Current and Voltage Excitations for the Eddy Current Model

Transcription

1 Eigenössische Technische Hochschule Zürich Ecole polytechnique féérale e Zurich Politecnico feerale i Zurigo Swiss Feeral Institute of Technology Zurich Current an Voltage Excitations for the Ey Current Moel R. Hiptmair an O. Sterz Research Report No July 2003 Seminar für Angewante Mathematik Eigenössische Technische Hochschule CH-8092 Zürich Switzerlan IWR, Universität Heielberg, Im Neuenheimer Fel 368, 6920 Heielberg, Germany

2 Current an Voltage Excitations for the Ey Current Moel R. Hiptmair an O. Sterz Seminar für Angewante Mathematik Eigenössische Technische Hochschule CH-8092 Zürich Switzerlan Research Report No July 2003 Abstract We present a systematic stuy of how to take into account external excitation in the ey current moel. Emphasis is put on mathematically soun variational formulations an on lumpe parameter excitation through prescribe currents an voltages. We istinguish between local excitation at known contacts, known generator current istributions an non-local variants that rely on topological concepts. The latter case entails the violation of Faraay s law at so-calle cuts an prevents us from reconstructing a meaningful electric fiel. IWR, Universität Heielberg, Im Neuenheimer Fel 368, 6920 Heielberg, Germany

3 Introuction In orer to obtain computationally tractable moels of electric evices often one consiers the full fiel equations only for parts, whereas a lumpe circuit escription is use for the remainer. This makes it necessary to couple both moels. In other wors, we have to figure out how to link the quantities use in circuit moels, currents an voltages, with electromagnetic fiels. We investigate this issue in the case of (weak) variational formulations of the magneto-quasistatic ey current moel. We can istinguish between two basically ifferent weak formulations, voltage or current excitation, an local or non-local excitation. We aim to give a systematic treatment of all these cases leaing to stable variational formulations that can serve as the basis for finite element iscretization schemes. Our presentation is part of brisk recent research in this area: without even pretening to give an exhaustive list of references, we woul like to mention the pioneering contributions of P. Dular [DGL99, DLN98, DHL99] an, in particular, [Dul0]. Other researchers have consiere the fiel equations after iscretization, viewing them as small-scale circuit equations, see e.g. [RTV02]. Moreover, it has been realize that non-local excitation is intimately linke to topological issues [Ket0, Bos00]. It is important to note, that a true lumpe parameter excitation is not feasible for the ey current moel, e.g. there is no canonical way to impose a current or voltage without aitional moeling information on the istribution of the sources. Even in the case of the so-calle nonlocal excitations, that rely on topological concepts, aitional information will be necessary if a coupling to a circuit moel shall be establishe.. Ey Current Moel In the sequel we consier an open an connecte computational omain R 3 with exterior unit normal vector fiel n on its bounary. Let be partitione into two isjoint open subsets, that is = I, where represents the conuctors an I the insulating air region. Generically, we have = R 3, but introucing an artificial cut-off bounary will rener it boune. Thus, throughout the remainer of the paper, we can restrict ourselves to boune. We aopt the notation Γ C = I an write Γ i for the bounaries of the connecte components of, Γ C = i Γ i. The complete ey current moel reas [AFV0] curl e = tb in (Faraay s law), (a) curl h = j in (Ampere s law), (b) iv b = 0 in (magnetic Gauß law), (c) b = µh in, () j = j G + σ e in (Ohm s law), (e) where e an h stan for the electric an magnetic fiel, respectively, b enotes the magnetic inuction, j the current ensity, an j G = j G (x,t) is an impose current ensity (generator current). We assume that supp j G is boune an that j G epens continuously on time. The material parameter σ L () escribes the electric conuctivity, an satisfies σ > σ 0 > 0 in, σ 0 in I. The coefficient µ L (), µ > µ 0 > 0 everywhere in, is the magnetic permeability. We will also make use of the ielectric constant ɛ L (), ɛ > ɛ 0 > 0 everywhere. These equations have to be supplemente by suitable bounary conitions on the (artificial)

4 bounary. Below we impose electric bounary conitions n e = f on the part e of the bounary, an magnetic bounary conitions n h = g on h, where = e h, e h =. Both f an g epen on space an time an are assume to be eligible tangential traces of fiels. We point out that in the case of unboune we have to eman a ecay of the fiels accoring to h(x) =O( x 2 ), e(x) =O( x ) as x. In variational formulations we nee to use suitable weighte spaces [Hip02b, Sect. 2]. Then all arguments employe in the case of boune remain vali. We also stress, that non-linear, in particular ferromagnetic, materials are amitte, that is, µ is allowe to epen on b. Though not incorporate in the above formulation, tensorial material parameters can be easily accommoate, too. We opte for a treatment in the time omain, but most results carry over to the frequency omain after replacing t with iω, ω > 0 the angular frequency. Equation (b) implies a compatibility conition for for j G in I : For any sub-omain V I must hol j G n S =0. V This implies iv j G = 0 in I. Note that (c) has to be enforce at initial time only, since iv b = 0 follows from (a). t The ey current moel represents a magneto-quasistatic approximation to Maxwell s equations: it is reasonably accurate for slowly varying fiels, for which the change in magnetic fiel energy is ominant, see [ABN00, Dir96]. Slowly varying, means that ɛ µ l δt (2) where l is the characteristic size of the region of interest an δt the smallest relevant time-scale. This means that has to be small compare to the wavelength of electromagnetic waves, which makes it possible to ignore wave propagation. If this cannot be one, the issue of voltage an current excitation becomes far more tricky an might not make much sense any more. This is beyon the scope of this paper. There is a secon conition for the valiity of the ey current approximation, which requires that the typical time-scale is long compare to the relaxation time for space charges, that is, the conuctivity must be large enough to make ɛ σδt (3) hol true. This implies that no space charges nee to be taken into account. We point out an important ifference between the full Maxwell equations an the ey current equations (). Whereas in the former there is a perfect symmetry between electric an magnetic quantities, this symmetry is broken in the case of the ey current moel. This is a consequence of ropping the isplacement currents. 2

