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Electromagnetism Zhu Jiongming Department of Physics Shanghai Teachers University.

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Electromagnet Electromagnet ism ism Zhu Jiongming Department of Physics Shanghai Teachers University
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ElectromagnetismElectromagnetism

Zhu Jiongming

Department of Physics

Shanghai Teachers University

ElectromagnetismElectromagnetism

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 Current

Chapter 9 Electromagnetic Waves

Chapter 8 Chapter 8 Alternating Current Alternating Current

§1. Alternating Current

§2. Three Simple Circuits

§3. Complex Number and Phasor

§4. Complex Impedance

§5. Power and Power Factor

§6. Resonance

§1.§1. Alternating Current Alternating Current

Steady current : magnitude and direction not changing

Varying currentmagnitude varying , not reversing

Alternating currentSinusoidal current : i = Imcos ( t + ) , u , Three important quantities : Amplitude Im

( or rms I = Im/ )2 Angular frequency ( = 2 /T = 2f ) Initial phase ( phase t + )

Im

o t

i

Alternating CurrentAlternating 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 m Im, Um

( rms : root-mean-square )

§2.§2. Three Simple Circuits Three Simple Circuits

1. Introduction

2. Pure Resistance

3. Pure Capacitance

4. Pure Inductance

1. Introduction1. 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 )

)cos(2 itI )cos(2 utU

To study :(1) U / I = ? Ratio of rms

(2) u - i = ? Difference of phase

2. Pure Resistance2. Pure Resistance

u(t) = i(t) R

or

R

iu

)cos(2 itIR )cos(2 utU

U = I R u = i

or U / I = R u - i = 0

0 t

iu

3. Pure Capacitance3. Pure Capacitance

Left plate q = Cu

iu)cos(2 itI

)]sin([2 utUC

I = UC i = u + / 2

or U / I = 1/C u - i = - / 2

Capacitive reactance : XC = 1/C

0 t

u

Ct

qi

d

d

t

uC

d

d

)2

cos(2 utCU

i

Pure Capacitance : current leads voltage by / 2

4. Pure Inductance4. Pure Inductance

i u

0 t

ui

L acts as an emf u(t) = - S(t)

)]sin([2 itIL

)2

cos(2 itIL

U = LI u = i + / 2

or U / I = L u - i = / 2

Inductive reactance : XL = L

t

iL

d

dS

)cos(2 utU

L

自t

iLu

d

d

Pure Inductance : current lags voltage by / 2

ExercisesExercises

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

§3.§3. Complex Number and Phasor 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

Expressions of Complex NumbersExpressions of Complex Numbers

Algebraic : = a + jb a = Re()

b = Im() )1( j

a

b

bar

1

22

tan

)

0 +1

+j(a,b)

a

br

Phasor : r = | | modulus

a = r cos

b = r sin

Trigonometric : = r cos + j rsin Exponential : = r e j

( Euler formula : e j = cos + j sin )

Multiplication : Division :

Calculations of Complex NumbersCalculations of Complex Numbers

Addition/Subtraction :1 2 = ( a1 a2 )+ j ( b1 b2 )

( parallelogram rule )

111 jba 1e1jr 222 jba 2e2

jr

21 ee 2121 jj rr )(

2121e jrr

)(

2

1

2

1 21e j

r

r

Information : rms, initial phase Steps of calculation : i, u calculating result of of i, u

real complex take real part 4 theorems :( Next page )

2. Complex Number Method2. Complex Number Method

Instantaneous : i = Imcos ( t + ) = Re[ Ime j ( t + ) ]

where Ime j ( t + ) jtj Iee 2 Ie tj 2

jIeI

UI , UI ,

Complex rms Definition :

Four TheoremsFour Theorems

Complex rms of (ki) ( k any real constant )Ik

21 II Ij jI /

]2Re[ Ie tj ]2Re[ )2/( jtj eIe

Ij

]/2Re[ )2/( jtj IeejI /

Complex rms of( i1 i2 )

Complex rms of di/dt Complex rms of idtPro. : 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 /

3. Phasors3. Phasors

complex rms phasorjIeI

0 +1

+j

I

21 II Ij

jI /

complex rms of di/dt

complex rms of idt

length = I ( rms ) angle = ( phase )

parallelogram rule

times of length , rotate counterclockwise /2

1/ times of length , rotate clockwise /2

4. Complex Form of 4. Complex Form of uu , , i i RelationsRelations

● Pure Resistance

● Pure Capacitance

● Pure Inductance

U

Pure ResistancePure Resistance

Instantaneous : u = i R

Complex rms :or

U = I R

u = i

RIU

RIeUe iu jj

I0

R

iu

Pure CapacitancePure Capacitance

Instantaneous : or

U = I /C u = i - / 2

U

ICj

U 1

iu jj IeCj

Ue

1

I0

t

uCi

d

d ti

Cu d

1

Complex rms :

or

iu

C

)2

(1

ijIe

C

Acturely, 1/jC includes all information

about relationship between u and i

( ratio of rms and difference of phase )Complex capacitive reactance : - j XC = - j /C

Pure InductancePure Inductance

i u

L

t

iLu

d

d

U = LI

u = i + / 2

Instantaneous :

Complex rms :

or

IjLU

iu jj LIejUe )

2(

ijLIe U

I0

Complex inductive reactance : j XL = jL

Example 1 Example 1 (( p.330p.330/[Ex./[Ex.11]])) (1)(1)

Series RL circuit, relation between u and i.

