On Passivity and Power-Balance Inequalities of Nonlinear RLC Circuits Dimitri Jeltsema, Student Member, IEEE, Romeo Ortega, Fellow, IEEE, and Jacquelien M

1174 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 50, NO. 9, SEPTEMBER 2003 On Passivity and Power-Balance Inequalities of Nonlinear RLC Circuits Dimitri Jeltsema, Student Member, IEEE, Romeo Ortega, Fellow, IEEE, and Jacquelien M. A. Scherpen

Abstract—Arbitrary interconnections of passive (possibly non- reactive power—our result unveils some sort of reactive linear) resistors, inductors, and capacitors define passive systems, power-balance. with power port variables the external source voltages and cur- The remainder of the paper is organized as follows. In Sec- rents, and storage function the total stored energy. In this paper, we identify a class of RLC circuits (with convex energy function and tion II, we briefly review some fundamental results in circuits weak electromagnetic coupling), for which it is possible to “add a theory, like the classical definition of passivity and Tellegen’s differentiation” to the port terminals preserving passivity—with a Theorem. The new passivity property for RL and RC circuits is new storage function that is directly related to the circuit power. To established in Section III. In Section IV,this result is extended to establish our results, we exploit the geometric property that volt- a class of RLC circuits using the classical Brayton–Moser equa- ages and currents in RLC circuits live in orthogonal spaces, i.e., Tel- legen’s theorem, and heavily rely on the seminal paper of Brayton tions. Finally, we conclude the paper with some remarks and and Moser published in the early sixties. comments on future research. Notation: Throughout the paper we will denote by Index Terms—Brayton–Moser equations, nonlinear circuits, passivity, Tellegen’s theorem. the partial derivative of a vector function with respect to a -dimensional column vector , i.e., I. INTRODUCTION ASSIVITY is a fundamental property of dynamical sys- P tems that constitutes a cornerstone for many major developments in circuit and systems theory, see, e.g., [3], [9], and Consequently, denotes the second partial derivative the references therein. It is well known that (possibly nonlinear) (Hessian), i.e., RLC circuits consisting of arbitrary interconnections of passive resistors, inductors, capacitors, and voltage and/or current sources are also passive with power port variables, the external source voltages and currents, and storage function of the total stored energy [2]. Our main contribution in this paper is the II. TELLEGEN’S THEOREM AND PASSIVITY proof that for all RL or RC circuits, and a class of RLC circuits, it Consider a circuit consisting of inductors, capacitors, is possible to “add a differentiation” to one of the port variables resistors, and voltage and/or current sources, called the (either voltage or current) preserving passivity with a storage branches of the circuit. Let and function which is directly related to the circuit power. The new , with , de- passivity property is of interest in circuit theory, but also has note the branch currents and voltages of the circuit, respectively. applications in control (see [7] for some first results regarding It is well known that Tellegen’s Theorem [8] states that the set stabilization). of branch currents (which satisfy Kirchhoff’s current law), say Since the supply rate (the product of the passive port vari- , and the set of branch voltages (that sat- ables) of the standard passivity property, as defined in, e.g., [3] isfy Kirchhoff’s voltage law), say , are orthogonal sub- and [9], is voltage current, it is widely known that the dif- spaces. As an immediate consequence of this fact we have ferential form of the corresponding energy-balance establishes the active power-balance of the circuit. As the new supply rate (1) is voltage the time derivative of the current (or current the time derivative of the voltage —quantities which are sometimes adopted as suitable definitions of the supplied which states that the total power in the circuit is preserved. Corollary 1: Voltages and currents in a (possibly nonlinear) RLC circuit satisfy Manuscript received December 18, 2002; revised April 25, 2003. This work was supported in part by the European Community Marie Curie Fellowship in the framework of the CTS (Control Training Site). This paper was recommended (2) by Associate Editor A. Ushida. D. Jeltsema and J. M. A. Scherpen are with the Delft Center of Systems and Control, Delft University of Technology, Delft 2600 GA, The Netherlands as well as (e-mail: [email protected]; [email protected]). R. Ortega is with the Laboratoire des Signaux et Systémes, CNRS-Supelec, 91192 Gif-sur-Yvette, France (e-mail: [email protected]). (3) Digital Object Identifier 10.1109/TCSI.2003.816332

