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# Leakage Impedance of Transformer Windings

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1. Leakage Impedance of Transformer Windings Dept. Of Information Engineering – DEI University of Padova Prof. Giorgio Spiazzi

2. Leakage Impedance of Transformer Windings Ref.: P.L. Dowell, “Effects of Eddy Currents in Transformer Windings,” Proc. of IEE, Vol.13, No.8, August 1966, pp.1387-1394. • High Frequency induced effects on transformer windings • Qualitative analysis of transformer winding leakage impedance • Quantitative analysis of transformer winding leakage impedance • Dowell curves • Examples Outline:

3. Simple Transformer Winding Arrangement Secondary winding Isolation gap Primary winding

4. Magneto-motive Force in the Core Window m.m.f. (dc) 0

5. Magneto-motive Force in the Core Window Secondary winding Isolation gap Primary winding m.m.f. (dc) 0

6. Leakage Flux Leakage flux Power loss in the winding resistance The leakage flux in the core window causes eddy currents in the windings

7. Leakage Flux Leakage flux Magnetic energy stored in the core window crossed by the leakage flux Magnetic energy stored in the transformer leakage inductance

8. Skin Effect DPEN = skin depth DW DPEN JREAL JEQUIVALENT Current lines High frequency currents in the conductor generate a variable magnetic field that induces voltages and, consequently, currents. The latter are directed in such a way to reinforce the current flowing close to the conductor surface

9. Proximity Effect Example: faced PCB tracks  + + + + + + + + + + PCB • • • • • • • • • • W • The current in a close path distributes itself in such a way so as to minimize the energy drawn from the source.

10. HF Current Distribution in Windings . . . . . . . . . . . . + + + + + + + + + + + +  F F g Inductor: single winding The current concentrates on winding inner surface

11. HF Current Distribution in Windings . . . . . . . . . + + + + + + + + + + + + . . . . + . . . + . . + . + . + . + F F . + . + . + + + + Transformer: single layer Primary: 4 turns - 3A / Secondary: 1 turn - 12A The magnetic field is almost zero outside the two windings but is high between them

12. LF Current Distribution in Windings F=NI Energy density [J/m] W [J] P1 P2 S3 S2 S1 Transformer with multiple layers Current homogeneously distributed inside conductors

13. LF Current Distribution in Windings F=NI Reduced leakage inductance Energy density [J/m] W [J] Multiple winding transformer with interleaved primary/secondary windings Pa Sa Sb Pb

14. HF Current Distribution in Windings S3 S2 S1 P1 P2 F=NI Multiple winding transformer Magnetic field only between layers Conductor thickness >> DPEN

15. HF Current Distribution in Windings I= +3 -2 +2 -1 +1 . . . + + + + + + S3 S2 S1 F=NI Secondary winding: 1A Magnetic field only between layers : different currents induced on layer surfaces Conductor thickness >> DPEN

16. Passive Layers I= +3 -3 +3 -2 +2 -1 +1 . . . . . . + + + + + + + + + S3 S2 S1 P F=NI High losses! Winding carrying zero current in a given instant (e.g. one primary winding in a push-pull transformer, one secondary winding in transformer with center tapped secondary, EMI shield)

17. Leakage Impedance When considering the leakage impedance due to a particular layer, it is necessary to consider the other layers of the windings insofar as they affect the flux in the layer being considered The leakage flux crossing a winding layer determines both its ac leakage resistance and inductance

18. Leakage Impedance From the behavior of the m.m.f. in a core window we can say : The leakage flux distribution across any layer depends only on the current in that layer and the total current between the layer and an adjacent position of zero m.m.f. P1 P2 S3 S2 S1 m.m.f. (dc) 0

19. Leakage Impedance Winding portions For leakage impedance calculation purposes we can consider the whole winding subdivided in parts containing each a position of zero m.m.f.. Such parts will be termed “winding portions”. m.m.f. (dc) 0

20. Leakage Impedance An intersection gap can be considered to be part of either of the adjacent portions; thus, the leakage impedance due to these gaps will be referred to the primary if they are associated to a primary winding portion or to the secondary if the gaps are associated to a secondary winding portion. m.m.f. (dc) 0 Gap

21. Interleaved Windings Winding portions Pa Sa Sb Pb A winding portion can contain a half layer (if the corresponding winding section is composed by an odd number of layers) Two cases: • The winding portion contains m full layers • The winding portion contains m full layers + a half layer m.m.f. (dc) 0

