GENERALIZATION TO NON-LINEAR NETWORKS OF A THEOREM DUE TO HEAVISIDE

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1 R 382 Philips Res. Repts 14, , 1959 GENERALIZAION O NON-LINEAR NEWORKS OF A HEOREM DUE O HEAVISIDE Summary by S. DUINKE~ A theorem enunciated by Heaviside and proved by. Lorentz for linear electro-magnetic systems, subjected to a constant electric force which is suddenly impressed, is extended to electrical networks comprising nonlinear reactances and linear. dissipances. he theorem states that the amount of work to be done by a constant-voltage source exclusively in favour of the transients bringing the network from the rest state to the steady state, is equal to the excess of the sum of electric energy and coenergy over the sum of magnetic energy and eo-energy. Rêsumê On généralise aux réseaux électriques comprenant des réactances nonlinéaires et des dissipances linéaires un théorème énoncé par Heaviside et démontré par Lorentz pour des systèmes électro-magnétiques linéaires sournis à un champ électrique constant brusquement appliqué. Ce théorèmè exprime que l'énergie qu'une source de tension constante doit fournir pour alimenter uniquement les effets transitoires amenant le réseau de I'état de repos à l'état stationnaire est égale à I'excès de la somme de l'énergie et de la co-énergie électriques sur la somme de I'énergie et de la co-énergie magnétiques. Zusammenfassung Ein durch Heaviside ausgesprochenes und durch Lorentz bewiesenes heorem für lineare elektromagnetische Systeme, die einer konstanten, plötzlich eingeschalteten elektrischen Kraft unterworfen sind, wird für elektrische Netzwerke, die nicht-!ineare Reaktanzen und!ineare Dissipanzen enthalten, ausgebreitet. Das heorem besagt, dab der eil der Energie, der von einer Quelle konstanter Spannung an ein Netzwerk ausschlieblich für den Ausgleichsvorgang vom Ruhezustand in den Beharrungszustand geliefert wird, genau dem ÜberschuB der Summe der elektrisch en Energie und Ko-energie über die Summe der magnetischen Energie und Ko-energie entspricht. 1. Introduetion In his "Electrical Papers", Heaviside 1) stated without proof for linear electromagnetic systems that: "he whole work done by impressed forces suddenly started exceeds the amount.representing the waste by Joule heating at the final rate (when there is any), supposed to start at once, by twice the excess of the electric over the magnetic energy of the steady field set up". An indication ofthe way in which this result presumably was obtained can be found at several places in Heaviside's "Electromagnetic heory" 2). Lorentz 3) gave a proof of the statement based on Maxwell's equations, which, although being confined to the Iinear case, can be readily generalized so as to comprise the case in which non-linear magnetics and dielectrics are

2 422 S. DUINKER present. o that end one has to introduce, in addition to the magnetic and the electric energy, viz., =JdVJ!ldB and U=JdVJ EdD, respectively, the notions of magnetic and electric eo-energy, viz., ' = J d V J BdH and U' = J d V J DdE, respectively. For linear networks ellegen 4) has proved the theorem in a different way and has further shown it to be related to other network theorems. Still other but closely connected theorems, linking the final steady state to the impedance oflinear networks, have been derived byvan der Pol 5), Parodi 6) and LuneIIi 7). hese theorems, however, do not lend themselves to a generalization towards non-linear networks. For networks comprising non-linear reactances but linear dissipances we shall give a proof of the Heaviside theorem which is shorter than the one presented by Lorentz. We shall start from the equations of generalized dynamics and use, after Cherry 8), the notions of magnetic co-energy ' and electric eo-energy U'. In the case of non-linear network elements the latter quantities are given by ' = J qj(i)di and U' = J q(e)de,. where qj(i) and q(e) represent single-valued flux-current and charge-voltage characteristics, respectively. 2~ Proof of the theorem Cherry has shown 8) that the behaviour of an electrical network comprising non-linear reactances but linear dissipances, if. considered as a dynamical system with n degrees of freedom, is described by the Lagrangian equations: d st.»t: DkXk = yk, k = 1,..., n. dt 'öxk 'öxk If the impressed forces yk correspond to voltage (current) sources, the coordinates Xk and their velocities x» correspond to mesh charges qk (node "fluxes" qjk) and their mesh currents Ïk = qk (node voltages Vk = CPk), respectively. he Lagrangian function L is given by L = '(ql,. "'. qn) - U(ql,..., qn), and the dissipance Dk by a resistance Rk, if yk corresponds to a voltage source ek, whereas L = U'(cpl,..., cpn) - (qjl,..., qjn), (2) and Dc corresponds to a conductance Gk, if y» represents a current source ik. I (1)

3 0 GENERALIZAION OF A HEOREM DUE O HEAVISIDE 423 We assume for simplicity that the network, which is originally at rest, contains only a single constant-voltage source in mesh n, which is started at the moment t = O. hen the analysis is based on mesh equations (1) with charges qk as the coordinates, while Yk = e» = 0, for k = 1,..., n - 1, and yn = en = O, ~ E, for for t ~O, t» O. At the time t = (tending to infinity), when the transients have died down, a steady state exists characterized by d.c. mesh currents which mayor may not be zero. he distribution of these currents is determined by the voltage equations: (3) in which asterisks denote the values of the various quantities in the steady state. he sum of the-electric energy and ditto co-energy in the steady state amounts to making use of eq. (3) and observing that a mesh charge qk* = I qkdt o assumes a constant final value only ifthe corresponding capacitive mesh voltage (?JL(öqk)* is equal to a non-zero constant, i.e., ifthe mesh contains a capacitor. 0 he energy delivered by the voltage source up to t = is found to be Hence (4) (5) he sum of the magnetic energy and ditto co-energy in the steady state amounts to n * + '* = ~ qé (?JL/?Jqk)* k=l 11 d?jl = ~ qk* I - - dt k= 1 0 dt?jqk when use is made of eqs (1). n?jl = ~ qk* I (ek Rk qk) dt, 0 k=l 0?Jqk (6)