The proof of this corollary is easily established noting that, if where . If the resistors are passive, the (respectively, ), then, clearly also circuit defines a passive system with power port variables (respectively, ), and then, invoking orthog- and storage function of the total resistors con- onality of and , see also [1], [8] and the references therein. tent. Another immediate consequence of Tellegen’s theorem is the Similarly, arbitrary interconnections of passive capacitors following, slight variation of the classical result in circuit theory, with the convex energy function , voltage-controlled see, e.g., [2, Sec. 19.3.3], whose proof is provided for the sake resistors, and sources, satisfy the power-balance inequality of completeness. Proposition 1: Arbitrary interconnections of inductors and (8) capacitors with passive resistors verify the energy-balance inequality where . If the resistors are passive, the circuit defines a passive system with power port variables and storage function the total resistors co-content. (4) Proof: The proof of the new passivity property for RL circuits is established as follows. First, differentiate the resistors where we have defined the total stored energy content with and the inductor fluxes and the capacitor charges, respectively. If, furthermore, (9) the inductors and capacitors are also passive, then, the network defines a passive system with power port variables Then, by using the fact that and storage function of the total energy. Proof: First, notice that , where we have used the fact that and , and the relations and . Then, by (1) we have that and by invoking Faraday’s law, i.e., , we ob- (notice that we have adopted the tain standard sign convention for the supplied power). Hence, noting (10) that for passive resistors, and integrating the latter equations form 0 to , we obtain (4). Passivity follows from where the nonnegativity stems from the convexity assumption. positivity of for passive inductors and capacitors. Finally, by substituting (9) and (10) into (2) of Corollary 1, with and , and integrating form 0 to yields the result. III. NEW PASSIVITY PROPERTY FOR RL AND RC CIRCUITS The proof for RC circuits follows verbatim, but now using In this section, we first consider circuits consisting solely of (3) of Corollary 1 instead of (2), the relation inductors and current-controlled resistors and sources, denoted and the definition of the co-content. by , and circuits consisting solely of capacitors and voltage- Remark 1: In some cases it is also possible to apply Proposi- controlled resistors and sources, denoted by . Furthermore, tion 2 to RL (respectively, RC) circuits containing voltage-con- to present the new passivity property, we need to define some trolled resistors in (respectively, current-controlled resistor additional concepts that are well known in circuit theory [6], in ) under the condition that the curves are invert- [8], and will be instrumental to formulate our results. ible. If, for example, contains a voltage-controlled resistor, Definition 1: The content of a current-controlled resistor is say , and its constitutive relation is invert- defined as ible, it should then be possible to rewrite the characteristic equation in terms of the current, i.e., . In the linear case, this means that instead of writing (or in (5) terms of the resistors co-content: ), we may write (Ohm’s law in its conventional while, for a voltage-controlled resistors, the function form), and hence its content reads and the new passivity property (7) can be established; see also, (6) Fig. 1. Remark 2: The new passivity properties of Proposition 2 is called the resistors co-content. differ from the standard result of Proposition 1 in the following Proposition 2: Arbitrary interconnections of passive induc- respects. First, while Proposition 1 holds for general RLC cir- tors with convex energy function , current-controlled cuits, the new properties are valid only for RL or RC systems. resistors and sources, satisfy the power-balance inequality Using the fact that passivity is invariant with respect to negative feedback interconnections it is, of course, possible to combine RL and RC circuits and establish the new passivity property for (7) some RLC circuits. A class of RLC circuits for which a similar 1176 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 50, NO. 9, SEPTEMBER 2003

where is the inductance matrix, is the capacitance matrix, is called the mixed-potential and is given by

(12)

where is a (full-rank) matrix that captures the (a) interconnection structure between the inductors and capacitors. If we add external sources2 , (11) can be written as

(13)

where , and with . Remark 4: Notice that the mixed-potential function contains (b) both the content and co-content which are, due to the topo- logical completeness assumption, described in terms of the in- Fig. 1. (a) Resistor characteristic with invertible @i Yv A curve. (b) Noninvertible resistor characteristic. ductor currents and capacitor voltages, respectively. In other words, for topologically complete circuits there exist a matrix property holds will be identified in Section IV. Second, the con- such that, for the resistors contained in ,we dition of convexity of the energy functions required for Proposi- have that , while for the resistors contained in tion 2 is sufficient, but not necessary for passivity of the dynamic we have , with . L and C elements. Hence, the class of admissible dynamic elements is more restrictive. B. Generation of New Storage Function Candidates Remark 3: It is interesting to remark that the supply rate Let us next see how the Brayton–Moser equations (13), can of the new passive systems defined by either the product be used to generate storage functions for RLC circuits. Suppose or , relates with an alternative we multiply (13) by , i.e., definition of reactive power. The interested reader is referred to, e.g., [5] and [11] for more details on this subject.