22. Frequency-Independent Components of Leakage Impedance a • Increasing the frequency will affect the current distribution across each conductor, but the total net current will remain unaltered. Consequently, the magnetic field H and its associated energy in the intersection gaps will be independent of the frequency. b g u m.m.f. (dc) Winding portion 0

23. Frequency-Independent Components of Leakage Impedance  D a • Hp: • Square section conductor having the same section of circular ones • Average turn length T • Uniform magnetic field in the core window

24. Leakage Inductance of Isolation Gap g a b g m.m.f. (dc) 0 Integer number m of layers N = Number of turns per layer Np = Number of turns of the whole winding portion

25. Leakage Inductance of Isolation Gap g Integer number m of layers +half layer a b Np = Number of turns of the whole winding portion g m.m.f. (dc) 0

26. Leakage Inductance of Gaps u Between Layers Integer number m of layers a pth Gap b u m.m.f. (dc) pth Gap 0

27. Leakage Inductance of Gaps u Between Layers (Total gap width) Overall inductance: a b u m.m.f. (dc) 0

28. Leakage Inductance of Gaps u Between Layers Integer number m of layers +half layer a p = 1m b u m.m.f. (dc) pth Gap 0

29. Leakage Inductance of Gaps u Between Layers (Total gap width) Overall inductance: a b u m.m.f. (dc) 0

30. DC Winding Inductance Since the flux in the intersection gaps have already been taken into account, the energy associated to the flux crossing the conductor layers of a given winding portion can be done considering the layers close to each other without gaps. Being the current density constant at dc, the current and the associated magnetic field vary linearly with position x as indicated in the figure (x=0 corresponds to the position of zero m.m.f.).

31. DC Winding Inductance Integer number m of layers a b h H dx 0 mh x

32. DC Winding Inductance 1 2 (m+ )h Integer number m of layers +half layer a b Same procedure, only substitute m with m+1/2 h H dx 0 x

33. DC Winding Resistance Form factor. It is equal to 1 when the turns of the same layer are close to each other. Integer number m of layers a b h

34. DC Winding Resistance Integer number m of layers +half layer The resistance of the winding portion is half of the resistance of the winding section (made up by 2m+1 layers) having that portion as half section a b h

35. AC Winding Leakage Impedance • Only the flux crossing the winding layers is considered. • The current density inside each layer is calculated. • The voltage developed across each layer is calculated as the sum of a resistive component plus an induced voltage due to the linked flux. • The total voltage across the winding portion is calculated summing the voltage across each layer. • The leakage impedance of the winding portion is calculated whose real and imaginary parts represent the leakage resistance and inductance, respectively.

36. AC Winding Leakage Impedance x 0 dx pth layer Integer number m of layers A generic layer p (p=1m) is considered, and inside it, an infinitesimal layer dx at position x (as respect to the edge of pth layer closer to the position at zero m.m.f.) a b g h u m.m.f. (dc) 0

37. AC Winding Leakage Impedance fb fc Magnetic field at position x: Flux linking elementary layer dx is fb+fc: a b g h u pth layer

38. AC Winding Leakage Impedance V is independent of x: fb fc Voltage across pth layer: a b g h u pth layer

39. AC Winding Leakage Impedance Solution: fb fc a b g h u pth layer

40. AC Winding Leakage Impedance fb fc Coefficients: a b g h u pth layer

41. Current Density Distribution Current density inside pth layer DPEN = skin depth

42. Normalized Current Density fs = 100kHz 1° layer 2° layer 3° layer 20 JN(x) JN(x) JN(x) 15 10 5 0 0 h

43. Normalized Current Density fs = 500kHz fs = 100kHz fs = 50kHz 3° layer JN(x) 30 20 10 fs = 10kHz 0 0 h

44. AC Winding Leakage Impedance Voltage across pth layer: Vp is independent of x. Thus, it is calculated at x = h: fb fc a b g h u pth layer

45. AC Winding Leakage Impedance Let’s calculate the flux in the generic pth layer: fb fc fc is the flux in all the winding layers beyond the pth layer edge at position x = h: a b g h u pth layer

46. AC Winding Leakage Impedance

47. AC Winding Leakage Impedance Total flux crossing pth layer: Total flux linking pth layer:

48. AC Winding Leakage Impedance Current density at the edge of pth layer far from the position at zero m.m.f.: Voltage across pth layer:

49. AC Winding Leakage Impedance Associated leakage impedance: Total voltage across the winding portion:

50. AC Winding Leakage Impedance AC inductance: AC resistance: