Memory Elements: a Paradigm Shift in Lagrangian Modeling of Electrical
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Memory Elements: A Paradigm Shift in Lagrangian Modeling of Electrical Circuits Dimitri Jeltsema Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands (Email: [email protected]) Abstract: Meminductors and memcapacitors do not allow a Lagrangian formulation in the classical sense since these elements are nonconservative in nature and the associated energies are not state functions. To circumvent this problem, a different configuration space is considered that, instead of the usual loop charges, consist of time-integrated loop charges. As a result, the corresponding Euler-Lagrange equations provide a set of integrated Kirchhoff voltage laws in terms of the element fluxes. Memristive losses can be included via a second scalar function that has the dimension of action. A dual variational principle follows by considering variations of the integrated node fluxes and yields a set of integrated Kirchhoff current laws in terms of the element charges. Although time-integrated charge is a somewhat unusual quantity in circuit theory, it may be considered as the electrical analogue of a mechanical quantity called absement. Based on this analogy, simple mechanical devices are presented that can serve as didactic examples to explain memristive, meminductive, and memcapacitive behavior. Keywords: Memristor, Meminductor, Memcapacitor, Memory elements, Lagrangian modeling. 1. INTRODUCTION The remainder of the paper is organized as follows. In Section 2, the mathematical properties of the memris- Memristors, meminductors, and memcapacitors constitu- tor, meminductor, and memcapacitor are discussed. The tive an increasingly important class of two-terminal circuit structure of the Lagrangian equations for a large class of elements whose resistance, inductance, and capacitance re- conventional circuits is briefly reviewed in Section 3 and it tain memory of the past states through which the elements will be shown that meminductors and memcapacitors do have evolved. All three elements are nonlinear and can be not fit into this framework a priori since these elements identified by their pinched hysteresis loop in the voltage are nonconservative in nature and the associated energies versus current plane, current versus flux plane, and voltage are not state functions. To circumvent this problem, in versus charge plane, respectively. While there are many Section 4 a different configuration space is considered that, discovered experimental realizations and applications of instead of the usual loop charges, consist of time-integrated systems that exhibit memristive behavior, ranging from loop charges. The Lagrangian is defined by the difference applications in non-volatile nano memory to intelligent arXiv:1201.1032v2 [math.DS] 1 May 2012 between two novel state functions in a fashion similar machines with learning and adaptive capabilities, the num- to the usual magnetic co-energy minus electric energy ber of systems showing memcapacitive and meminductive setup, but having the dimensions of energy times time- behavior is still somewhat limited. Nevertheless, several squared which, in turn, is equivalent to action times time. applications of these concepts are foreseen in the field As a result, the corresponding Euler-Lagrange equations of logic and arithmetic operations using memristive and provide a set of integrated Kirchhoff voltage laws (iKVL’s) memcapacitive devices, and field-programmable quantum in terms of the element fluxes. Memristive and resistive computation using meminductive and memcapacitive de- losses can be included via the introduction of a second vices. See Di Ventra et al. (2009) for an introduction into scalar function that has the dimension of action. Further- the memory elements, and Pershin and Di Ventra (2011) more, a dual variational principle follows by considering for a fairly complete overview of the current state-of-the- variations of the integrated node fluxes and yields a set art and an extensive list of references. of integrated Kirchhoff current laws (iKCL’s) in terms In the past century, a significant amount of research has of the element charges. Although time-integrated charge, been devoted to the description and analysis of conven- which in SI units is measured in Coulomb times seconds, tional electrical circuits using Lagrangian and Hamilto- is a somewhat unusual quantity in circuit theory, it may nian methods; see Jeltsema and Scherpen (2009) and the be considered as the electrical analogue of a mechanical references therein. Since these classical frameworks are quantity called absement. Based on this analogy, in Sec- important not just for their broad range of applications, tion 5, simple mechanical devices are presented that can but also for their role in advancing deep understanding of serve as didactic examples to explain meminductive and physics, the next step is to consider circuits made from memcapacitive behavior. Finally, the paper is concluded memristors, meminductors, and memcapacitors. with some final remarks in Section 6. 