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Chapter 6. Capacitance and in-ductance

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2Contents

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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41. Capacitors• Capacitance is defined to be the ratio of charge to voltage difference.• Used to store charges• Used to store electrostatic energy

0V

0Qq

SV SV

SV SVV EIf 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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7Type of capacitors

t

0

)(1

)( dttiC

t

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8Frequently used formulas on capacitors

dt

dC

dt

dqi

t

t

t

t

tt

diC

t

diC

diC

diC

0

0

0

)(1

)(

)(1

)(1

)(1

0

2

2

1)()()()()( C

dt

d

dt

tdtCtittp

22 )]([2

1

2

1)()()( tCdC

d

dditW

tt

q

C

Energy :

Power :

Voltage :

Capacitance :

Current :

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9

i

t

t0

0

tdi

C )(

1

iυ relation of capacitors

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)(ti

Example 6.2

The voltage across a 5-μF capacitor has the waveform shown in Fig. 6.4a. Deter-mine 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 instantaneously due 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

0dt

dCi DC

)()( 00 tt

0t

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12Example 6.3

Determine the energy stored in the electric field of the capacitor in Example 6.2 at t=6 ms.

][1440)]([2

1)( 2 JtCtW

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The current in an initially uncharged 4μF capacitor is shown in Fig. 6.5a. Let us de-rive 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

dxxptWt

40

421012810648

202

)()( 1292

4

0

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142. Inductors

dt

diLtL )(

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15Two 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. (Faraday’s law)

Current

Current

B-field

B-field

V

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Biot-Savart Law Faraday’s Law

rrRRr

B

,4

ˆ2C R

Id

LidS

aB

dt

dV

ind

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17Self 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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18Frequently 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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19Properties 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 instantaneously due to finite current supply.

0t

)()( 00 titi

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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

LC

L

][44.1)2.1)(102(2

1 23

1mJWL

][48.6)8.1)(104(2

1 23

2mJWL

][62.2)2.16)(1020(2

1 26

1mJWC

][92.2)8.10)(1050(2

1 26

2mJWC

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21Example 6.6

The current in a 10-mH inductor has the waveform shown in Fig. 6.9a. Determine the voltage waveform.

mst

mstt

mstt

ti

40

421040102

1020

20102

1020

)( 33

3

3

3

mst

mstmV

mstmV

t

40

42][100102

1020)1010(

20][100102

1020)1010(

)(3

33

3

33

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22Example 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

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23Example 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 3

0

33

t

tmAtedxexitt x

00

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

6

t

tWetttittp

t

00

0][5.2)]([

2

1)(

622

t

tJettiLtW

t

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24Capacitor and inductor specifications

Standard tolerance values are ; 5%, ; 10%, and ; 20%.

Tolerances aretypically 5% or 10% of the speci-fied value.

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25Example 6.10The capacitor in Fig. 6.11a is a 100-nF capacitor with a tolerance of 20%. If the volt-age waveform is as shown in Fig. 6.11b, let us graph the current waveform for the minimum and maximum capacitor values.

dt

dCi

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266.3 Capacitor and Inductor Combinations

=

N

N

i iS CCCCC

11111

211

N

N

iiP CCCCCC

3211

=

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=

=

N

N

i iP LLLLLL

111111

3211

N

N

iiS LLLLLL

3211

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286.4 RC Operational Amplifier Circuits

Op-amp differentiator

Ci

iRdt

dC o

211 )(

0,0 i

dt

tdCRo

)(112

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29Op-amp integrator

i

dt

dC

R o )(21

1

0,0 idt

tdC

Ro )(

21

1

0)0(

)(1

)0()(1

)(1

0 121

0 121

121

o

t

o

tt

o dxxCR

dxxCR

dxxCR

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30Example 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 kΩ and C1=2 μF, determine the waveform at the output of the op-amp.

mstV

mstV

dt

td

dt

tdCRo 105][4

50][4)(10)2(

)( 13112

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31Example 6.18

If the integrator shown in Fig. 6.25b has the parameters R1=5 kΩ and C2=0.2μF, de-termine 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

0 121 stt

stttdxx

CR

t

o