Geometric modeling of nonlinear RLC circuits

396 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 2, FEBRUARY 2005

Geometric Modeling of Nonlinear RLC CircuitsGuido Blankenstein

Abstract—In this paper, the dynamics of nonlinear RLC circuitsincluding independent and controlled voltage or current sourcesis described using the Brayton–Moser equations. The under-lying geometric structure is highlighted and it is shown that theBrayton–Moser equations can be written as a dynamical systemwith respect to a noncanonical Dirac structure. The state variablesare inductor currents and capacitor voltages. The formalism canbe extended to include circuits with elements in excess, as wellas general noncomplete circuits. Relations with the Hamiltonianformulation of nonlinear electrical circuits are clearly pointed out.

Index Terms—Brayton–Moser equations, circuit theory, dissipa-tive circuits, excess elements, modeling, noncanonical Dirac struc-tures, noncomplete networks, nonlinear circuits.

I. INTRODUCTION

THE mathematical modeling of nonlinear RLC circuits hasa long and rich history in the study of electrical networks.

A lot of effort has been put into finding suitable Euler–La-grangian or Hamiltonian descriptions of these networks. Thereason stems from the wealth of theoretical investigationsbased on these models conducted in classical mechanics, e.g.,with respect to stability issues. Euler–Lagrangian descriptionsof RLC circuits were obtained in e.g., [1]–[3]. In [1], the La-grangian coordinates were taken as a subset of capacitor fluxesand inductor charges, which might be regarded as unnatural.The Lagrangian had the classical “mechanical” form of elec-tromagnetic coenergy (corresponding to a subset of inductorsand capacitors in the circuit) minus electromagnetic energy(corresponding to the complementary inductors and capaci-tors), plus a suitable cross product term. The disadvantage ofhaving capacitor fluxes and inductor currents was resolved in[2] at the expense of going to a generalized Euler–Lagrangianformalism. Here, the generalized coordinates were defined asa subset of capacitor charges and inductor fluxes. The gener-alized velocities however were not defined as their ordinarytime derivatives, but instead a general differential operatorwas used to relate generalized coordinates and generalizedvelocities. This led to the generalized velocities being a subsetof capacitor voltages and inductor currents, complementary tothe set of generalized coordinates. In [3], an Euler–Lagrangiandescription of complete RLC circuits was given, deduced fromthe Brayton–Moser equations [4]. As in [1], the generalizedcoordinates consisted of (all) capacitor fluxes and inductor

Manuscript received September 5, 2003; revised July 28, 2004. This work wassupported by the European sponsored project GeoPlex under Reference CodeIST-2001-34166. This paper was recommended by Associate Editor M. Green.

The author was with the Department of Mechanical Engineering, KatholiekeUniversiteit Leuven, Leuven B-3001, Belgium. He is now with the Departmentof Electrical Energy, Systems and Automation, Ghent University, B-9052 Zwi-jnaarde, Belgium (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCSI.2004.840481

charges, with the generalized velocities being their ordinarytime derivatives. In all the Euler–Lagrangian representationsmentioned so far, the resistive elements and independent orcontrolled sources were included by adding a generalizedforce vector in the description. Furthermore, the canonicalHamiltonian formulation of the network was obtained out ofthe Euler–Lagrangian formulation by applying the standardLegendre transform. The Hamiltonian represents the totalelectromagnetic energy of all inductors and capacitors in thecircuit. A drawback of the proposed descriptions is that theyare generally nonminimal since their order is necessarily evenwhile the order of the network may be odd. This is translatedinto the need for additional constraint equations or unspecifiedparameters in the Lagrangian [1], and additional compatibilityequations and the introduction of artificial “quasi-coordinates”[2]. The order of the representation proposed in [3] is in facttwice the order of the network, and the constitutive relationshave to be reused in order to return to a minimal representation.

Adecisivesteptowardagenerallyvalidminimalrepresentationwas taken in [5], [6] where nonlinear LC circuits were describedby applying a noncanonical Hamiltonian formulation deduceddirectly out of Kirchhoff’s laws. These Hamiltonian systems aredefined with respect to a geometric structure called a Poissonbracket [6], [7]. The full-state variables of the system consistof all capacitor charges and all inductor fluxes and, hence,the order of the system equals the (possibly odd) order of thenetwork. Since the Poisson tensor as defined by the bracket isconstant, canonicalcoordinatescanbe foundwhich transformthePoisson tensor into a canonical symplectic form plus a numberof zero rows. The coordinates corresponding to the canonicalsymplectic form are called the active variables, whereas thecoordinates corresponding to the zero rows constitute a set ofintrinsic conserved quantities for the network called Casimirfunctions or nonactive variables. In the active variables, thedynamical equations have the form of a reduced order canonicalHamiltonian system. In [5], a method was presented to determinethese active and nonactive variables. This method was given agraphical interpretation in [6] by giving a bond graph realizationof the Poisson structure using the symplectic gyrator element.The active variables are explicitly represented by their timederivatives as the flow variables at the ports of the gyrators, whilethe nonactive variables are given by integrating the balanceequations at the 0-junctions.

