A geometric modelling of nonlinear RLC networks

Delia Ionescu ∗

Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700, Bucharest, Romania

A new approach to nonlinear RLC circuits, which is based on the geometric Birkhoffian formalism, is described in this note.The configuration space and a special Pfaffian form, called Birkhoffian, are obtained from the constitutive relations of theinvolved resistors, inductors and capacitors and from Kirchhoff’s laws. No assumptions are placed upon the topology of thenetwork. Properties of the corresponding Birkhoffian such as its regularity, or its dissipativeness, are discussed in this context.

1. Introduction The mathematical modelling of nonlinear circuits has a long and rich history (see, for example, [2], [3], [8],[9], [10] and the references therein). For the dynamics of these networks, a large variety of Euler-Lagrangian or Hamiltonianformulations, which differ in the choice of variables and of Lagrangian or Hamiltonian functions, and in the underlyinggeometric structure (Riemannian or symplectic or Poisson or Dirac) of the state space, have been considered. An alternativeapproach to the study of dynamical systems is the Birkhoffian formalism, a global formalism of implicit systems of secondorder ordinary differential equations on a manifold. The Birkhoffian formalism has the ability to deal naturally with the studyof the dynamics of LC (see [6]) and RLC circuits (see [7]); in particular, it includes networks which contain closed loopsformed by capacitors, as well as inductor cutsets. In section 2 of the note at hand we describe the Birkhoffian formulationof the dynamic equations of a nonlinear RLC circuit, with no assumptions placed on its topology. The regularity and thedissipativeness of the corresponding Birkhoffian, are discussed.

2. Birkhoffian formulation of the dynamics of RLC circuits A simple electrical circuit provides us with an orientedconnected graph. The graph will be assumed to be planar. Let b be the total number of branches in the graph, n be one lessthan the number of nodes and m be the cardinality of a selection of loops that cover the whole graph. By Euler’s polyhedronformula, b = m + n. We choose a reference node and a current direction in each l-branch of the graph, l = 1, ..., b. Wealso consider a covering of the graph with m loops, and a current direction in each j-loop, j = 1, ..., m. We assume thatthe associated graph has at least one loop, meaning that m > 0. An oriented connected graph can be described by matriceswhich contain only 0, ±1, these are: the incidence matrix B ∈ Mbn(R), rank(B) = n, and the loop matrix A ∈ Mbm(R),rank(A) = m. For the fundamentals of electrical circuit theory, see, for example, [4].

Let us now consider an RLC electrical circuit consisting of r resistors, k inductors and p capacitors, such that to eachbranch of the associated graph it corresponds just one electrical device, that is, b = r + k + p. Using the matrices A and B,Kirchhoff’s current law and Kirchhoff’s voltage law can be expressed by the equations

BT I = 0 (KCL), AT v = 0 (KV L) (1)

where I = (IΓ, Ia, Iα) ∈ Rr × Rk × Rp � Rb is the current vector and v = (vΓ, va, vα) ∈ Rr × Rk × Rp � Rb is thevoltage drop vector.Tellegen’s theorem establishes a relation between the matrices AT and BT : the kernel of the matrix BT is orthogonal to thekernel of the matrix AT (see e.g., [3] page 5).

We consider the voltage-current laws for nonlinear devices given by

vΓ = RΓ(IΓ), Γ = 1, ..., r, va = La(Ia)dIa

dt, a = r + 1, ..., r + k, vα = Cα(Qα), α = r + k + 1, ..., b,

(2)

RΓ, La, Cα : R −→ R\{0} being smooth functions, Qα denote the charges of the capacitors, with Iα = dQα

dt .Summing up, the equations governing the network are⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

BT

⎛⎝

IΓIa

dQα

dt

⎞⎠ = 0

AT

⎛⎝

RΓ(IΓ)La(Ia) dIa

dtCα(Qα)

⎞⎠ = 0

(3)

∗ Delia Ionescu: e-mail: Delia.Ionescu@imar.ro , Phone: +00 40 21 3196506, Fax: +00 40 21 3196505

PAMM · Proc. Appl. Math. Mech. 6, 813–814 (2006) / DOI 10.1002/pamm.200610386

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The classical book by Birkhoff [1], contains in Chapter I many interesting ideas about classical dynamics from the view-point of differential geometry. In order to present these ideas in a coordinate free fashion, one considers the formalism of 2-jets(see [5]). The space of configurations M is a smooth m-dimensional differentiable connected manifold and the covariant char-acter of the Birkhoff generalized forces is obtained by introducing the notion of elementary work, called Birkhoffian, a specialPfaffian form ω defined on the 2-jets manifold J2(M). A local coordinate system (q) on M induces natural local coordinates(q, q, q) on J2(M). Locally, a Birkhoffian ω is given by ω = Qj(q, q, q)dqj with certain functions Qj : J2(M) → R. Thedynamical system associated to ω is the subset of the 2-jets manifold given by the following implicit second order ODE’s:

Qj(q, q, q) = 0, for all j = 1, ..., m. (4)

The notion of Birkhoffian allows to formulate the concepts of reciprocity, regularity, affine structure in the accelerations,conservativeness, dissipativeness, in an intrinsic way (see [5], [6], [7]).