2. DEFINITION OF MEMORY ELEMENTS ation with respect to time, yields q = CM (ϕ)V, (7) From a mathematical perspective, the behavior of a two- where C (ϕ) := dσ/dϕ denotes the incremental capaci- terminal resistor, inductor, and capacitor, whether linear M tance. The memory aspect of a flux-modulated memcapac- or nonlinear, is described by a relationship between two itor stems from the fact that it ‘remembers’ the amount of the four basic electrical variables, namely, voltage V , of voltage that has been applied to it. current I, charge q, and flux ϕ, where t A meminductor (resp. memcapacitor) that depends on q(t)= I(τ)dτ, (1) the history of its flux (resp. charge) can be formulated Z−∞ t by starting from a constitutive relationship of the form ϕ(t)= V (τ)dτ. (2) q =q ˆ(ρ) (resp. ϕ =ϕ ˆ(σ)). Z−∞ In the special case that the constitutive relationship of A resistor is described by a constitutive relationship be- a memristor is linear, a memristor becomes an ordinary tween current and voltage; a capacitor by that of voltage linear resistor. Indeed, in such case (3) reduces to ϕ = Rq, and charge; and an inductor by that of current and flux with constant memristance M (the slope of the line), or linkage. equivalently, V = RI, which precisely equals Ohms law. Based on logical and symmetry reasonings, Chua (1971) Hence it is not possible to distinguish a two-terminal linear postulated the existence of a fourth element that is char- memristor from a two-terminal linear resistor. The same acterized by a constitutive relationship between the re- holds for a linear meminductor and linear memcapacitor, maining two variables, namely charge and flux. This el- where (6) reduces to ρ = Lq, or equivalently, ϕ = LI, and ement was coined memristor (a contraction of memory (7) to σ = Cϕ, or equivalently, q = CV , respectively. and resistor) referring to a resistor with memory. The memory aspect stems from the fact that a memristor ‘re- 3. SELF-ADJOINTNESS OF CIRCUIT DYNAMICS members’ the amount of current that has passed through it together with the total applied voltage. More specifically, The dynamical behavior of any electrical circuit consisting if q denotes the charge and ϕ denotes the flux, then a of conventional, possibly nonlinear, resistors, inductors, charge-modulated memristor is defined by the constitutive and capacitors is basically determined by three types relationship ϕ =ϕ ˆ(q). Since flux is defined by the time of equations: those arising from Kirchhoff’s voltage law integral of voltage V (like in Faraday’s law), and charge is (KVL), those arising from Kirchhoff’s current law (KCL), and the constitutive relationships of the elements. In many the time integral of current I, or equivalently, V =ϕ ˙ and 1 I =q ˙, we obtain cases this leads to differential equations of the form j V = RM (q)I, (3) Aij (x ˙)¨x + Bi(x, x˙)=0, (8) where RM (q) := dϕ/dq is the incremental memristance. where i, j = 1,...,n, and x ∈ Rn represents a column Note that (3) is the definition of a memristor in impedance vector of loop charges and node fluxes. form. The admittance form I = GM (ϕ)V , with incremen- The system of differential equations (8) allow a Lagrangian tal memductance GM (ϕ) := dq/dϕ, is obtained by starting description if we can find a Lagrangian L(x, x˙) satisfying from a constitutive relationship q =q ˆ(ϕ). d ∂L ∂L − ≡ A (x ˙)¨xj + B (x, x˙). (9) In addition to the memristor, it is shown in Di Ventra dt ∂x˙ i ∂xi ij i et al. (2009) that the memory-effect can be associated to In mechanics it is known that the existence of a Lagrangian inductors and capacitors as well. For that, let σ and ρ L(x, x˙) relies on the fact that the system of differential denote the time-integrals of charge and flux, equations (8) is self-adjoint, which for the present form is t tantamount to the following set of integrability conditions σ(t)= q(τ)dτ, (4) (Santilli, 1978): Z−∞ t Aij = Aji, (10) ρ(t)= ϕ(τ)dτ, (5) Z−∞ ∂Aik ∂Ajk = , (11) respectively. Then, a memory inductor, or meminductor ∂x˙ j ∂x˙ i for short, is a two-terminal element defined by a constitu- ∂B ∂B 1 ∂ ∂B ∂B i − j = i − j x˙ k, (12) tive relation ρ =ρ ˆ(q). Indeed, differentiation of the latter ∂xj ∂xi 2 ∂xk ∂x˙ j ∂x˙ i with respect to time yields ∂Bi ∂Bj ϕ = LM (q)I. (6) + =0. (13) ∂x˙ j ∂x˙ i Since LM (q) := dρ/dq relates flux with current, its values clearly correspond to the units of inductance. Bearing 3.1 Conventional Conservative Circuits in mind that charge is the time integral of current, the memory aspect of a charge-modulated meminductor stems From a Lagrangian perspective, it is well-known that from the fact that it ‘remembers’ the amount of current for a large class of circuits that are made from current- that has passed through it.