chapter6c2

8/10/2019 chapter6c2

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EMLAB

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Chapter 6. Capacitance andinductance


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Contents

1. Capacitors

2. Inductors

3. Capacitor and inductor combinations

4. RC operational amplifier circuits

5. Application examples


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Power supply board

PC motherboard

inductor capacitor

inductor

capacitor

Cell phone

Usage of inductors and capacitors


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1. Capacitors

Capacitance is defined to be the ratio of charge to voltage difference.

Used to store charges

Used to store electrostatic energy

0V

0Q

q

SV SV

SVSVVE

If the voltage difference between the

terminals of the capacitor is equal to

the supply voltage, net flow of

charges becomes zero.

V

QC


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Electrons(-) are absorbed.(+) charges are generated

Electrons(-) are generated.(+) charges are absorbed.

Generation of charges : battery

e2ZnZn 2

234 HNH222NH

e

Electrons are generated via

electro-chemical reaction.


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Used to store charges

Used to store electrostatic energy

Slow down voltage variation

Usage of capacitors


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Type of capacitors

t

0

)(1

)( dttiC

t
http://www.google.co.kr/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=6ZZX4Racl1o-sM&tbnid=ZOMpPYoONnjwqM:&ved=0CAUQjRw&url=http://variable-capacitors.blogspot.com/&ei=X9HvUtbPCOmSiQem_IGQBQ&psig=AFQjCNF7BjRZ_I3oSPZ6jAia_i8E2D2F_g&ust=1391534331654366


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9

i

t

t0

0

t

diC

)(1

irelation of capacitors


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10

)(ti

Example 6.2

The voltage across a 5-F capacitor has the waveform shown in Fig. 6.4a. Determine

the current waveform.

dt

dCi

mst

mstt

mstt

t

80

8696102

24

60106

24

)( 3

3

mst

mstmA

mstmA

ti

80

8660

6020

)(


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Capacitor voltage cannot change instantaneouslydue to finite current supply.

Properties of capacitors

SV

i

t

dt

dCi

SR

CC

In steady state, capacitor behaves as if open circuited.

SV

iSR

0

dt

dCi DC

)()( 00 tt

0t


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Example 6.3

Determine the energy stored in the electric field of the capacitor in Example 6.2 at

t=6 ms.

][1440)]([2

1)(

2JtCtW


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The current in an initially uncharged 4F capacitor is shown in Fig. 6.5a. Let us

derive the waveforms for the voltage, power, and energy and compute the energy

stored in the electric field of the capacitor at t=2 ms.

Example 6.4

mst

mst

mstt

ti

40

428

20102

16

)(

3

mst

msttdx

msttxdx

tt

t

40

42824)8(4

1

2010001084

1

)(0

0

23

mst

mstt

mstt

ttitp

40

426416

208

)()()(

3

mst msttt

mstt

dxxptW

t

40421012810648

202

)()(

1292

4

0


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2. Inductors

dt

diLtL )(


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Two important laws on magnetic field

Current generates magnetic field(Biot-Savart Law)

inducedV

Time-varying magnetic field generates

induced electric field that opposes the

variation. (Faradayslaw)

Current

Current

B-field

B-field

V


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Biot-Savart Law Faradays Law

rrRRr

B

,4

2C R

Id

LidS

aB

dt

dV

ind


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Self induced voltage

dt

diL

dt

dV

ind

The induced voltage is generated such that it opposes the applied magnetic flux.

The inductor cannot distinguish where the applied magnetic flux comes from.

If the magnetic flux is due to the coil itself, it is called that the induced voltage

is generated by self-inductance.

=


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Frequently used formulas on inductors

dt

diL

t

t

t

t

tt

dL

ti

dL

dL

dL

i

0

0

0

)(1

)(

)(1

)(1

)(1

0

2

2

1)(

)()()()( Li

dt

dti

dt

tdiLtittp

22)]([

2

1

2

1)()()( tiLdLi

d

dditW

tt

i

NL

Energy :

Power :

Voltage :

Inductance :

Current :


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Properties of inductors

SV

i

t

i

dt

diL

SR

LL

In steady state, inductor behaves as if short circuited.