Sol. : u = u1 + u2

i

u

u2

u1 R

L

21 UUU ILjRI ILjR )( jzeLjR exponential :

where

222 LRz

R

L 1tan

0 +1

+j

R

L

zIzeU j 222/ LRzIU

iu

)( iu jj zIeUeIf i = Imcost is known, can get u = zImcos(t+ )

ILjU 2

IRU 1

Example 1 Example 1 (( p.330p.330/[Ex./[Ex.11]])) (2)(2)

Phasor :first draw

I0

U

2U

1U

then I

same phase with I

leads by / 2I

and U2 /U1 = L/R

then 21 UUU

get 22

21 UUU

iu

ILR 222

1

21tanU

UR

L1tan

= 3 110 2

Example 2Example 2 (( p.332p.332/[Ex./[Ex.22]]))Fluorescent lamp ( daylight lamp ) : tube R , ballast L , in series , emf 220 V , tube U1=110 V. Find U2 of ballast.

Sol. :u

u2

u121 UUU

U22 = U 2 - U1

2

22

21 UUU

R

L~

31102 U )(V190

= 220 2 - 110 2

Example 3Example 3 (( p.332p.332/[Ex./[Ex.33]]))RC in parallel. Find relation between i1and i2 .

Sol. : phasor

in parallel , draw first

i

ui2i1

RUI /1

UCjI 2

and I2 /I1 = CR

i2 leads i1 by / 2

R C

same phase with U

leads by / 2U

U0

I

2I

1I

U

Example 4Example 4 (( p.332p.332/[Ex./[Ex.44]]))Continue Ex.3, find phase difference between i and i1 .

Sol. : I2 /I1 = CR

i2 leads i1 by /2

R = 138 k = 1.38 10 5 C = 1000 pF = 10 -9 F

= 2f = 2 2000

CR 1.73

/3 ( i leads i1 )

1

21tanI

I

U0

I

2I

1I

CR1tan

§4.§4. Complex Impedance Complex Impedance

1. Three Ideal Elements

2. Two-Terminal Net without emf

3. Exponential Formula and Algebraic Formula

●Exponential Formula

●Algebraic Formula

●Impedance Triangle

1. Three Ideal Elements1. Three Ideal Elements

Resistor u = Ri

Capacitor

Inductor

IRU

tiC

u d1

IC

jU 1

t

ILu

d

d ILjU

RZ

CjZ1

LjZ

introduce Complex Impedance Z so that IZU Z determined by R 、 L 、 C and , not U 、 I Z represents relation between i and u

( U/I and u - i )

2. Two-Terminal Net without emf2. Two-Terminal Net without emf

orIZU IUZ /

u

i2i1

i

i

uILjRU )(

LjRIUZ /

Ex.2 : RC in parallel i = i1+ i2

Ex.1 : RL in series

21 III

RUI /1

Cj

UI

/2

UCj

)1

( CjR

UI

1)1

( CjR

Z CRj

R

1

222

2

1 RC

CRjR

jxr

3. Exponential and Algebraic Formulae3. Exponential and Algebraic Formulae

Exponential Formula : Z = ze j

z impedance —— modulus of Z

phase constant —— angle of Z

I

U

)( iujeI

U

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

IUz /

iu

Z represents relations for i and u ( U/I and u - I )

Impedance TriangleImpedance Triangle

Z = ze j

Z = r + j x

22 xrz

r

x1tan 0 +1

+j

r

jx Zz

r

xz

Complex Form of LawsComplex Form of Laws

1. Ohm’s Law

DC : U = IR U = - IR

AC : ZIU ZIU

21

111

ZZZ

0)( I )()( ZI

2. Kirchhoff’s Rules

DC : ( I ) = 0 ( ) = ( IR )

AC :

series connection : Z = Z1 + Z2 + ···

parallel connection :

ExampleExample (( p.336p.336/[Ex.]/[Ex.] ))Condition for balancing an AC bridge.