5 It is important to note that the equations () o not completely etermine the electric fiel in I. There it is unique only up to a curl-free electrostatic component. In orer to restore uniqueness of e everywhere in we may eman the gauge conition iv(ɛ e) = 0 in I, an that all total conuctor charges will vanish. For an in-epth iscussion in the time-harmonic case an with homogeneous electric (n e = 0 on ) an magnetic (n h = 0 on ) bounary conitions on we refer to [AFV0]. However, the electric fiel in I is of little interest in many applications. It might not even make much physical sense, for instance close to inuctor coils that are escribe by a given current ensity istribution. In this case an ungauge formulation makes perfect sense..2 Variational Formulations Corresponing to primal an ual ways to cast secon-orer elliptic bounary value problems into weak form, we can istinguish between two ifferent variational formulations of (), which use one of the equations (a) or (b) in strong form ( pointwise ) an the other in weak form ( average ) [Bos85]. h-base formulation This formulation retains the magnetic fiel h as primary unknown. The variational problem relies on the function spaces an V(j g, g) := {h H(curl; ), curl h = j g in I, n h = g on h } (4) V 0 := V(0, 0). (5) For efinitions an properties of function spaces like H(curl; ) the reaer is referre to [GR86, Ch. ]. Then, the variational formulation escribing the evolution of the magnetic fiel from t = 0 to t = T, T>0 fixe, reas: Fin h C ([0,T], V(j g, g)), h(0) = h 0 V(j g (0), g(0)), such that for all h V 0 σ curl h curl h x + t (µh) h x = σ j G curl h x + (n }{{ e } ) h S. (6) =f This equation contains Ampere s law (b) in strong form, Faraay s law (a) in weak form. e a-base formulation Since (c) is true in all of R 3, we can write b = curl a. Then e = ta gra v with v staning for a scalar potential. To fix a we can rely on the so-calle temporal gauge, which sets v = 0, that is e = ta. We introuce the space W(f) := {a H(curl; ), n a = f t on e } The requirement that h is continuously ifferentiable w.r.t. time can be relaxe, cf. [RR93, Sect. 0.]. Then excitations that jump in time can be accommoate. 3

6 an arrive at the following variational formulation: Fin a C ([0,T], W(f)), a(0) = a 0 W(f(0)), such that for all a W(0) µ curl a curl a x + σ t a a x = j G a x (n }{{ h } ) a S. (7) =g The reaer shoul be aware that this is an ungauge formulation, since extra conitions are neee to etermine a in I uniquely. Such ungauge formulations will be use throughout this paper. Coulomb type gauges enforcing the orthogonality of a to all graients in I are straightforwar. It turns out that these two variational formulations possess vastly ifferent properties as far as lumpe parameter excitation is concerne. Therefore, we will iscuss ifferent kins of current an voltage excitation separately for the h- an a-base equations. To begin with, we will ruminate about the meaning of current an voltage in the next section. Then, we treat ifferent cases of excitations an we will show how to incorporate them into the variational equations (6) an (7). As soon as a variational formulation is establishe a conforming Galerkin finite element iscretization base on a given mesh of is reaily available: H(curl; )-conforming ege elements [Hip02a, Sect. 3] can be use to approximate both a an h an the corresponing test spaces, whereas piecewise polynomial globally continuous Lagrangian finite elements have to be use for scalar potentials. Skirting issues of bounary fitting an numerical quarature, this instantly yiels systems of equations that represent the iscretize problem. In light of this canonical proceure, we will not aress iscretization below, unless special provisions have to be taken. h 2 Funamentals of Excitation It is straightforwar how to obtain the global current I flowing in a loop of the conuctor or through a contact from the ey current solution: the current through an oriente surface Σ is reaily available from I = j n S. Σ Conversely, it is ifficult to give a meaning to the concept of voltage. To begin with, it is only meaningful relate to an oriente path γ via the formula U γ = e s. (8) γ Obviously, U is not only a function of the enpoints of γ, because curl e 0. On the other han, circuit moeling assumes a unique voltage between two noes. Using (8), coupling of fiels an circuits cannot be accomplishe. We recall, that the voltage between two noes of a circuit measures the ifference in potential energy of charge carriers. This suggests that we efine voltage base on power flux. Seen from the circuit the fiel omain is a network component with one or more ports. If there is only one port an P enotes the total power fe into the fiel omain, we can use the formula P := U I 4