IV. PASSIVITY OF BRAYTON–MOSER CIRCUITS which, after reorganizing the terms, yields the following equa- The previous developments show that, using the content and tion: co-content as storage functions and the reactive power as supply rate, we can identify new passivity properties of RL and RC circuits. In this section, we will establish similar properties for RLC (14) circuits. Toward this end, we strongly rely on some fundamental results reported in [1]. Furthermore, we assume that the That is, consists of the sum of a quadratic term current-controlled resistors , with , plus the inner product of the source port variables in the desired are contained in and the voltage-controlled resistors form (compare with the left-hand side , with , are contained in . The of (7) of Proposition 2). Unfortunately, even under the reason- class of RLC circuits considered here is then composed by an able assumption that the inductor and capacitor have convex interconnection of and . energy functions, the presence of the negative sign in the first main diagonal block of makes the quadratic form sign-in- A. Brayton and Moser’s Equations definite, and not negative (semi-)definite as desired. Hence, we In the early 1960s, Brayton and Moser [1] have shown that the cannot establish a power-balance inequality from (14). More- dynamic behavior of a topologically complete1 circuit (without over, to obtain the passivity property an additional difficulty external sources) is governed by the following differential equa- stems from the fact that is also not sign-definite. tions: To overcome these difficulties we borrow inspiration from [1] and look for other suitable pairs, say and , which we call admissible, that preserve the form of (13). More precisely, we want to find matrix functions , (11) with , verifying

1A circuit is called “topologically complete” if it can be described by an in- (15) dependent set of inductor currents and capacitor voltages such that Kirchhoff’s laws are satisfied. For a detailed treatment on topologically completeness, see 2Restricting, for simplicity, to circuits having only voltage sources in series [10]. with the inductors. JELTSEMA et al.: ON PASSIVITY AND POWER-BALANCE INEQUALITIES OF NONLINEAR RLC CIRCUITS 1177 and scalar functions (if possible, positive semi- must ensure that the transformed dynamics (16) can be ex- definite), such that the circuit dynamics (13) can be (re)written pressed in the form (13), which is equivalent to requiring that as . This naturally restricts the freedom in the choices for and in Proposition 3. (16) Theorem 1: Consider a (possibly nonlinear) RLC circuit sat- If we multiply (16) by like before, we have that isfying (13). Assume the following. A.1. The inductors and capacitors are passive and have strictly convex energy functions. A.2. The voltage-controlled resistors in are passive, linear, and time-invariant. Also, , and thus by taking the sum of (6) we have that Hence, if the symmetric part of is negative semi-definite, for all . that is, if (15) is satisfied and thus , we may state A.3. Uniformly in ,wehave (noting that ) that

where denotes the spectral norm of a matrix. from which we obtain a power-balance inequality with the de- Under these conditions, we have the following power-balance sired port variables. Furthermore, if is positive semi-def- inequality: inite we are able to establish the required passivity property. In the proposition below, we will provide a complete char- acterization of the admissible pairs and . For that, (19) we find it convenient to use the general form (11), i.e., where the transformed mixed-potential function is defined as , where for the case considered here . Proposition 3: For any and any constant symmetric matrix

(17) If, furthermore (18) A.4. The current-controlled resistors are passive, i.e., . Proof: A detailed proof of (17) and (18) can be found in Then, the circuit defines a passive system with power port [1, p.19]. variables and storage function the transformed An important observation regarding Proposition 3 is that, for mixed-potential . suitable choices of and , we can now try to generate a ma- Proof: The proof consists in first defining the parameters trix with the required negativity property, i.e., and of Proposition 3 so that, under the conditions A.1–A.4 . of the theorem, the resulting satisfies (15) and is a pos- Remark 5: Since has the units of power and itive semi-definite function. a quadratic term in the gradient of (see (18)), First, notice that under assumption A.2 the co-content is also has the units of power. A similar discussion holds linear and quadratic. To ensure that is linear in ,asis for , which is the mixed-potential without the sources. required to preserve the desired port variables, we may select The difference between and the original mixed-potential and . Now, using (17) we obtain is that we have “swapped” the resistive terms. However, after some straightforward calculations the solution of the differential equation (13) precisely coincides with the solution of (16), i.e.,

