1. Magnetostatics

2. Recall: Magnetostatics
Current in an inductor
Bio-Savart’s Law
Right-hand rule
Not conservative
Magnetic Flux Density
Magnetic Field
Induction machines
Transformers?
Magnets
Inductance..
Emf

3. Magnetic Force on a Current Element

6. Magnetic Force on a Current Loop

9. Ampère’s Law

10. Internal Magnetic Field of Long Conductor
For r < a
Cont.

11. External Magnetic Field of Long Conductor
For r > a

12. Magnetic Field of Toroid
Applying Ampere’s law over contour C:
Ampere’s law states that the line integral of H around a closed contour C is equal to the current traversing the surface bounded by the contour.
The magnetic field outside the toroid is zero. Why?

13. Recall: Inductance Types? Where did you hear about inductance?
What is it?
Units
Any other info?

14. Coils of wire
Stores electromagnetic field
Oppose the change in current
Electromagnetic waves
Mutual inductance
Units Henry
Using transformers to do something to voltage
Motors or generators
Step up or step down
Isolation

18. There are circular rings of magnetic-field around all currents

20. Right Hand Rule Number of magnetic field lines unit is Weber

21. What affects the # of magnetic field lines?
Current. (double I -> double # lines)
Length of wire ( l increases -> #lines increases)
Cross section of the wire (CS increases -> # lines decreases)
Other currents in the vicinity (can increase or decrease # of lines)
Metal that the wire is made of if ferromagnetic

22. Inductance is # of H lines around the conductor per amp of current through it
L=N/I
Unit Henry
Inductance is related to the geometry of conductors, and magnetic material properties

24. Types of Inductances Self-Inductance Mutual Inductance Loop inductance
Partial Inductance
Total Inductance / Net Inductance / Effective inductance

25. Self-Field Line Rings and Mutual-Field Line Rings

26. Self Inductance
Self-Field line rings – magnetic field rings due to current in the wire
Number of field-line rings around a wire per amp of current in its own wire

27. Mutual Inductance
Mutual-Field line rings- magnetic field rings around a wire due to a current from another wire.
Number of field-line rings around one wire, per Amp of current in another wire.

28. Practice

29. Induced Voltage
When the number of magnetic field lines changes around one wire, the voltage is induced across the wire.

30. Partial Inductance We can’t have a partial current
Currents always flow in loops
However, sometimes we don’t know how the rest of the loop looks like so:
We define inductance of a small part of the loop as partial inductance
Two types of partial inductance
Partial self-inductance
Partial mutual-inductance

31. Partial Self-Inductance of a Rod
30-gauge wire, 1mm long -> 25nH/inch or 1nH/mm
The more spread out the current distribution, the lower the partial self-inductance.

32. Estimate the inductance of a via-hole through a substrate 64mils thick.

34. Partial Mutual Inductance between two round wires
In general partial mutual inductance between two wires is much smaller than partial self-inductance of either wire.

35. Rule of thumb for partial inductance
If the spacing between two conductors is farther apart than their length:
For example two 20mil via holes spaced 20 mils apart have no coupling between them

36. What will happen here?

37. Effective, total or Net Inductance
The total number of field lines around just this section per amp of current in the loop
It included the contribution of field line rings from all the current segments in the loop

38. Ground Bounce - Voltage between two points in the return path due to a changing current

39. Ground Bounce To decrease ground-bounce:
Decrease the partial self-inductance of the return path (short lengths and wider interconnects)
Increase the mutual inductance of the two legs (by brining them closer together)

40. Example: What if the wires carry current in the same direction?
Total inductance is??

42. Loop Self and Mutual Inductance
Total number of field lines surrounding the entire loop, per amp of current in the loop.

43. Loop Inductance of a circular loop
A 30 gauge wire (10mils=d)
Bent in a 1 inch diameter
Has 85nH inductance

44. Inductance formulas on web

45. Loop inductance of two adjacent straight and round wires

46. Loop inductance per square of planes
Parallel-plate waveguide capacitance

47. Spreading Inductance The planes are not connected like this
But through vias

48. Internal and External Self-Inductance

49. Current Distributions and Skin Depth

50. Skin depth vs. ½ oz copper trace

51. How can you calculate inductance?

52. How do we find H? How do we find E? Ampere’s law Bio-Savart’s Law
Gauss’ Law
Coulomb’s Law

53. Maxwell’s Equations

54. What do we know about motors/generators
Generator is a motor in reverse
Stator and rotor typically
We use coupled coils to transfer magnetic energy