Chapter 5: Static Magnetic Fields

Transcription

1 Chapter 5: Static Magnetic Fields 5-8. Behavior of Magnetic Materials 5-9. Boundary Conditions for Magnetostatic Fields Inductances and Inductors

2 5-8 Behavior of Magnetic Materials Magnetic materials can be roughly classified into three main groups in accordance with their r values. A material is said to be Diamagnetic, if r 1 ( Paramagnetic, if r 1 ( Ferromagnetic, if r >> 1 ( is a very small negative number) is a very small positive number) is a large positive number) Ferromagnetism can be explained in terms of magnetized domains.

3 5-8 Behavior of Magnetic Materials Domain structure of a polycrystalline ferromagnetic specimen: (Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.197)

4 5-8 Behavior of Magnetic Materials Hysteresis loops in the B H plane for ferromagnetic material: (Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.197)

5 5-8 Behavior of Magnetic Materials For weak applied fields, say up to point P 1 on the B H magnetization curve in Fig 5-12 domain-wall movements are reversible. When an applied field becomes stronger (past P 1 ), domain-wall movements are no longer reversible, and domain rotation toward the direction of the applied field will also occur. If an applied field is reduced to zero at point P 2, the B H relationship will not follow the solid curve P 2 P 1 O, but will go down from P 2 to P 2, along the broken curve in the figure.

6 5-8 Behavior of Magnetic Materials This phenomenon of magnetization lagging behind the field producing it is called hysteresis. The curve OP 1 P 2 P 3 on the B H plane is called the normal magnetization curve. If the applied magnetic field is reduced to zero from the value at P 3, the magnetic flux density does not go to zero but assumes the value at B r. This value is called the residual or remanent flux density (in Wb/m 2 ) and is dependent on the maximum applied field intensity.

7 5-8 Behavior of Magnetic Materials The existence of a remanent flux density in a ferromagnetic material makes permanent magnetic. To make the magnetic flux density of a specimen zero, it is necessary to apply a magnetic field intensity H c in the opposite direction. This required H c is called coercive force, or coercive field intensity. Hysteresis loss: The energy lost in the form of heat in overcoming the friction encountered during domain-wall motion and domain rotation.

8 5-9 Boundary Conditions for Magnetostatic Fields From the divergenceless nature of the B field in Eq. 95-6) we may conclude directly that the normal component of B is continuous across an interface : (5-68) For linear and isotropic media, B 1 = H 1 and B 2 = H 2, Eq. (5-68) becomes (5-69)

9 5-9 Boundary Conditions for Magnetostatic Fields (Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.199)

10 5-9 Boundary Conditions for Magnetostatic Fields In letting the sides bc = da = h approach zero. (5-70) where is the surface current density on the interface normal to the contour abcda The more general form for Eq. (5-70) is (5-71) where is the outward unit normal from medium 2 at the interface.

11 5-9 Boundary Conditions for Magnetostatic Fields When the conductivities of both media are finite, currents are specified by volume current densities and free surface curres are not defined on the interface. J s equals zero, and the tangential component of H is continuous across the boundary of almost all physical media; it is discontinuous only when an interface with an ideal perfect conductor or a superconductor is assumed. Thus, for magnetostatic fields, we normally have: (5-72)

12 5-10 Inductances and Inductors Let us designate the mutual flux. We have: (5-73) Fig. 5-14: Two magnetically coupled loops (Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.201)

13 5-10 Inductances and Inductors B 1 is directly proportional to I 1 ; hence, is also proportional to I 1 : (5-74) where the proportionality constant L 12 is called the mutual inductance between loops C 1 and C 2, with SI unit Henry(H). In case C 2 has N 2 turns, the flux linkage due to is (5-75)

14 5-10 Inductances and Inductors Equation (5-74) then generalizes to (5-76) (5-77)

15 5-10 Inductances and Inductors The mutual inductance between two circuits is then the magnetic flux linkage with one circuit per unit current in the other Some of the magnetic flux produced by I 1 links only with C 1 itself, and not with C 2. The total flux linkage with C 1 caused by I 1 is (5-78) The self-inductance of loop C1 is defined as the magnetic flux linkage per unit current in the loop itself for a linear system: (5-79)

16 5-10 Inductances and Inductors A conductor arranged in an appropriate shape to supply a certain amount of self-inductance is called an inductor. The procedure for determining the self-inductance of an inductor is as follows: 1. Choose an appropriate coordinate system for he given geometry. 2. Assume a current I in the conducting wire. 3. Find B from I by Ampere s circuital law, eq.(5-10), if symmetry exists; if not, Biot-Savart law, eq.(5-31) must be used.

17 5-10 Inductances and Inductors 4. Find the flux linking with each turn,, from B by integration: B ds S 5. Find he flux linkage by multiplying by the number of turns. 6. Find L by taking the ratio L = / I.