7 to efine the voltage rop U between the two contacts. Summing up, we rely on the conservation of current an power to establish the coupling of the ey current moel with circuit equations. If there are several ports, the situation is not so clear, because it will be only the total power consume in the fiel region that is accessible. However, it will turn out that the methos evise to realize voltage excitation possess some inherent locality, which makes it easy to relate a voltage to contacts on inuctor loops. Using a pragmatic superposition principle, we give a meaning to iniviual voltages for each of the several ports of the fiel region. From () we euce the following power balance, cf. Poynting s theorem, where P mag := P := P mag + P Ohm = P = P + P (9) t b h x, P Ohm := e j G x, P = σ e 2 x (e h) n S. We see that two types of power sources occur in the ey current moel, which compensate the Ohmic losses P Ohm an the magnetic power P mag, that inclues a change in the magnetic energy an magnetic losses: Firstly, sources P ue to impresse currents, which push charges against the electric fiel e. Seconly, sources P ue to bounary conitions. We have chosen the signs in a way that reners P,P positive, if the sources fe power into the ey current moel. Remark. In the frequency omain power balance (9) has to be state as iω 2 µ h 2 x + 2 σ e 2 x = 2 e j G x 2 (e h ) n, (0) where all fiels are complex amplitues (phasors) an labels complex conjugates. The real part of eq. (0) states the balance of the so calle active power, which is the mean value of the power with respect to one perio that compensates losses. The imaginary part of eq. (0) states the balance of the so calle reactive power which is the maximum value of the change in the magnetic energy. Topological preliminaries Here, we give a cursory escription of topological concepts inispensable for the unerstaning of lumpe parameter excitation. More information can be foun in [Kot87, Bos98a, Bos00, Ket0] an textbooks about homology theory. First, we recall the notion of an orientable l-imensional piecewise (p.w.) smooth manifol in a generic open boune omain R 3. For l = these are irecte paths, for l = 2 the reaer may think of surfaces equippe with a crossing irection. We appeal to geometric intuiting to introuce the concept of a bounary Σ of an l-imensional piecewise smooth manifol Σ. The bounary carries an inuce orientation an is itself an l -imensional p.w. smooth manifol. We call an l-imensional piecewise smooth oriente manifol an l-cycle, l =, 2, if its bounary is empty. Two l-cycles are sai to be homologous, if their union (after a possibly 5

8 change of orientation) is the bounary of an l + -imensional manifol, which is a proper subset of. This efines an equivalence relation on the set Z l () of l-cycles. An l-imensional manifol Σ is calle a relative l-cycle, if Σ. Two relative l-cycles are homologous, if their union supplemente by a part of bouns an l + -imensional manifol. This introuces an equivalence relation on the set Z l (, ) of relative l-cycles. The equivalence classes of homologous, non-bouning -cycles in are calle loops. There are β () such loops, where β () enotes the first Betti number of, which is a funamental topological invariant. Cruely speaking, β () is equal to the number of holes piercing. A loop can be escribe by a representative in the form of a close non-bouning oriente path. Moreover, by cuts of we refer to the homology equivalence classes in Z 2 (, ), which can be represente by β () isjoint p.w. smooth oriente surfaces. These cuts are also known as Seifert surfaces. We remark that cuts an loops are ual to each other (Poincaré uality): we can fin representatives for loops an cuts such that they form pairs of paths/surfaces that intersect each other. If Σ,..., Σ N, N := β () stans for a complete set of cuts, then all -cycles in \(Σ... Σ N ) are bouning. In particular, every curl-free vectorfiel in \(Σ... Σ N ) agrees with a graient of a scalar function. If we consier a curl-free vectorfiel in all of, it nee not be a graient. However there is a finite-imensional co-homology space H () H(curl; ) that satisfies im H () =β () such that {u H(curl; ), curl u =0} = gra H () H (). 3 Excitation by Generator Current Distributions The generator currents j G play the role of right han sies in the ey current equations. As such they are natural caniates for realizing an excitation. Prescribe generator currents j G can be put into two ifferent categories: (a) (see Fig., left) Close current loops (strane inuctors in the parlance of [Dul0]) in I, that is supp j g I an β (supp j g ) =, which moel coils with known currents. (b) (see Fig., right) Current sources ajacent to conuctors, i.e., supp j g. Note that supp j g shoul have an aitional loop compare to. In both cases we en up with the same variational formulations. We mainly restrict ourselves to the case of one port so that we have to take into account either one total current I or one voltage U. In aition, we make the assumption suppj g I an use homogeneous electric bounary conitions n e = 0 on. The case supp j G is not completely irrelevant, as it represents a source with internal resistivity, but it can be easily moele on the circuit level. 3. h-base formulation Current excitation A prescribe total current is implie by the choice of j G. In the situation (b), see Fig., this fixes the total current in the conucting loop as a consequence of Ampere s law, which is present 6

9 (a) (b) j G I I j G Figure : Two typical situations encountere in the case of prescribe j G : left (a): close inuctor loop etache from ; right (b): current source ajacent to. in the variational formulation in strong form. Choose h G C ([0,T], H(curl; )) such that curl h G = j G in I for all times. In situation (a) we coul use the Biot-Savart law to compute such a h G, but for practical computations the support of h G shoul be as small as possible, which can be achieve by techniques, presente, for instance, in [DHR + 97]. Please note that Ampere s law usually rules out supph G = supp j G. Given h G we get: Seek h h G + C ([0,T], V 0 ) 2 such that for all h V 0 σ curl h curl h x + t (µh) h x =0. () Obviously the choice of h G has no impact on the solution h, because for two ifferent excitation fiels h G an h G we have h G h G V 0. As explaine above, the voltage is efine through power. Accoring to (9) the power injecte into the ey current omain is equal to P = σ curl h 2 x + = t (µh) h x σ curl h curl(h G + h ) x + = σ curl h j G x + t (µh) (h G + h ) x t (µh) h G x = t (µh) h G x, where h := h h G. The bottom equality is justifie, because we assume suppj G =. Thus, the voltage is given by U I = 2 For the sake of brevity initial conitions will be roppe in the sequel. t (µh) h G x. (2) 7

