Lesson 13 Inductance, Magnetic energy /force /torque


 Donald Garrett
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1 Lesson 3 nductnce, Mgnetic energy /force /torque 楊 尚 達 ShngD Yng nstitute of Photonics Technologies Deprtment of Electricl Engineering Ntionl Tsing Hu Uniersity, Tiwn
2 Outline nductnce Mgnetic energy Mgnetic force Mgnetic torque
3 Sec. 3 nductnce. Self & mutul inductnces. Elution procedures
4 Definition Closed loop C crrying current will crete Φ B ds, flux: flux linge: B S f, by r B Λ μ dl 4π ' C B NΦ B rb, Φ B ds rφ, Λ rλ Selfinductnce of the loop C : S L Only depend on geometry Λ
5 Definition n the presence of nother loop C, will pss through C, mutul flux linge: where Φ B ds S B B L B Λ N Φ Mutulinductnce between the loops: Λ Depend on geometry & mteril.
6 Comment μ dl A( r), ' 4π C ( r, r ) L NΦ N S B A μ N ( ) A ds A L dl 4π C N C A dl μnn dl dl 4π C C μnn dl dl 4π C C L L
7 Elution of inductnce (Method ). Assume current flowing on the loop.. Find B by Ampere s lw or BiotSrt lw: μ dl H dl, B ' C C 4π 3. Find Λ( ) by Λ N B ds 4. Find L by L Λ, independent of S
8 Elution of inductncereference figure B C μ 4π dl dl
9 Elution of inductnce (Method ). Assume current flowing on the loop.. Find H nd B by Method 3. Find the stored energy W m V ( ) H B d 4. Find L by Wm L
10 Exmple 3: Solenoid inductor () Consider hollow solenoid with crosssectionl re S, n turns per unit length. Find the inductnce per unit length L.. Assume current flowing on the loop.. By Ampere s lw: μ n B
11 Exmple 3: Solenoid inductor () 3. For unit length (l), 4. By definition: Λ Λ n Φ n n ( n ) ( μ n ) S μ S n μ S L
12 Exmple 3: Two concentric coils () Consider two coils C, C with N, N turns nd lengths l, l. They re wound concentriclly on thin cylindricl core of rdius with permebility μ. Find the mutul inductnce L.. Assume C, C he currents,
13 Exmple 3: Two concentric coils () N. By Ampere s lw, uniform field B μ l 3. Flux linge of C due to C : Λ N Φ N B S 4. By definition: L N N l μ π
14 Sec. 3 Mgnetic Energy. Energy of ssembling current loops. Energy of mgnetic fields
15 Energy of ssembling current loopsone loop () Closed loop C with selfinductnce L. f the loop current i increses from to slowly (qusisttic), n emf of: dφ dt L di dt will be induced on C to oppose the chnge of i (Frdy s lw, Lenz s lw).
16 Energy of ssembling current loopsone loop () The wor done to oercome the induced nd enforce the chnge of i is: W di dt ( t) i ( t) dt L idt L i di L which is stored s mgnetic energy: W one loop L
17 Energy of ssembling current loopstwo loops () nsert loop C with selfinductnce L, mutul inductnce L. f we mintin i, while i increses from to slowly, n emf of: dφ di L dt dt will be induced on C in n ttempt to chnge i wy from
18 Energy of ssembling current loopstwo loops () The wor done to mintin i is: W di dt t) dt L dt L ( di L
19 Energy of ssembling current loopstwo loops (3) Menwhile, n emf of: dφ di L dt dt will be induced on C to oppose the chnge of i (from to ). The wor done to oercome nd enforce the chnge of i is: W L
20 Energy of ssembling current loopstwo loops (4) The totl mgnetic energy stored in the system of two current loops is: W L + L + two loops L
21 Energy of ssembling current loopsn loops The totl mgnetic energy stored in the system of N current loops crrying currents,,.., N, is: W m N N j By L Λ, the flux (linge) of loop C due to ll the N current loops: Φ N j L j j, L j j N W m Φ
22 Energy of continuous current distributions Decompose system of continuous current distribution J (r ) in olume V' into N elementry current loops C, ech hs current Δ nd filmentry crosssectionl re Δ B, A Φ C B ds S A dl
23 Energy of continuous current distributions Δ Φ N C N m dl A W ( ) ( ) J dl J dl J dl Δ Δ Δ Δ, Δ N C m J A W ( ) V m d J A W
