COMPLEX INDUCTANCE AND ITS COMPUTER MODELLING

Transcription

1 Journal of ELECTRICAL ENGINEERING, VOL. 53, NO. 1-2, 22, COMPLEX INDUCTANCE AND ITS COMPUTER MODELLING Daniel Mayer Bohuš Ulryh The paper introdues the onept of the omplex indutane as a parameter of a oil loaded by sinusoidal urrent and plaed near an eletrially well ondutive body in whih the magneti field of the oil indues eddy urrents. Studied will be only axi-symmetri ylindrial oils and the paper shows how easily their omplex indutanes an be determined by means of professional programs for eletromagneti field analysis. The suggested methodology is illustrated on omputation of the indutane of a ring oil plaed near a ondutive slab. Investigated is also the dependene of the omplex indutane on frequeny. K e y w o r d s: oil indutane, eddy urrents, quasistationary eletromagneti field 1 INTRODUCTION Methods of omputation of the self- as well as mutual indutanes of oils and lines of various geometries an be found in a lot of referenes. Analytial solutions, for example, are dealt with in [1 4], while numerial solutions in [5 8]. The paper is aimed at the influene of near massive eletrially ondutive bodies on the oil indutane. It is well known that the presene of a near permeable body inreases the oil indutane while the presene of a ondutive non-magneti body dereases it the more the higher is the frequeny of the field urrent. The physial bakgrounds of these hanges are obvious. Ferromagneti parts lower the relutane of paths of the magneti flux linked with the oil, whih leads to an inrease of the magneti flux linked with partiular turns of the oil and, therefore, to a growth of its indutane. In ase the oil arries alternating urrent, its magneti field in a near eletrially ondutive body indues eddy urrents that (by the Lenz rule) redue the flux linked with the oil and lower its indutane. Building a mathematial model for an eletri iruit ontaining a oil with near metal bodies requires investigation of the magneti field in the neighbourhood of the oil, whih is not quite easy. In order to simplify the iruit equations we an haraterize the oil by a parameter let us all it a omplex self-indutane with whih we will work in the same manner as with the usual selfindutane defined in non-ondutive and non-magneti media. Diffiulty of solution of the problem will be then transferred to the question of investigation of this omplex self-indutane, whose determination assumes omputation of the magneti field of the oil. Suh a task may easily be proessed by means of any suitable professional ode for the magneti field analysis. These onsiderations may analogously be extended to a ouple of oils with onsequent introdution of the onept of a omplex mutual indutane. This, however, will not be dealt with in this paper. R 1 U ~ 1 R 1 U 1 L 1 L 12 =L 21 I 1 R 2 I 2 ~ 1 R 1 U L ~ 2 L 2 a) b) ) Fig. 1. a) loop above the ondutive half-spae, b) two indutively linked loops, ) equivalent loop in the free spae. University of West Bohemia, Faulty of Eletrial Engineering, Sady Pětatřiátniků 14, Pilsen, Czeh Republi mayer@kte.zu.z, ulryh@kte.zu.z ISSN FEI STU

2 Journal of ELECTRICAL ENGINEERING VOL. 53, NO. 1-2, r I - r N I - J r -eddy S z J eddy - S p V z Fig. 2. A loop near an eletrially ondutive body. Fig. 3. A oil with N turns near an eletrially ondutive body. 2 MOTIVATION FOR INTRODUCTION OF THE COMPLEX SELF INDUCTANCE A thin loop 1 arrying alternating urrent is plaed near an eletrially ondutive plane (Fig. 1a). The influene of this body may be replaed by another shortiruited loop 2 (Fig. 1b). Loop 1 will be haraterized by only one omplex parameter L, see Fig. 1. The system in Fig. 1b is desribed by matrix equations [ R1 + jωl 1 jωl 12 jωl 21 R 2 + jωl 2 Elimination of urrent I 2 provides where ] [ ] I1 = I 2 [ ] U. (1) ( R1 + jωl ) I1 = U (2) L = L 1 ωl 12L 21 ( ) R2 2 + ω2 L 2 ωl2 + jr 2. (3) 2 Quantity L will be denoted as the omplex self-indutane. The voltage on it an be expressed as U = jωl I 1 (4) This onsideration (but not the orresponding equations) holds even in the ase that the indued urrents are not onentrated in loop 2 but distributed in a near ondutive body. The omplex self-indutane depends on angular frequeny ω of the field urrent (L = L (ω)); with higher frequeny the module of L dereases and argument of L reahes negative values. 3 MATHEMATICAL FORMULATION OF THE PROBLEM 3.1 Starting assumptions Solution to the task starts from the following assumptions: The oil as well as the ondutive bodies are axisymmetri (ylindrial or ring oils), see Figs. 2 and 3. The magneti field is, therefore, of 2D harater. The oil has N turns and arries sinusoidal urrent. The oil and its neighbourhood represent a magnetially linear and hysteresis-free medium ( µ = onst) that is also eletrially linear (γ = onst). Only one oil is onsidered. Extension of the methodology to two indutively linked oils would be, however, relatively easy. The last assumption is a good professional program that solves the harmonially variable eletromagneti field in a linear medium desribed by the Helmholtz equation (see [8] or [9]) for phasor A A + k 2 A = µ J (5) where k 2 = jωγµ, A = α A α (r, z) and J = α J α is the phasor of the urrent density in the field oil. The magneti flux density phasor may be determined from relation B = rot A that provides two omponent equations B r = A α z, B z = 1 r r (ra α ). (6) Information about the available programs for the magneti field analysis is given in [1]. 3.2 Calulation of L from the stati definition Let an eletrially ondutive body be plaed in the neighbourhood of a thin loop, forming irle with radius r. Its omplex self-indutane may be expressed by means of the stati definition extended on phasors: L = φ I 1 = φi 1 I 2 1, (7) where φ is the phasor of magneti flux φ(t) linked with loop, I 1 is the phasor of the urrent in the oil and I 1 its omplex onjugate. Phasor φ may be expressed as φ = φ 1 + φ 12, (8)

3 26 D. Mayer B. Ulryh: COMPLEX INDUCTANCE AND ITS COMPUTER MODELLING 1 2 r D F C H r min = r max = 52 A E G B z Fig. 4. To the example: a ring oil above an eletrially ondutive slab. Fig. 5. The definition area of the solved boundary problem. where φ 1 is the phasor of the magneti flux indued by urrent I 1 and φ 12 the phasor of the magneti flux indued by eddy urrents in the near ondutive body. Phasors φ 1 and I 1 are in phase, but phasor φ 12 is shifted and gives a omplex harater to the indutane L defined by (7). Using the Stokes theorem, the total magneti flux may be expressed as φ = B ds = Adl = 2πA α r (9) S where S is the area of the irle bounded by urve and B = r B r (r, z) + z B z (r, z), ds = z ds, dl = α dl, A = α A z. Let us onsider a irular oil formed by N thin turns with radii r i, i = 1,..., N. The oil has inner and outer radii R 1 and R 2 and height h. Its omplex selfindutane is given by (7), where φ = φ i = 2π A αi r i. (1) Vetor potential A αi is indued in the i-th thin turn by urrent I 1 in the oil and eddy urrents in the near ondutive body. Integration of the vetor potential over volume V provides V A α dv = R 2 2π h r=r 1 α= z=. = 2π A α rdrdαdz = 2π A αi r i S p = 2π S p N S p A α rds A αi r i, (11) where S p is the ross-setion of one turn. Denoting S p the ative setion of the whole oil, then S p = N S p and J α = I 1 S p = NI 1 S p (12) so that, in aordane with (7), the omplex self-indutane is given as L = N I 1 S p A α dv. (13) V Computation of integral A α dv is usually implemented in the postproessor menu of most professional V programs see eg [11]). Definition (7) has a sense in the ase that the phasor of the voltage indued in the oil is expressed by ommon relation (4). Indeed, for definition (7) we get equation (4): U = jω φ i = jωl I Calulation of L from the eletromagneti field energy Two thin loops (Fig. 1b) arry urrents i 1 = I m1 sin(ωt + ϕ 1 ) and i 2 = I m2 sin(ωt + ϕ 2 ). The magneti energy of the system is W m (t) = 1 2 L 1i L 2i 2 2+L 12 i 1 i 2 = 1 2 L 1I 2 m1 sin 2 (ωt+ϕ 1 ) L 2I 2 m2 sin 2 (ωt + ϕ 2 ) L 12I m1 I m2 [ os(ϕ1 ϕ 2 ) os(2ωt + ϕ 1 + ϕ 2 ) ]. (14) The average value of this magneti energy may be expressed by means of phasors W m = 1 2 Re W m, (15) where W m is the omplex magneti energy. As W m = 1 2 L 1I 1 I L 2I 2 I 2 + L 12I 1 I 2, (16)

