ET 162 Circuit Analysis

Capacitors

Electrical and Telecommunication Engineering Technology

Professor Jang

AcknowledgementAcknowledgement

I want to express my gratitude to Prentice Hall giving me the permission to use instructor’s material for developing this module. I would like to thank the Department of Electrical and Telecommunications Engineering Technology of NYCCT for giving me support to commence and complete this module. I hope this module is helpful to enhance our students’ academic performance.

OUTLINESOUTLINES

Introduction to Capacitors

The Electric Field

Capacitance

Transients in Capacitive Networks: Charging Phase

Capacitors in Series and Parallel

Key Words: Capacitor, Electric Field, Capacitance, Transient

ET162 Circuit Analysis – Capacitors Boylestad 2

Introduction to Capacitors

Thus far, the only passive device appearing in the class has been the resistor. We will now consider two additional passive devices called the capacitor and the inductor, which are quite different from the resistor in purpose, operation, and construction.

Unlike the resistor, both elements display their total characteristics only when a change in voltage or current is made in the circuit in which they exist. In addition, if we consider the ideal situation, they do not dissipate energy as does the resistor but store it in a form that can be returned to the circuit whenever required by the circuit design.

ET162 Circuit Analysis – Capacitors Boylestad 3

The Electric Field

ET162 Circuit Analysis – Capacitors Boylestad 4

DA

flux unit area= ψ ( / )

The electric field is represented by electric flux lines, which are drawn to indicate the strength of the electric field at any point around any charged body; that is, the denser the lines of flux, the stronger the electric field. In Fig. 10.1, the electric field strength is stronger at point a than at position b because the flux lines are denser at a than b.

The flux per unit area (flux density) is represented by the capital letter D and is determined by

FIGURE 10.1 Flux distribution from an isolated positive charge.

The Electric FieldThe Electric Field

ET162 Circuit Analysis – Capacitors Boylestad 5

The electric field is represented by electric flux lines, which are drawn to indicate the strength of the electric field at any point around the charged body. The electric field strength at any point distance r from a point charge of Q coulombs is directly proportional to the magnitude of the charge and inversely proportional to the distance squared from the charge.

FIGURE 10.2 Electric flux distribution: (a) like charges; (b) opposite charges.

Electric flux lines always extend from a positively charged body to a negatively charged body, always extend or terminate perpendicular to the charged surfaces, and never intersect.

ψ ≡=

=

Q coulombs CE F Q newtons coulomb N C

EkQr

N C

( , )/ ( / , / )

( / )12

CapacitanceCapacitanceA capacitor is constructed simply of two parallel conducting plates separated by insulating material (in this case, air). Capacitance is a measure of a capacitor’s ability to store charge on its plates.

FIGURE 10.3 Electric flux distribution between the plates of a capacitor: (a) including fringing; (b) ideal.

VQC =

A capacitor has a capacitance of 1 farad if 1 coulomb of charge is deposited on the plates by a potential difference of 1 volt cross the plates.

C = farad (F)Q = coulombs (C)V = volts (V)

ET162 Circuit Analysis – Capacitors Boylestad 6

ET162 Circuit Analysis – Capacitors Boylestad 7

FIGURE 10.4 Effect of a dielectric on the field distribution between the plates of a capacitor: (a) alignment of dipoles in the dielectric; (b) electric field components between the plates of a capacitor with a dielectric present..

ε

ε ε ε

ε

ε ε ε

=

=

=

= = × −

DE

farads meter F M

C dA

C Ad

Ad

F

r

r r

( / , / )

. ( )

0

012885 10

E : Electric field (V/m)D : Flux densityε : Permittivity (F/m)C : Capacitance (F)Q : Charge (C)A : Area in square meters d : Distance in meters between the plates

Ex. 10-1 Determine the capacitance of each capacitor on the right side of Fig.10.5.

FIGURE 10.5

a C F F

b C F F

c C F F

d C pF

pF F

. (5 )

. ( . ) .

. . ( )

. (5)( / )

( )

( )( ) .

= =

= =

= =

=

= =

3 1512

01 0 05

2 5 20 504

1 81000

160 1000 016

µ µ

µ µ

µ µ

µ

ET162 Circuit Analysis – Capacitors Boylestad 8

Ex. 10-2 For the capacitor of Fig. 10.6:a. Determine the capacitance.b. Determine the electric field strength between the plates if 450 V are applied across the plates.c. Find the resulting charge on each plate.

( )( ) Fm

mmFdAC oo

12

3

212

100.59105.1

01.0/1085.8 −−

−

×=×

×==

εa.

mVm

VdV /10300

105.1450 3

3×≅

×==

−ε

( )( )nCC

VCVQVQC

55.2610550.26450100.59

9

12

=×=

×==

=

−

−

b.

c.

FIGURE 10.6

ET162 Circuit Analysis – Capacitors Boylestad 9

FIGURE 10 7 Summary of capacitive elements

Transients in Capacitive Networks: Charging Phase

ET162 Circuit Analysis – Capacitors Boylestad 11

FIG. 10.8 Basic charging network. FIG. 10.9 ic during charging phase. FIG 10.10 Vc during charging phase.

FIG. 10.11 Open-circuit equivalent for a capacitor following the charging phase.

FIG. 10.12 Short-circuit equivalent for a capacitor (switch closed, t=0).

ET162 Circuit Analysis – Capacitors Boylestad 22

A capacitor can be replaced by an open-circuit equivalent once the charging phase in a dc network has passed.

i ER

eCt RC= − /

t RC ond s= (sec , )

Figure 10.13 The e-x function (x ≥ 0).

Figure 10.14 ic versus t during the charging phase.

Figure 10.15 vc versus t during the charging phase. Figure 10.16 Effect of C on the charging phase.

v E eCt RC= − −( )/1

v EeRt RC= − /

Figure 10.17 vR versus t during the charging phase.ET162 Circuit Analysis – Capacitors Boylestad 13

ET162 Circuit Analysis – Capacitors Boylestad 14

Ex. 10-3 a. Find the mathematical expressions for the transient behavior of vC, iC, and vR for the circuit of Fig. 10.18 when the switch is moved to position 1. Plot the curves of vC, iC, and vR.b. How much time must pass before it can be assumed, for all practical purposes, that iC≈ 0 A and vC≈ E volt?

a RC Fs ms

v E e e

i ER

eV

ke

e

v Ee eb ms ms

Ct t

Ct t

t

Rt t

. (8 )( )

( ) ( )

(5 )

. ( )

/ /( )

/ /( )

/( )

/ /( )

τ

τ

τ

τ

τ

= = × ×

= × =

= − = −

= =

= ×

= == =

−

−

− − ×

− − ×

− − ×

− − ×

−

−

−

−

10 4 1032 10 32

1 40 1808

10

405 5 32 160

3 6

3

32 10

32 10

3 32 10

32 10

3

3

3

3

Ω

Ω

FIGURE 10.18

FIGURE 10.19

Transients in Capacitive Networks: Discharging Phasev Ee disch ing

i ER

e disch ing

v Ee disch ing

Ct RC

Ct RC

Ct RC

=

= −

= −

−

−

−

/

/

/

: arg

arg

arg

Figure 10.20 Demonstrating the discharge behavior of a capacitive network.

Figure 10.21 The charging and discharging cycles for the network of fig. 10.19.

ET162 Circuit Analysis – Capacitors 15

τ =

= =

= − = − ×

= − = −

− − ×

− − − ×

− − ×

−

−

−

32

40

10

40

32 10

3 32 10

32 10

3

3

3

ms

v Ee e

i ER

e e

v Ee e

Ct RC t

Ct RC t

Ct RC t

/ /

/ /

/ /

( )

(5 )

Ex. 10-4 After vC in Example 10.3 has reached its final value of 40 V, the switchis shown into position 2, as shown in Fig. 10.21. Find the mathematical expressions for the transient behavior of vC, iC, and vR after the closing of the switch. Plot the curves for vC, iC, and vR using the defined directions and polarities of Fig. 10.18. Assume that t = 0 when the switch is moved to position 2.

FIGURE 10.23

FIGURE 10.21

Capacitors in Series and Parallel

ET162 Circuit Analysis – Capacitors Boylestad 17

Q Q Q Q

C C C C

CC C

C C

T

T

T

= = =

= = =

=+

1 2 3

1 2 3

1 2

1 2

1 1 1 1

Figure 10.24 Series capacitors.

