# Electromagnetism Zhu Jiongming Department of Physics Shanghai Teachers University.

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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 Chapter 7 Magnetic Materials Chapter 8 Alternating CurrentAlternating Current Chapter 9 Electromagnetic Waves

Chapter 8 Alternating Current §1. Alternating CurrentAlternating Current §2. Three Simple CircuitsThree Simple Circuits §3. Complex Number and PhasorComplex Number and Phasor §4. Complex ImpedanceComplex Impedance §5. Power and Power FactorPower and Power Factor §6. ResonanceResonance

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

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

§2. Three Simple Circuits §2. Three Simple Circuits 1. IntroductionIntroduction 2. Pure ResistancePure Resistance 3. Pure CapacitancePure Capacitance 4. Pure InductancePure Inductance

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

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

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

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

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

§3. Complex Number and Phasor 1. Complex Numbers ● ExpressionsExpressions ● CalculationsCalculations 2. Complex Number MethodComplex Number Method 3. PhasorsPhasors 4. Complex Form of Relations between u and iComplex Form of Relations between u and i ● Pure Resistance ● Pure Capacitance ● Pure Inductance 5. ExamplesExamples

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

F Multiplication ： F Division ： Calculations of Complex Numbers  Addition /S ubtraction ：  1   2 = ( a 1  a 2 )+ j ( b 1  b 2 ) （ parallelogram rule ）

 Information ： rms, initial phase F Steps of calculation ： i, u   calculating result of  of i, u real complex take real part F 4 theorems ：（ Next page ） 2. Complex Number Method F Instantaneous ： i = I m cos (  t +  ) = Re[ I m e j (  t +  ) ] where I m e j (  t +  ) F Complex rms  Definition ：

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

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

4. Complex Form of u, i Relations ● Pure ResistancePure Resistance ● Pure CapacitancePure Capacitance ● Pure InductancePure Inductance

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

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

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

Example 1 （ p.330 /[Ex. 1 ] ） (1) Series RL circuit, relation between u and i. Sol. ： u = u 1 + u 2 i u u2u2 u1u1 R L exponential ： where 0 +1 +j+j R LL  z If i = I m cos  t is known, can get u = zI m cos(  t+  )

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

= 3  110 2  Example 2 （ p.332 /[Ex. 2 ] ） Fluorescent lamp ( daylight lamp ) ： tube R ， ballast L ， in series ， emf 220 V ， tube U 1 =110 V. Find U 2 of ballast. Sol. ： u u2u2 u1u1  U 2 2 = U 2 - U 1 2 R L ~ = 220 2 - 110 2

Example 3 （ p.332 /[Ex. 3 ] ） RC in parallel. Find relation between i 1 and i 2. Sol. ： phasor in parallel ， draw first i u i2i2 i1i1 and I 2 / I 1 =  CR i 2 leads i 1 by  / 2 R C same phase with leads by  / 2 0 

Example 4 （ p.332 /[Ex. 4 ] ） Continue Ex.3, find phase difference between i and i 1. Sol. ： I 2 / I 1 =  CR i 2 leads i 1 by  / 2 R = 138 k  = 1.38  10 5  C = 1000 pF = 10 - 9 F  = 2  f = 2   2000  CR  1.73     / 3 （ i leads i 1 ） 0 

§4. Complex Impedance 1. Three Ideal ElementsThree Ideal Elements 2. Two-Terminal Net without emfTwo-Terminal Net without emf 3. Exponential Formula and Algebraic FormulaExponential Formula and Algebraic Formula ● Exponential Formula ● Algebraic Formula ● Impedance Triangle

1. Three Ideal Elements Resistor u = Ri Capacitor Inductor introduce Complex Impedance Z so that F Z determined by R 、 L 、 C and  ， not U 、 I F Z represents relation between i and u （ U / I and  u -  i ）