4 Journal of ELECTRICAL ENGINEERING VOL. 53, NO. 1-2, Here potential A 11 in region V 1 is indued by urrent I 1, potential A 22 in region V 2 is indued by the eddy urrents, potential A 12 in region V 1 is also indued by the eddy urrents and finally potential A 21 in region V 2 is indued by urrent I 1. The omplex self-indutane of the oil may be defined from extended energeti definition a) L = 2W m1 I 1 I 1. (19) After substituting for W m1 from (17) and using we easily get again eq. (13). J 1 = α J = onst, (2) b) It is not ompliated to prove that phasor of the voltage on omplex self-indutane L is also expressed by (4). This may be performed by means of equations (19), (11), (2) and (12). 4 ILLUSTRATIVE EXAMPLE A RING COIL ABOVE A CONDUCTIVE SLAB ) Fig. 6. Fore lines of the magneti field: a) a opper slab, f = 1 Hz, b) an iron slab, f = 1 Hz, ) a opper slab, f = 1 Hz. it is obvious that the omplex harater of W m is aused by the third term expressing the linkage energy. We use the relation for magneti field energy W m = 1 2 V AJdV, where integration is performed over region V ontaining the oil and near eletrially ondutive body. In ase vetors A(t) and J(t) vary sinusoidally in time, the average value W m of the magneti energy may be expressed by means of omplex energy W m aording to (15), where the omplex energy of oil 1 (region V 1 ) and near ondutive body 2 (region V 2 ) is given by formula W m = W m1 + W m2 = 1 A 2 1 J 1 dv + 1 A 2 2 J 2dV (17) V 1 V 2 where both phasors of vetor potentials A 1 and A 2 (omplex amplitudes) onsist of two omponents: A 1 = A 11 + A 12, A 2 = A 21 + A 22. (18) The suggested algorithm of omputation of the omplex self-indutane is illustrated on the following example: a ring oil is plaed above a ondutive slab (Fig. 4). Number of turns of the oil N = 17 and material of the slab is a) opper (γ = S/m), b) aluminium (γ = S/m), ) steel (γ = S/m, µ r = 1), d) non-ondutive and non-magneti material. Frequeny f, 1 4 Hz. Current I = 2 A and its density determined from the sizes of the oil J = Am 2. Cross-setion S p of the oil is m Mathematial model The ontinuous mathematial model is defined as a boundary problem on domain Ω = Ω 1 Ω 2 Ω 3 Ω 4 in ylindrial o-ordinates r, α, z (Fig. 5). Distribution of the phasor of vetor-potential A = α A α in partiular sub-regions follows from equations Ω 1 (oil): A α + k 2 A α = µ J, where k 2 = jωγµ, Ω 2 (slab): A α + k 2 A α =, (21) Ω 3,4 (air): A α =, where J is the phasor of the hosen urrent density. As the medium is linear, its seletion an be quite arbitrary. The boundary onditions read: AB: A α = (antisymmetry), BCDA: A α = (fore line, ontinuity).

5 28 D. Mayer B. Ulryh: COMPLEX INDUCTANCE AND ITS COMPUTER MODELLING Current density J (1 6 A/m 2 ) E J*.5.1 L (m) Fig. 7. Distribution of the amplitude of the eddy urrent density on surfae EF (see Fig. 5), a opper slab, I = 1 A, f = 1 Hz G B z (1-3 T) b a r min F r max r(m) Fig. 8. Distribution of the amplitude of the magneti flux density in plane GH (see Fig. 5), I = 1 A, a) a opper slab, f = 1 Hz, b) an iron slab, f = 1 Hz, ) a opper slab, f = 1 Hz. 4.2 Results and their disussion The mathematial model was solved by the FEMbased SW produt QuikField [11]. Distribution of the eletromagneti field in the neighbourhood of the oil is H shown in Fig. 6. Figure 7 depits the distribution of the eddy urrents on the surfae of the slab and Fig. 8 an analogous magneti flux density distribution in the plane parallel with the surfae of the slab at the distane of 18 mm. The indutane of the oil itself, without presene of any ondutive slab, was alulated by means of the approximate algorithm desribed in [3] and its value is H. The dependene of the omplex indutane on angular frequeny ω (it is L = L (ω) / ϕ(ω)) is shown in Figs. 9 and 1. The module of L varies with frequeny in aordane with expetations. The self-indutane of the oil without presene of a ondutive slab does not depend on frequeny, while the presene of suh a slab leads to a monotoni derease of the indutane with inreasing frequeny. The dependene on frequeny f is negligible in the domain of low frequenies. The presene of an iron slab leads to a growth of the indutane. 5 CONCLUSION The omplex self-indutane was introdued as a harateristi parameter of an AC urrent arrying oil plaed in eletrially ondutive media. This parameter may relatively easily be determined by means of existing professional odes for the analysis of eletromagneti fields. The methodology is fully usable even in the ase that the oil is wound from massive ondutors with non-negligible skin effet. Despite this fat, there exists an open question whether this oneption may be generalized even for periodial, but non-harmoni waveforms of urrents or for general waveforms of urrents ourring in the analysis of transients. Another question is assoiated with using the methodology for other shapes (non-ylindrial) of the field oils. APPENDIX The omplex indutane represents a useful tool for determining the losses due to eddy urrents indued in a 1.4x x1-3 1.x1-3 8.x1-4 6.x1-4 4.x1-4 2.x1-4 1 L(H) b a f(hz) ( ) f(hz) a b Fig. 9. Dependene of the module of L on frequeny for a) a opper slab b) aluminium slab, ) an iron slab. Fig. 1. Dependene of the argument of L on frequeny for a) a opper slab, b) an aluminium slab, ) an iron slab

