1. 1 Magnetic circuits Summary for EM ELEC 3105 BASIC EM AND POWER ENGINEERING

2. MAGNETIC CIRCUIT PRELIMINARIES 2 I Iron core µ rc =5000 Air µ ra =1.000 Want to examine how B and H are related: Through the side wall of the core Through the end face of the core

3. MAGNETIC CIRCUIT PRELIMINARIES 3 I Iron core µ rc =5000 Air µ ra =1.000 Through the side wall of the core Use boundary conditions for tangential components B primarily in the iron core

4. MAGNETIC CIRCUIT PRELIMINARIES 4 I Iron core µ rc =5000 Air µ ra =1.000 Through the end face of the core Use boundary conditions for normal components B preserved through end face

5. MAGNETIC CIRCUIT PRELIMINARIES 5 I Iron core µ rc =5000 Air µ ra =1.000 In core region all B field lines are inside the core At the end all B field lines are continuous and exit into air

6. MAGNETIC CIRCUIT PRELIMINARIES 6 I D L L L D/2 Conservation of magnetic flux at a branch point Similar to conservation of current at a node

7. MAGNETIC CIRCUIT PRELIMINARIES 7 I The magnetic flux is produced by the current in the wires D L and Just like in electrical circuits Q? What plays the role of resistance in magnetic circuits?

8. MAGNETIC CIRCUITS 8 We can build a torus of ferromagnetic material and wrap N turns of wire carrying a current I around it. The iron core magnetizes, giving a large bound current on the surface of the torus, thousands of times larger than I.

9. MAGNETIC CIRCUITS 9 The B field inside the torus is much larger than that outside, so most of the field is inside the core. Boundary Conditions Since magnetic field lines form closed loops, the fields cannot vary as we go around the torus core. There may be minor variations at different radii within the core though. constant inside the core Gaussian volume Flux through top equals flux through bottom.

10. MAGNETIC CIRCUITS 10 For closed loop Closed loop inside the core material and encloses a current NI, the number of wires passing through the loop times the current in each Approximate length of core

11. MAGNETIC CIRCUITS 11 Can read B from B versus H curve for the core material.

12. MAGNETIC CIRCUITS 12 We now introduce a small gap in the core material. Assuming minimal spreading of the field lines, B in the core is the same as B in the gap. Small gap continuous across gap

13. MAGNETIC CIRCUITS 13 We now introduce a small gap in the core material. Assuming minimal spreading of the field lines, B in the core is the same as B in the gap. Small gap not continuous across gap

14. MAGNETIC CIRCUITS 14 For closed loop Substituting for H gap gives an expression for B and H core which are also related by the magnetization curve of the core. Thus Solving for B

15. MAGNETIC CIRCUITS 15 H core B Straight line Magnetization curve The intersection gives B and H fields in the core, and the H field in the gap can be found from B.

16. MAGNETIC CIRCUITS 16 H core B Straight line Magnetization curve The intersection gives B and H fields in the core, and the H field in the gap can be found from B. Linear approximation to magnetization curve Some error introduced using a linear fit for B-H curve. constant

17. MAGNETIC CIRCUITS 17 Then H core B In the denominator, the two terms may be of the same order even though since Similar to the formula for a current through two resistors in series

18. MAGNETIC CIRCUITS 18 Similar to the formula for a current through two resistors in series

19. MAGNETIC CIRCUITS 19 Similar to the formula for a current through two resistors in series Magnetic circuitElectric circuit FluxCurrent MMFEMF ReluctanceResistance PermeanceConductance PermeabilityConductivity

20. ELECTRIC AND MAGNETIC CIRCUITS “COMPARISON” 20

21. MAGNETIC CIRCUITS 21 Similar but only approximately Flux tends to leak out of a magnetic circuit since Magnetic fields take the path of least reluctance. Current takes the path of least resistance.

22. MAGNETIC CIRCUITS 22 Magnetic shielding Cavity has very small B inside Iron core Magnetic field bypasses the cavity

23. MAGNETIC CIRCUITS 23 Magnetic shielding Long thin cavity magnetized iron core OR with or higher

24. EXAMPLE 1: MAGNETIC CIRCUIT 24 I µ 1,L 1,A 1 µ 2,L 2,A 2 µ 3,L 3,A 3 N Determine electrical equivalent circuit Example completed in class

25. EXAMPLE 1: MAGNETIC CIRCUIT 25 I µ,L 1,A 1 µ,L 2,A 2 µ,L 3,A 3 N Determine electrical equivalent circuit Example completed in class

26. 26 Summary for EM ELEC 3105 BASIC EM AND POWER ENGINEERING

27. 27 (1)

28. 28 (2)

29. 29 (3)