Is memristor a dynamic element? - IEEE Xplore

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Nov 21, 2013 - gation of whether the memristor is a dynamic element. Introduction: It is well known that there are three fundamental circuit elements: resistor ...
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.

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 ⎧ x˙ = a(z − W (w)x) ⎪ ⎪ ⎨ y˙ = gy − bz (8) z˙ = −x + y ⎪ ⎪ ⎩ w˙ = 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

ELECTRONICS LETTERS 21st November 2013 Vol. 49 No. 24 pp. 1523–1525

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 ⎧ ⎨ x˙ = a(z − ax − bx3 ) (13) y˙ = gy − bz ⎩ z˙ = −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: [email protected] 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

ELECTRONICS LETTERS 21st November 2013 Vol. 49 No. 24 pp. 1523–1525