Chapter 1 Magnetic Circuits

Transcription

1 Principles of Electric Machines and Power Electronics Third Edition P. C. Sen Chapter 1 Magnetic Circuits

2 Chapter 1: Main contents i-h relation, B-H relation Magnetic circuit and analysis Property of magnetic materials Inductance core loss Sinusoidal excitation Fig_1-1

3 Source of magnetic field Magnetic field around a current-carrying conductor Right-hand (Thumb) rule: thumb direction i; fingertip direction - H Fig_1-1

4 Magnetic field intensity H Ampere s law: the line integral of the magnetic field intensity H around a closed path is equal to the total current linked by the contour ර H dl = I H 2πρ = I H = I 2πρ Unit: A/m Fig_1-2

5 Flux density B B = μh = μ r μ 0 H Unit: Tesla Wb/m 2 μ 0 = 4π 10 7 henry/m, is the permeability of free space μ r is the relative permeability of the media μ is the permeability of the medium Relative permeability of medium 1. air, conductor, insulator Ferromagnetic: iron, steel, nickel 10 2 to Electric machine: 2000~6000 Fig_1-4

6 Magnetic flux Φ Φ m = ඵ B ds = ඵ B n ds = B n A Unit: Weber (Wb)

7 Magnetomotive force (mmf) ---ampere-turn H 2πr = Ni H = Ni 2πr Magnetomotive force: F = Ni F = H 2πr = Hl Magnetic reluctance: B = μh F = Bl μ Φ = BA F = Φl Aμ R = l Aμ F = ΦR Fig_1-4

8 Analogy between magnetic circuit and electric circuit V V = IR F = ΦR V V

9 Analogy between magnetic circuit and electric circuit Apply DC circuit analysis method: Ohm s Law, KCL, KVL, equivalent circuit, superposition Case1: What if permeability increases to infinity? Case2: air core inductor

10 Magnetic circuit analysis- rectangular core Mean core length: Reluctance: l = 2(l 1 + l 2 ) R rec = l Aμ Flux: φ = F R rec = NI Aμ l Flux density: B = φ A = NI μ l

11 Magnetic circuit analysis- solenoid core Core reluctance: Air reluctance: Flux: Flux density: R core = l Aμ R air = l air A air μ = 0 φ = B = φ A F R core = = NI μ l NI Aμ l Flux path: core and air

12 Magnetic circuit analysis-rectangular core with air gap Core reluctance: R c = l Aμ 0 μ r Air gap reluctance: Flux: R g = l g Aμ 0 φ = F R c + R g = = NI Aμ 0μ r l + l g μ r NI l + l g Aμ 0 μ r Aμ 0 NI Aμ = l + l g μ r Flux density: B = φ A = NI μ l + l g μ r * Flux density of air gap and core: same Fig_1-9

13 Example(text book P11): Magnetic circuit with two sets of current carrying conductors. Neglect leakage and fringing. If magnetic μ r is1200, cross-section is square. Determine air gap flux, B and H. (all dimension in centimeter) FigE_1-3

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15 Example: Synchronous AC machine Air gap length g = 1cm, I = 10 A, N = 1000 turns. Rotor pole face area Ar = 0.2 m 2. Assume that the rotor and the stator of the synchronous machine have negligible reluctance (infinite permeability) and neglect fringing. (a.) Draw the magnetic circuit. (b.) Determine the magnetomotive force. (c.) Determine the reluctance of each air gap. (d.) Determine the total magnetic flux in each air gap. (e.) Determine the magnetic flux density in each air gap.

16 Calculate inductance Faraday s law of induction: Emf= rate of change of flux V = dφ dt V = N dφ dt φ = F R = Ni Aμ l V = N V = L di dt d(ni Aμ l ) dt = N 2 Aμ l di dt L = N 2 Aμ l = N2 R

17 Flux linkage V = N dφ dt Flux linkage: V = dnφ dt λ = N V = dλ dt V = dli dt λ = Li Inductance: flux linkage per ampere of current: L = λ i What is the inductance of non-uniform core device? L = λ i L = Nφ i = NNi/(R c + R ag ) i N 2 = R c + R ag Fig_1-11

18 Nonlinear BH curve and Hysteresis loop B = μh = μ r μ 0 H B H

19 Hysteresis phenomenon Fig_1-12

20 Magnetic core loss: hysteresis loss n max p K B f h h Fig_1-14

21 Magnetic core loss: eddy current loss 2 2 max p K B f e e Fig_1-15

22 Magnetic core loss: total Determined by: volume, material and frequency p p p c h e Fig_1-14

23 Sinusoidal Excitation Faraday s law: Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. t = max sin(ωt) v t = N d (t) dt v max = 2πfN max = N max ωcos(wt) v rms = 4.44fN max Fig_1-17