ECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson Dept. of ECE. Notes 31 Inductance

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1 ECE 3318 Applied Electricity nd Mgnetism Spring 018 Prof. Dvid R. Jckson Dept. of ECE Notes 31 nductnce 1

is chosen from right hnd rule for the inductor flux : The fingers re in the direction of the current reference

2 nductnce ˆn S Single turn coil The current produces flux though the loop. Definition of inductnce: Mgnetic flux through loop: ψ ψ S B nˆ ds [ Wb] ˆn unit norml to loop The unit norml is chosen from right hnd rule for the inductor flux : The fingers re in the direction of the current reference direction, nd the thumb is in the direction of the unit norml. [ H] Note: Plese see the Appendix for summry of right-hnd rules. Note: is lwys positive

3 N-Turn Solenoid N Λ totl flux Nψ We ssume here tht the sme flux cuts though ech of the N turns. ψ flux through one turn The definition of inductnce is now Λ Nψ 3

4 Exmple Find z s µ r N ψ Note: We neglect fringing here nd ssume the sme mgnetic field s if the solenoid were infinite. RH rule for inductor flux From previous notes: ( ) H zˆ n, ρ < 0, ρ > N turns n N / s : nˆ zˆ so ( π ) ψ B nˆ ( π ) ( ) B π µµ H 0 ( π ) µµ 0 r ( n ) ( π ) µµ r ( ) N 0 n / z r z 4

5 Exmple (cont.) s z µ r N turns Finl result: N π µµ s ( ) [ ] 0 r H Note: is incresed by using high-permebility core! 5

6 Exmple Toroidl nductor N turns µ r R Find H inside toroid Find Note: This is prcticl structure, in which we do not hve to neglect fringing nd ssume tht the length is infinite. 6

7 Exmple (cont.) 7

8 Exmple (cont.) N turns µ r R The rdius R is the verge rdius (mesured to the center of the toroid). µ r A H Assume ˆ φ H φ A is the cross-sectionl re: A π 8

9 Exmple (cont.) ρ C C H dr ( ˆ φρ φ) encl H d encl µ r C r so N turns π 0 Hφ ρdφ encl H φ Hence encl πρ We then hve RH rule in Ampere's lw : encl N H N φ πρ ˆ [ A/m] 9

10 Exmple (cont.) RH rule for the inductor flux : N turns ˆn φˆ N Nψ ( ˆ) B n A A π H R µ r ˆ N φ [ A/m] πρ N B N N ( A ) φ ρ R ( µ ) 0 µ rh A φ ρ R N A π R µµ 0 r Hence N µµ 0 r A [H] π R 10

11 Exmple Coxil cble gnore flux inside wire nd shield (PEC, f > 0) z b h Find l (inductnce per unit length) A short is dded t the end to form closed loop. - z Center conductor + S h Note: We clculte the flux through the surfce S shded in green. 11

12 Exmple (cont.) RH rule for the inductor flux : ˆn φˆ + z - Center conductor S ˆn h ψ ψ B nˆ ds Bφ ds µ Hφ ds S S S 1

13 Exmple (cont.) + z - Center conductor S ˆn h ρ ψ µ S H ds φ ρ b b b ψ µ h Hφ dρ µ h dρ µ h ρ πρ π [ ln ] b 13

14 Exmple (cont.) z b h [ ln ] b ψ µ h ρ π b µ h ln π Hence ψ 1 b µ h ln π Per-unit-length: l µµ 0 r π b ln [ H/m] 14

15 Exmple (cont.) Recll: Z 0 C l l l µµ b ln [ H/m] π 0 r From previous notes: Note: µ r 1 for most prcticl coxil cbles. Hence: Z C l πε 0ε r b ln [F/m] η µ b π ε 0 r 0 ln [ Ω] r η µ 0 0 Ω ε [ ] (intrinsic impednce of free spce) 15

16 Voltge w for nductor A B + - v t ( ) ( ) B B v t E dr E dr A PEC wire: C Note: There is no electric field inside the wire. C ˆn C Apply Frdy s lw: B dψ E dr nˆ ds t dt S ψ S v t B nˆ ds Note: The unit norml is chosen from the righthnd rule in Frdy s lw, ccording to the direction of the pth C. ( ) Hence dψ dt 16

