Chapter 15 Magnetic Circuits and Transformers

Transcription

1 Chapter 15 Magnetic Circuits and Transformers Chapter 15 Magnetic Circuits and Transformers 1. Understand magnetic fields and their interactio with moving charges. 2. Use the right-hand rule to determine the directi of the magnetic field around a current-carrying wire or coil. 1

2 3. Calculate forces on moving charges and current carrying wires due to magnetic fields. 4. Calculate the voltage induced in a coil by a changing magnetic flux or in a conductor cutting through a magnetic field. 5. Use Lenz s law to determine the polarities of induced voltages. 6. Apply magnetic-circuit concepts to determine the magnetic fields in practical devices. 7. Determine the inductance and mutual inductance of coils given their physical parameters. 8. Understand hysteresis, saturation, core loss, and eddy currents in cores composed of magnetic materials such as iron. 2

3 9. Understand ideal transformers and solve circuits that include transformers. 10. Use the equivalent circuits of real transformers to determine their regulations and power efficiencies. 3

4 MAGNETIC FIELDS Magnetic flux lines form closed paths that are close together where the field is strong and farther apart where the field is weak. Flux lines leave the north-seeking end of a magnet and enter the south-seeking end. When placed in a magnetic field, a compass indicates north in the direction of the flux lines. 4

5 Right-Hand Rule 5

6 Forces on Charges Moving in Magnetic Fields f = qu f = B qub sin( θ ) 6

7 Forces on Current-Carrying Wires df = idl B f = ilb sin( θ ) φ = A Flux Linkages and Faraday s Law B da Faraday s law of magnetic induction: e = dλ dt λ = Nφ 7

8 Lenz s Law Lenz s law states that the polarity of the induced voltage is such that the voltage would produce a current (through an external resistance) that opposes the original change in flux linkages. 8

9 Voltages Induced in Field-Cutting Conductors e = Blu 9

10 Magnetic Field Intensity and Ampère s Law 7 B = µh µ = 4π 10 Wb Am 0 µ = r µ µ 0 Ampère s Law: H dl = i 10

11 Magnetic Field Around a Long Straight Wire B = µ H = µ I 2πr 11

12 Flux Density in a Toroidal Core µ NI B = 2πR 12

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14 MAGNETIC CIRCUITS In many engineering applications, we need to compute the magnetic fields for structures that lack sufficient symmetry for straightforward application of Ampère s law. Then, we use an approximate method known as magnetic-circuit analysis. magnetomotive force (mmf) of an N- turn current-carrying coil I = N I reluctance of a path for magnetic flux R = µa I = Rφ 14

15 Advantage of the Magnetic-Circuit Approach The advantage of the magnetic-circuit approach is that it can be applied to unsymmetrical magnetic cores with multiple coils. 15

16 Fringing We approximately account for fringing by adding the length of the gap to the depth and width in computing effective gap area. 16

17 A Magnetic Circuit with Reluctances in Series and Parallel 17

18 INDUCTANCE AND MUTUAL INDUCTANCE L = λ i L = N R 2 e = L di dt 18

19 Mutual Inductance L 1 = λ i 11 1 L 2 = λ i 22 2 M = λ 21 = i 1 λ i

20 Dot Convention Aiding fluxes are produced by currents entering like marked terminals. Circuit Equations for Mutual Inductance λ1 = L1i 1 ± Mi2 λ2 = ± Mi1 + L2i2 dλ1 di1 di2 e1 = = L1 ± M dt dt dt dλ2 di1 di e2 = = ± M + L2 dt dt dt 2 20

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22 MAGNETIC MATERIALS The relationship between B and H is not linear for the types of iron used in motors and transformers. 22

23 Energy Considerations W v = W Al = B 0 H db 23

24 Core Loss Power loss due to hysteresis is proportional to frequency, assuming constant peak flux. 24

25 Eddy-Current Loss Power loss due to eddy currents is proportional to the square of frequency, assuming constant peak flux. 25

26 Energy Stored in the Magnetic Field W B = 0 2 B B db µ 2µ v = 26

27 IDEAL TRANSFORMERS N N 2 () t v () t v2 = 1 1 I = N 1 2rms I1rms N 2 ( t) p ( t) p2 = 1 27

28 Transformer Summary 1. We assumed that all of the flux links all of the windings of both coils and that the resistance of the coils is zero. Thus, the voltage across each coil is proportional to the number of turns on the coil. N N 2 () t v () t v2 =

29 2. We assumed that the reluctance of the core is negligible, so the total mmf of both coils is N N zero. () 1 t i () t i2 = A consequence of the voltage and current relationships is that all of the power delivered to an ideal transformer by the source is transferred to the load. 29

30 Analysis of a Circuit Containing an Ideal Transformer 30

31 31 Impedance Transformations L Z L N N Z = = I V

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33 REAL TRANSFORMERS 33

34 Variations of the Transformer Model 34

35 Regulation and Efficiency V percent regulation = V no -load load V load 100% power efficiency Pload Ploss = 100% = 1 100% Pin P in 35