6 Journal of ELECTRICAL ENGINEERING VOL. 53, NO. 1-2, near ondutive body. Let the oil arry sinusoidal urrent i(t) = 2 I sin ωt whih phasor (omplex RMS value) I = I. The phasor of voltage on the oil is The omplex power is U = jω(l r + jl i )I. S = U I = P + jq = jω(l r + jl i )I 2. The losses due to eddy urrents are P = ω L i I 2 and as L i <, then P >. Aknowledgement The finanial support of GA CR (grant projet No. 12 /1/141) is highly aknowledged. Referenes [1] HAK, J. : Iron-Free Coils, Koehler, Leipzig, (in German) [2] GROVER, F. W. : Indutane Calulations: Working Formulas and Tables, New York, Dover, [3] KALANTAROV, P. L. CEJTLIN, L. A. : Computation of Indutanes, Energija, Leningrad, 197. (in Russian) [4] DOLEŽEL, I. VERDYCK, D. BELMANS, R. GEYSEN, W. : Analytial and FE Open Boundary Calulations of the Self and Mutual Indutanes in a System of Conentri Cylindrial Air-Core Coils, Ata Tehnia ČSAV 35 (199), [5] MAYER, D. ULRYCH, B. : Numerial Computation of Indutanes of Air-Core oils., Elektrotehn. obzor 7 No. 3 (1981), (in Czeh) [6] REIF, J. MAYER, D. : Computation of Indutanes for Iron-Free Coils of General Shapes, Ata Tehnia ČSAV 41 (1996), [7] MAYER, D. ULRYCH, B. : Numerial Computation of Indutanes of Coils of Finite Cross-Setions with Ferromagneti Core, Proeedings of XXIV IC-SPETO, Vol. 1, pp [8] MAYER, D. POLÁK, J. : Methods of Solution of Eletri and Magneti Fields, SNTL/ALFA, Praha, (in Czeh) [9] HAŇKA, L. : Eletromagneti Field Theory, SNTL/ALFA, Praha, (in Czeh) [1] MAYER, D. ŠKOPEK, M. ULRYCH, B. : Solution of the Eletromagneti Fields by Modern SW Produts, EE Journal 7 No. 1-2 (21), (in Czeh) [11] Reeived 27 June 21 Daniel Mayer (Prof,Ing, DrS) was born in Pilsen (Czeh Republi) on August 8, 193. He reeived the Ing, PhD and DS degrees in eletrial engineering from Tehnial University of Prague in 1952, 1958 and 1979, respetively. In 1956 he began his professional areer as a Senior Leturer and later as a Assoiate Professor at the University of West Bohemia of Pilsen. In 1968 he was appointed Full Professor of the Theory of Eletrial Engineering. For many years he has been head of the Institute of the Theory of Eletrial Engineering. His main teahing and researh interests inlude iruit theory, eletromagneti field theory, eletrial mahines and apparatus, methodology of eletrial engineering eduation and history of eletrial engineering. He has published 6 books and over 19 sientifi papers in these fields of study. Professor Mayer is a Fellow of the Institution of Eletrial Engineering (UK). He is a member of editorial advisory boards of many international journals. Bohuš Ulryh, (Do, Ing, PhD), born in 1937, works at the Department of the Theory of Eletrial Engineering at FEL WBU in Pilsen. His professional interests are aimed at modern numerial methods of solution of eletromagneti and oupled problems. Author and o-author of about 1 papers and several textbooks. Co-investigator of several grant projets (GA CR and MSMT). Author of a lot of user s SW for the alulation of eletromagneti fields and oupled problems in different dispositions. SLOVART G.T.G. s.r.o. GmbH E X P O R T - I M P O R T E X P O R T - I M P O R T of periodial s and of non-periodi ally printed matters, book s and CD - ROM s Krupinská 4 PO BOX 152, Bratislava 5,Slovakia tel.: , fax.: gtg@internet.sk, SLOVART G.T.G. s.r.o. GmbH E X P O R T - I M P O R T