2. Electromagnetic fundamentals

Transcription

1 2. Electromagnetic fundamentals Prof. A. Binder 2/1

2 AMPERE s law: Excitation of magnetic field by electric current Examle: Two different currents I 1, I 2 with two different numbers of turns 1 and N and two different flow directions: C r H r ds Θ Amere turnsθ : Θ N I 1 - I 2 The integration of magnetic field strength H along closed loo (curve C), which sans the area A, is equal to the resulting current flow (Amere turns Θ) enetrating through the area A. Positive field direction is connected to ositive current flow direction by RIGHT HAND RULE. Prof. A. Binder 2/2

3 Law of magnetic flux on closed surfaces The total magnetic flux Φ on closed surface A of volume V is always ZERO! A r r B da Φ 0 B:Magnetic flux density Normal comonent of B-vector on both sides of surface A is identical: B n, 1 Bn,2 Magnetic field has always north- AND south oles: NO magnetic monooles! Minimum ole number is 2: One north and one south ole (Examle: Earth magnetic field! Number of magnetic oles 2 (ole air number 1, 2, 3,... means 2, 4, 6,... oles). Prof. A. Binder 2/3

4 Magnetic field of current excited coil in air ga AMPERE s law: C r r H ds 2H Fe Fe + 2Hδ δ 2Hδδ Θ B δ B Fe H Fe B Fe /µ Fe 0 (µ Fe ) und H δ B δ /µ 0 (µ 0 4π 10 Vs/Am) Field vectors H r, B r in air ga: only dominating radial comonents considered! Θ Number of turns of coil N c, coil current I c : µ µ µ N c I c B δ 0Hδ 0 0 2δ 2δ 7 Prof. A. Binder 2/4

5 Magnetomotive force V(x) and current layer A(x) As H Fe 0 (µ Fe ): field lines of H δ start and end at iron surfaces: V magnetomotive force V" in air ga: Vδ Hδ δ δ µ δ ( x) B ( x) 0 δ Θ current layer" A(x): A lim in slot region, A 0 in tooth region b 0 b b: slot width: here b 0 for simlification! Calculation of B δ with use of current layer A(x): B δ x µ 0 µ 0 µ 0 Hδ ( x) A( x) dx ) δ 0 ( V ( x ) ( x) δ V 0 Total magnetic flux at closed surface A H surrounding rotor in air ga is zero This determines integration constant V 0. AH r r B da l Fe 2 τ B δ x 0 ( x) dx 0 Prof. A. Binder 2/5

6 Magnetic air ga field of grou of coils Coil grou: The windings er ole are given by more than one coil. Coils are connected in series (q coils er grou). Coil grous distanced by one ole itch τ distributed along machine circumference. Concentrated" Amere-turns er coil is Θ. Magnetic air ga field of coil grou is symmetrical to abscissa field curve B δ (x) above and below abscissa x is identical. Flux er ole and er axial length Area beneath field curve: ositive & negative areas are equal: north ole flux south ole flux. Examle: Number of coils er ole and hase" q 2. Prof. A. Binder 2/6

7 Magnetic alternating field (AC field) Feeding the coil grous with sinusoidal alternating current i c : Amlitude Î c, frequency f, angular frequency ω 2π f, T 1/f : eriod of oscillation i c ( t) Iˆ cosωt c B δ ( x, t) B δ ( x) cosωt Air ga field oscillates also sinusoidal with time, BUT maintains its satial distribution (its shae its distribution along x)! The amlitude of (radial) field comonent at locus x changes with time between ositive and negative maximum value. Prof. A. Binder 2/7

8 TESLA s idea for rotating (moving) magnetic air ga field THREE windings ( hases ) U, V, W with ositive and negative current flow direction 6 zones with notation +U, -W, +V, -U, +W, -V form a WINDING BELT. Zones with ositive current flow direction chosen so, that hase V is shifted with resect to hase U by 2τ /3, and hase W by 4τ /3. Winding belt hases U, V, W fed with 3 sinus currents: Each AC current time-shifted with T/3 hase shift: i U (t), i V (t), i W (t) ( symmetrical 3-hase AC CURRENT SYSTEM). i U ( t) Iˆcos( ω t + ϕ) ω T i V ( t) Iˆcos( ω t + + ϕ) 3 ω 2T i W ( t) Iˆcos( ω t + + ϕ) 3 We use comlex hasor calculus for sinusoidal AC currents & voltages: ωt jϕ jωt { I 2 e } Re{ I e 2 e } Iˆcos( ω + ) j jϕ i( t) Re t ϕ I I e Prof. A. Binder 2/8

9 Magnetic moving field Field curve moves with increasing time t to the left! After time T the field curve has assed the distance 2τ Velocity of linear movement is called 2τ vsyn 2 fτ T synchronous velocity! Synchronous rotational seed n syn in case of rotating field arrangement: ω syn vsyn vsyn 2 πnsyn d / 2 τ / π si 2πf n syn f Prof. A. Binder 2/9

