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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 I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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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