Finally, in [8] and [9] a minimal Hamiltonian formulationof LC circuits containing elements in excess was given. Thestate variables are defined as the set of all capacitor chargesand all inductor fluxes, including the excess elements. Thedynamical equations consist of a set of differential and algebraicequations, where the latter ones describe the constraints set by thecapacitor loops and the inductor cutsets. The geometric structure

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BLANKENSTEIN: GEOMETRIC MODELING OF NONLINEAR RLC CIRCUITS 397

underlying these equations is a Dirac structure [10], [11]. Itis the geometric representation of the internal interconnectionstructure of the circuit according to Kirchhoff’s laws. Thecanonical coordinates for this Dirac structure were determined,slightly adapting the method in [5], [6], yielding the active andnonactive variables as well as the principal algebraic constraints.Interaction ports, consisting of the dual port variables currentand voltage, describing the controlled sources in the system, canbe added to obtain a port-controlled Hamiltonian formulationof LC networks containing elements in excess, with interactionports.

In this paper, we wish to study another minimal representa-tion of the dynamics of nonlinear RLC circuits by consideringthe Brayton–Moser equations [4]. In this case, the state variablesare given by the capacitor voltages and the inductor currents.In Section II we show that we can model the Brayton–Moserequations for a topologically complete RLC network withsources as a two-terminal multiport. The geometry underlyingthe Brayton–Moser equations is studied in Section III. It isshown that the Brayton–Moser equations can be written as adynamical system with respect to a newly defined geometricstructure, called a noncanonical Dirac structure. In Sections IVand V the formalism is extended to include circuits with ele-ments in excess and general noncomplete circuits. This removesthe common topological conditions on the network, i.e., thatof being (topologically) complete. In Section VI a transforma-tion theorem of [4] is restated within the geometrical settingintroduced in this paper. Finally, Section VII contains someconcluding remarks and some outlooks on future investigations.

II. BRAYTON–MOSER’S EQUATIONS

Recall that a RLC network is called complete if the inductorcurrents and capacitor voltages form a complete set of variables,i.e., they can be chosen independently without violating Kirch-hoff’s laws and they determine either the voltage or the current(or both) in every branch of the network. Notice that this rulesout capacitor loops and inductor cutsets, i.e., excess elements.Brayton and Moser [4] showed that the dynamics of a completeRLC network can be described in terms of its inductor currentsand capacitor voltages as the set of first order differential equa-tions1

(1a)

(1b)

Here, is the inductance matrix andis the capacitance matrix, both assumed to be positive

definite, and is a scalar functioncalled the mixed potential function. (See also [12], or recently[13], for a closely related differential geometric exposition.) Acomplete network is called “topologically complete” if it has agraph possessing a tree containing all capacitors and none of

1Throughout, we assume that the resistors are such that the equilibriumproblem can be solved, i.e., the voltage and current in any branch of the circuitcan be (not necessarily uniquely) determined from the complete variables i

and v [4, pp. 7–8].

the inductors2, where each resistor in the tree is a current-con-trolled resistor and each resistor in the corresponding co-treeis a voltage-controlled resistor and, finally, the resistors are lo-cated in such a way that there exists no fundamental loop whichcontains both tree resistors and co-tree resistors. For the class oftopologically complete RLC networks the mixed potential func-tion can be explicitly constructed as [4]

(2)

Here, is the total current potential (content) of all cur-rent-controlled resistors and is the total voltage poten-tial (co-content) of all voltage-controlled resistors. The matrix

, with elements in , is fully determinedby the topology of the network. (From now on, we will use theshorthand notation .) See also [14] for aclear overview.

A. Brayton–Moser’s Equations for RLC Networks WithSources

In this section, the Brayton–Moser equations are deduced fortopologically complete RLC networks containing independent,respectively controlled, voltage and current sources. As sug-gested in [4], Section III, we will principally look at sourcesas special kind of (nonpassive) resistors. Independent, respec-tively controlled, voltage sources are considered to be specialtypes of current controlled resistors (producing a fixed, respec-tively, time-dependent voltage level independently of the valueof the current). Analogously, independent or controlled currentsources are considered to be special types of voltage controlledresistors. In order for the network to be topologically complete,the sources have to be located in such a way that the topolog-ical constraints described in the previous section are satisfied. Inparticular, the voltage sources should be located in the tree andthe current sources should be located in the co-tree, and thereshould be no fundamental loops containing both voltage sourcesand co-tree resistors (including current sources), or both currentsources and tree resistors (including voltage sources). More-over, the equilibrium problem should be solved, which meansthat the currents corresponding to the voltage sources have tobe determined from the inductor currents, and the voltages cor-responding to the current sources have to be determined fromthe capacitor voltages. This is possible if every voltage source isincluded in at least one cutset for which the only other elementsare inductors3 and, dually, every current source is included in atleast one loop for which the only other elements are capacitors.

From now on, we will assume that all the above conditionsare satisfied, such that the network with sources is topologi-cally complete and the equilibrium problem is solvable. Con-sider the vector of independent voltage sources and denote theirconstant voltage values by the vector . The vectorof corresponding currents can be determined from the in-ductor currents, by considering the appropriate cutsets, as

, where is a constant matrix

2This is implied by the network being complete and, hence, the absence ofexcess elements (notice that the tree is not assumed to be connected).

3Notice that a special case is given by voltage sources which are in series withinductors. These networks have been considered in [15].


determined by the topology of the network. The current poten-tial corresponding to the independent voltage sources is thengiven by . Analogously, let

denote the (time-dependent ) voltage values of the con-trolled voltage sources, and the vector of corre-sponding currents, . Then, the cur-rent potential corresponding to the controlled voltage sourcesis given by . Dually, considerthe vector of independent current sources and denote their con-stant current values by the vector . The vector of cor-responding voltages can be determined from the capacitorvoltages, by considering the appropriate loops, as ,where is a constant matrix deter-mined by the topology of the network. The voltage potentialcorresponding to the independent current sources is then givenby . Analogously, letdenote the (time-dependent ) current values of the controlledcurrent sources, and the vector of correspondingvoltages, . Then the voltage poten-tial corresponding to the controlled current sources is given by

.Since the network is assumed to be topologically complete,

the dynamical equations are given by (1) with the mixed poten-tial (2) given by

(3)

These equations can be rewritten as

(4a)

(4b)

where

(5)

Notice that we explicitly extracted the controlled sources outof the mixed potential and left the independent (i.e., constant)sources to be modeled by the mixed potential function. Thereason is that we wish to model the system as a network withinteraction ports consisting of controllable voltage and currentsources. In fact, just as the network elements like inductors,capacitors, resistors and sources are modeled as two-terminalelements, we wish to model the complete circuit as a two-ter-minal multiport itself. Therefore we need to define a vector ofoutput variables next to the input vector . The most nat-ural choice is to consider the power-conjugate variables and

, defined by

(6)

as output variables for the system (4). Later on, we will seethat the input and output variables thus defined are very closelyrelated to what we could call the proper port variables for thenetwork.

In conclusion, a topologically complete RLC network withsources can be modeled as a two-terminal multiport whose dy-

namics is defined by (4) and (5), with representing the con-trolled voltage sources and the controlled current sources(being the input variables). The output variables are the powerconjugate variables defined by (6).

III. GEOMETRY OF BRAYTON–MOSER’S EQUATIONS

In [6], it was shown that the Hamiltonian formulation for thedynamics of LC circuits leads to a Hamiltonian system definedon a Poisson manifold. A Poisson manifold is a smooth manifold

endowed with a Poisson bracket, being a skew-symmetricbilinear map from4 into , satisfyingthe Jacobi identity and Leibnitz’ rule, see e.g., [7]. In particular,the dynamics are shown to have the form

(7)

Here, the state variables consist of inductor fluxesand capacitor charges, and denotes the total energy of thecircuit. The Poisson bracket is defined by the skew-sym-metric matrix

(8)

with determined by the fun-damental loop matrix corresponding to the capacitor’stree. By the skew-symmetry property it follows that

, which isthe mathematical statement for the fact that the total energy ofthe system is conserved, i.e., the circuit is lossless.

On the contrary, the Brayton–Moser equations (1) can bewritten in the form

(9)

The state variables are the inductor currents andthe capacitor voltages, is the mixed potential function, and

is the symmetric matrix

(10)

Since is not skew-symmetric it does not endow the state man-ifold with a Poisson bracket. This is also clearfrom the physics of the system since the mixed potential is notconserved , not even for circuits without resistive ele-ments.5 However, is symmetric and invertible and thereforedefines a pseudo-Riemannian metric on the state spaceby (see also [4], [12])

(11)

(For simplicity we assumed that and are diag-onal matrices, and

.) Recall that a

4C (M) denotes the space of smooth functions onM .5Recall that P does not represent energy, but instead has units of power.

Hence, it is not conserved, not even for LC circuits.


pseudo-Riemannian metric is defined as a smooth tensor fieldsuch that for each the

restriction defines a symmetricnondegenerate bilinear form on the tangent space , see,e.g., [16]. The gradient of a smooth function on ,endowed with the pseudo-Riemannian metric , is defined as

(12)

Hence, comparing with (9) we see that we can write theBrayton–Moser equations (1) as the gradient system ([12])

(13)

A. Dirac Structures

In the previous section it was shown that the Hamiltonianformulation of the dynamics of LC networks leads to a geo-metric structure called Poisson structure, while the dynamics ofRLC networks described by Brayton–Moser’s equations leadsto consider pseudo-Riemannian structures. However, in bothcases we did not consider networks with interaction ports suchas controlled current or voltage sources, nor networks with el-ements in excess. In the latter case there will be algebraic con-straints in the system defined by Kirchhoff’s loop and nodelaws, so that the network’s dynamics can certainly not be writtenin the form of the explicit differential equations (7) or (13).(Of course, one could try to transform the circuit to an equiv-alent circuit without capacitor loops or inductor cutsets, butin case some of the elements are not bijective this may notalways be possible.) In the former case the system has to beaugmented with interaction ports such as inputs and outputswhich, again, are clearly not included in (7) or (13). We canconclude that the Poisson structure and the pseudo-Riemannianstructure we have seen above are not flexible enough to handlethe description of general RLC circuits. In [8] the problem ofadding interaction ports as well as elements in excess was han-dled simultaneously by considering a new geometric structurecalled Dirac structure [10], [11]. Dirac structures form a di-rect generalization of Poisson structures and as such retain the“skew-symmetry” property. That is, Hamiltonian systems de-fined with respect to these Dirac structures necessarily have con-served Hamiltonian, i.e., the corresponding circuit is lossless.In [8] it was shown how to describe the dynamics of LC net-works with interaction ports and elements in excess as Hamil-tonian systems with respect to these Dirac structures. On theother hand, as we already saw previously the Brayton–Moserequations do not constitute a set of “lossless” equations, in thesense that the mixed potential is not conserved, not even forlossless circuits. Therefore we need to generalize the concept ofDirac structures in order to be able to describe these “nonloss-less” equations. The proper generalization is given by the con-cept of noncanonical Dirac structures [17]. We will describe theBrayton–Moser equations for topologically complete networkswith interaction ports and elements in excess as dynamical sys-tems with respect to these new geometric structures. In doingso, it will turn out that these noncanonical Dirac structures caneven handle the description of noncomplete networks, which

