in (11) is
just the flux ϕ
M
(t).
3
This confirms the results in [2] that
the (ϕ
M
(t),q
M
(t)) are the sole electrical variables useful to
characterize memristors and memristor circuits as well.
2
It is worth to observe that two initial conditions have to be specified in order
to completely determine the evolution of electrical variables in the M − C
circuit, i.e., the order of complexity of the M − C circuit is two (see Theorem 5
in [1]).
3
Similar considerations hold in circuits composed of one memristor con-
nected to an inductor. In such a case the key variable is the charge q
M
(t) in the
ideal charge-controlled memristor.
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CORINTO AND FORTI: MEMRISTOR CIRCUITS: FLUX—CHARGE ANALYSIS METHOD 3
Second, (10)—which yields (11)—is derived from the in-
tegration of the Kirchhoff Current Law (KCL) over (t
0
,t),
thus it can be formulated as “the sum of the capacitor incre-
mental charge q
C
(t) − q
C
(t
0
)=C(v
C
(t) − v
C
(t
0
)) and the
memristor incremental charge q
M
(t) − q
M
(t
0
)=f(ϕ
M
(t)) −
f(ϕ
M
(t
0
)) is zero.”
It turns out that the fundamental step in reducing by one
the number of ODEs in the IVP (7), (8) is the integral of the
KCL in (t
0
,t), referred to as KqL. The KqL states that “the
algebraic sum of the incremental charge in a closed surface
is zero” (Charge Conservation Law). With reference to the
M − C circuit in Fig. 1(a), we can rewrite (10) as follows:
q
C
(t; t
0
)+q
M
(t; t
0
)=0 (12)
where q
C
(t; t
0
)=q
C
(t) − q
C
(t
0
) and q
M
(t; t
0
)=q
M
(t) −
q
M
(t
0
) are the capacitor and memristor incremental charges,
respectively.
By duality we can also introduce the integral of the Kirchhoff
Voltage Law (KVL) in (t
0
,t), referred to as KϕL, in terms of
the incremental fluxes (Flux Conservation Law).
In conclusion, the illustrative example makes clear that the
pillars of the new analysis method are:
• the use of KqLandKϕL in terms of the incremental flux
and charge
ϕ
k
(t; t
0
)˙= ϕ
k
(t) − ϕ
k
(t
0
)=
t
t
0
v
k
(τ)dτ (13)
q
k
(t; t
0
)˙= q
k
(t) − q
k
(t
0
)=
t
t
0
i
k
(τ)dτ (14)
for any t≥t
0
,wherev
k
(t) is the voltage across and i
k
(t)
is current through any two-terminal element in LM,
respectively;
• the use of Constitutive Relations (CRs) expressed ac-
cording to the same electrical variables, that is, CRs of
resistors, capacitors, inductors, and memristors specified
in terms of incremental charges and fluxes.
The main advantage of the proposed method is that it enables
to describe memristor-based circuits in the class LM by means
of IVPs for a reduced number of ODEs compared to current ap-
proaches available in literature (e.g., [8] and [15]). This permits
to simplify the investigation of nonlinear dynamic behavior and
bifurcation phenomena in memristor circuits and to make clear
the influence of initial conditions. Section IV will present the
application of the proposed flux-charge analysis method for a
comprehensive study of nonlinear dynamics and bifurcations in
the M − C circuit of Fig. 1(a).
The key idea that allows us to develop the method is the
conceptual difference between charge and flux defined in (2)
and (3) and incremental charge and flux given in (13) and (14).
The following property makes clear such concept.
Property 1: The incremental charge and flux in (13) and (14)
reduce to the charge and flux in (2) and (3) if and only if t
0
→
−∞, i.e., the circuit topology is invariant for any t∈(−∞, +∞).
Property 1 follows directly from the definition of ϕ
k
(t, t
0
)
and q
k
(t, t
0
). On the other hand, the past dynamics over
(−∞,t
0
) has to be considered in order to set independent initial
conditions of a circuit switching its topology at the finite instant
Fig. 2. Two-port network where the charge generator q(t
0
) and the flux
generator ϕ(t
0
) take into account the initial conditions at a finite instant t
0
.
t
0
. The two-port network in Fig. 2 provides a symbolic circuit
representation of (13) and (14) including a charge generator
q(t
0
) and a flux generator ϕ(t
0
) to take into account initial
conditions for charge and flux at t
0
.