10 Linear machines Linear movement, e.g. drive system for magnetically levitated Hi-seed train (MagLev) Cruising seed of MAGLEV train TRANSRAPID : Data: τ 258 mm, f 270 Hz (Maximum frequency of feeding inverter) v 2 fτ m/s km/h syn Rotating field machines Rotating art of machine ( Rotor): Two-ole machine (2 2): Magnetic field rotates with n syn 50 Hz 3000/min Sixty-ole hydro generator (2 60): Magnetic field rotates with n syn 120/min f 50 Hz n syn 1/min f 60Hz n syn 1/min Changing direction of rotation of magnetic field by changing connection of two terminals! Prof. A. Binder 2/10

11 Rotating waves - Travelling waves Rotating wave: x is stator circumference co-ordinate (rotating machine) Travelling wave: x is stator linear co-ordinate (linear machine) x π B δ1( x, t) Bˆ δ1 cos 2πf t τ Wave velocity: Observer, who is moving with the wave, sees constant argument of cos() const. v syn dx dt d dt τ ( const. + 2πft) 2 π Wave in oosite direction: ˆ x π B 1( x, t) B 1 cos( + 2πf t) v δ τ Examle: At frequency f 50 Hz: v syn in m/s is SAME number as ole itch in cm: 2 fτ δ syn e.g. 2-ole turbine generator (2 2) in thermal ower lant: n syn 3000/min: - stator bore diameter d si 1.2 m, ole itch τ 1.2π/ m 188 cm - v syn 188 m/s 676 km/h rotor surface velocity, as rotor is sinning synchronously with rotating stator magnetic field wave (synchronous machine!) fτ [ m / s] [ cm v τ ] syn Prof. A. Binder 2/11

12 FOURIER-Analysis: Determining fundamental & harmonic waves FOURIER-series: A eriodical function V(γ) with eriod 2π may be described by an infinite sum of sine & co-sine functions with decreasing wave length. Ordinal numbers: Amlitudes:, Average value: V ( γ ) V 0 + ν, a ν, b ) ν 1,2,3,... ν 1, 2, 3,... 2π [ Vˆ cos( ν γ ) + Vˆ sin( ν γ ] ˆ 1 1 Vν, a V ( γ ) cos( ν γ ) dγ Vˆ ν, b π V ( γ ) sin( ν γ ) dγ π 0 2π 0 1 V0 V ( γ ) dγ 2π 0 Magnetomotive force (MMF) of air ga field: a) NO UNIPOLAR flux: V 0 0 b) MMF V symmetrical to abscissa: NO even ordinal numbers c) By choosing origin so, that MMF V is even function V(γ) V(-γ): NO sine-wave functions occur in FOURIER sum. 2π Prof. A. Binder 2/12

13 Fundamental and harmonic air ga field waves V ( x, t) V ν ν 1, 5,7,... ( x, t) ν 1, 5,7,... 2 π m N k w, ν ν I νπx cos( τ ωt) ν 1, -5, 7, -11, 13, -17,... Phase number m: is usually 3 Positive and negative ordinal numbers: Velocity of harmonic waves decreases with 1/ ν : ν 1 + 2mg g 0, ± 1, ± 2, ± 3,..., ν fτ / ν Wave amlitudes (% of fundamental): Bˆ / (%) Underlined: slot harmonics! δν Bˆ δ1 v syn 2 ν q 1, W/τ 2/3, Q/ q 2, W/τ 5/6, Q/ q 3, W/τ 7/9, Q/ Prof. A. Binder 2/13

14 FARADAY s law of induction a) b) Each change of flux Φ, which is linked to conductor loo C, causes an induced voltage u i in that loo; the induced voltage is the negative rate of change of the linked flux. r r If coil is used instead of loo with N series connected turns, so u i is N-times bigger: u i N dφ / dt Flux linkage u i dφ / dt Flu: Φ B da Ψ N Φ dψ / dt Changing of Ψ : a) B is changing, b) Area A is changing with velocity v u i A Prof. A. Binder 2/14

15 Law of induction: also called: "LENZ s rule" Lenz s rule: A change of flux linkage induces voltage u i, which drives a current i in the loo, which excites a magnetic field B e, whose direction is oosite to the original change of flux linkage. Examle: Induction in short circuited loo at rest: - The change of external field B causes an increase of flux density with orientation from bottom to to. This causes increase of flux in loo area A and induces electrical field E Wi. B r / t - E Wi is left hand oriented to and drives in loo C a current i. - Current i excites (Amere s law!) a right hand oriented magnetic field B e. B r / t - Orientation of B e is oosite to change of original flux density. Result: The reaction field B e acts AGAINST the original flux density change! Prof. A. Binder 2/15