10 Let Σ be a cut associate with a loop in the conuctor in the sense of Poincaré uality. Using a normalize generator current scale by the yet unknown total current j G = I j 0, j 0 n S =, (3) we can write (2) U = Σ t (µh) h 0 x, (4) provie that curl h 0 = j 0. Here, the normal of Σ has to be oriente in such a way that we get I U > 0 if power is fe into the ey current moel. Remark 2. In the case of multiple ports, we assume that the associate generator currents have isjoint supports. Thus, (4) gives a voltage for each port. Voltage Excitation Voltage excitation can be achieve by imposing (4) as constraint an using the total current as Lagrangian multiplier, which yiels the variational formulation: Seek h C ([0,T], V 0 ) an I C ([0,T]) such that for all h V 0 σ curl h curl h x + t (µ(h + I h 0)) h x =0 C t (µh) h 0 x = U. Here h 0 C ([0,T], H 0 (curl; )) has to satisfy curl h 0 = j 0 with j 0 as in (3). In the case of multiple ports localization is performe as in Rem. 2 above. (5) 3.2 a-base formulation Current Excitation As in the case of the h-base formulations, the total current I is fixe by specifying j G. In situation (b), see Fig., Ampere s law, which is weakly incorporate into the a-base formulation, ensures that the total current I will flow in the entire conucting loop. The ungauge variational equation is straightforwar: Seek a C ([0,T], H 0 (curl; )) such that for all a H 0 (curl; ) µ curl a curl a x + j G a x. (6) Again, we use the power P = µ curl a curl t a x + to efine the voltage U = P/I = σ t a a x = σ t a t a x = j G t a x = I j 0 t a x j 0 ta x. (7) 8

11 Voltage Excitation Again, a constraint equation can be use to impose a voltage: Seek a C ([0,T], H 0 (curl; )) an I C ([0,T]) such that for all a H 0 (curl; ) µ curl a curl a x + σ t a a x I j 0 a x =0 (8) j 0 t a x = U. 4 Excitation through Bounary Conitions (Contacts at ) Another kin of excitation is supplie by currents impose at contacts Σ = on the bounary. For the sake of simplicity we will only consier two contacts Σ = Σ + Σ on, as sketche in Fig. 2. Two situations can be istinguishe. (a) The contacts are locate where meets the exterior artificial bounary, see Fig. 2, left. (b) The contacts Σ +, Σ an Θ boun a volume, the electromotive region, in which the laws of electroynamics are not in effect, cf. [Dul0, Sect. 3.3]. Aing to creates a new loop an both Σ + an Σ are cuts for this loop, which are suppose to be isjoint. In particular Σ + Σ is the relative bounary of in. (a) (b) I Σ + Σ Σ + Θ Θ Figure 2: Situation (a): excitation through exterior bounary conitions. Situation (b) Bounary conitions on surface of hole in the universe, = R 3 \, Θ = I 4. h-base formulation Voltage excitation Voltage excitation is realize by means of bounary conitions for the electric fiel n e = 0 on \ Θ (9) n e = U(t) gra Γ v on Θ, (20) 9

12 where v Σ + =, v \(Θ Σ + ) =0, v H 2 ( ). (2) Hence, the zone Θ ( I ) is the surface through which power is injecte. Incorporating conitions (9) (2) into the bounary term in (6) yiels (n e) h S = U gra Γ v (n h ) S = = U Θ v curl h n S + U Θ v h s = U γ + h s, where γ + = Σ +. Here we have integration by parts on the surface. The variational formulation taking into account voltage excitation reas: Seek h C ([0,T], V 0 ) such that for all h V 0 σ curl h curl h x + t (µh) h x = U γ + h s. (22) Note that by Ampere s law we have I = h s, γ + as the conuctor is penetrating the loop γ +. Moreover, γ + I an curl h = 0 in I, which reners h γ + h s a continuous functional on V 0. Now, replacing h in (22) with h reveals that P = UI. This means that the voltage U introuce via the electric bounary conitions (9) matches the efinition of voltage base on power, see Sect. 2. Remark 3. The right han sie of (22) is of the form U f(h ), where f is a continuous functional on V 0 measuring the total current through a contact. Remark 4. Note that in situation (a) of Fig. 2 the geometry of Θ oes not enter the variational formulation at all: it has no impact on h. Yet, the electric fiel e in I will epen on Θ. Another observation is that in situation (b), cf. Fig. 2, the path γ + can be any close path wining aroun that loop of the conuctor that is cut by Σ + /Σ. In a sense, the excitation possesses a non-local character. On the other han, the location of the contacts still enters the variational formulation via the omain. Remark 5. Consier situation (b) an assume that Σ + an Σ are parallel with istance δ. We point out that a limit of the magnetic fiel when δ 0 will not agree with the solution of the variational problem when is ignore altogether. To see this note that even for a general geometry of the loop there is never a magnetic flux through Σ ±, no matter how close they are. It is also worth noting that the energy of the electric fiel solution will blow up as δ 0. Remark 6. The generalization of (22) to the multiport setting is obvious, because all occurring surfaces an paths are clearly associate with a particular contacts. In situation (a) one contact woul be assigne the role of groun an the others woul be treate like Σ + in the previous consierations. 0

