EE 470 Electric Mchines Review of Mxwell s Equtions in Integrl Form Objectives: 1. To discuss clssifiction of mterils 2. To discuss properties of homogeneous, liner, isotropic, nd time-invrint mterils 3. To review Mxwell s equtions in integrl form. pecificlly: 4. To discuss Guss Lw for the Electric Field 5. To discuss Guss Lw for the Mgnetic Field 1
lssifiction of Mterils 1. onductors { Good Bd 2. emiconductors 3. Dielectrics (Insultors) { Perfect Imperfect 4. Mgnetic Mterils 2
Mgnetic lssifiction of Mterils 1. Prmgnetic (µ r 1+) 2. Dimgnetic (µ r 1 ) 3. Nonmgnetic metls (µ r = 1) 4. uperconducting (µ r = 0) 5. Ferromgnetic (µ r = 100-1,000,000) 6. Ferrimgnetic (hrd nd soft mterils with low electric conductivities) 3
Properties of Mterils Homogeneous: hrcteristics of mteril do not depend on the loction in the mteril medium (Opposite: Non-Homogeneous) Isotropic: hrcteristics re not dependent on the direction of excittion fields (Opposite: Anisotropic) Liner: hrcteristics re not dependent on the mgnitude of excittion fields (Opposite: Nonliner) Time-Invrint: hrcteristics do not chnge with time (Opposite: Time-Vrying) Notes: In this course, we will del with: 1. Homogeneous, isotropic, time-invrint mterils. 2. Liner nd nonliner mterils. 4
oncepts E Electric Field Intensity Vector (V/m) D Displcement Flux Density Vector (/m 2 ) B Mgnetic Flux Density Vector (Wb/m 2 or T) H Mgnetic Field Intensity Vector (A-turn/m) J onduction urrent Density Vector (A/m 2 ) ρ hrge Density (/m 3 ) ϵ Mteril Permittivity (F/m) ϵ r Reltive Permittivity ϵ o Permittivity of Free pce (F/m) µ Mteril Permebility (H/m) µ r Reltive Permebility µ o Permebility of Free pce (H/m) σ onductivity (/m) c Velocity of Light in Free pce (m/s) 5
Mxwell s Equtions in Integrl Form 1. Guss Lw for the Electric Field: D d = V ρ dv = Q enclosed 2. Guss Lw for the Mgnetic Field: B d = 0 3. Ampère Lw: H dl = J d + d D d 4. Frdy s Lw: E dl = d B d 6
onstitutive Equtions for Liner Isotropic Mterils D = ϵ E = ϵr ϵ o E (F/m) B = µ H = µr µ o H (H/m) J = σ E (A/m 2 ) onstnts ϵ o = 8.854 10 12 (F/m) µ o = 4π 10 7 (H/m) ϵ o µ o c 2 = 1 c = 2.9979 10 8 3 10 8 (m/s) ϵ o 1 36π 10 9 (F/m) 7
Liner Isotropic Dielectric Mteril: D = ϵo E (In Free pce) P: Polriztion Vector χ e : Electric usceptibility P = ϵo χ e E D = ϵo E + P = ϵo (1 + χ e ) E = ϵ o ϵ r E D = ϵ E (In Mteril Medium) ϵ = ϵ r ϵ o (F/m) ϵ o = 8.854215 10 12 (F/m) 8
Liner Isotropic Mgnetic Mteril: B = µo H (In Free pce) M: Mgnetiztion Vector χ m : Mgnetic usceptibility M = χ m 1+χ m Bµo H = Bµo χ m 1+χ m Bµo = B µ o (1+χ m ) = B µo µ r B = µ H (In Mteril Medium) µ = µ r µ o (H/m) µ o = 4π 10 7 (H/m) 9
Guss Lw for the Electric Field d D urfce hrge Density ρ Q enclosed in Volume V D d = V ρ dv = Q enclosed The totl electric flux out of closed surfce is equl to the totl chrge enclosed. Note 1: The differentil surfce vector d is oriented outwrds. Note 2: D = ϵ r ϵ o E for liner isotropic dielectric mteril. 10
Exmple 1: Find the electric field E due to point chrge q t distnce x from the chrge. Gussin urfce : phere of rdius x centered t the point chrge q x d E D d = Qenclosed D d = q D d = q ϵ o E(4πx 2 ) = q q E = 4πϵ o x 2 11
Exmple 2: Find the electric field E due to chrge Q uniformly distributed on hollow sphere of rdius r t distnce 0 x < r. Q Gussin urfce : phere of rdius x concentric to the r x d E sphere of rdius r. D d = Qenclosed D d = 0 D d = 0 ϵ o E(4πx 2 ) = 0 E = 0 12
Exmple 3: Find the electric field E due to chrge Q uniformly distributed on hollow sphere of rdius r t distnce x > r. Q Gussin urfce : phere of rdius x concentric to the sphere of rdius r. r x d E D d = Qenclosed D d = Q D d = Q ϵ o E(4πx 2 ) = Q E = Q 4πϵ o x 2 13
Guss Lw for the Mgnetic Field d o 1 d o B d i B 2 B d o = 0 The totl mgnetic flux out of closed surfce is equl to zero. Note 1: The differentil surfce vector d o is oriented outwrds. Note 2: B = µ r µ o H for liner isotropic mgnetic mteril. 14