Assumption A.1 ensures that and are positive definite. Hence, a Schur complement analysis [4] proves that, under Remark 6: Some simple calculations show that a change of Assumption A.3, (19) holds. This proves the power-balance in- coordinate , on the dynam- equality. Passivity follows from the fact that, under Assumption ical system (11) acts as a similarity transformation on . A.2 and A.4, the mixed-potential function is posi- Therefore, this kind of transformation is of no use for our pur- tive semi-definite for all and . This completes the proof. poses where we want to change the sign of to render the quadratic form sign-definite. Remark 7: Assumption A.3 is satisfied if the voltage-controlled resistances are “small.” Recalling that these C. Power-Balance Inequality and the New Passivity Property resistors are contained in , this means that the coupling be- Before we present our main result, we first remark that tween and , that is, the coupling between the inductors in order to preserve the port variables ,we and capacitors, is weak. 1178 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 50, NO. 9, SEPTEMBER 2003

Remark 8: We have considered here only voltage sources. Some preliminary calculations suggest that current sources can be treated analogously using an alternative definition of the mixed potential. Furthermore, it is interesting to under- score that from (14) we can obtain, as a particular case with , the new passivity property for RL circuits of Proposition 2, namely

However, the corresponding property for RC circuits Fig. 2. Simple RLC circuit with nonlinear current-controlled resistor. does not follow directly from (14), as it requires the utilization VI. CONCLUDING REMARKS of (3) instead of (2), as done above. Our main motivation in this paper was to establish a new passivity property for RL, RC, and a class of RLC circuits. We have V. E XAMPLE proven that for this class of circuits it is possible to “add a differ- Consider the RLC circuit depicted in Fig. 2. For simplicity, entiation” to the port variables preserving passivity with respect assume that all the circuit elements are linear and time-invariant, to a storage function which is directly related to the circuit’s except for the resistor . The voltage-current relation of power. The new supply rate naturally coincides with the defini- is described by . The interconnection matrix tion of reactive power. , the content and the co-content are readily Instrumental for our developments was the exploitation of found to be , and Tellegen’s theorem. Dirac structures, as proposed in [9], pro- , respectively, and thus, the mixed- vide a natural generalization to this theorem, characterizing in potential for the circuit is an elegant geometrical language the key notion of power preserving interconnections. It seems that this is the right notion to try to extend our results beyond the realm of RLC circuits, e.g., to mechanical or electromechanical systems. A related question is whether we can find Brayton–Moser like models for this class of systems. Hence, the differential equations describing the dynamics of the There are close connections of our result and the shrinking circuit are given by dissipation Theorem of [12], which is extensively used in analog very large-scale integration circuit design. Exploring the rami- fications of our research in that direction is a question of signif- icant practical interest.

ACKNOWLEDGMENT The authors would like to thank R. Griño for his useful re- The new passivity property is obtained by selecting and marks regarding the interpretation of the new storage function , yielding that if and only as a reactive power. R. Ortega would like to express his grati- if tude to B. E. Shi, along with whom this research was started.

(20) REFERENCES [1] R. K. Brayton and J. K. Moser, “A theory of nonlinear networks—I,” Quart. App. Math., vol. 22, no. 1, pp. 1–33, 1964. Under the condition that and , positivity [2] C. A. Desoer and E. S. Kuh, Basic Circuit Theory. New York: Mc- of is easily checked by calculating (18), i.e., Graw-Hill, 1969. [3] D. Hill and P. Moylan, “The stability of nonlinear dissipative systems,” IEEE Trans. Automat. Contr., pp. 708–711, Oct. 1976. [4] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [5] N. LaWhite and M. D. Ilic´, “Vector space decomposition of reactive power for periodic nonsinusoidal signals,” IEEE Trans. Circuits Syst. I, vol. 44, pp. 338–346, Apr. 1997. [6] W. Millar, “Some general theorems for nonlinear systems possesing resistor,” Phil. Mag., vol. 42, pp. 1150–1160, 1951. [7] R. Ortega, D. Jeltsema, and J. M. A. Scherpen, “Stabilization of non- In conclusion, if (20) is satisfied, then the circuit of linear RLC circuits via power-shaping,” presented at the Latin American Conf. on Automatic Control, Guadalajara, México, Dec. 2002. Fig. 2 defines a passive system with power port variables [8] P. Penfield, R. Spence, and S. Duinker, Tellegen’s Theorem and Elec- and storage function . trical Networks. Cambridge, MA: MIT Press, 1970. JELTSEMA et al.: ON PASSIVITY AND POWER-BALANCE INEQUALITIES OF NONLINEAR RLC CIRCUITS 1179