The following notation is used henceforth to denote electrical
variables and parameters of any circuit in LM. We assume the
circuit has fixed topology for t ≥ t
0
and is made of:
• n
C
capacitors C
j
(j =1,...,n
C
). The capacitor voltages
v
C
j
(t) and currents i
C
j
(t) are organized in the vectors
v
C
(t)=
v
C
1
(t),...,v
C
n
C
(t)
i
C
(t)=
i
C
1
(t),...,i
C
n
C
(t)
.
The corresponding incremental flux and charge vectors
for t ≥ t
0
are obtained by means of (13) and (14), that is,
ϕ
C
(t; t
0
)=ϕ
C
(t) − ϕ
C
(t
0
) and q
C
(t; t
0
)=q
C
(t) −
q
C
(t
0
),where
ϕ
C
(t)=
ϕ
C
1
(t),...,ϕ
C
n
C
(t)
q
C
(t)=
q
C
1
(t),...,q
C
n
C
(t)
.
• n
L
inductors L
m
(m =1,...,n
L
). The inductor voltages
v
L
m
(t) and current i
L
m
(t) are organized in the vectors
v
L
(t)=
v
L
1
(t),...,v
L
n
L
(t)
i
L
=
i
L
1
(t),...,i
L
n
L
(t)
.
The corresponding incremental flux and charge vec-
tors for t ≥ t
0
are ϕ
L
(t; t
0
)=ϕ
L
(t) − ϕ
L
(t
0
) and
q
L
(t; t
0
)=q
L
(t) − q
L
(t
0
),where
ϕ
L
(t)=
ϕ
L
1
(t),...,ϕ
L
n
L
(t)
q
L
(t)=
q
L
1
(t),...,q
L
n
L
(t)
.
• n
R
ideal resistors R
s
(s =1,...,n
R
). The resistor volt-
ages v
R
s
(t) and current i
R
s
(t) are organized as
v
R
(t)=
v
R
1
(t),...,v
R
n
R
(t)
i
R
=
i
R
1
(t),...,i
R
n
R
(t)
.
The corresponding incremental flux and charge vectors for
t ≥ t
0
are ϕ
R
(t; t
0
)=ϕ
R
(t) − ϕ
R
(t
0
) and q
R
(t; t
0
)=
q
R
(t) − q
R
(t
0
),where
ϕ
R
(t)=
ϕ
R
1
(t),...,ϕ
R
n
R
(t)
q
R
(t)=
q
R
1
(t),...,q
R
n
R
(t)
.
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4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS
• n
M
memristors M
p
(p =1,...,n
M
). The memristor
voltages v
M
p
(t) and currents i
M
p
(t) are organized as
v
M
(t)=
v
M
1
(t),...,v
M
n
M
(t)
i
M
=
i
M
1
(t),...,i
M
n
M
(t)
.
The corresponding incremental flux and charge vec-
tors for t ≥ t
0
are ϕ
M
(t; t
0
)=ϕ
M
(t) − ϕ
M
(t
0
) and
q
M
(t; t
0
)=q
M
(t) − q
M
(t
0
),where
ϕ
M
(t)=
ϕ
M
1
(t),...,ϕ
M
n
M
(t)
q
M
(t)=
q
M
1
(t),...,q
M
n
M
(t)
.
• n
E
ideal independent voltage sources e
w
(t)(w =1,...,
n
E
). The voltages e
w
(t) are organized in a vector e(t)=
(e
1
(t),...,e
n
E
(t)). The corresponding incremental flux
vector for t ≥ t
0
is
ϕ
e
(t; t
0
)=
ϕ
e
1
(t; t
0
),...,ϕ
e
n
E
(t; t
0
)
.
• n
A
ideal independent current sources a
z
(t)(z =1,...,
n
A
). The currents a
z
(t) are organized in a vector
a(t)=(a
1
(t),...,a
n
A
(t)). The corresponding incre-
mental charge vector for t ≥ t
0
is
q
a
(t; t
0
)=
q
a
1
(t; t
0
),...,q
a
n
A
(t; t
0
)
.
• = n
C
+ n
L
+ n
R
+ n
M
+ n
E
+ n
A
is the number of
two-terminal elements (k =1,...,) composing any
circuit in LM. It follows that we can arrange v =
(v
1
,...,v
)
, i =(i
1
,...,i
)
, ϕ =(ϕ
1
,...,ϕ
)
and
q =(q
1
,...,q
)
as follows:
v(t)=(v
C
(t), v
L
(t), v
R
(t), v
M
(t), v
E
(t), v
A
(t))
i(t)=(i
C
(t), i
L
(t), i
R
(t), i
M
(t), i
E
(t), i
A
(t))
ϕ(t)=(ϕ
C
(t), ϕ
L
(t), ϕ
R
(t), ϕ
M
(t), ϕ
E
(t), ϕ
A
(t))
q(t)=(q
C
(t), q
L
(t), q
R
(t), q
M
(t), q
E
(t), q
A
(t))
where (·)
denotes the transpose vector.
