12.1 Static Magnetic Field in the Presence of Magnetic Materials

Transcription

1 Electroagnetics P1-1 Lesson 1 Magnetostatics in Materials 1.1 Static Magnetic Field in the Presence o Magnetic Materials Concept o induced agnetic dipoles An aterial has an icroscopic agnetic dipoles (i.e., tin current loops) arising ro: (1) orbiting electrons, () electrons and nucleus o an ato spinning on their own axes. Howeer, a aterial bulk ade up o a large nuber o randol oriented olecules tpicall has no acroscopic dipole oent in the absence o external agnetic ield. As shown in Fig. 1-1a, consider an ato consisting o a nucleus o positie charge q and an electron o negatie charge e oing along a circular path o radius r with a constant angular elocit ω in counterclockwise sense (corresponding to a elocit u = a φ rω ). The Coulob s orce experienced b the electron centriugal orce a r eω r q = ee ( E = ar ) proides the 4πε r F e ( e eans the ass o electron) to support the orbiting otion. When a agnetic ield F Fig Classic odel to interpret diaagetis. B a = B is applied, the orbiting electron experiences an extra orce = eu B [eq. (11.1)], where u = a φ rω (Fig. 1-1b). The total orce F e + F increases, proiding a stronger centriugal orce a ω r r e and a larger angular elocit

2 Electroagnetics P1- ω ( > ω). Since the loop current equals ue eω I = = (in clockwise sense), the presence πr π o B changes the agnetic dipole oent o each ato b an aount o ( ΔI r ) Δ = a π, where e( ω ω ΔI = ). As a result, a net dipole oent eerges, π contributing to a agnetic ield in opposite direction with the applied one. This classical odel can be used to interpret diaagnetis. As shown in Fig. 1-a, consider a circular current loop on the x-plane with a agnetic dipole oent a =. When a agnetic ield B = a B is applied, the positiel charged particles located on the upper seicircle ( x > ) will experience a agnetic orce F in the a direction, while those located on the lower seicircle ( x < ) will experience a agnetic orce F in the + a direction. The resulting torque will lip the current loop until the agnetic dipole oent is aligned with the applied agnetic ield a = (Fig. 1-b). Since all the icroscopic agnetic dipole oents are aligned with B, a net oent eerges, contributing to a agnetic ield in the sae direction with the applied one. This odel can be used to interpret paraagnetis. Fig Classic odel to interpret paraagetis. <Coent> Quantu echanical odel (spin) is required to properl interpret dia- and para-agnetis.

3 Electroagnetics P1-3 Magnetiation ector and equialent current densities To anale the eect o induced dipoles, we deine a (icroscopic) agnetiation ector M as the olue densit o agnetic dipole oent: k M li (1.1) Δ Δ where k denotes the k-th agnetic oent inside a dierential olue Δ ν. I the agnetiation ector M is inhoogeneous (i.e., aries with position) soewhere, there ust exist net agnetiation current at that position. This phenoenon can be illustrated in two cases. Fig The odel to deduce the agnetiation surace current densit. 1) The agnetiation ector M is discontinuous on the air-aterial interace, where there ust exist net agnetiation current (Fig. 1-3a). To quantitatiel odel this phenoenon, consider a dierential olue Δ V = dxdd adjacent to the interace = (whose unit noral ector is a n = a ). As shown in Fig. 1-3b, assue the olue has a rectangular current loop with area counterclockwise sense, corresponding to a agnetic dipole oent Δ S = dd and current I lowing in ΔM = axiδs and a ΔM I agnetiation ector M = = a x. Since the currents ro neighboring olues ΔV dx will cancel with each other except or the coponents lowing on the interace, there is a current I lowing along a oer a cross-length dx. The corresponding surace current

4 Electroagnetics P1-4 densit is J s I = a, which can be generalied to: dx J = M (A/) (1.) s a n I I or M an = ax a = a in this particular case. dx dx ) Consider two adjacent agnetic dipoles with agnetiation ectors M a = M (x) and a M ( x + dx), where M ( x) I( x) ΔS I( x) = = (Fig. 1-4). The net current passing ΔS d d through a surace bounded b contour C (dashed) between the two agnetic dipoles is: ( x) I( x + dx) = [ M ( x) M ( x + dx ]d I ). The corresponding olue current densit is: I( x) I( x + dx) M ( x) M ( x + dx) J = a = a = a dxd dx which can be generalied to: J M x = M (A/ ) (1.3). or M = a x M x x a M a M = a x x a a M = a in this particular case. x M Fig The odel to deduce the agnetiation olue current densit. <Coent> 1) Eq. (1.) can be regarded as a special case o eq. (1.3) on the surace, where the curl o