will be shown in Section V. We continue this section with re-calling the definition of Dirac structures and defining noncanon-ical Dirac structures. The Brayton–Moser equations (4) will bedescribed with respect to these noncanonical Dirac structuresin Section III-B. Excess elements will be handled in Section IV,whereas noncomplete networks will be considered in Section V.

Let denote a vector space and let denote its dual vectorspace. Recall that the dual space consists of all linear func-tionals on , and is itself a vector space again. The dual paring,or duality product, between an element and , de-noted by , is defined as the action of the linear functional

on and yields a real number, that is

(14)

There exists an intrinsic nondegenerate symmetric bilinear formon defined by

(15)

where . This is called the canon-ical bilinear form (which explains the subscript can). A vectorsubspace is called a constant Dirac structure withrespect to the canonical bilinear form if it satisfies thecondition , see, e.g., [18]. Here, denotes the or-thogonal complement of with respect to the canonical bilinearform (15), i.e.,

(16)

It can be shown that a constant Dirac structure can be equiv-alently defined as a vector subspace such that

and (see [18], [19]).Generally, however, we will not be interested in constant

Dirac structures and in fact we would like our linear subspaceto change depending on the values of some underlying statespace variables. This leads to the following notion ([18], [19]):A Dirac structure is a map depending smoothlyon , assigning to each a vector subspace

such that , with theorthogonal subspace of with respect to .

Notice that so far we have defined the Dirac structures to be“isotropic” (i.e., ) with respect to the canonical bi-linear form (15). Therefore they are also called canonical Diracstructures. As explained above they are not suitable howeverto describe the dynamics of “nonlossless” equations. Therefore,we need to generalize the notion of Dirac structures in order toconsider more flexible structures. These are called noncanonicalDirac structures and will be defined next.

Definition 1: Consider an arbitrary nondegenerate sym-metric bilinear form on denoted by . A vectorsubspace is called a constant Dirac structurewith respect to the bilinear form if , wheredenotes the orthogonal complement of with respect to .

Next, suppose that the bilinear form depends smoothlyon some state variables , and defines for every anondegenerate symmetric bilinear form . A smooth map

is called a Dirac structure with respect to


if for every , with denotingthe orthogonal complement of with respect to .

Notice that if the bilinear form equals the canonical bilinearform in (15), then, the above definition comes down to the onefor canonical Dirac structures. In order to make a clear dis-tinction, a Dirac structure with respect to a noncanonical bi-linear form will be called a noncanonical Dirac structure. No-tice that since the bilinear form is nondegenerate, also in thiscase , and yields6

(17)

B. Dynamical Formalism

We start this section by reviewing the formalism used in [8]to describe the dynamics of nonlinear LC networks containingelements in excess together with interaction ports, using thenotion of canonical Dirac structures. This outline will helpus to understand the analogous developments we will presentafterwards for the Brayton–Moser equations. The state vari-ables of the system are the inductor fluxes and thecapacitor charges (including the excess elements),which makes the state space . These are calledthe energy variables since they define the circuit’s total energy

. The vector space is defined as the Cartesianproduct of the space of flow variables and the space ofinput port variables. The flow variables are defined as the rateof change of the energy variables, i.e., and ,so that . The input port variables are defined bythe controlled voltage sources and the controlledcurrent sources , so that . The dualvector space is defined as the Cartesian product of the space

of effort variables and the space of output port variables.The effort variables are defined as the coenergy variables, i.e.,

and , so that .Notice that the flow and effort variables are conjugate variablesin the sense that , the rate of change ofthe energy function. Thus , the dual space of , withduality product being the common inner product on .The output port variables are given by the currents corre-sponding to the controlled voltage sources and the voltagescorresponding to the controlled current sources .Clearly, represents the total power suppliedto the network through its ports, hence, with dualityproduct the common inner product on . Thus, thevector space is given by

(18)

In [8], a constant canonical Dirac structure is de-fined which is fully determined by the network graph of the cir-cuit. In fact, the Dirac structure exactly describes the intercon-nection structure of the circuit as defined by Kirchhoff’s loopand node laws. The precise form of is not important here andcan be found in [8]. Then, it is shown that the dynamics of thecircuit can be described as a “port-Hamiltonian system” with

6In case of the canonical bilinear form this is equivalent to 2hw j vi = 0.

respect to this Dirac structure and the Hamiltonian , definedas the set of differential and algebraic equations

(19)

where represents time. Since the Dirac structure is canonical,it follows that , which yields

(20)

i.e.,

(21)

