Constitutive relations of three fundamental circuit elements: If w and q
are basic and independent physical attributes, the constitutive relations
of resistor R, inductor L and capacitor C can be rewritten as follows:
Is memristor a dynamic element?
B.C. Bao, Z. Liu and H. Leung
resistor: w(t) = Rq(t)
Recently, it was found that flux w and charge q are a pair of complementary basic physical attributes generating elementary circuit
elements. With such a finding, two basic electrical circuit laws and
constitutive relations of three fundamental circuit elements are
rebuilt, upon which new dynamical equations describing dynamical
circuits containing memristors are established. This leads to investigation of whether the memristor is a dynamic element.
Introduction: It is well known that there are three fundamental circuit
elements: resistor, inductor and capacitor. By considering four circuit
variables, electric current i, voltage v, charge q and magnetic flux w,
Chua in 1971 [1] defined the fourth circuit element, the memristor,
which was then implemented by HP Labs in 2008 [2]. The implementation triggered a wide interest in memristors, including dynamic
chaotic memristive circuits. It is generally thought that the chaotic
circuits containing the memristor introduce an additional circuit dimension, implying that the memristor is a dynamic element [3, 4].
However, recently, Wang [5] thought that magnetic flux w and charge
q are two basic physical attributes that generate elementary circuit
elements. In his work, the definition of a two-terminal circuit element
is given as: ‘an elementary electronic circuit element should link two
physical attributes, at least one of which should be basic’. For this
reason, the resistor and the memristor belong to the basic element
and the memclass linking w and q with a unit of Ω or Ohm, the inductor
inductor belong to the basic element class linking w and q with a unit
of H or Henry and the capacitor
and the memcapacitor belong to the
basic element class linking q and w with a unit of F or Farad. Under
this premise, we rethink Kirchhoff’s circuit laws and the constitutive
relations of three classic fundamental circuit elements, on which new
dynamic equations of chaotic memristive circuits can be established.
It is found that the memristor is not a dynamic element.
Two new electrical circuit laws: For a lumped parameter circuit, an
elementary electronic circuit element characterised by two basic physical attributes is shown in Fig. 1. Conveniently, the reference directions
of flux w and charge q marked in Fig. 1 are the associated reference
directions.
inductor: w(t) = Li(t) = L
(3)
dq
dt
(4)
dw
dt
(5)
capacitor: q(t) = Cv(t) = C
where i(t) = dq/dt and v(t) = dw/dt.
For these new constitutive relations, q is the state variable of inductor
L and w is the state variable of capacitor C. In addition, (3)–(5) show that
the inductor L and the capacitor C are dynamic elements, whereas the
resistor R is not a dynamic element. In a circuit consisting of these
three fundamental circuit elements, the order of the circuit is determined
by the number of two dynamic elements of inductor and capacitor.
For a charge-controlled (or flux-controlled) memristor, its constitutive
relation can be described as
w(t) = f (q) or q(t) = g(w)
(6)
where f (·) and g(·) are two nonlinearities. Equation (6) indicates that the
memristor and the resistor both belong to the same basic element class.
Dynamic equations of dynamic circuits containing memristors: A
memristor-based oscillator shown in Fig. 2 is used as an example to
illustrate that the memristor cannot be a dynamic element [3]. The memristor is flux-controlled and characterised by a smooth continuous cubic
nonlinearity
q(w) = aw + bw3
(7)
where a, b > 0.
i3
–G
C2
+i
L
+
+
v2
–
v1
–
C1
v
M
–
Fig. 2 Canonical Chua’s oscillator containing memristor
element
q
a
+
j
–
b
Fig. 1 Circuit element and associated reference direction
Kirchhoff’s current law (KCL) states that the algebraic sum of all the
currents entering and leaving a node must be equal to zero. Integrating
the identity relation of KCL with respect to time t yields
N
qn = 0
(1)
n=1
where N is the number of branches connected to the node and qn is the
nth charge entering (or leaving) the node.
Equation (1) is commonly known as the conservation of charge,
which states that the algebraic sum of all the electric charges entering
a node (or a closed boundary) is zero. For this law, charges entering a
node may be regarded as positive, whereas charges leaving the node
may be taken as negative or vice versa.
Kirchhoff’s voltage law (KVL) states that the algebraic sum of all the
voltages drops around a loop must be equal to zero. This idea by
Kirchhoff is known as the conservation of energy. Integrating the
identity relation of KVL with respect to time t yields
M
wm = 0
(2)
m=1
where M is the number of fluxes in the loop (or the number of branches
in the loop) and wm is the mth flux.
Equation (2) states that the algebraic sum of all the fluxes around a
closed path (or loop) in a circuit is zero.
Techset CompositionLtd, Salisbury
Consider four state variables in Fig. 2, capacitor C1 voltage v1, capacitor C2 voltage v2, inductor L current i3 and memristor M flux w. By
letting x = v1, y = v2, z = i3, w = w, α = 1/C1, β = 1/C2, γ = G/C2 and L =
1, the dynamic equations of Fig. 2 can be expressed as
⎧
ẋ = a(z − W (w)x)
⎪
⎪
⎨
ẏ = gy − bz
(8)
ż = −x + y
⎪
⎪
⎩
ẇ = x
where W(w) = dq(w)/dw = a + 3bw 2 and the memristor is regarded as a
dynamic element.