13 Remark 7. As has become clear, topological concepts pervae the iscussion. A comprehensive iscussion of excitation through bounary conitions from a topological point of view is given in the lanmark paper [Bos00]. It is worth mentioning that also in this paper a consistent energy balance is a key point. Current excitation In the situations epicte in Fig. 2 a total current I C ([0,T]) is impose by prescribing the normal component of a suitable current ensity j on the contacts I + = j n S, I = j n S. Σ + Σ Charge conservation requires I + = I. The orientation of Σ + an Σ is inuce by the orientation of. To obtain I U > 0 if power is fe into the ey current moel we set I := I = I +. Furthermore, we chose h jn C ([0,T], H(curl; )) such that iv Γ (h jn n) =curl h jn n =(j n)/i on, curl h jn = 0 in I an efine V + 0 := {h H(curl; ); curl h = 0 in I, iv Γ (h n) = 0 on }. This leas to the following variational formulation: Seek h I h jn + C ([0,T], V + 0 ) such that for all h V + 0 σ curl h curl h x + t (µh) h x =0. (23) Note that power is injecte through Θ Σ. Besies the prescribe normal components of the current ensity on the contact, the variational formulation implies the bounary conition t b n =0. This can be seen by testing (23) with graients. Parallel to (4) voltage has to be efine through power. Thus, we get U = P I = σ curl h curl h j n x + t (µh) h j n x. (24) Remark 8. Another option for enforcing a particular total current through the contacts is by means of a constraint. This approach coul be pursue in the case of the h-base formulation, but we will eluciate it for the a-base scheme in the next section.

14 4.2 a-base formulation Voltage excitation Again, we use the electric bounary conitions from (9) base on the potential v from (2). We enote by ṽ a H ()-extension of v an point out that gra ṽ/ H 0 (curl; ). These bounary conitions lea to the following variational formulation: Seek a U t gra ṽ + C ([0,T], H 0 (curl; )) such that for all a H 0 (curl; ) µ curl a curl a x + σ t a a x =0. (25) Looking at the power balance, we fin that the voltage satisfies P = UI: P = µ curl a t curl a x + = U σ t a t a x curl a curl gra ṽx + U µ because the functional F (a) = the contact Σ +, cf. Rem. 3. Σ + σ t σ ta gra ṽx = U σ t a n S = U I, Σ + a n S = I actually provies the total current through Current Excitation We first stuy prescribe normal components j n = j n of the current ensity at contacts. We introuce the space W + 0 := {a H(curl; ); iv Γ (a n) = 0 on }. Then, any a W + 0 can be written as a = a 0 + gra v with a 0 H 0(curl; ),v H (), (26) since has been assume to be simply connecte. In contrast to the h-base formulation, the bounary conition b n = 0 on is incorporate strongly by virtue of the construction of W + 0, because curl a = b. On the other han the normal component j n = j n of the current ensity is impose only weakly: using (26), the surface integral on the right han sie of the variational formulation (7) can be expresse by (n h) a S = (h n) gra Γ v S = v iv Γ (h n) S = v j n S. Hence, an a-base variational formulation using prescribe j n to effect current excitation reas: Seek a C ([0,T], W + 0 ) such that for all a 0 H 0(curl; ) an all v H () µ curl a curl a x + v j n S. (27) σ t a a x = 2

15 In the spirit of Sect. 2 voltage is efine through power. To this en, consier a splitting of the solution of (27) a = a 0 + gra v accoring to (26). This will be plugge into the expression for the power. which means P = µ curl a curl t a x + σ t a t a x = vj n S vj n S, U = P I =. I As far as finite element iscretization of (27) is concerne, we recommen that the ecomposition (26) is taken into account by iscretizing a 0 by means of ege elements in H 0(curl; ), whereas v is approximate in the space of continuous scalar shape functions associate with noes on the bounary. This will yiel a irect splitting in the semi-iscrete setting. Secon, we consier a contact touching an exterior PEC bounary an want to prescribe a total current I. Temporarily, we abanon the temporal gauge an set for the electric fiel e = ta U gra ṽ, where ṽ is an arbitrary H ()-extension of v H 2 ( ). As in the case of voltage excitation, we eman that v = on Σ +, v = 0 on Σ, an a H 0 (curl; ). This means that the current through a suitably oriente cut Σ is given by I = j n S = ṽ j n S. Σ Σ + Since j n = 0 on Γ C an iv j = 0 in hols in a weak sense, we arrive at I = ṽ j n S = j gra ṽx = σ( t a + U gra ṽ) gra ṽx. Using this expression for the current, an a-base variational formulation with current excitation can be recast into a formulation with unknown voltage U an a constraint enforcing the current: Seek a C ([0,T], H 0 (curl; )) an U C ([0,T]) such that for all a H 0 (curl; ) µ curl a curl a x + σ t a a x + U σ gra ṽ a x =0 C a gra ṽx + U σ gra ṽ 2 x = I. σ t (28) Checking the power balance confirms that U gives the voltage as efine in Sect. 2. Theorem 9. For the variational problems (25) an (28) we fin b = curl a an, thus, e C inepenently of the choice of ṽ. Proof. Plug ṽ = ṽ an ṽ = ṽ 2 into (25) an (28), respectively, an set δv := U t (ṽ ṽ 2 ) in. 3

16 Σ Σ + I Figure 3: A nonzero current through a separate contact violates Ampere s law. Then there is δa H 0 (curl; ), such that δa C = gra δv. As δv Σ + = δv Σ = 0, we fin an H ()-extension δv of δv C. Setting δa := gra δv we can conclue: if a is a solution belonging to ṽ, then a 2 = a + δa is a solution for the choice ṽ 2. This shows curl a = curl a 2. Remark 0. It is tempting to remove the contacts from an put contact layers on parts of. However, the ey current moel cannot accommoate a current flowing out of into I as in Fig. 3. The reason is that this will violate Ampere s law: The assumption I>0leas to a contraiction as can be seen by integrating (b) over Σ +. In short, there is no meaningful way to take into account separate contacts insie in ey current computations. 0 I = j n S = h s = h s = j n S =0 Σ + Σ + Σ Σ 5 Non-Local Excitations In the case of ey current simulation it is not unusual that the geometric CAD moel oes not contain information on the location of contacts on the surface of massive inuctors. Nevertheless, there are ways to impose total currents an voltages. 5. h-base formulation Voltage Excitation Examining voltage excitation in Sect. 4. we saw, that in situation (a) (contacts on, see Fig. 2) the magnetic fiel solution h turne out to be inepenent of the location of the surface 4

