Prof. David R. Jackson ECE Dept. Spring 2016 Notes 30 ECE 3318 Applied Electricity and Magnetism 1

Magnetic Materials Because of electron spin, atoms tend to acts as little current loops, and hence as electromagnetics, or bar magnets. NS NS NS NS When a magnetic field is applied, the little atomic magnets tend to line up. This effect is what causes the material to have a relative permeability. B 2

Magnetic Materials 3 Type of MaterialProperty Nonmagnetic  r = 1 Diamagnetic  r < 1 Paramagnetic  r > 1 Ferrimagnetic  r >> 1 Ferromagnetic  r >> 1 For more information: Please see the textbooks to learn more about magnetic materials and permeability.

Magnetic Materials 4 Material Relative Permeability  r Vacuum 1 Air Water Copper Aluminum Silver Nickel 600 Iron 5000 Carbon Steel 100 Transformer Steel 2000 Mumetal 50,000 Supermalloy 1,000,000 Note: Values can often vary depending on purity and processing.

Boundary Conditions Note: If there is no surface current: 5 The unit normal vector points towards region 1. JsJs Ht1Ht1 Ht2Ht2 Bn1Bn1 Bn2Bn2 Region 1 Region 2 (Please see the textbooks for a derivation.)

Boundary Conditions (cont.) JsJs PEC Assume zero magnetic field inside the PEC (This is true for any f > 0, please see the note at the end on skin depth.) HtHt Note: The surface current is directly related to the tangential magnetic field at the surface. 6

Boundary Conditions (cont.) Magnetic field lines must bend around a perfect electric conductor (PEC). This is the opposite behavior of electric field lines. B PEC E 7

Boundary Conditions (cont.) 8 Note: The “skin depth”  of a metal determines how the magnetic field will vary within a good conductor (discussed in ECE 3317). The field penetrates with distance z into the conductor as z ,  Conductor DC limit: Finite conductivity and zero frequency:  =  (there is a uniform field inside the conductor) PEC limit: Infinite conductivity and nonzero frequency:  = 0 (zero fields inside conductor)

Magnetic Stored Energy Hence we can write We also have 9

Example z I a N turns LsLs Find U H inside solenoid 10 Assume infinite solenoid approximation.

Finite-Length Solenoid 11 In a practical finite-length solenoid, the magnetic flux density immediately outside the solenoid is the same as that inside the solenoid. From BCs: Note: It is the strength of the B field immediately outside the solenoid that determines the lifting strength of the magnet. I z a N turns LsLs The solenoid is long but finite. (from BCs)

Hysteresis 12 Hysteresis is a nonlinear effect that many magnetic materials exhibit. New sample (has never been magnetized) (saturation)

Hysteresis (cont.) 13 Hysteresis causes power loss for an AC magnetic field in a material (such as in a transformer core). f = frequency [ Hz ] z N turns LcLc AcAc + -

Hysteresis (cont.) 14 Power going into coil: z N turns LcLc AcAc + -

Hysteresis (cont.) 15 Energy going into the coil in one cycle of the waveform (period T ): A h = area inside hysteresis curve z N turns LcLc AcAc + - Note: Note: The energy is not zero (there is loss)!

16 The power loss (in watts) due to hysteresis is: We then have A h = area inside the hysteresis curve [ T A/m ] f = frequency [ Hz ] V c = volume of the core [ m 3 ] Hysteresis (cont.) where