This is the mathematical statement for the energy balance ofthe system, stating that the rate of change of the total internalenergy equals the power supplied to the system through its ports.In other words, the system is conservative (lossless). Indeed, ifthe power supplied through the ports is zero then it follows that

and the energy is conserved.So far, we have reviewed the Hamiltonian formulation of

LC circuits described in [8]. From now on we will considerthe Brayton–Moser equations for a topologically completeRLC network as given in (4). We show that they can bewritten in the same formalism as above, using noncanon-ical Dirac structures. For clarity, excess elements are notyet considered, they will be handled in Section IV. We startby noticing that in the Brayton–Moser case the “generatingfunction” for the dynamics is not the total energybut rather the mixed potential function , whichis a function of the inductor currents and the capacitor volt-ages. Moreover, as can easily be checked, the mixed potentialfunction has units of power, instead of energy. This leads usto define the power variables to be and , which yieldsthe state space . The flow variables are de-fined to be the rate of change of the power variables, i.e.,

. As in the above case,the input port variables are .The effort variables are defined as the copower vari-ables , respectively,

, which equal minus the in-ductor voltages and the capacitor currents up tosome constant terms. Thus . Again, the flowand effort variables are conjugated variables in the sense that

,the rate of change of the mixed potential function. Sowith duality product the common inner product on .Recall that in (6) we defined the output variables of the systemto be the currents of the controlled voltage sources to-gether with the voltages of the controlled current sources.Although these variables are power conjugated to the inputs

and , i.e., has units of power, they arenot conjugated with respect to which has units of(i.e., power per second). In order to be consistent we need toredefine the output port variables to be the time derivatives

, respectively, of (6) (notice the minussign!). This yields the proper output port variables for the


networkwith duality product the common inner product on .Thus, as in the previous case, the vector space is definedas , which equals (18). Notice, however, thatthe flow, effort and output port variables are not defined in thesame way as in the Hamiltonian formulation!

The next step is to define a Dirac structure on . Considerthe smooth map defined as

(22)

Lemma 2: The map defined in (22) isa Dirac structure with respect to the bilinear form

(23)

where is defined as in (10).The lemma can easily be proved by checking that

, where is the orthogonalcomplement of with respect to . Nowthe next proposition follows easily.

Proposition 3: The Brayton–Moser equations (4) canbe described as a dynamical system with respect to thenoncanonical Dirac structure in (22) by setting

, and .In other words, the Brayton–Moser equations (4) can equiv-

alently be written as the set of differential equations

(24)

Since is a noncanonical Dirac structure, (17) is satisfied,which yields the “balance equation”

(25)i.e.,

(26)

Here, . Equation (26) shows that is not con-served, not even when (i.e., when the power supplied tothe network through its ports is zero).

IV. NETWORKS WITH ELEMENTS IN EXCESS

In this section, we will show how to write the dynamics ofa class of nonlinear RLC circuits with elements in excess as adynamical system with respect to a noncanonical Dirac struc-ture. Basically, the class of networks we consider are the onesthat can be made topologically complete by adding voltage orcurrent sources, after which we can apply the Brayton–Moserdescription. It turns out that the introduced voltage or currentsources are exactly the circuit’s interpretation of the “Lagrangemultipliers” used in [20].

Consider a nonlinear RLC circuit with elements in excess, thatis, capacitor-only loops and/or inductor-only cutsets. Let

denote the currents of all the inductors and thevoltages of all the capacitors (including the excess elements).Suppose there are capacitor loops, then Kirchhoff’s loop lawtells us that there exists a constant matrixsuch that the voltages have to satisfy the condition

(27)

Analogously, suppose there are inductor cutsets, then ac-cording to Kirchhoff’s node law there exists a constant matrix

such that the currents have to satisfy thecondition

(28)

Clearly the considered network is not complete since andcannot be chosen independently (because they have to satisfythe constraints (27) and (28)) and, hence, do not form a completeset of variables.

Consider the th capacitor loop defined by , wheredenotes the th row of . Add a controlled current source

in series with one of the capacitors in the loop. Then thecapacitor-only loop disappears. The voltage correspondingto the current source is defined by Kirchhoff’s loop law:

. Doing so for every capacitor loop, we introduce extracontrolled current sources in the network denoted in vector no-tation by with corresponding voltages defined by

(29)

Next, consider the th inductor cutset defined by . Adda controlled voltage source in parallel over one of theinductors in the cutset. Then the inductor-only cutset disappears.The current corresponding to the voltage source is definedby Kirchhoff’s node law: . Doing so for everyinductor cutset, we introduce extra controlled voltage sourcesin the network denoted in vector notation by withcorresponding currents defined by

(30)

Notice that after these operations, the network has no capac-itor loops or inductor cutsets anymore, i.e., there are no elements


in excess. The assumption that we make is that the extended net-work obtained in this way is topologically complete.7 Under thisassumption, we can calculate the Brayton–Moser equations forthe extended network. Indeed, the introduced current sourcesyield an extra voltage potential term ,while the introduced voltage sources yield an extra current po-tential term . According to Section II.Athe Brayton–Moser equations get the form (4), including theextra sources

(31a)

(31b)

Here, is the mixed potential of (5).Next, we turn back to the original circuit with excess ele-

ments by short-circuiting every controlled current sourceand open-circuiting every controlled voltage source . Thatis, we add the conditions , and

. Notice that these conditions are equivalent tothe constraints (27) and (28). The source then becomes atime-dependent current source, determined by the condition thatthe constraints (27) have to be satisfied at all times. The source

becomes a time-dependent voltage potential determined bythe condition that the constraints (28) have to be satisfied at alltimes. In other words, the dynamics of the nonlinear RLC cir-cuit with elements in excess is given by the differential equa-tions (31) together with the constraints (27) and (28), whereand act as the Lagrange multipliers determined by the con-dition that the constraints (27) and (28) have to be satisfied atall times. Notice that these equations were also reported in [20],deduced however via the Hamiltonian description of the circuitand without the circuit’s construction of the Lagrange multi-pliers as time-dependent current and voltage sources.