4 424 S. DUINKER If a mesh (e.g., k) carries a current in the final state, i.e. if qk* =ft 0, it cannot contain a capacitor and consequently bl/bqk must be equal to zero. Conversely, if the mesh k does contain capacitors, i.e. if bl/bqk =ft 0, it cannot maintain a current in the steady state. Hence the expression (6) becomes where (7) represents the "pseudo" heat dissipation which would have occurred if the constant final current distribution would have been present all the time from t = to t = 7. From eqs (5) and (7) we find by subtraction U* + U'* -:- (* + '*) = A- W/, (8) which is the desired result. Since the excess of the energy A delivered by the voltage source over the "pseudo" heat dissipation W/ equals exactly the amount of work that has to be done by the source exclusively in favour of the transients which bring the network from the initial rest state to the final steady state, the result expressed by (8) can be formulated as follows: heorem. In a network comprising non-linear reactances and linear resistances the amount of work, to be done by a constant-voltage source exclusively in favour of the transients bringing the network from the rest state to the steady state, is equal to the excess of the sum of electric energy and eo-energy over the sum of magnetic energy and co-energy, all energies as stored in the steady state. 3. Remarks (i) he energy A delivered by the voltage source up to t = 7, instead of being expressed by eq. (4), can alternatively be written as follows: (9)" if we denote the actual dissipation up to t = 7 by W' he elimination of A from eqs (8) arid (9) yields an expression for the excess of the actual dissipation over the pseudo dissipation, viz.,. U'*- (2* + '*) =. W,- W/. (10).It is to be noted that the quantities A, W and W/, if not equal to zero, depend on the length of the interval 7. However, the differences of these quanti-

5 GENERALIZAION OF A HEOREM DUE O HEA VISIDE 425 ties (i.e., A": - W./, A - W and W - Wor') attain a constant final value as soon 'as the transients have died down. (U) It is easy to see that the results expressed by eqs (8) and (10) hold good if a number of constant-voltage sources is simultaneously connected to the network. Equivalent results are also obtained for a network, initially at rest, when it is connected to a number of constant-current sources suddenly started at t = O. In this latter case the analysis has to be based on nodal equations (1), using a Lagrangian function of the form (2) which is expressed in terms of node voltages VK and their time integrals, the node "fluxes" epk = J Vkdt. (iii) he validity of the theorems expressed by eqs (8) and (10) is easily demonstrated for a simple network with a single degree of freedom, comprising a linear resistance R in series with an inductor whose flux-current relationship is given by a non-linear characteristic, ep= ep(i). he differential equation corresponding to this network is for which we shall write dep/dt + ir = E, - CP/R= i- E/R _: i- i*, where i* = E/R denotes the current in the final steady state. he excess' of the work A over the pseudo dissipation Wor' up to the time t = -;. 00 will consequently be A - W./ = J (Ei- i*2r)dt = - (E/R) f cp dt o 0 = -i*cp* = -(* + '*), which confirms eq. (8), since U* = U'* = 0 owing to the absence of capacitors. For the excess of the actual dissipation over the pseudo dissipation we find Wor- W./ = J (i2r - i*2r)dt = J (Ei - i cp - i*2r)dt o 0 = - (* + '*) - f i cp dt = - (2* + '*), o which confirms eq. (10). he fact that, without knowing the transient behaviour in detail (i.e., without knowing the general solution of the non-linear network equations which is hard, to find, if at all), the total work and the total dissipation involved with the transients can be written down immediately, is rather peculiar. (iv) he exclusion of non-linear dissipances is not a matter of convenience, but a necessity; For example, non-linear resistors find expression in eq. (1) by a

6 426 S. DUINKER term Vk = Vk(qk) representing the non-linear characteristic, instead of by the term Rkqk. Consequently eq. (5) becomes and eq. (7) assumes the form * + '* = W./ - ~ qk* f Vk dl, k=1 0 so that we have to write, instead of eq. (8), U* + U'* - (* + '*) = A - W/ + ~ f (Vkqk* - vk*qk)dt. k=1 0 Obviously, the integral vanishes only if Vk = Rkqk. n n Eindhoven, June 1959 REFERENCES 1) O. Heaviside, "Electrical Papers", Vol. Il, p. 412.,2) O. Heaviside, "Electro-magnetic heory", Benn Bros Ltd., London 1922, Vol. I, pp.113, 310; Vol. III, pp. 95, 127,456. 3) H. A. Lorentz, Proc. Nat. Acad. Sci. 8, 333, 1922 = "Collected Papers", M. Nijholf, he Hague 1936, Vol. III, p ) B. D. H. ellegen, Philips Res. Repts 7, 259, ) B. van der Pol, Physica 4,585, ) M. Parodi, "Introduction à l'étude des réseaux électriques", S.E.D.E.S., Paris 1948, p ) L. Lunelli, Alta Frequenza 24,246, ) E. C. Cherry, Phi). Mag.42, 1161, 1951.

Philips Research Reports':

VOL.6 R 2 APRIL 1951 Philips Research Reports': EDITED BY THE RESEARCH LABORATORY OF. V. PHILIPS' GLOEILAlIIPEFABRIEKE, EIDHOVE, ETHERLADS R 160 Philips Res. Rep. 6, 81-85, 1951 THE DISCHARGE OF A SERIES