5 Electroagnetics P1-5 the agnetiation ector M is ininite. ) Equialent agnetiation current densities J s, J can be used in collaboration with eq. (11.11) to ealuate the ector potential (and then agnetic ield) contributed b agnetied aterial. Exaple 1-1: A uniorl agnetied clinder (peranent agnet) has radius b, length L, and M a = M (Fig. 1-5a). Find B along the -axis. Ans: B eq s (1.), (1.3), there is unior agnetiation surace current densit J s = a M a = a r φ M on the side wall, and no agnetiation current densit elsewhere. Consider a circular ring o height d centered at (,, ), where there is a dierential current di J d s = M d = lowing along a φ. B Exaple 11-4, the current ring contributes to a dierential agnetic lux densit: db = a at an obseration point P (,, ) μm d 1 + b b 3 /. The total agnetic lux densit becoes: = = L B db = a μ M L + b ( L) + b = Fig. 1-5b shows the noralied agnitude o agnetic lux densit B as a unction o longitudinal position. The two cures represent the results due to two clinders o the sae olue but dierent lengths and cross-sectional areas, respectiel. It is eident that longer clinders (with saller cross section) can produce stronger axial agnetic ield..

6 Electroagnetics P1-6 Fig (a) A uniorl agnetied clinder (ater DKC), and (b) the resulting B () ersus axial position or two dierent clinders o the sae olue. Magnetic ield intensit In the presence o agnetic aterials, the total agnetic ield would be deterined b both ree and agnetiation currents. The undaental postulate eq. (11.3) is thus odiied as: B = μ. ( J + ) B eq. (1.3), B = μ J + ( M ), B μ ( M ) = J μ J B μ, M = J, μ H = J, (1.4) B H = M (A/) (1.5) μ The ter H in eq. (1.4) is deined as the agnetic ield intensit, totall deterined b the ree currents. The integral or o eq. (1.4) gies the Apere s circuital law: C H dl = I (1.6) <Coent> 1) The agnetic ield induced b a agnetic dipole (current loop) is in the sae direction as the dipole oent M (Fig. 1-6a). The total ield ( ~ B ) is the suation o ields

7 Electroagnetics P1-7 caused b ree current ( ~ H ) and agnetiation ( ) H = B M μ [eq. (1.5)]. ~ M, B = μ H + M, i.e., μ ) The electric ield induced b an electric dipole (separated charges) is in opposite direction as the dipole oent P (Fig. 1-6b). Total ield ( ~ E ) is the suation o ields caused b ree charge ( ~ D ) and polariation ( P ~ ), ε E = D P, i.e., D ε E + P [eq. (7.9)]. = Fig Coparison between (a) agnetic, and (b) electric dipoles. For linear, hoogeneous, and isotropic agnetic aterials, the agnetiation ector is proportional to the applied agnetic ield: M = χ H, (1.7) where χ is a diensionless quantit independent o the agnitude (linear), position (hoogeneous), and direction (isotropic) o H. Eq. (1.5) becoes B μ ( H + M ) = = μ 1+ ( H + χ H ) = μ ( χ )H, B = μh, (1.8) where the pereabilit o the ediu μ is deined as: μ = μ 1+ ( ) χ (1.9)

8 Electroagnetics P1-8 <Coent> The strateg is using a single constant μ to replace the tedious induced agnetic dipoles, agnetiation ector, and equialent current densities in deterining total agnetic ield. Exaple 1-: Consider a wound-coil inductor. (1) H is related to the surace densit o ree current on the coil. () M is related to the surace densit o agnetiation current (the direction is shown b assuing paraagnetic). (3) B μ corresponds to the surace densit o total current or uncopensated ree current. Fig Phsical eanings o agnetic lux densit B, agnetiation ector M, and agnetic ield intensit H illustrated in the exaple o a wound-coil inductor (ater C. C. Su). Exaple 1-3: A stead current I lows in N turns o wire wound around a erroagnetic toroidal core o pereabilit μ (agnetic source). The toroid has a large ean radius r, a sall cross-sectional radius a ( << ) r, and a narrow air gap o length l g (Fig. 1-8a). Find B, H in the core and air gap, respectiel. Ans: Assue the lux has no leakage and nor ringing eect in the air gap, total lux (thus agnetic lux densit B ) is constant throughout the agnetic loop, i.e., B eq s (1.6), (1.8), B = B = a B. g φ

9 Electroagnetics P1-9 B l B + lg = NI, where l = π r lg μ μ B =, l NI μ + l g μ H B =, μ H g. Bg =. μ Fig (a) Magnetic toroid with air gap. (b) The corresponding equialent circuit (ater DKC). <Coent> 1) B = Bg, but H << H g, or a sall H can induce a strong M in the erroagnetic aterial ( μ >> μ ), proiding a ajorit o agnetic lux in the erroagnetic core. ) Total lux Φ = BS = NI ( l μ S ) + ( l μ S ) g, which can be deined as: V Φ R + R g, where S = πa is the cross-sectional area, V = NI is the agnetootie orce (), R i l i = ( i, g) μs = is the reluctance ( R is the internal reluctance o the agnetic source, R g is the load reluctance). This is analogous to V I = R in electric circuits (Fig. 1-8b). For a closed path with ultiple agnetic sources and reluctances: j N I = R Φ (1.1) j j k k k 3) B H cure o erroagnetic aterial is usuall nonlinear and depends on the histor o