SV

iSR

0dt

diL DC

Inductor current cannot change instantaneouslydue to finite current supply.

0t

)()( 00 titi

20

6


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Find the total energy stored in the circuit of Fig. 6.8a.

Example 6.5

0815254273

09

36

9

111

11

CCC

CC

VVV

VV

][8.109

62.16],[2.16

21VVVV CC

][8.19

],[2.16

91

2

1

1A

VIA

VI

C

L

C

L

][44.1)2.1)(102(2

1 231 mJWL

][48.6)8.1)(104(2

1 232

mJWL

][62.2)2.16)(1020(2

1 261

mJWC

][92.2)8.10)(1050(2

1 262 mJWC


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22

E l 6 7


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Example 6.7

The current in a 2-mH inductor is

][)377sin(2)( Atti

Determine the voltage across the inductor and the energy stored in the inductor.

][)377cos(508.1)]377sin(2[

)102()( 3

Vtdt

td

dt

diLtL

][)377(sin004.0)]377sin(2)[102(2

1)]([

2

1)( 2232 JtttiLtWL

23

E l 6 8


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Example 6.8

The voltage across a 200-mH inductor is given by the expression

00

0][)31()(

3

t

tmVett

t

Let us derive the waveforms for the current, energy, and power.

00

0][5)31(200

10 30

33

t

tmAtedxexi

tt

x

00

0][)31(5)()()(

6

t

tWetttittp

t

000][5.2)]([

21)(

62

2

ttJettiLtW

t

24

C it d i d t ifi ti


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Capacitor and inductor specifications

Standard tolerance

values are ; 5%, ;

10%, and ; 20%.

Tolerances are

typically 5% or

10% of the

specified value.

25

E l 6 10


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Example 6.10

The capacitor in Fig. 6.11a is a 100-nF capacitor with a tolerance of 20%. If the

voltage waveform is as shown in Fig. 6.11b, let us graph the current waveform for the

minimum and maximum capacitor values.

dt

dCi

26

6 3 C it d I d t C bi ti


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EMLAB

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6.3 Capacitor and Inductor Combinations

=

N

N

i iS CCCCC

11111

211

N

N

i

iP CCCCCC

321

1

=

27


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EMLAB

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=

=

N

N

i iP LLLLLL

111111

3211

N

N

i

iS LLLLLL

321

1

28

6 4 RC O ti l A lifi Ci it


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EMLAB

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6.4RC Operational Amplifier Circuits

Op-amp differentiator

Ci

iRdt

dC o

2

11 )(

0,0 i

dt

tdCRo

)(112

29

O i t t


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EMLAB

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Op-amp integrator

i

dt

dC

R o )(2

1

1

0,0 idt

tdC

R

o )(2

1

1

0)0(

)(1

)0()(1

)(1

0 1

210

1

21

1

21

o

t

o

tt

o dxxCR

dxxCR

dxxCR

30

Example 6 17


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EMLAB

Example 6.17

The waveform in Fig. 6.26a is applied at the input of the differentiator circuit shown

in Fig. 6.25a. If R2=1 kand C1=2 F, determine the waveform at the output of the

op-amp.

mstV

mstV

dt

td

dt

tdCRo

105][4

50][4)(10)2(

)( 13112

31

Example 6 18


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Example 6.18

If the integrator shown in Fig. 6.25b has the parameters R1=5 k and C2=0.2F,

determine the waveform at the op-amp output if the input waveform is given as in Fig.

6.27a and the capacitor is initially discharged.

][1.020

][1.002010)20(10)(

1 33

0 1

21 stt

stttdxx

CR

t

o

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