17 Voltge w for nductor (cont.) so v v or v dψ dt d ( i) dt di dt B i + - v t ( ) C ˆn From the inductor definition we hve: ψ i Note: The direction of the current is chosen to be consistent with the direction of the flux, ccording to the RH rule for the inductor flux. Note: We re using the ctive sign convention here. 17

18 Voltge w for nductor (cont.) To gree with the usul circuit lw (pssive sign convention), we chnge the reference direction of the voltge drop. (This introduces minus sign.) i B - v + Pssive sign convention (for loop) v di dt Note: This ssumes pssive sign convention. 18

19 Energy Stored in nductor i i ( 0) 0 + v - t v( t) di v( t) dt di P ( t) vi i dt t di W ( t) P( t) dt i dt dt 0 0 ( ) it ( 0) ( ) it 1 1 i di i i i t i ( 0) Hence we hve 1 U H t i t ( ) ( ) [ J] U For DC we hve H 1 [ J] 19

20 Energy Formul for nductor We cn write the inductnce in terms of stored energy s: U H Next, we use U H 1 B H dv V We then hve 1 V B H dv This gives us n lterntive wy to clculte inductnce. 0

21 Exmple Find using the energy formul N z Solenoid s From previous exmple, we hve Energy formul: 1 N UH U H µµπ 0 r s Hence, we hve N µµπ 0 r H s [ ] 1

22 Exmple Find l using the energy formul 1 V B H dv z b Coxil cble Note: We ignore the mgnetic stored energy from fields inside the conductors. (We would hve fields inside the conductors t DC, but not t high frequency.) h 1 µµ 0 r H dv 1 µµ 0 V r V Hφ dv h π b 1 0 r πρ 0 0 µµ ρdρdφdz 0 r 1 1 µµ ( h)( π) dρ π ρ b

23 Exmple (cont.) z b h 0 r 1 1 µµ 0 r ( h)( π) dρ π ρ µµ h b ln π b Per-unit-length: l µµ b ln π 0 r [ H/m] Note: We could include the inner wire region if we hd solid wire t DC. We could even include the energy stored inside the shield t DC. These contributions would be clled the internl inductnce of the cox. 3

24 Appendix n this ppendix we summrize the vrious right-hnd rules tht we hve seen so fr in electromgnetics. 4

25 Summry of Right-Hnd Rules Stokes s theorem: ( ) ˆ C V d r V n ds S Fingers re in the direction of the pth C, the thumb gives the direction of the unit norml. Frdy s lw: (sttionry pth) C E dr ψ S dψ dt B nˆ ds Fingers re in the direction of the pth C, the thumb gives the direction of the unit norml (the reference direction for the flux). Ampere s lw: C H dr encl ˆ encl J n ds S Fingers re in the direction of the pth C, the thumb gives the direction of the unit norml (the reference direction for the current enclosed). 5

Summry of Right-Hnd Rules (cont.) Mgnetic field lw: For wire or current sheet or solenoid, the thumb is in the direction of the current nd the fingers give the direction of the mgnetic field.

26 Summry of Right-Hnd Rules (cont.) Mgnetic field lw: For wire or current sheet or solenoid, the thumb is in the direction of the current nd the fingers give the direction of the mgnetic field. (For current sheet or solenoid the fingers re simply giving the overll sense of the direction.) nductor flux rule: ψ ψ S B nˆ ds Fingers re in the direction of the current nd the thumb gives the direction of the unit norml (the reference direction for the flux). 6

Chapter 7 Steady Magnetic Field. september 2016 Microwave Laboratory Sogang University

Chapter 7 Steady Magnetic Field. september 2016 Microwave Laboratory Sogang University Chpter 7 Stedy Mgnetic Field september 2016 Microwve Lbortory Sogng University Teching point Wht is the mgnetic field? Biot-Svrt s lw: Coulomb s lw of Mgnetic field Stedy current: current flow is independent