17 Θ through which exciting power was supplie (see Fig. 2). Also in situation (b) excitation gave rise to a right han sie functional that i no longer contain geometric information about the electromotive region. a) b) I C Ξ Ξ Σ Σ I Figure 4: Non-local excitation: (a) Conuctor touching. Cutting surface Ξ in I is epicte. (b) Conucting loop away from, close by Seifert surface Ξ in I an cut by surface Σ insie. This suggests that in both situations (a) an (b) Θ is completely remove from the variational formulation (22). In situation (b) of Fig. 2 this means that we incorporate into, that is, we alter the computational omain (see Fig. 4). Thus, we arrive at the following variational formulation that is compliant with Rem. 3. Seek h C ([0,T], V 0 ) such that for all h V 0 σ curl h curl h x + t (µh) h x = U γ + h s. (29) Note that U plays the role of a circulation voltage, as will be explaine in Sect An extension to the multiport case is straightforwar, if ports are associate with loops of. Current excitation The constraint that curl h = 0 in I built into the space V 0 efine in (5) can be taken into account by means of scalar potentials together with so-calle co-homology vector fiels: V 0 h I = gra φ + L q i, φ H ( I ), (30) i= where q i is a basis of the first co-homology space H ( I ), an L stans for the first Betti number of I, see Sect. 2. Using the Seifert surfaces of I we can easily construct the fiels q i [ABDG98]. For the situation illustrate in Fig. 4, that is, for L =, a possible co-homology fiel q := q can be obtaine as the (generalize) graient of a function in H ( I \ Σ) that has a jump of height across the cut Ξ q := gra θ, θ H ( I \ Ξ), [θ] Ξ =. (3) 5

18 Let q be an extension of q H(curl; I ) to a function in H(curl; ). Then, base on the variational formulation (6), the total current through Σ + can be prescribe by fixing the contribution to V 0 from q. Accoringly, we efine Ṽ 0 := {h V 0, h s =0}, where γ := Σ (see Fig. 4). Then we get the following variational formulation that inclues non-local current excitation by a temporally varying total current I: Seek h I q + C ([0,T], Ṽ 0) such that for all h Ṽ0 σ curl h curl h x + t (µh) h x =0. (32) Recovery of voltages can formally be one accoring to (24), where h jn has to be replace with the extene co-homology vector fiel q, that is, U = P I = curl h curl q x + σ t (µh) q x. (33) Again, U plays the role of a circulation voltage, see 5.3. If a current shoul be impose on more than one loop, the contributions of other basis functions of the co-homology space to h have to be fixe. Assignment of voltages to loop is straightforwar then by plugging ifferent extene co-homology vector fiels into (33). γ Inconsistencies in the case of non-local excitations In Sect. 2 we pointe out that power can be elivere to the ey current omain either by a generator current j G or by non-homogeneous bounary conitions on. In (29) an (32) none of these possibilities is implemente. The question is where the power necessary to sustain currents in comes from. This motivates some oubts on the physical relevance of non-local excitation. It turns out that the variational problems (29) an (32) are not completely consistent with the ey current moel: they o not allow the recovery of a vali electric fiel in I that fits h. In other wors, there is no electric fiel e that solves n e = n e C on Γ C, (34a) n e = 0 on \ Γ C (only situation (a)), (34b) curl e = t (µ h) in I, (34c) iv(ɛ e) = 0 in I, (34) Γ i ɛ e n S =0, (34e) where e C = σ curl h in. The reason is that, by Stokes theorem applie to Ξ, the right han sies in (34a), (34b), an (34c) have to satisfy the compatibility conition e C s = t (µ h) n S. (35) Ξ Ξ 6

19 This is Faraay s law on the Seifert surface Ξ. However, the fiels use in the variational formulation are not regular enough (e is merely in L 2 ()) to allow the evaluation of the path integral (35). Hence, we have to consier (35) in weak form (see [AFV0]): (curl h n) q S + σ Theorem. Solutions of (29) or (32) violate (36). I t (µ h) q x = 0 (36) Proof. Testing (29) with h V 0 an (32) with h Ṽ0, respectively, where { h h C C 0 ( ) = h, I 0 we see that every solution h satisfies curl σ curl h + t (µh) = 0 in in the sense of istributions. Integration by parts applie to (32) yiels σ (curl h n) h S + t (µ h) h x =0. Since q / Ṽ0 the equality (36) is not implie! For (29) we may use h = q V 0 an obtain (curl h n) q S + σ t (µ h) q x = U 0, which contraicts (36). The bottom line is that we have to sacrifice some of the conitions in (34) in orer to obtain a caniate for the electric fiel in I. For instance, we can give up the continuity [n e] ΓC = 0, which means that we conone e / H(curl; ). If we o so, it will become clear where the energy comes from that fees the ey current problem: The jump [n e] ΓC results in a jump of the normal component of the Poynting vector S := e h along Γ C. The right han sie of the power balance (9) has to be augmente by the term P ΓC := h [n e] ΓC S. Γ C I I 5.2 a-base formulation Again, we consier the arrangement of Fig. 4. As explaine above, non-local excitations may rule out e H(curl; ). Therefore, with a function U : [0,T] R, we set where we have use an extension by zero e = ta U p, (37) p := { p in, 0 in I. 7