System (8) is a four-dimensional (4D) system, which indicates that the
memristor-based canonical Chua oscillator is a fourth-order circuit. The
equilibrium points of (8) are infinite and given by set A = {(x, y, z, w)|x =
y = z = 0, w = c} (c is a real constant) [3]. The characteristic equation at
any equilibrium point has a zero eigenvalue. Hence, system (8) is a particular dynamical system and different from the conventional dynamical
systems [3].
If all the circuit elements in Fig. 2 are described by flux w and charge
q, there exist the following relations by (1) and (2):
q + q 1 − q3 = 0
(9)
q2 − Gw2 + q3 = 0
(10)
w1 − w2 + w3 = 0
(11)
where q, q1, q2 and q3 stand for the charges of M, C1, C2 and L, respectively, w1, w2 and w3 represent the fluxes of C1, C2 and L, respectively,
and q = aw + bw3 = aw1 + bw31 .
Substitute (4) and (5) into (9)–(11). For the state variables of capacitor
C1 flux w1, capacitor C2 flux w2 and inductor L charge q3, we have a set
Doc: {EL}ISSUE/49-24/Pagination/EL20132788.3d
Circuits and systems
of three first-order differential equations described by
⎧
dw1
⎪
⎪
= −aw1 − bw31 + q3
C
⎪
⎪ 1 dt
⎪
⎨
dw2
C
= Gw2 − q3
⎪ 2 dt
⎪
⎪
⎪
⎪
⎩ L dq3 = −w + w
1
2
dt
fundamental circuit variables, the memristor is not a dynamic element
and the system model of the memristor-based circuit is 3D and conventional. Furthermore, this example verifies that flux w and charge q are a
pair of complementary basic physical attributes.
(12)
Let x = w1, y = w2, z = q3, α = 1/C1, β = 1/C2, γ = G/C2 and L = 1.
Equation (12) can then be rewritten as
⎧
⎨ ẋ = a(z − ax − bx3 )
(13)
ẏ = gy − bz
⎩
ż = −x + y
System (13) is a 3D system, implying that the memristor-based canonical Chua oscillator is a third-order circuit. The equilibrium points of
(13) are finite and determined by
S0 = (0, 0, 0)
g
a
g
a
g g
a
S1, 2 = +
− , +
− , +
−
bb
bb
b
b
b bb
b
and the eigenvalues of the corresponding characteristic equations are
nonzero. Thus, system (13) is a conventional dynamical system and
the memristor is not a dynamic element.
It is remarkable that on modelling the memristor-based canonical
Chua’s oscillator by utilising flux w and charge q, the dimensionality
of the corresponding dynamical system reduces and the memristor
behaves like a special nonlinear resistor.
Conclusion: By the example of the memristor-based canonical Chua
oscillator, it can be found that when we consider i, v, q and w to be
four fundamental circuit variables, the memristor is a dynamic
element and the system model of the memristor-based circuit is 4D
and particular, whereas when we consider q and w to be two
Acknowledgments: This work was supported by the National Natural
Science Foundation of China (grant 51277017) and the Natural
Science Foundation of Jiangsu Province, China (grant BK20120583).
© The Institution of Engineering and Technology 2013
27 August 2013
doi: 10.1049/el.2013.2788
B.C. Bao (School of Information Science and Engineering, Changzhou
University, Changzhou 213164, Jiangsu, People’s Republic of China)
E-mail: mervinbao@126.com
Z. Liu (Department of Electrical Engineering, Nanjing University of
Science and Technology, Nanjing 210094, Jiangsu, People’s Republic
of China)
H. Leung (Department of Electrical Engineering, University of Calgary,
Calgary, Alberta, Canada T2N 1N4)
References
1 Chua, L.O.: ‘Memristor – the missing circuit elements’, IEEE Trans.
Circuit Theory, 1971, CT-18, (5), pp. 507–519
2 Strukov, D.B., Snider, G.S., Stewart, D.R., and Williams, R.S.: ‘The
missing memristor found’, Nature, 2008, 453, pp. 80–83
3 Bao, B.C., Liu, Z., and Xu, J.P.: ‘Steady periodic memristor oscillator
with transient chaotic behaviours’, Electron. Lett., 2010, 46, (3),
pp. 228–230
4 Iu, H.H.C., Yu, D.S., Fitch, A.L., Streeram, V., and Chen, H.:
‘Controlling chaos in a memristor based circuit using a twin-T notch
filter’, IEEE Trans. Circuits Syst. I, Reg. Pprs., 2011, 58, (6),
pp. 1337–1344
5 Wang, F.: ‘A triangular periodic table of elementary circuit elements’,
IEEE Trans. Circuits Syst. I, Reg. Pprs., 2013, 60, (3), pp. 616–623