16 Induction of voltage in stator coil B x, t) Bˆ cos( xπ / τ ω ) 1( 1 t Sinusoidal moving wave δ δ causes changing coil flux Φ(t) τ / 2 Φ(t) l B (x,t)dx 2 τ lbˆ cosωt δ1 δ1 π τ / 2 Induced AC voltage in coil is sinusoidal: Uˆ i, c ωncφ c 2πfN c 2 τ ˆ lb π flux linkageψ(t) ( t) dψc( t) / dt Uˆ i, δ1 N c Φ(t) sinωt Voltage amlitude: (full-itched coil) u i, c c Prof. A. Binder 2/16

17 Induction of voltage in itched coil Pitched coil: Coil san is only W instead of τ : Φ cµ ( t) l W / 2 W / 2 Bˆ δµ µπx cos( τ µωt) dx 2 τ ˆ lb π δµ π W sin( µ 2 τ ) cosωt Linked coil flux is smaller by itch coefficient k,µ, comared to full-itched coil. k, µ π W sin µ 2 τ Prof. A. Binder 2/17

18 Induction of voltage in grou of coils µ 1: The induced sinusoidal AC voltage er coil grou is the sum of comlex hasors of the q coils. The coil voltage hasors are hase shifted by angle between adjacent coils: α Q,µ α Q, µ µ 2π /(2mq) Distribution coefficient: k d, µ Uˆ i, quˆ gr, µ i, c, µ αq, 2sin q 2 αq q 2sin 2 µ, µ π sin µ 2m π q sin µ 2mq Prof. A. Binder 2/18

19 Induced voltage er hase Machine with 2 oles, two-layer winding: One hase consists of 2 coil grous with q itched coils er grou. Induced voltage er hase (r.m.s. value): Fundamental: 2 U ˆ i1 2πf N kw1 τ lbδ 1 N π 2 qn, c / a 2 τ µ-th harmonic: U µ πµ f N k µ lbˆ i, 2 w, δµ π µ Examle: 12-ole synchronous generator: n 500/min, 2 12, f50 Hz - Stator winding: N c 2, q 2, W 5/6τ, a 2, τ 0.5 m, l 1 m - Number of turns er hase: N 2 qnc / a / 2 24 µ Bˆ / f µ Φ cµ i, µ U i µ U Bˆ δµ δµ Bˆ δ1 k w1 d1 1 U, / i, 1 - T % Hz mwb V % Facit: By itching and by coil grou arrangement voltage harmonics are reduced drastically. k k Prof. A. Binder 2/19

20 Three hase winding: Self induction leads to magnetizing inductance Stator air ga field waves, excited by stator current I, induce in stator winding by self induction the voltage u i! B δν x x t Bˆ νπ t m k (, ) δν cos ω B N I τ ˆ µ 0 2 w, ν δν ν 1, 5, 7, 11,13, δ π ν L Stator air ga field waves B δν ( x, t) : Seed n ν is n syn /ν. Hence stator field fundamental and field harmonics induce in stator coils ALL with the same frequency f. f ν ν ( nsyn / ν ) nsyn f r.m.s. of self-induced voltage er hase for each ν-th field harmonic: U i, ν 2 πf N kw, ν 2 τ lbˆ π ν Magnetizing inductance er hase: L hν forν-th air ga field harmonic wave. δν U ωl i, ν hν I L 2 2 kw, ν 2 hν µ 0N 2 2 ν π m lτ δ Prof. A. Binder 2/20

21 Stray inductance of stator winding er hase L + L L Air ga field: Fundamental wave Magnetizing field (subscrit h): L h L h, ν 1 Magnetizing inductance L h Magnetic field in slots (slot stray field) and around the winding overhang is NOT linked with rotor winding. It does NOT roduce any forces with rotor current. Hence it does NOT contribute to electromechanical energy conversion, and is thus called stray field (subscrit σ). Stray flux induces in stator winding additional voltage by self induction. Hence we define: Slot stray inductance L σq, overhang stray inductance L σb : U ( L L i σ, Q+ b ω σq + σb ) Air ga field harmonic waves induce stator winding with voltage U i,ν with the same frequency f. So they are summarized as total harmonic voltage : L σ σ, Q σ, b + σ, o U i, ν ν 1, 5,7,... Ui, ν 2 ν 1, 5,7,..., Lh, total Lhν ( 1+ σ o) Lh, ν 1 ωi kw ν σ o 1 ν 1, 5,7,... ν 1, 5,7,.. ν kw, 1 σ o : harmonic stray coefficient (is small: ca ). Harmonic field waves are linked to rotor, but disturbe basic machine function; hence they are summarized in harmonic stray inductance L σo : U ωl I L σ L iσ, o σ, o, σ, o o h I Prof. A. Binder 2/21

22 Active and reactive ower in load reference frame Active ower mui cos ϕ P Reactive ower Q mui sin ϕ ϕ < 90 P < 0, Generator Q < 0, caacitive load ϕ < 0 P > 0, Motor Q < 0, caacitive load 3. 0 ϕ < 90 P > 0, Motor Q > 0, inductive load ϕ < 180 P < 0, Generator Q > 0, inductive load Prof. A. Binder 2/22