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Electromagnetism. Zhu Jiongming Department of Physics Shanghai Teachers University. Electromagnetism. Chapter 1 Electric Field Chapter 2 Conductors Chapter 3 Dielectrics Chapter 4 Direct-Current Circuits Chapter 5 Magnetic Field Chapter 6 Electromagnetic Induction

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Electromagnetism


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    1. Electromagnetism Zhu Jiongming Department of Physics Shanghai Teachers University

    2. Electromagnetism • Chapter 1 Electric Field • Chapter 2 Conductors • Chapter 3 Dielectrics • Chapter 4 Direct-Current Circuits • Chapter 5 Magnetic Field • Chapter 6 Electromagnetic Induction • Chapter 7 Magnetic Materials • Chapter 8Alternating Current • Chapter 9 Electromagnetic Waves

    3. Chapter 8 Alternating Current • §1. Alternating Current • §2. Three Simple Circuits • §3. Complex Number and Phasor • §4. Complex Impedance • §5. Power and Power Factor • §6. Resonance

    4. i Im o t §1. Alternating Current • Steady current:magnitude and direction not changing magnitude varying ,not reversing Alternating current Varying current • Sinusoidal current:i = Imcos (  t + ) , u ,  • Three important quantities: • Amplitude Im • (or rms I = Im/ ) • Angular frequency  • ( = 2 /T = 2f ) • Initial phase  • ( phase  t + )

    5. Alternating Current • Features of sinusoidal quantities: • derivative and integral are still sinusoidal • any periodic quantities can expand as a sum of sinusoidal functions with different frequency • Denotation: • instantaneous:little case i , u • rms: capital I , U • amplitude: subscript mIm, Um • ( rms:root-mean-square )

    6. §2. Three Simple Circuits • 1. Introduction • 2. Pure Resistance • 3. Pure Capacitance • 4. Pure Inductance

    7. 1. Introduction • DC R act on current • L short circuit(ideal,no resistance) • C open circuit(ideal,no current) • AC R、L、C all act on current • L self-induced emf • C charge/discharge • Relationship between i and u • i = Imcos (  t + i) • u = Umcos (  t + u) To study: (1) U / I = ? Ratio of rms (2) u- i = ? Difference of phase

    8. i u R u i t 0 2. Pure Resistance • u(t) = i(t) R • or •  U = I R u = i • or U / I = R u - i = 0

    9. C i u u t 0 i 3. Pure Capacitance • Left plate q = Cu •  I = UCi = u +  / 2 • or U / I = 1/C u - i = -  / 2 • Capacitive reactance:XC= 1/C Pure Capacitance :current leads voltage by  / 2

    10. L u i 自 u i t 0 4. Pure Inductance • L acts as an emf u(t) = -S(t) •  U = LIu = i +  / 2 • or U / I = L u - i =  / 2 • Inductive reactance :XL= L Pure Inductance :current lags voltage by  / 2

    11. Exercises • p.361 / 8 - 2 - 1, 2, 3

    12. §3. Complex Number and Phasor • 1. Complex Numbers • Expressions • Calculations • 2. Complex Number Method • 3. Phasors • 4. Complex Form of Relations between u and i • Pure Resistance • Pure Capacitance • Pure Inductance 5. Examples

    13. a = Re() b = Im() +j (a,b) b r +1 0 a Expressions of Complex Numbers • Algebraic:= a + jb • Phasor:r = |  | modulus • a = r cos • b = r sin  • Trigonometric:  = r cos + j rsin • Exponential:  = r ej • ( Euler formula:ej = cos + j sin )

    14. Calculations of Complex Numbers • Addition/Subtraction: • 1 2 = ( a1 a2 )+ j ( b1  b2 ) • ( parallelogram rule) • Multiplication: • Division:

    15. 2. Complex Number Method • Instantaneous:i = Imcos ( t + ) = Re[ Ime j ( t + ) ] • where Ime j ( t + ) • Complex rms • Definition: • Information:rms, initial phase • Steps ofcalculation: • i, u  calculating result of  of i, u • real complex take real part • 4 theorems:( Next page )