20 In situation (a) p is given by p := gra θ, θ H ( ), θ Σ + =, θ Σ =0. (38) Please note that there is no curl-free extension of p to H 0 (curl; ). In situation (b) we chose p as a representative of the first co-homology space H ( ) constructe accoring to p := gra θ, θ H ( \ Σ), [θ] Σ =. (39) Thus, (37) efines a fiel e / H(curl; ) whose tangential components are iscontinuous across Γ C. After we have given up the continuity of tangential components of e, we can state a-base variational formulations that allow non-local excitations. Remark 2. For practical computations it is more appropriate to chose a p with small support, see [Dul0]. Voltage Excitation In this case we can state the problem: seek a C ([0,T], H 0 (curl; )) such that for all a H 0 (curl; ) µ curl a curl a x + σ t a a x = U σp a x. (40) Current Excitation σ p a x =0 The appropriate variational problem reas: seek a C ([0,T], H 0 (curl; )) an U C ([0,T], R) such that for all a H 0 (curl; ) µ curl a curl a x + σ t a a x + U C σ p 2 x = I. (4) σ t a p x + U The expression for the power fe into the ey current moel in case of voltage as well as current excitation reas P = UI, an U is a circulation voltage. The (weak) expression for the current in the conuctor is given in both cases by the secon equation in (4). 5.3 Physical Interpretation of Non-Local Excitations The bottom line is that the non-local excitations iscusse above lea to electric fiels that o not belong to H(curl; ), since their tangential components have a jump across Γ C. How can this flaw of the mathematical moel be reconcile with physics? We propose the following explanation: the jumps [n e] ΓC can be regare as the effect of very thin ieal coils that o not create an exterior magnetic fiel. What we have in min is epicte in Fig. 5. Letting the thickness of the coils ten to zero, they will only affect the electric fiel, because any impact on the magnetic fiel is rule out by the esign of the coils. The fiels insie the coils are completely ignore. This causes inconsistency with Faraay s law, because 8

21 γ 0 ɛ j G G I Figure 5: Thin coil enowe with current that oes not create a magnetic fiel outsie the coil (To illustrate the current istribution the coil has been cut open.). the variation of magnetic flux insie the thin coils is neglecte. Eventually, it is a given generator current in an extremely thin coil that is responsible for the non-local excitation. Coils that o not generate any external magnetic fiel can easily be constructe theoretically: their associate vector potential a G is given in analogy to the co-homology fiel q from (32) (see also [Bos98b, Exercise 8.5]): let G be the omain occupie by the coil an G its complement G := \ G. Denote by Ξ the corresponing Seifert surface such that the co-homology space H ( G \ Ξ) becomes trivial. Define a G G := gra θ, θ H ( G \ Ξ), [θ] Ξ = an perform a H(curl; )-extension to G. Then, we have with b G := curl a G b G 0 in G. On the other han, given a temporally varying a G, the circulation γ t a G s along the close path γ in Fig. 5 oes not vanish. Hence, we get an inuctive excitation of currents through a fiel t b G that vanishes outsie the coil. We conclue that the exact location of the thin coil oes not affect the magnetic fiel. This accounts for the term non-local excitation. However, in orer to reconstruct the electric fiel in the insulating region I we nee to know the location of the coil. If we want to get e, the excitation has to be localize. This highlights the following fact: in orer to recover the electric fiel from the ey current moel everywhere in we nee extra information compare to what is require for the computation of the magnetic fiel. Besies information about possible charge ensities in I (which are usually assume to be zero) we nee to know the position of iealize sources in the case of non-local excitation. Remark 3. As in Sect. 3 one might try to start from the generic excitation by a thin coil as escribe above, an then to erive variational formulations by passing to the limit ɛ 0. However, this will lea to a sequence of vector potentials a G (ɛ) that oes not converge in H(curl; ). Coupling to Circuit Equations The interpretation of the excitation as an inuctive one has an important consequence concerning coupling to a circuit moel: there is no reason for the terminal current in the circuit moel I term 9

