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Fundamentals of Static Magnetic Fields in Electrical Systems

Understand Lorentz's force equation for static electric and magnetic fields, permeability in free space, conservation of magnetic flux, Ampere's circuital law, magnetic circuits, and behavior of magnetic materials. Learn boundary conditions and inductances in electrical systems.

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Fundamentals of Static Magnetic Fields in Electrical Systems

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  1. CH6 Static Magnetic Field

  2. Electric force • Magnetic force • Elecmagnetic fore (N) ~ Lorentz’s force equation

  3. Static Electric Field Static Magnetic Field Free space Steady current Permeability of free space

  4. No magnetic flow sources • Magnetic flux lines always close • Low of conservation of magnetic flux • Each magnets has a north south • Magnetic poles cannot be isolated pole

  5. Ampere’s circuital law summary

  6. EX 6-1 • Inside conductor • Outside conductor C2 outside conductor encloses I

  7. b If 空心圓柱(柱座標)

  8. EX 6-2 (Toroidal Coil) A circular contour C with radius r (b - a) < r < (b + a) (b - a) < r < (b + a) (No source)

  9. (a) Direct application of Ampere Law (b) Special case of torid Ex 6-2, EX 6-3 (Solenoid Coil)

  10. 6-3 Vector Magnetic Potential : magnetic potential [Vector] c.f : electric potential [Scalar] 取 Vector Poisson’s equation Coulomb gauge

  11. In Cartesian coordinates, Vector

  12. Magnetic Flux through a given area S which is bounded by contour C

  13. 6-4 Biot-Savart Law and applications Vector source Magnetic: Current distribution Current element

  14. 0 Biot-Savart Law：[Valid in steady current] 0 c.f. 因次分析

  15. Biot-Sarvart Law： Action at a distance： field

  16. Gauss thm. 特殊式 Biot-Sarvart Law Poission Eqe. Stoke Thm. Ampere’s Law

  17. PF: =0 =0 N 封閉磁迴路 S C.F 孤立電單極

  18. From 0 其中

  19. From Steady current Static magnetic field 其中： Steady state Coulomb Gauge

  20. 其中： Ampere’ Law Poission Equ.

  21. Gauss Thm.

  22. Example 6-4 (a)

  23. Example 6-5 From 6-4

  24. Example 6-6 is canceled due to cylindrical sym.

  25. 6-5 Magnetic Dipole 2’ pole = 0 20 pole = 0

  26. c.f.

  27. c.f.

  28. Scalar Magnetic Potential : Scalar Magnetic Potential (not physical) : Non conservative (path dependent)

  29. 磁性物質 Components source Conductor free electron Non-conductor polarized ion 6-6 Magnetization and Equivalent Current Density 定義

  30. c.f.

  31. EX : 6-8

  32. 靜磁學 (包含導體與磁性材料) C.F.

  33. Static Magnetic Source : (permeativity) Conductor : Magnetic material : Copy 電學 數學上等效，無物理

  34. Ex 6-9 電場 與（Dipole 類似）

  35. 6-7 Ampere’s circuital law ; relative permeability

  36. 6-8 Magnetic Circuits Electric circuit : Voltage / Current source ; V, I, … Magnetic circuit : Transformer / Generator / Motor … ; closed path c to enclose N turns of I (m.m.f) magnetomotive force [Amp]

  37. Ex 6-10 Sol : is identical in different material ; f : ferromagneticg : gap Ampere law

  38. Magnetic Flux ; S : cross-section : length of ferromagnetic core. :ferromagnetic core :air gap : Reluctance Analog to : [Electric circuit]

  39. Magnetic circuit Electric circuit mmf mag. flux reluctance Permeability emf electric current , resistance , conductivity ,

  40. An exact analysis of magnetic circuits is difficult ◎Leakage Fluxes ◎Fringing effect ◎ 2 conditions must be satisfied Similar to Kirchhoff’s voltage Law Kirchhoff’s current Law

  41. EX.6-11 K.V.L. (Time Independent) Loop1: Loop2:

  42. 6-9 Behavior of Magnetic Materials ◎Diamagetic: ◎Paramagnetic: ◎Ferromagnetic:

  43. 6-10 Boundary Conditions for Magnetostatic Field

  44. Ex 6-12 Bn component component or Similar to E-field Magnitude of In ferromagnetic parallel interface Originates in a ferromangeitc , Flux perpendicular to interface

  45. Ex 6-13 Surface current Example 6-8 [p246] Center End at interface quantity

  46. 6-11 Inductances & Inductors Mutual flux ： mutual inductance between loops C1 and C2 If loop C2 has N2 turns , Generalizes to

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