    16. Four Theorems • Complex rms of (ki)( k any real constant) • Complex rms of( i1 i2 ) • Complex rms of di/dt • Complex rms of idt • Pro.: i = Imcos ( t + ) • di/dt = Imcos ( t +  + /2 ) •  complex rms of di/dt = I e j e j/2 •  idt = (1/)Imcos ( t +  - /2 ) •  complex rms of idt = I e j e -j/2 /

    17. +j +1 0 3. Phasors • complex rms phasor • length = I(rms) • angle = (phase)  • parallelogram rule • complex rms of di/dt •  times of length,rotate counterclockwise /2 • complex rms of idt • 1/ times of length,rotate clockwise /2

    18. 4. Complex Form of u , i Relations • Pure Resistance • Pure Capacitance • Pure Inductance

    19. i u R 0 Pure Resistance • Instantaneous: u = i R • Complex rms: • or •  U = I R • u = i

    20. C i u 0 Pure Capacitance • Instantaneous: or • Complex rms : • or • U = I /C u = i-  / 2 • Acturely, 1/jC includes all information • about relationship between u and i • ( ratio of rms and difference of phase) • Complex capacitive reactance:- j XC= - j /C

    21. L u i 自 0 Pure Inductance • Instantaneous : • Complex rms : • or •  U = LI • u = i +  / 2 • Complex inductive reactance:j XL= jL

    22. i u1 R +j u u2 L +1 0 L R Example 1 (p.330/[Ex.1])(1) • Series RL circuit, relation between u and i. • Sol.:u = u1+ u2 exponential: where If i = Imcost is known, can get u = zImcos(t+) z 

    23. same phase with leads by  / 2 0  Example 1 (p.330/[Ex.1])(2) • Phasor: • first draw • then • and U2 /U1= L/R • then • get

    24. u1 u2 R u ~ L Example 2(p.332/[Ex.2]) • Fluorescent lamp ( daylight lamp ):tube R,ballast L,in series,emf 220 V,tube U1=110 V. Find U2 of ballast. • Sol.:  U22 = U2- U12 = 220 2- 110 2 = 3  110 2 

    25. i i2 i1 u same phase with leads by  / 2 R C  0 Example 3(p.332/[Ex.3]) • RC in parallel. Find relation between i1and i2 . • Sol.:phasor • in parallel,draw first and I2 /I1 = CR i2leadsi1by  / 2

    26. 0 Example 4(p.332/[Ex.4]) • Continue Ex.3, find phase difference between i and i1 . • Sol.:I2/I1= CR • i2leads i1by /2 R = 138 k = 1.38  10 5  C= 1000 pF = 10-9 F  = 2f = 2  2000 CR 1.73   /3(ileads i1)

    27. §4. Complex Impedance • 1. Three Ideal Elements • 2. Two-Terminal Net without emf • 3. Exponential Formula and Algebraic Formula • Exponential Formula • Algebraic Formula • Impedance Triangle

    28. 1. Three Ideal Elements • Resistoru = Ri • Capacitor • Inductor • introduce ComplexImpedance Z so that • Z determined by R、L、C and ,not U、I • Z representsrelation between i and u • ( U/I andu-i)

    29. i i i2 u u i1 2. Two-Terminal Net without emf • or • Ex.1:RL in series • Ex.2:RC in parallel i = i1+i2

    30. 3. Exponential and Algebraic Formulae • Exponential Formula :Z = ze j • z impedance —— modulus of Z •  phase constant —— angle of Z • Z representsrelations for i and u( U/I andu-I) • Algebraic Formula : Z = r + j x • r effective resistance > 0, not necessarily = R Ex. • x effective reactance > 0 for inductive net < 0 for capacitive net = 0 for resistive net

    31. Z +j jx +1 0 r Impedance Triangle • Z = ze j • Z = r + j x z  z x  r

    32. Complex Form of Laws • 1. Ohm’s Law • DC: U = IR U = - IR • AC: • series connection: Z = Z1 + Z2 + ··· • parallel connection: • 2. Kirchhoff’s Rules • DC:  ( I ) = 0  (  ) =  ( IR ) • AC:

    33. i2 i3 i1 C L R2 R1 A B C R3 i4 G R4 D ~ Example(p.336/[Ex.]) • Condition for balancing an AC bridge. • Sol.:uAC = uAD and or • Maxwell Bridge , for • measuring L

    34. Exercises • p.362 / 8 - 4 - 3, 5, 6, 15

    35. §5. Power and Power Factor • 1. Instantaneous Power, Average Power • and Power Factor • 2. Significance of Raising Power Factor • 3. Method to Raise Power Factor

    36. 1. Power and Power Factor • DC:P = IU keepconstant • AC:p(t) = i(t)u(t) instantaneous power • ( for AC with f = 50 Hz,average is important ) • Average power • Pure Resistance • Pure Inductance • Pure Capacitance • Two-Terminal Net without emf

    37. p P I u i 0 t T Pure Resistance • Resistance: i = Imsin t u = iR p = iu = i2R • Resistor:non-energy-storing, energy  heat

    38. p i u T 0 t External energy M Field energy Never dissipated at all! Pure Inductance • Inductance:voltage leads current by /2 • i = Imsin t u = Umsin( t + /2) = Umcos t • 0 T/4 and T/2  3T/4: • p > 0, absorb energy and store it in M field • T/4  T/2 and 3T/4T: • p < 0, release energy, field disappear ( i: Im0 )

    39. p i u t 0 T External energy E Field energy Never dissipated at all! Pure Capacitance • Capacitance:current leads voltage by /2 • u = Umsin t i = Imcos t • 0 T/4 and T/2  3T/4: • p > 0, absorb energy and store it in E field • T/4  T/2 and 3T/4T: • p < 0, release energy, field disappear ( u:  Um0)

    40. p u t 0 T i Two-Terminal Net without emf • u = Umsin t i = Imsin( t -) • ( Trigonometric:cos(- )- cos(+ ) = 2sin sin ) • Resistor:  = 0 P = IU • Inductor:  =  /2 P = 0 • Capacitor: =-  /2 P = 0 cos—— Power factor

    41. I R ~ Z R 2. Significance of Raising Power Factor • S = IU visual power • P = IUcos work power • Q = IUsin workless power • Lost on cable (1) voltage U’ = IR (2) power P’ = I2R • Reduce lost: • R ,thick wire,cost more • I,not decrease consumer’s power P = IUcos —— increase power factor cos • Ex.:inductive load, i lags u by   • Workless current:I Q= I sin • work current: I P= I cos

    42. i’ iC i u ’  3. Method to Raise Power Factor • Workless current:I Q= I sin • work current: I P= I cos • then P = IUcos = I PU • I Q= I sin useless to P , • but a part of total current I , • and a part of energy lost on cable P’ = I2R • increase cos to reduce I Q • inductive net add capacitace • cos’ > cos • P = IUcos = I’Ucos’

    43. Exercises • p.365 / 8 - 5 - 1, 5

    44. L C R uR uL uC u Resonance §6. Resonance • Resonance: • series RLC circuit UL = UC u and i in phase resistive UL > UC u leads i Inductive UL < UC u lags i capacitive

    45. Resonance in Series Circuit (1) • Current: • Complex impedance: Impedance: Current: Resonance: maximumcurrent: Resonance frequency of RLCcircuit: when  = 0,I = I 0 maximum ,resonance

    46. Resonance in Series Circuit (2) • Voltages: UL = IL UC = I/C • Resonance:UL0 = I00L UC0= I0/0C Let thenUL0 = QU UC0 = QU • if R << 0L, Q very large ~ 10 2( good,bad) • UL0 = UC0 = QU > U • Quality factor:

    47. I I0 0  0 ~ ~ ~ Resonance in Series Circuit (3) • Resonance curve: • Relation for I ~  • keep R、L、C、U constant • Selectivity: • to select the wanted program • —— modulate for a radio • adjust C change • when 0 matches 1ofa signal • for example ( 0 = 1 ) • then I1 >> Ii( i  1)

    48. Exercises • p.367 / 8 - 7 - 1