Lecture 23 Flux Linkage and Inductance

Transcription

1 Lecture 3 Flux Linkage and nductance Sections: 8.10 Homework: See omework file

2 te sum of all fluxes piercing te surfaces bounded by all turns (te total flux linking te turns) Λ= NΦ, Wb Flux Linkage in Coils ideal scenario: eac turn creates te same flux Φ and tis flux links all turns (no leakage) if flux density B is uniform inside te coil te flux linkage is Λ= NBS, Wb B N Λ N 1 tis is te coil s self-flux linkage, i.e., te flux links to te current wic generates it LECTURE 3 slide Φ= B S Λ= NΦ N

3 Self and Mutual Flux Linkage flux linkage may be self-flux linkage and mutual-flux linkage self-flux linkage links current wic it is due to every coil as self-flux linkage mutual-flux linkage is due to currents in oter inductors mutual-flux linkage implies two coupled coils Λ =Λ +Λ, Λ =Λ +Λ N1 11 N1 1 Λ = Φ Λ = NΦ Λ = Φ Λ = NΦ NN N N N N B1 B 1 N 1 Λ 1 Λ Λ 11 Λ 1 1 N LECTURE 3 slide 3

4 Self and Mutual nductance self inductance L sows te amount of self-flux linkage due to unit current Λself L =, H = Wb/A mutual inductance M sows te amount of mutual flux linkage due to unit current producing tis mutual flux M = Λ mutual, H self inductance L (or simply inductance) is defined for a single inductor mutual inductance M is defined for a pair of inductors LECTURE 3 slide 4

5 Example Self Flux Linkage and nductance of Toroid N H φ =, H/m, ρ1 ρ ρ(see L16) πρ ρ N ρ Φ= µ Hφ dρ dz = µ ln 0 ρ π ρ 1 1 Bφ N ρ Λ= NΦ= µ ln, Wb π ρ 1 N ρ L = µ ln, H π ρ 1 if toroid is tin, field is mostly uniform N H NA N A φ = Φ= µ L= µ πρ πρ πρ B ρ ρ 1 d LECTURE 3 av slide 5 ρ 0 C ρ w z B ρ0 = 0.5( ρ1+ ρ) A= ( ρ ρ ) 1 w

6 Example for Mutual Flux Linkage and Mutual nductance two coaxial solenoids (assume uniform field is in cross-sections) solenoid #1: radius a, lengt D 1, N 1 turns solenoid #: radius b, lengt D, N turns b<< ad, << D 1 solenoid # is smaller tan solenoid #1 and is positioned in its middle N Nπb H H b = Φ 1 = µ 1 π = µ D1 D1 NN 1 1π b Λ 1 = NΦ 1 = µ D Λ1 NN 1 π b M 1 = = µ D LECTURE 3 slide 6, H a b

7 Pysical Significance of Flux Linkage: Faraday's Law Wy is te flux linkage important rater tan te flux itself? Te answer is given by Faraday s law of EM induction dφ = e, V dt f te magnetic flux canges in time, electromotive force e (V) is induced in eac turn of te coil. Eac turn acts like an AC voltage source. Tese equivalent voltage sources are connected in series, tus producing an overall voltage at te coil s terminations as d d E Ne N Φ Λ = = =, V dt dt t is te rate of cange in time of te flux linkage, wic determines te induced voltage of a coil: dλ d dk V = E = = L, V or Vj = M jk dt dt dt LECTURE 3 slide 7

8 nductance per Unit Lengt: Coaxial Cable linkage per unit lengt determines inductance per unit lengt µ 0 B φ =, T (Ampère s law) πρ µ 0 dλ= dφ= Bd φ ρ = dρ πρ b µ 0 µ 0 b Λ= dρ = ln, Wb/m πρ π a a µ 0 b L = ln, H/m π a L = Λ, H/m c b a H LECTURE 3 slide 8

9 nductance per Unit Lengt: Parallel-plate Line express flux linkage PUL in terms of H field w Λ=Φ= B( l) = µ H ( l) Λ = µ H area express H field in terms of current using Ampère s law H d L = Hw = find inductance PUL Lꞌ C l H Ampère s contour Λ L = = µ 0, H/m w LECTURE 3 slide 9

10 Obtaining nductance from Capacitance Expressions compare te capacitance and inductance PUL expressions for a coaxial line L = µ ln ( b/ a), H/m π and for a parallel-plate line L = µ w LC = µε C = w C = ε it can be sown tat for any infinitely long TL πε, F/m ln( b/ a) te inductance PUL formula can be obtained from te reciprocal of te capacitance formula were ε is replaced by 1/μ LECTURE 3 slide 10

11 nductance per Unit Lengt: Twin-Lead Line πε C =, F/m ln + 1 r r if r µ 0 µ 0 µ L = ln = arccos, H/m L ln π r r π r π r B A r s y 0 A z = 0 ρ 1 s equipotentials Az + LECTURE 3 slide 11 P ρ r A x B

12 You ave learned: tat te inductance of a structure is proportional to te flux linkage tat te flux linkage is proportional not only to te number of turns generating te flux but also to te number of turns intercepting tis flux tat te self inductance of a coil is proportional to N tat te mutual inductance of a pair of coils is proportional to N 1 N ow to calculate te inductance per unit lengt of transmission lines LECTURE 3 slide 1