Using the first set of equations (3), we are going to define a family of m-dimensional affine-linear configuration spacesMc ⊂ Rb, parameterized by a constant vector c in Rn which corresponds to initial values of certain state variables of thecircuit. A Birkhoffian ωc on the configuration space Mc arises from a linear combination of the second set of equations (3).

Since the matrix B is constant, if we integrate the first set of equations (3), we get BT x = c, with I = x, c a constant vectorin Rn. We define

Mc := {x ∈ Rb|BT x = c} (5)

Mc is an affine-linear subspace in Rb, its dimension is m = b − n, because rank(B) = n.We denote local coordinates on Mc by q = (q1, .., qm). Solving the system in (5), we express any of the x-variables in termsof q-s, namely, x = N q + K, where N ∈ Mbm(R) and K ∈ Rb. Differentiating the last relation we get x = N q.Using Tellegen’s theorem and a fundamental theorem of linear algebra, we get that the kernel of the matrix AT is equal to thekernel of the matrix N T . Then, we define a Birkhoffian ωc of Mc such that the differential system (4) is the linear combinationof the second set of equations in (3) obtained by replacing AT with the matrix N T .In the natural coordinate system (q, q, q) on J2(Mc), we get

ωc = Qj(q, q, q)dqj = [Fj(q)q + Hj(q) + Gj(q)] dqj (6)

with Fj , Hj and Gj smooth functions given by Fj(q)q =∑k+r

a=r+1

∑mi=1 N a

j N ai La

(∑ml=1 N a

l ql)qi,

Hj(q) =∑r

Γ=1 NΓj RΓ

(∑ml=1 NΓ

l ql), Gj(q) =

∑bα=r+k+1 N α

j Cα

(∑ml=1 Nα

l ql + Kα).

Further, we can discuss in this context the concepts of regularity ([5], [6]), conservativeness ([5], [6]), dissipativeness ([7]).

A Birkhoffian ω is regular if det[

∂Qj

∂qi (q, q, q)]

i,j=1,...,m�= 0 and there exists (q, q, q) ∈ J2(M) such that Qj(q, q, q) =

0, j = 1, ..., m.A Birkhoffian ω is conservative if there exists a smooth function Eω : TM → R such that Qj(q q, q)qj = ∂Eω

∂qj qj + ∂Eω

∂qj qj .

A Birkhoffian ω is dissipative if there exists a smooth function E0ω : TM → R and a dissipative 1-form D on TM ,that is, a vertical 1-form on TM locally given by D = Dj(q, q)dqj , with Dj(q, q)qj > 0, such that Qj(q q, q)qj =∂E0ω

∂qj qj + ∂E0ω

∂qj qj + Dj(q, q)qj . We prove that• For an RLC eletric network with properly described nonlinear resistors, nonlinear inductors and capacitors, each Birkhoffian of the family(here referred as c-Birkhoffian) is not conservative; in fact it is dissipative.• The ”restricted classes” of networks involving capacitors loops and inductor cut-sets are captured precisely by the general Birkhoffianformalism. If there exists in the network some loop which contains only capacitors or only resistors, or only resistors and capacitors, thec-Birkhoffian is never regular.• If there exists in the network some loop which contains only capacitors we can give a procedure by means of which the original configu-ration space can be reduced to a lower dimensional one, thereby regularizing the Birkhoffian.

Acknowledgements The participation at the conference was partially supported by IMAR through the contract of excellency CEx05-D11-23/2005.

References[1] G. D. Birkhoff, Dynamical Systems, American Mathematical Society Colloquium Publications, Vol. IX, New York, (1927).[2] A.M. Bloch and P.E. Crouch in: Differential Geometry and Control, Proceedings of Symposia in Pure Mathematics 64, 103-117,

American Mathematical Society, (1999).[3] R.K. Brayton and J.K. Moser, Quarterly of Applied Mathematics 22, I (1–33), II (81–104) (1964).[4] L. O. Chua, C. A. Desoer, D. A. Kuh, Linear and Nonlinear Circuits, (McGraw-Hill Inc., 1987).[5] M.H. Kobayashi and W.M. Oliva, Resenhas IME-USP 6 (1), 1–71 (2003).[6] D. Ionescu, J. Scheurle, Z. Angew. Math. Phys., submitted for publication, 1–31 (2004).[7] D. Ionescu, J. Geom. Phys., to appear, 1–28 (2006).[8] B.M. Maschke and A.J. van der Schaft, Archiv fur Elektronik und Ubertragungstechnik 49, 362–371 (1995).[9] A.J. van der Schaft, Rep. Math. Phys. 41, 203–221 (1998).

[10] S. Smale, J. Differential Geometry 7, 193–210 (1972).

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Section 20 814