22 an the current I in the conuctor to be equal. The circuit current has to represent the current in the coil that is not inclue in the ey current moel with a non-local excitation. This means that coupling by conservation of current is rule out here. However, the change of the magnetic flux Φ hien behin the jump [n e] ΓC is a goo caniate for a coupling quantity because it can be uniquely efine in both moels. For the change of magnetic flux insie a thin coil along any curve γ Γ C circulating aroun the conuctor loop we get t Φ = [n e] ΓC s = U, γ which motivates the name circulation voltage for U. The fact that it oes not epen on the path γ is a consequence of the very esign of the coil an is necessary, since otherwise it woul not be meaningful to talk about the circulation voltage. Thus, the inuce terminal voltage of the coil will be equal to the circulation voltage U in the ey current problem (apart from the number of winings N): U in = N t Φ = NU The terminal current I term can be compute by the conservation of power if the self-inuctance L of the coil or the change of the magnetic energy W coil insie the coil are known: P term = U in I term = t W coil + P = t ( 2 LI2 term)+u I, where P is the (computationally available) power fe into the ey current problem. We conclue that even for coupling reasons we nee aitional information, reflecting the fact that we can use ifferent coils inucing the same resulting circulation voltage but with ifferent terminal voltages. References [ABDG98] C. Amrouche, C. Bernari, M. Dauge, an V. Girault. Vector potentials in three imensional nonsmooth omains. Math. Meth. Appl. Sci., 2(9): , 998. [ABN00] H. Ammari, A. Buffa, an J.-C. Néélec. A justification of ey currents moel for the Maxwell equations. SIAM J. Appl. Math., 60(5): , [AFV0] A. Alonso-Roriguez, P. Fernanes, an A. Valli. Weak an strong formulations for the time-harmonic ey-current problem in general omains. Report UTM 603, Dipartimento i Matematica, Universita egli Stui i Trento, Povo, Trento, Italy, September 200. [Bos85] A. Bossavit. Two ual formulations of the 3D ey currents problem. COMPEL, 4(2):03 6, 985. [Bos98a] A. Bossavit. Computational Electromagnetism. Variational Formulation, Complementarity, Ege Elements, volume 2 of Electromagnetism Series. Acaemic Press, San Diego, CA, 998. [Bos98b] [Bos00] A. Bossavit. On the geometry of electromagnetism IV: Maxwell s house. J. Japan Soc. Appl. Electromagntics & Mech., 6(4):38 326, 998. A. Bossavit. Most general non-local bounary conitions for the Maxwell equation in a boune region. COMPEL, 9(2): ,

23 [DGL99] [DHL99] P. Dular, C. Geuzaine, an W. Legros. A natural metho for coupling magnetoynamic h-formulations an circuit equations. IEEE Trans. Mag., 35(3): , 999. P. Dular, F. Henrotte, an W. Legros. A general an natural metho to efine circuit releations associate with magnetic vector potential formulations. IEEE Trans. Mag., 35(3): , 999. [DHR + 97] P. Dular, F. Henrotte, F. Robert, A. Genon, an W. Legros. A generalize source magnetic fiel calculation metho for inuctors of any shape. IEEE Trans. Mag., 33(2):398 40, 997. [Dir96] [DLN98] [Dul0] [GR86] [Hip02a] [Hip02b] [Ket0] [Kot87] H.K. Dirks. Quasi stationary fiels for microelectronic applications. Electrical Engineering, 79:45 55, 996. P. Dular, W. Legros, an A. Nicolet. Coupling of local an global quantities in various finite element formulations an its application to electrostatics, magnetostatics an magnetoynamics. IEEE Trans. Mag., 34(5): , 998. P. Dular. Dual magnetoynamic finite element formulations with natural efinitions of global quantities for electric circuit coupling. In U. van Rienen, M. Günther, an D. Hecht, eitors, Scientific Computing in Electrical Engineering. Proceeings of a workshop hel at Rostock, Germany, Aug , volume 8 of Lecture Notes in Computer Science an Engineering, pages Springer-Verlag, Berlin, 200. V. Girault an P.A. Raviart. Finite element methos for Navier Stokes equations. Springer, Berlin, 986. R. Hiptmair. Finite elements in computational electromagnetism. Acta Numerica, pages , R. Hiptmair. Symmetric coupling for ey current problems. SIAM J. Numer. Anal., 40():4 65, L. Kettunen. Fiels an circuits in computational electromagnetism. IEEE Trans. Mag., 37(5): , 200. P.R. Kotiuga. On making cuts for magnetic scalar potentials in multiply connecte regions. J. Appl. Phys., 6(8): , 987. [RR93] M. Renary an R.C. Rogers. An Introuction to Partial Differential Equations. Springer Verlag, New York, 993. [RTV02] G. Rubinacci, A. Tamburrino, an F. Villone. Circuits/fiels coupling an multiply connecte omains in integral formulations. IEEE Trans. Magnetics, 38(2):58 584,

24 Research Reports No. Authors Title R. Hiptmair, O. Sterz Current an Voltage Excitations for the Ey Current Moel A.-M. Matache, P.-A. Nitsche, C. Schwab Wavelet Galerkin Pricing of American Options on Lévy Driven Assets M. Becheanu, R.A. Toor On the Set of Diameters of Finite Point-Sets in the Plane C. Schwab, R.A. Toor Sparse finite elements for stochastic elliptic problems - higher orer moments R. Sperb Bouns for the first eigenvalue of the elastically supporte membrane on convex omains F.M. Buchmann Computing exit times with the Euler scheme 03-0 A. Toselli, X. Vasseur Domain ecomposition preconitioners of Neumann-Neumann type for hpapproximations on bounary layer meshes in three imensions M. Savelieva Theoretical stuy of axisymmetrical triple flame D. Schötzau, C. Schwab, A. Toselli Mixe hp-dgfem for incompressible flows III: Pressure stabilization F.M. Buchmann, W.P. Petersen A stochastically generate preconitioner for stable matrices A.W. Rüegg, Generalize hp-fem for Lattice Structures A. Schneebeli, R. Lauper L. Filippini, A. Toselli hp Finite Element Approximations on Non- Matching Gris for the Stokes Problem 02-2 D. Schötzau, C. Schwab, A. Toselli Mixe hp-dgfem for incompressible flows II: Geometric ege meshes A. Toselli, X. Vasseur A numerical stuy on Neumann-Neumann an FETI methos for hp-approximations on geometrically refine bounary layer meshes in two imensions 02-9 D. Schötzau, Th.P. Wihler Exponential convergence of mixe hp- DGFEM for Stokes flow in polygons 02-8 P.-A. Nitsche Sparse approximation of singularity functions 02-7 S.H. Christiansen Uniformly stable preconitione mixe bounary element metho for low-frequency electromagnetic scattering