Next, we show that analogously to Proposition 3, also theequations described in this section can be written as a dynamicalsystem with respect to a noncanonical Dirac structure. Considerthe smooth map definedas

(32)

Here, “ ” is used to denote the image of a matrix , that is,the vector space spanned by its columns. It is easy to proof that

is a Dirac structure with respect to the bilinear form (23).It is now immediately clear that the constrained Brayton–Moserequations (27), (28), (31) can be equivalently written as the setof differential end algebraic equations (24), where is replaced

7If this is not the case, one could make it topologically complete by using themethod in Section V.

by . Notice that the algebraic equation are given by thetime derivatives of (27) and (28).

V. BRAYTON–MOSER EQUATIONS FOR

NONCOMPLETE NETWORKS

Now assume that the network under consideration is not com-plete (but has no elements in excess). Of course, one can alwaysadd inductors in series and capacitors in parallel to the circuit tomake it complete [4, p. 17]. (Sometimes, the network can evenbe made topologically complete in this way. This idea is usedin [14].) Assume we add the inductors andcapacitors to make the network complete,i.e., the inductor currents and thecapacitor voltages form a com-plete set of variables of the circuit. Do this in such a way thatno inductor cutsets or capacitor loops appear in the augmentedcircuit. Then the Brayton–Moser equations for the augmentednetwork are given by (4) where

(33)

and

(34)

As in Proposition 3, the Brayton–Moser equations can bewritten as a dynamical system employing the noncanonicalDirac structure (22) with respect to the bilinear form (23),where is given as

(35)

with and defined in (33) and (34), respectively.Now turn back to the original circuit by taking the induc-

tance limits and the capacitancelimits . The corresponding ma-trix above becomes singular. However, the fact thatwith respect to (23) is independent of the regularity of and,in particular, holds also for the limiting case of inductancesand capacitances going to zero. Therefore, also in the case ofnoncomplete networks (without elements in excess) the equa-tions can be described in the form of Proposition 3. Notice thatthe method introduces the new variables and

whose dynamics is determined by the con-straints corresponding to the zero rows of

(36a)

(36b)

Under suitable invertibility conditions it is possible to explicitlysolve the above constraints for the augmented inductor-currentsand capacitor-voltages and substitute the result into to ob-tain an explicit description of the original circuit in the variables

. See also [4], [14] for this algorithm.The equations obtained in this section are closely related to thedifferential-algebraic equations reported in [13].


VI. STRUCTURE PRESERVING TRANSFORMATIONS

Equivalence between nonlinear circuits is a well studiedsubject; the reader is refered to [21] for a recent overview. Inthis section structure preserving coordinate transformations arestudied. We recall a theorem of [4] and re-interpret this resultusing concepts as defined for Dirac structures [22].

In [4] the equations (1) are considered, which we repeat herefor ease of reference

(37)

Here,is the matrix defined in (10) andthe mixed potential function. Transfor-

mations , are consideredwhich preserve the form (37), i.e., for which (37) implies

(38)

where . Notice that the matrix isleft unchanged under the coordinate transformation.8 Braytonand Moser proved the following theorem [4, p. 94].

Theorem 4: A coordinate transformation pre-serves the form (37) if and only if it preserves the pseudo-Rie-mannian metric (11), which is the case if and only if theJacobian matrix satisfies

(39)

Actually, the proof is quite straightforward, calcu-lating the transformed dynamics and observing that

shouldbe equal to .

Our goal here is to interpret the above theorem into a moregeometric setting using the notion of noncanonical Dirac struc-tures. First recall that the noncanonical Dirac structure corre-sponding to (37) is given by

(40)

This is a noncanonical Dirac structure with respect to the non-canonical bilinear form

(41)

(This follows directly from (22) and (23) by settingand omitting .) Now consider a coordinate transforma-tion . Then is called a Dirac automorphism if

, for all . Inother words, leaves the Dirac structure invariant. In [22] thisis called a symmetry of . Consider the Dirac structure (40),then it follows that the coordinate transformation is a Diracautomorphism if and only if

(42)

8In fact, any coordinate transformation y = (x) will transform (37) into theform B(y) _y = (@Q=@y)(y), but, in general, it is not true that B(y) = A(y).

for all such that . Clearly, this is a neces-sary and sufficient condition for (39) to hold. Furthermore, wehave shown the following in [22] (in a slightly more general set-ting): Let be a Dirac automorphism of . Then, is a so-lution of the dynamics , i.e., (37),if and only if is a solution of the dynamics

with , whichis (38). This yields the following geometric interpretation of theresult in Theorem 4.