Sol. : uAC = uAD

3311 ZIZI

21 II

and

4

3

2

1

Z

Z

Z

Z

ADAC UU

4422 ZIZI

43 II

or)1/( 44

3

2

1

CRjR

R

R

LjR

4

3

2

1

R

R

R

R 32RCRL

Maxwell Bridge , for

measuring L

A B

C

D

LR2

R3

R4

R1

CG

~

i1

i4

i2

i3

ExercisesExercises

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

§5.§5. Power and Power Factor Power and Power Factor

1. Instantaneous Power, Average Power

and Power Factor

2. Significance of Raising Power Factor

3. Method to Raise Power Factor

Average power

Pure Resistance Pure Inductance Pure Capacitance Two-Terminal Net without emf

1. Power and Power Factor1. 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 )

T

ttpT

P0

d)(1

Pure ResistancePure Resistance

Resistance : i = Imsin t u = iR p = iu = i 2R

T

tRtIT

P0

22m dsin

1

T

ttRIT 0

22m dsin

1

T

tt

RIT 0

2m d

2

2cos11

RI 2m2

1 RI 2 IU

Resistor : non-energy-storing , energy heat

m2

1rms II :

0 tT

iuP

I

p

0 T/4 and T/2 3T/4 : p > 0, absorb energy and store it in M field

Pure InductancePure Inductance

Inductance : voltage leads current by /2

i = Imsin t u = Umsin( t + /2 ) = Umcos t

tUIiup 2sin2

1mm

T

tpT

P0

d1

T

ttUIT 0mm d2sin

2

11 0

2m2

1LIW

p

0 tT

T/4 T/2 and 3T/4 T : p < 0, release energy, field disappear ( i : Im 0 )

External energy M Field energy

Never dissipated at all !

iu

Pure CapacitancePure 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 = Umsin t i = Imcos t

tUIiup 2sin2

1mm

T

tpT

P0

d1

0

T/4 T/2 and 3T/4 T : p < 0, release energy, field disappear ( u : Um 0)

p

2m2

1CUW

0 tT

External energy E Field energy

Never dissipated at all !

Two-Terminal Net without emfTwo-Terminal Net without emf

u = Umsin t i = Imsin( t - )

)]2cos([cos2

1mm tUIiup

T

tpT

P0

d1

Resistor : = 0 P = IU Inductor : = /2 P = 0 Capacitor : =- /2 P = 0

0cos2

10mm T

dtUIT

cos2

1mmUI cosIU

( Trigonometric : cos(- )- cos(+ ) = 2sin sin )

u

0 tT

i

p

cos —— Power factor

Lost on cable (1) voltage U’ = IR (2) power P’ = I 2R

Reduce lost : R , thick wire , cost more I , not decrease consumer’s power P = IUcos

2. Significance of Raising Power Factor2. Significance of Raising Power Factor

S = IU visual power P = IUcos work power Q = IUsin workless power

~ ZR

RI

—— increase power factor cos

Ex. : inductive load, i lags u by U

I

PI

QIWorkless current : I Q= I sin

work current : I P= I cos

3. Method to Raise Power Factor3. 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’ = I 2R

increase cos to reduce I Q

inductive net

u

iCi

i’

U

I

'I

CI

cos’ > cos

P = IUcos = I’Ucos’

add capacitace

ExercisesExercises

p.365 / 8 - 5 - 1, 5

§6.§6. Resonance Resonance

Resonance : series RLC circuit

I

U

CU

LU

CL UU

RUI

U

CU

LU

CL UU

RUI

U

CU

RU

LU

UL > UC

u leads i

Inductive

UL = UC

u and i in phase

resistive

UL < UC

u lags i

capacitive

CL

1

C

L

1

CL

1

Resonance

R L C

uuR uL uC

Resonance in Series Circuit Resonance in Series Circuit (( 11 )) Current : Complex impedance : )

1(

CLjRZ

22 )1

(C

LRz

Impedance :

Current :z

UI

22 )/1( CLR

U

Resonance :C

L

1

R

UI 0 maximum current :

LC

10 Resonance frequency of RLC circuit :

when = 0 , I = I 0 maximum , resonance

Resonance in Series Circuit Resonance in Series Circuit (( 22 )) Voltages : UL = IL UC = I/C

Resonance : UL0 = I00L UC0= I0/0C

Let

then    UL0 = QU UC0 = QU   if R << 0L , Q very large ~ 10 2 ( good , bad )    UL0 = UC0 = QU > U   Quality factor :        

LR

U0

CR

U

0

1

R

LQ 0

CR0

1

C

L

RQ

1

Resonance in Series Circuit Resonance in Series Circuit (( 33 )) Resonance curve : Relation for I ~

keep R 、 L 、 C 、 U constant

LC

10

Selectivity : to select the wanted program

—— modulate for a radio adjust C change

when 0 matches 1of a signal

for example ( 0 = 1 )

then I1 >> I i ( i 1 )

0

II0

0

~ ~ ~

ExercisesExercises

p.367 / 8 - 7 - 1


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