24 Comments W e N Q V N W m Φ W e V ( ρv ) d W m V ( A J ) d Electrosttics Mgnetosttics
25 Energy of mgnetic fields n rel pplictions (especilly electromgnetic wes), sources re usully fr wy from the region of interest, only the fields re gien source O J (r ) H, B
26 Energy of mgnetic fields () W m J (r ) contin ll the source currents V V V J H ( A J ) d A ( H ) () W By ector identity: A H H H B A H m d ( ) ( A) A ( H ) ( ) V V ( H B) d ( A H ) d
27 Energy of mgnetic fields3 Q A ds A d, S V ( ) ( A H ) d ( A H ) V S ds J (r ) V S (3) W m ( ) ( ) H B d A H ds V S
28 Energy of mgnetic fields4 S S Obs. pt. H A
29 Energy of mgnetic fields5 ) m (J 3 B H energy density ( ) 4 ) ( ) ( H A ds H A S π V m m d r w W ) (
30 Energy of mgnetic fields6 W m V H B d V dw m H B d
31 Exmple 33: Coxil cble inductor () Cylindricl symmetry, Ampere s lw, Find the stored mgnetosttic energy nd inductnce per unit length of: μ B H < < < b r r r r B if, if, π μ π μ φ φ
32 Exmple 33: Coxil cble inductor () Energy density: < < < b r r r r B H w m, 8, 8 4 π μ π μ dr r d π Differentil olume (L):
33 Exmple 33: Coxil cble inductor (3) Totl stored energy: r dr r W m <, π μ π μ b r b dr r W b m < <, ln 4 4 π μ π μ, L W m + + b W W L m m ln 8 ) ( π μ π μ internl externl
34 Sec. 33 Mgnetic Force. Force on current loops. Exmple: force between prllel wires
35 Force on currentcrrying loops Consider n elementl currentcrrying wire of crosssectionl re S, represented by differentil displcement ector dl Free chrges within the wire of chrge density ρ moe with elocity u( // dl ), experiencing force of: F ρs dl u B d m ( )
36 Force on currentcrrying loops dl u u dl J ρu df m ρs dl ρs u dl B df m ( u B) JS ( dl B) ( dl B) For current loop C : F m dl B C
37 Force on currentcrrying loops3 f B is creted by nother closed loop C crrying current, the force exerted on the loop C crrying current is: F B B dl B C μ μ dl 4π ' C 4π C dl
38 Force on currentcrrying loops4 F μ C C 4π dl ( dl ) F Counterprt in electrosttics: Coulomb s force between two chrges F q q 4πε
39 Exmple 34: Force between two long wires Find the force per unit length between two infinitely long, prllel wires seprted by d, crrying currents, in the sme direction. B μ x, F dl πd B y ( dz) z μ πd x μ πd ttrction force
40 Sec. 34 Mgnetic Torque. Exmple: mgnetic force & torque exerted on current loop
41 Exmple 35: Force & torque on currentcrrying loops () Consider circulr loop on the xyplne with rdius b, current in clocwise sense, nd is plced uniform mgnetic filed: B B + z B B y B
42 Exmple 35: Force & torque on currentcrrying loops () The force exerted on differentil current element on the loop due to : dl φ bdφ B b φ df m df r bb ( dl B ), ( bdφ ) ( B ) φ dφ z
43 Exmple 35: Force & torque on currentcrrying loops (3) df The force exerted on differentil current dl φ bdφ B : element on the loop due to df m bb d bb z ( dl B ), ( ) ( ) φbdφ yb φ( sinφ cosφ) ( ) x sinφdφ y y
44 Exmple 35: Force & torque on currentcrrying loops (4) The totl force exerted on the loop due to B : df rbb dφ, φ F bb π df π r ( φ) dφ
45 Exmple 35: Force & torque on currentcrrying loops (5) df The totl force exerted on the loop due to B : bb sinφd, z φ F z π bb df π sinφ dφ
46 Exmple 35: Force & torque on currentcrrying loops (6) The totl torque exerted on the loop due to : B dφ, bb F d r φ ( ) π π r r r B b b df T
47 Exmple 35: Force & torque on currentcrrying loops (7) The totl torque exerted on the loop due to : df bb sinφd, z φ B π T df rb b b B B φ π π [ ( )] z ( ) φ x sinφ ( ) r sinφdφ y cosφ sinφdφ
48 Exmple 35: Force & torque on currentcrrying loops (7) T π π b B x sin φdφ y sinφ cos x ( πb ) B mb T T + T m ) B n generl, x cosφ T x ( z y sin φ φ dφ T m B
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