Q Q Q QC C C C

T

T

= + += + +

1 2 3

1 2 3

Figure 10.25 Parallel capacitors.

ET162 Circuit Analysis – Capacitors Boylestad 18

Ex. 10-5 For the circuit of Fig. 10.26:a. Find the total capacitance.b. Determine the charge on each plate.c. Find the voltage across each capacitor.

aC C C C

F F F

C F

T

T

.

.

.

1 1 1 1

1200 10

150 10

110 10

0125 101

0125 108

1 2 3

6 6 6

6

6

= + +

=×

+×

+×

= ×

=×

=

− − −

µ

b Q Q Q QC E F V

C

T

T

.(8 )( )

= = =

= = ×=

−

1 2 3610 60

480 µ

c V QC

CF

V

V QC

CF

V

VQC

CF

V

and E V V V V

. .

.

.

11

1

6

6

22

2

6

6

33

3

6

6

1 2 3

480 10200 10

2 4

480 1050 10

9 6

480 10100 10

48 0

60

= =××

=

= =××

=

= =××

=

= + + =

−

−

−

−

−

−

Figure 10.26

Ex. 10-6 For the circuit of Fig. 10.27:a. Find the total capacitance.b. Determine the charge on each plate.c. Find the total charges.

a C C C CF F F

Fb Q C E F V mC

Q C E F V mCQ C E F V mC

c Q Q Q Q mC

T

T

.

. (800 )( ) .( )( ) .( )( ) .

. .

= + += + +=

= = × =

= = × =

= = × == + + =

−

−

−

1 2 3

1 16

2 26

3 16

1 2 3

800 60 12002060

10 48 38 460 10 48 2 881200 10 48 57 6

9888

µ µ µµ

Figure 10.27

ET162 Circuit Analysis – Capacitors Boylestad 19

Ex. 10-7 Find the voltage across and charge on each capacitor for the network of Fig. 10.28.

C C C F F F

CC C

C CF FF F

F

Q C E F V C

T

TT

T

T T

'

'

'

( )( )

( )( )

= + = + =

=+

=+

=

= = × =−

2 3

1

16

4 2 63 63 6

2

2 10 120 240

µ µ µ

µ µµ µ

µ

µ

Q Q QQ C

V QC

CF

V

Q C

T T

T

= ==

= =×

×=

=

−

−

1

1

11

1

6

6

240240 103 10

80

240

'

'

µ

µ

V QC

CF

V

Q C V F V CQ C V F V C

TT

T

T

T

''

'

'

'

( )( )( )( )

= =××

=

= = × =

= = × =

−

−

−

−

240 106 10

40

4 10 40 1602 10 40 80

6

6

2 26

3 36

µµ

Figure 10.28

Figure 10.29

ET162 Circuit Analysis – Capacitors Boylestad 20

ET162 Circuit Analysis – Capacitors Boylestad 21

Ex. 10-8 Find the voltage across and charge on capacitors C1 of Fig. 10.30after it has charged up to its final value.

V V V

Q C VF V

C

C

C

=+

=

=

= ×=

−

(8 )( )

( )( )

ΩΩ Ω

244 8

16

20 10 16320

1 16

µ

Figure 10.30

Figure 10.31

Ex. 10-9 Find the voltage across and charge on each capacitor for the network of Fig. 10.32. after each has charged up to its final value.

VV

V

V V V

Q C V F V C

Q C V F V C

C

C

C

C

2

1

1

2

7 727 2

76

2 727 2

16

2 10 16 32

3 10 1681 1

6

1 26

=+

=

=+

=

= = × =

= = × =

−

−

( )( )

( )( )

( )( )

( )(56 )

ΩΩ ΩΩΩ Ω

µ

µ

Figure 10.32

ET162 Circuit Analysis – Capacitors Boylestad 22

CapacitorsAcknowledgementOUTLINESIntroduction to CapacitorsThe Electric FieldThe Electric FieldFIGURE 10.3 Electric flux distribution between the plates of a capacitor: (a) including fringing; (b) ideal.FIGURE 10.7 Summary of capacitive elements.Transients in Capacitive Networks: Charging PhaseTransients in Capacitive Networks: Discharging PhaseCapacitors in Series and Parallel