Theorem 5: A coordinate transformation pre-serves the form (37) if and only if is a Dirac automorphismof in (40).

Actually, this theorem holds for any matrix , not nec-essarily of the form (10), and function . Furthermore, thetheorem can be generalized to general Dirac structures asindicated above, yielding the general statement: A coordinatetransformation preserves the form of the dynamicalequations , and turns them into

, if and only if is a Dirac auto-morphism of .

The interesting thing about the last statement is that it givesa synthesis between the transformation theorem of Brayton andMoser and the one known in classical mechanics, where onelooks for structure preserving transformations of Hamiltoniansystems. In fact, these systems can be written with respect to acanonical Dirac structure and it can be shown that being aDirac automorphism is equivalent to leaving the symplecticform invariant (also called a canonical transformation); see [22]for more details. In fact, this similarity was also pointed out in([4], footnote p. 95], but its deeper geometric meaning was notobserved.

VII. CONCLUSION

This paper is concerned with the description of nonlinearRLC circuits including independent and controlled voltage orcurrent sources and possibly having elements in excess. It isshown that a topologically complete RLC network with sourcescan be modeled as a two-terminal multiport whose dynamicsis described by the Brayton–Moser equations. The geometry ofthe Brayton–Moser equations is studied in detail. The notionof noncanonical Dirac structures is defined, being the basic un-derlying geometric structure for the dynamical equations. Em-ploying the concepts of flow and effort variables used in theHamiltonian formulation of electrical circuits [8], it is shownthat the Brayton–Moser equations can be written as a dynam-ical system with respect to a noncanonical Dirac structure. Thepower variables are the inductor currents and the capacitor volt-ages, defining the mixed potential function and constituting thestate variables of the dynamical system. The flow variables arethe time derivatives of the power variables, whereas the effortvariables are defined as the copower variables. Moreover, it isshown that for circuits with elements in excess the dynamics ofthe network is described by constrained Brayton–Moser equa-tions, yielding a set of differential and algebraic equations. TheLagrange multipliers corresponding to the constraints are iden-tified as additional time-dependent current and voltage sources.


Again, the underlying geometric structure is that of a noncanon-ical Dirac structure. Also noncomplete networks are shown tobe described using noncanonical Dirac structures. Finally, it isshown that structure preserving transformations are equivalentto Dirac automorphisms.

The balance equation (26) can be conveniently used to deductpassivity properties of the system and has applications in sta-bility or stabilizability (see e.g., [23]). It is important to no-tice that these passivity properties are (partly) determined bythe sign of the matrix which is defining the bilinearform (23). The stabilization of RLC networks using passivitybased on a Brayton–Moser description has been described in[15]. The identification of the geometry underlying the dynam-ical equations allows to transfer many control techniques as re-cently developed for Hamiltonian systems described by canon-ical Dirac structures [23]–[26], to new techniques based on theBrayton–Moser description presented in this paper. These ideasand their results will be presented elsewhere.

Another interesting topic for further research is the construc-tion of canonical coordinates. These coordinates have been con-structed for the Hamiltonian formulation of LC networks in[5], [6], [8], [9] and reveal some interesting dynamical prop-erties. In particular, in these coordinates the dynamics is splitinto a canonical (i.e., symplectic) Hamiltonian part in the ac-tive variables, a set of conserved quantities, also called Casimirfunctions or nonactive variables, and finally a set of algebraicconstraints expressed in terms of the gradient of the Hamil-tonian. Contrary to the Hamiltonian formulation, however, thenoncanonical Dirac structures defined in this paper are state de-pendent. Hence, suitable conditions for the existence of canon-ical coordinates have to be found, together with constructivemethods to find them. Interestingly, this study should combineconcepts from both symplectic and Riemannian geometry.

ACKNOWLEDGMENT

The author would like to thank the Department of MechanicalEngineering, Katholieke Universiteit Leuven, Leuven, Belgium,for its hospitality and support.

REFERENCES

[1] L. O. Chua and J. D. McPherson, “Explicit topological formulation ofLagrangian and Hamiltonian equations for nonlinear networks,” IEEETrans. Circuits Syst., vol. CAS-21, no. 2, pp. 277–286, Mar. 1974.

[2] H. G. Kwatny, F. M. Massimo, and L. Y. Bahar, “The generalized La-grange formulation for nonlinear RLC networks,” IEEE Trans. CircuitsSyst., vol. CAS-29, no. 4, pp. 220–233, Apr. 1982.

[3] L. Weiss and W. Mathis, “A Hamiltonian formulation for complete non-linear RLC networks,” IEEE Trans. Circuits Syst. I, Fundam. TheoryAppl., vol. 44, no. 9, pp. 843–846, Sep. 1997.

[4] R. Brayton and J. Moser, “A theory of nonlinear networks—I/II,” Q.Appl. Math., vol. 22, no. 1/2, pp. 1–33/81–104, 1964.

[5] G. M. Bernstein and M. A. Lieberman, “A method for obtaining a canon-ical Hamiltonian for nonlinear LC circuits,” IEEE Trans. Circuits Syst.,vol. 36, no. 3, pp. 411–420, Mar. 1989.

[6] B. M. Maschke, A. J. van der Schaft, and P. C. Breedveld, “An intrinsicHamiltonian formulation of the dynamics of LC circuits,” IEEE Trans.Circuits Syst. I, Fundam. Theory Appl., vol. 42, no. 2, pp. 73–82, Feb.1995.