Another Form of Guss Lw for the Mgnetic Field d o 1 d o B d i B 2 B d o + 1 B d i + 1 B d o = 0 B d o = 0 2 B d o = 0 2 B d i = B d o } 1 {{ }} 2 {{ } ϕ in = ϕ out 15
Ampère s Lw I d J d D dl H H dl = J d + d D d Note: The differentil surfce vector d is oriented ccording to the right-hnd rule by following the direction of the contour. 16
Ampère s Lw Mgnetic-Field ystem I d J dl H H dl = J d = Ienclosed An electric current induces mgnetic field. Note: For mgnetic-field systems excited with low frequencies: d D d 0 17
Ampère s Lw Mgnetic-Field ystem (ont d) I d J dl H ross ection A H dl = J d H dl = J d = J A ( I ) H dl = JA = A A H dl = I = Ienclosed A d 18
Ampère s Lw Mgnetic-Field ystem (ont d) I d θ J ross ection A dl H A cos θ H dl = H dl = J d J cos θ d = J cos θ A/ cos θ ( A ) H dl = (J cos θ) cos θ H dl = I = Ienclosed ( I = JA = A A) A/ cos θ 19 d
Ampère s Lw Mgnetic-Field ystem (ont d) I 1 I 2 I 3 d J 1 A 1 d A 2 d J 3 A 3 dl J 2 H dl = J 1 d + J 2 d + J 3 d A 1 A 2 A 3 H dl = J 1 d J 2 d + J 3 d A 1 A 2 A 3 H dl = J1 d J 2 d + J 3 d A 1 A 2 A 3 H dl = J1 A 1 J 2 A 2 + J 3 A 3 H dl = I1 I 2 + I 3 = I enclosed 20
Infinitely-Long Wire I d J dl H H dl = Ienclosed H dl = I H dl = I H(2πx) = I H = I 2πx, B = µ oh = µ oi 2πx 21
Frdy s Lw d B dl E E dl = d B d A time-vrying mgnetic flux induces voltge. Note: The differentil surfce vector d is oriented ccording to the right-hnd rule by following the direction of the contour. 22
Frdy s Lw for n Electrosttic Field (No Mgnetic Field) c b dl E b c E dl = 0 E dl + } {{ } E dl + } b {{ } E dl } c {{ } = 0 KVL: v b + v bc + v c = 0 23
Frdy s Lw (ont d) d B =NA, A=Are of One Loop N turns dl Mgnetic Flux ϕ (Unit: weber, Wb): ϕ = B d 1 loop Mgnetic Flux Linkges λ (Unit: weber-turn, Wb-t): λ = B d λ = B d N loops λ = N B d 1 loop λ = Nϕ 24
Grvittionl Potentil P m=1 kg g h h b dl g c h b e level g P = P b = P P b = P = b b g dl g dh = g(h h b ) = g h Electrosttic Potentil V V = V b = V V b = b E dl 25
Frdy s Lw (ont d) b d B dl E i dl E o + Region of negligible flux Region of considerble flux b d E dl = = dλ B d E i dl + E o dl } {{ }} b {{ } v b,in + v b,out = dλ 0 v b,out = dλ v = v b,out = dλ 26
Frdy s Lw with Moving ontour c d l B v V d + ril sliding conductor b x E dl = d B d b c d E dl + E dl + b c E dl + d E dl = dλ 0 + 0 + 0 + V d = dλ V d = dλ V d = d (Blx) V d = Blv 27
Ohm s Lw for Resistor dl E b J = σ E + v = v b Ohm s Lw: J = σe b b v = E dl = E dl ( J v = El = l = σ) 1 ( i l ( ) σ A) 1 l Ohm s Lw: v = i = Ri σ A 28
Ohm s Lw for pcitor +q Are A ε l E E E dl E E q Are A b Guss Lw: ϵea = q b b v = E dl = ( q ) v = El = l ( ϵa ϵa ) Ohm s Lw: q = v = v l i(t) = dq = dv E dl 29
Ohm s Lw for n Inductor + d λ v = d B N turns φ d Depth d ross ection A = wd Men Pth Length l µ w Wih w Frdy s Lw: v(t) = dλ λ = Nϕ = NBA Ampère s Lw: Hl = Ni λ = N(µH)A = L = N2 R = N 2 (l/µa) v(t) = dλ = L di µn2 A i = Li l 30
Mxwell s Equtions for Engineers 1. Guss Lw for the Electric Field: DA = ϵea = Q enclosed 2. Guss Lw for the Mgnetic Field: ϕ in = ϕ out 3. Ampère Lw for Mgnetic-Field ystem: H k l k = I enclosed k 4. Frdy s Lw: v(t) = dλ = N dϕ 31
Ampère s Lw of Force I 2 I 1 ^ 12 ^ 21 dl 2 dl 1 R df 1 = I 1dl1 df 2 = I 2dl2 µ o I 2 µ o I 1 dl2 â 21 4πR 2 dl1 â 12 4πR 2 32
Ampère s Lw of Force I 2 I 1 ^ 12 ^ 21 dl 2 dl 1 R where df 2 = I 2dl2 µ o I 1 dl1 â 12 4πR 2 = I 2 dl2 db 1 db 1 = µ oi 1dl1 â 12 4πR 2 33
Mgnetic Field of n Infinitely Long Wire rrying urrent I z dz ^ z α ^ R I R φ r (r, φ,0) ^ φ y x db = µ oidz â z â R 4πR 2 = µ oidz sin α 4πR 2 = µ oir dz 4πR 3 â µ o Ir dz ϕ = 4π(z 2 + r 2 ) â ϕ 3/2 âϕ 34
Mgnetic Field of n Infinitely Long Wire rrying urrent I z dz ^ z α ^ R I R φ r (r, φ,0) ^ φ y x B = = µ oir 4π z = db = z r 2 z 2 + r 2 z = z = µ o Ir dz 4π(z 2 + r 2 ) = µ oi 2πr âϕ 3/2 âϕ 35