10 Electroagnetics P1-1 agnetiation, μ changes with H and its histor (hsteresis), odiication is required in inding B (DKC p5). 4) Due to er liited contrast o μ aong dierent agnetic aterials, conineent o agnetic ield is usuall uch worse than that o current densit, lux leakage and ringing eect are norall non-negligible, aking the odel o agnetic circuit less accurate. 1. General Boundar Conditions or M-ields Deriation As in electrostatics, we appl integral ors o the two undaental postulates on dierential contour and thin pill box ( Δh ) across the interace o two agnetic edia to derie the boundar conditions (BCs) or tangential and noral coponents o agnetic ield. 1) Tangential BC: B eq. (1.6), H dl H = Δw + H Δw = H 1 ( ) 1t Δw H abcda where ( i = 1, ) t Δw = J sn Δw, H it eans the H i coponent in the ab -direction, J sn is the surace current densit coponent in the right thub-direction a thub when the our ingers ollow the direction o the path. H = 1 t H t J sn. In general, a n ( H1 H ) = J s (1.11) where a n is the unit noral ector directed ro ediu to ediu 1.

11 Electroagnetics P1-11 Fig Dierential contour used to derie tangential BC o static agnetic ields (ater DKC). <Coent> Note that the projections o H 1 and H on the interace are generall not in parallel. Fig. 1-1 illustrates a special exaple when the projections o H 1 and H on the interace (x-plane) are perpendicular with each other. Fig Surace current densit and boundar agnetic ields. ) Noral BC: BC: B eq. (11.4), ( )( ) B ds = B S B n Bn 1 an B an ΔS =, 1 = (1.1) <Coent> 1) Onl ree surace current J s counts in eq. (1.11). I none o the two interacing edia is perect conductor, J =, H1 t = H t. s ) Eq s (1.11), (1.1) reain alid een the ields are tie-aring (Lesson 14).

12 Electroagnetics P Behaior o Magnetic Materials (*) Categoriation 1) Diaagnetic aterials ( μ < μ ): As shown in Fig. 1-1, an external agnetic ield can change the angular elocities o orbiting electrons, resulting in a net agnetic dipole oent in opposite direction o the applied ield (Len s law). B eq s (1.7), (1.9), χ <, μ < μ. Diaagnetis is present in all aterials, but is usuall er weak 5 ( ~ 1 ) χ and asked in paraagnetic/erroagnetic aterials. It disappears when the external ield is withdrawn. ) Paraagnetic aterials ( μ > μ ): As shown in Fig. 1-, an external agnetic ield tends to align the dipole oents o orbiting and spinning electrons such that the net agnetic dipole oent is in the sae direction o the applied ield. χ >, μ > μ. 5 Paraagnetis is usuall er weak ( χ ~ 1 ), reduced b theral ibration (randoiing the dipole oents), and disappears when the external ield is withdrawn. 3) Ferroagnetic aterial ( μ >> μ ): This tpe o aterials consists o an doains with linear diensions in the range between 1 μ and 1. Each doain has ull aligned dipole oents due to strong coupling orces aong spinning electrons (b quantu theor). Doain walls o ~1 atos thick exist between adjacent doains. The doains are disorganied (aterial is deagnetied) i the teperature is aboe a critical alue (curie teperature) when theral energ exceeds the coupling energ. When an external agnetic ield is applied, the doains with dipole oents aligned with the applied ield will expand, largel increasing the total agnetic lux (iron has χ ~ 4 ). Howeer, the relation between the total agnetic lux (~B-ield) and the applied ield (~H-ield) is uch ore inoled than a siple linear unction like eq. (1.8).

13 Electroagnetics P1-13 Fig Doains o a erroagnetic aterial (ater DKC). Hsteresis cure o erroagnetic aterials 1) Reersible agnetiation: I the external agnetic ield is weak (up to P 1 in Fig. 1-1), doain wall oeent is reersible. The B H cure is a unction, i.e., one H-alue corresponds to a unique B-alue. ) Hstersis: I the external agnetic ield is stronger (sa P in Fig. 1-1), the B H cure is not a unction. For exaple, when the H-alue is decreased ro H ( > ) to H ( < ), the B-alue will change along the upper branch o the broken lines connecting P and P. When the H-alue is increased ro H ( < ) to H ( > ), the B-alue will change along the lower branch o the broken lines. In other words, one H-alue a correspond to two B-alues, depending on the histor o the external ield. 3) Saturation: I the external agnetic ield is er strong (sa P 3 in Fig. 1-1), all the doains are aligned, and urther increasing external ield does not increase the lux densit. Fig Hsteresis o a erroagnetic aterial (ater DKC).

14 Electroagnetics P1-14 <Coent> 1) The hsterisis behaior o B H cure akes that μ depends on: (1) agnitude o H, () histor o the aterial s agnetiation (sae H a correspond to dierent B s), seriousl coplicating the analsis o agnetic circuit. ) Applications in generators, otors, transorers preer tall, narrow hsteresis loop, while peranent agnets preer at hsteresis loop. 3) The area o hsteresis loop is the energ loss per unit olue per ccle in the or o heat in oercoing the riction o doain rotation (DKC, Proble P.6-9).