[9] A. J. van der Schaft, v -Gain and Passivity Techniques in Nonlinear Romeo Ortega (S’76–M’80–SM’98–F’99) was born Control. London, U.K.: Springer-Verlag, 2000. in Mexico. He received the B.Sc. degree in electrical [10] L. Weiss, W. Mathis, and L. Trajkovic, “A generalization of Brayton- and mechanical engineering from the National Uni- Moser’s mixed potential function,” IEEE Trans. Circuits Syst. I, pp. versity of Mexico, Mexico city, Mexico, the Master 423–427, Apr. 1998. of Engineering degree from the Polytechnical Insti- [11] J. L. Wyatt and M. D. Ilic´, “Time-domain reactive power concepts for tute of Leningrad, Leningrad, U.S.S.R., and the Doc- nonlinear, nonsinusoidal or nonperiodic networks,” Proc. IEEE Int. teur D’Etat degree from the Politechnical Institute of Symp. Circuits Syst., vol. 1, pp. 387–390, May 1990. Grenoble, Grenoble, France in 1974, 1978, and 1984, [12] J. L. Wyatt, “Little-known properties of resistive grids that are useful in respectively. analog vision chip designs,” in Vision Chips: Implementing Vision Algo- He then joined the National University of Mexico, rithms with Analog VLSI Circuits, C. Koch and H. Li, Eds. Piscataway, where he worked until 1989. He was a Visiting Pro- NJ: IEEE Computer Science Press, 1995. fessor at the University of Illinois Urbana, in 1987–1988 and at the McGill Uni- versity, Montreal, QC, Canada, in 1991–1992. Currently, he is with the Labora- toire de Signaux et Systemes (SUPELEC), Paris, France. His research interests are in the fields of nonlinear and adaptive control, with special emphasis on applications. Dr. Ortega is a member of the IFAC Technical Board and chairman of the IFAC Coordinating Committee on Systems and Signals, and has been a member of the French National Researcher Council (CNRS) since June 1992, and a Fellow of the Japan Society for Promotion of Science in 1990–1991. He is an Associate Editor of Systems and Control Letters, the International Journal of Adaptive Control and Signal Processing, the European Journal of Control and the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY.

Dimitri Jeltsema (S’03) received the B.Sc. degree Jacquelien M. A. Scherpen received the M.Sc. and in electrical engineering from the Rotterdam School Ph.D. degrees in applied mathematics from the Uni- of Engineering, Rotterdam, The Netherlands, and versity of Twente, Twente, The Netherlands, in 1990 the M.Sc. degree in systems and control engineering and 1994, respectively. from the University of Hertfordshire, Hertford, Currently, she is an Associate Professor at the U.K., in 1996 and 2000, respectively. He is currently Delft Center for Systems and Control, Delft Uni- working toward the Ph.D. degree at the Delft versity of Technology, Delft, The Netherlands. She Center of Systems and Control, Delft University of has held visiting research positions at the Universite Technology, The Netherlands. de Compiegne, Compiegne, France, Laboratoire de During his studies, he worked as an Engineer in Signaux et Systemes (SUPELEC), Gif-sur-Yvette, several electrical engineering companies. During France, the University of Tokyo, Tokyo, Japan, the 2002, he was a visiting student at the Laboratoire de Signaux et Systemes Old Dominion University, Norfolk, VA, and the University of Twente. Her (SUPELEC), Paris, France. His research interests are nonlinear circuit theory, research interests include nonlinear model reduction methods, realization power electronics, switched-mode networks, and physical modeling and theory, nonlinear control methods, with in particular modeling and control of control techniques. physical systems with applications to electrical circuits. Mr. Jeltsema is a student member of the Dutch Institute of Systems and Con- Dr. Scherpen is an Associate Editor of the IEEE TRANSACTIONS ON trol (DISC). AUTOMATIC CONTROL.