[7] V. I. Arnold, Mathematical Methods of Classical Mechanics. NewYork: Springer-Verlag, 1978.

[8] B. M. Maschke and A. J. van der Schaft, “Note on the dynamics of LCcircuits with elements in excess,” in Memorandum Faculty of AppliedMathematics 1426, Enschede, The Netherlands: Univ. of Twente, Jan.1998.

[9] A. M. Bloch and P. E. Crouch, “Representations of Dirac structures onvector spaces and nonlinear LC circuits,” in Proc. Symp. Pure Math.,Diff. Geom., Control Theory, vol. 64, 1999, pp. 103–117.

[10] T. Courant, “Dirac manifolds,” Trans. Amer. Math. Soc., vol. 319, pp.631–661, 1990.

[11] I. Dorfman, Dirac Structures and Integrability of Nonlinear EvolutionEquations. Chichester, U.K.: Wiley, 1993.

[12] S. Smale, “On the mathematical foundations of electrical circuit theory,”J. Diff. Geom., vol. 7, pp. 193–210, 1972.

[13] W. Marten, L. O. Chua, and W. Mathis, “On the geometrical meaning ofpseudo hybrid content and mixed potential,” Arch. Elekt. Übertragung.,vol. 46, no. 4, pp. 305–309, 1992.

[14] L. Weiss, W. Mathis, and L. Trajkoviæ, “A generalization ofBrayton–Moser’s mixed potential function,” IEEE Trans. CircuitsSyst. I, Fundam. Theory Appl., vol. 45, no. 4, pp. 423–427, Apr. 1998.

[15] R. Ortega, D. Jeltsema, and J. Scherpen, “Power shaping: A new para-digm for stabilization of nonlinear RLC circuits,” IEEE Trans. Autom.Control, vol. 48, no. 10, pp. 1762–1767, Oct. 2003.

[16] R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis,and Applications, 2nd ed. New York: Springer-Verlag, 1988.

[17] G. Blankenstein, “A joined geometric structure for Hamiltonian and gra-dient control systems,” in Prepr. IFAC Workshop LagrangHamiltonianMethods Nonlinear Control, Seville, Spain, Apr. 3–5, 2003, pp. 61–66.

[18] A. J. van der Schaft and B. M. Maschke, “The Hamiltonian formulationof energy conserving physical systems with ports,” Arch. Elek. Übertra-gung., vol. 49, no. 5/6, pp. 362–371, 1995.

[19] M. Dalsmo and A. J. van der Schaft, “On representations and inte-grability of mathematical structures in energy-conserving physicalsystems,” SIAM J. Control Optim., vol. 37, no. 1, pp. 54–91, 1999.

[20] D. Jeltsema and J. M. A. Scherpen, “On nonlinear RLC networks: Port-controlled Hamiltonian systems dualize the Brayton–Moser equations,”in Proc. IFAC World Congress Autom. Control, Barcelona, Spain, July2002.

[21] W. Mathis, “Transformation and equivalence,” in The Circuits and Fil-ters Handbook, ch. 36, W. K. Chen, Ed.. Boca Raton, FL, Dec. 2002, pp.1009–1029.

[22] G. Blankenstein, “Implicit Hamiltonian Systems: Symmetry and Inter-connection,” Ph.D. dissertation, Univ. of Twente, Enschede, The Nether-lands, Nov. 2000.

[23] A. J. van der Schaft, L -Gain and Passivity Techniques in NonlinearControl, 2nd ed. New York: Springer-Verlag, 2000.

[24] R. Ortega, A. J. van der Schaft, I. Mareels, and B. Maschke, “Puttingenergy back in control,” IEEE Control Syst. Mag., vol. 21, pp. 18–33,2001.

[25] R. Ortega, A. J. van der Schaft, B. Maschke, and G. Escobar, “Intercon-nection and damping assignment passivity-based control of port-con-trolled Hamiltonian systems,” Automatica, vol. 38, no. 4, pp. 585–596,Apr. 2002.

[26] G. Blankenstein, “Matching and stabilization of constrained systems,” inProc. Int. Symp. Math. Theory Networks Syst. (MNTS’02), Notre Dame,IN, Aug. 2002.

Guido Blankenstein received the degree in en-gineering and the Doctorate degree in appliedmathematics from the University of Twente,Enschede, The Netherlands, in 1996 and 2000,respectively.

From January 2001 to July 2001, he held aPost-Doctoral position at the Laboratoire des Sig-naux et Systèmes, CNRS-Supélec, Gif sur Yvette,France, and from September 2001 to September2002 he was a Post-Doctoral Assistant in the Depart-ment of Mathematics, EPFL, Lausanne, Switzerland.

He worked as a Researcher within the European sponsored project GeoPlex(further information is available at www.geoplex.cc). in the Department ofMechanical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium,from September 2002 to August 2004. Currently, he is a Researcher in theDepartment of Electrical Energy, Systems and Automation, Ghent University,Zwijnaarde, Belgium. His research interests are in mathematical systems andcontrol theory, mainly directed toward Hamiltonian dynamics, systems withsymmetry, modeling, and nonlinear control.

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Geometric modeling of nonlinear RLC circuits