2. Two-Terminal Net without emf or u i2i2 i1i1 i i u Ex.2 ： RC in parallel i = i 1 + i 2 Ex.1 ： RL in series

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

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

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

Example （ p.336 /[Ex.] ） Condition for balancing an AC bridge. Sol. ： u AC = u AD and or Maxwell Bridge, for measuring L A B C D L R2R2 R3R3 R4R4 R1R1 C G ~ i1i1 i4i4 i2i2 i3i3

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

§5. Power and Power Factor 1. Instantaneous Power, Average Power and Power Factorand Power Factor 2. Significance of Raising Power FactorSignificance of Raising Power Factor 3. Method to Raise Power FactorMethod to Raise Power Factor

Average power F Pure ResistancePure Resistance F Pure InductancePure Inductance F Pure CapacitancePure Capacitance F Two-Terminal Net without emfTwo-Terminal Net without emf 1. Power and Power Factor DC ： P = IU keep constant AC ： p(t) = i(t)u(t) instantaneous power （ for AC with f = 50 Hz ， average is important ）

Pure Resistance Resistance ： i = I m sin  t u = iR p = iu = i 2 R Resistor ： non-energy-storing ， energy  heat 0 t T i u P I p

 0  T / 4 and T / 2  3T / 4 ： p > 0, absorb energy and store it in M field Pure Inductance Inductance ： voltage leads current by  / 2 i = I m sin  t u = U m sin(  t +  / 2 ) = U m cos  t p 0 t T  T / 4  T / 2 and 3T / 4  T ： p < 0, release energy, field disappear ( i ：  I m  0 ) External energy M Field energy Never dissipated at all ！ i u

Pure Capacitance Pure Capacitance u i  0  T / 4 and T / 2  3T / 4 ： p > 0, absorb energy and store it in E field Capacitance ： current leads voltage by  / 2 u = U m sin  t i = I m cos  t  T / 4  T / 2 and 3T / 4  T ： p < 0, release energy, field disappear ( u ：  U m  0) p 0 t T External energy E Field energy Never dissipated at all ！

Two-Terminal Net without emf Two-Terminal Net without emf u = U m sin  t i = I m sin(  t -  )  Resistor ：  = 0 P = IU  Inductor ：  =  / 2 P = 0  Capacitor ：  = -  / 2 P = 0 ( Trigonometric ： cos(  -  ) - cos(  +  ) = 2sin  sin  ） u 0 t T i p cos  —— Power factor

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

3. Method to Raise Power Factor Workless current ： I Q = I sin  work current ： I P = I cos  then P = IUcos  = I P U I Q = I sin  useless to P, but a part of total current I, and a part of energy lost on cable P’ = I 2 R increase cos  to reduce I Q inductive net u iCiC i i’  ’’ cos  ’ > cos  P = IUcos  = I’Ucos  ’ add capacitace

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

§6. Resonance F Resonance ： series RLC circuit U L > U C u leads i Inductive U L = U C u and i in phase resistive U L < U C u lags i capacitive Resonance R L C u uRuR uLuL uCuC

Resonance in Series Circuit （ 1 ） F Current ： Complex impedance ： Impedance ： Current ： Resonance ： maximum current ： Resonance frequency of RLC circuit ： when  =  0 ， I = I 0 maximum ， resonance

Resonance in Series Circuit （ 2 ）  Voltages ： U L = I  LU C = I /  C Resonance ： U L0 = I 0  0 LU C0 = I 0 /  0 C Let then U L0 = QUU C0 = QU if R <<  0 L ， Q very large ～ 10 2 （ good ， bad ） U L0 = U C0 = QU > U F Quality factor ：

Resonance in Series Circuit （ 3 ） F Resonance curve ： Relation for I ~  keep R 、 L 、 C 、 U constant F Selectivity ： to select the wanted program —— modulate for a radio adjust C  change when  0 matches  1 of a signal for example (  0 =  1 ) then I 1 >> I i （ i  1 ） 0  I I0I0 00 ~ ~~

Exercises p.367 / 8 - 7 - 1