• n is the number of nodes in any circuit in LM.
II. A
NALYSIS METHOD IN THE (ϕ, q)-DOMAIN
We consider in what follows a circuit in the class LM of
nonlinear memristor circuits defined in the previous section and
suppose it has a fixed topology for t ≥ t
0
,where−∞ <t
0
<
+∞. Our goal is to show that we can develop a method enabling
to analyze the circuit dynamics for t ≥ t
0
in the (ϕ, q)-domain,
i.e., we can obtain circuit equations describing the dynamics no
longer using the current and voltage variables. To this end we
will address the following main issues:
• how to write Kirchhoff laws using the incremental charge
and flux;
• how to write the CR of each element in the class LM and
IVP for a system of ODEs in the (ϕ, q)-domain.
A. Incremental Kirchhoff Laws
Since is the number of two-terminal elements in LM,andn
is the number of nodes, we can write n − 1 fundamental cutset
equations in the form Ai(t)=0 (A ∈ R
(n−1)×
is the reduced
incidence matrix) and − n +1 fundamental loop equations
Bv(t)=0 (B ∈ R
(−n+1)×
is the reduced loop matrix). By
integrating between t
0
and t ≥ t
0
we obtain
Aq(t)=Aq(t
0
) (15)
Bϕ(t)=Bϕ(t
0
). (16)
These equations are expressed in the (ϕ, q)-domain and
involve all the initial conditions q(t
0
), ϕ(t
0
). It is important
to stress the following key points. Among q(t
0
), ϕ(t
0
), the ini-
tial conditions q
C
k
(t
0
)=C
k
v
C
k
(t
0
) and ϕ
L
k
(t
0
)=L
k
i
L
k
(t
0
)
can be obtained for any circuit in LM by the measurement
at the instant t
0
of voltages v
C
k
(t
0
) across capacitors and
currents i
L
k
(t
0
) through inductors by means of a voltmeter or
an ammeter. Instead, q
L
k
(t
0
)=
t
0
−∞
i
L
k
(t)dt and ϕ
C
k
(t
0
)=
t
0
−∞
v
C
k
(t)dt cannot be obtained via measurements at t
0
(see
also footnote 11 in [16]). Rather, to evaluate q
L
k
(t
0
) and
ϕ
C
k
(t
0
) we would require the specific knowledge of the past
circuit history for t<t
0
[see for example Fig. 1(b)], an in-
formation that is usually unavailable or difficult to obtain in
practice. In addition, it may happen that q
a
(t
0
)(ϕ
e
(t
0
)) are
not finite for some current (voltage) ideal generators.
4
The Kirchhoff Laws can be made independent of initial
conditions by means of the incremental fluxes and charges
defined in (13) and (14). Indeed, using the incremental charge
q(t, t
0
), the Kirchhoff Charge Law (KqL) takes the simpler
form
Aq(t; t
0
)=0 (17)
which no longer involves the initial charges q(t
0
),butjust
expresses the constraints on the incremental charges due to the
topology.
Similarly, the Kirchhoff Flux Law (KϕL) using the incre-
mental flux ϕ(t, t
0
) can be written as
Bϕ(t; t
0
)=0. (18)
It is known that these equations give in overall indepen-
dent topological constraints on q(t; t
0
), ϕ(t; t
0
) in the (ϕ, q)-
domain.
The necessity to introduce Kirchhoff Laws independent of
initial conditions in turn implies that all circuit elements have
to be described by means of incremental fluxes and charges at
their terminals. The CRs of circuits elements in terms of the
incremental flux and charge at their terminals are given in the
next section.
Remark 1: The common assumption made in the literature to
ensure that Kirchhoff Laws result to be independent of initial
conditions is that q(t
0
)=0and ϕ(t
0
)=0. This assumption is
not true in general as shown in the circuit of Fig. 1(b).
4
It is readily derived that if a(t)=A (e(t)=E) for all t ∈ (−∞,t
0
], with
A ∈ R (E ∈ R) constant, then q
a
(t
0
)=
t
0
−∞
Adτ (ϕ
e
(t
0
)=
t
0
−∞
Edτ)
tends to infinity.
