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Chapter 9 – The Basic Capacitor

electronics fundamentalscircuits, devices, and applications

THOMAS L. FLOYDDAVID M. BUCHLA

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The Basic Capacitor

Capacitor:Electrical device composed of two parallel conductive plates separated by an insulating material (dielectric).

The ability to store charge is the definition of capacitance.

DielectricConductors

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Die le c tric

Pla te sLe a d s

Ele c tro ns

BA

+

+

+

+

+

+

+

+

Initially uncharged

+ BA

+

+

+

+

+

+

+

Charging

+ BA

V S

+

+

+++++++++

Fully charged

BA

VS

+

+

+++++++++

Source removed

The charging process…

A capacitor with stored charge can act as a temporary battery.

The Basic Capacitor

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Capacitance is the ratio of charge to voltage

QC

V

Capacitance

C = Capacitance (ability to store charge) Q = Charge (coulombs)V = Volts

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Capacitance (farad)

FaradAmount of capacitance when one coulomb of charge is stored with one volt across the plates.

QC

V

The amount of charge on a capacitor is determined by the size of the capacitor (C) and the voltage (V).

Q CV

Capacitance

If a 22 mF capacitor is connected to a 10 V source, the charge is 220 mC

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Capacitance

An analogy:

Imagine you store rubber bands in a bottle that is nearly full.

You could store more rubber bands (like charge or Q) in a bigger bottle (capacitance or C) or if you push them in more (voltage or V). Thus,

Q CV

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A capacitor stores energy in the form of an electric field that is established by the opposite charges on the two plates. The energy of a charged capacitor is given by the equation

Energy Storage

2

2

1CVW

where

W = the energy in joulesC = the capacitance in faradsV = the voltage in volts

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Electrical Characteristics

Voltage rating – amount of voltage that can be withstood across the plates

Breakdown or working voltage - voltage at which physical damage is done to the capacitor.

Dielectric strength – A measure of a compounds ability to serve as an insulator. V/mil (.001” or 2.54 * 10-5 meter)

See Table 9-2 – Page 392

Temperature coefficient – indicates the amount and direction of change in capacitance value with temperature

Leakage – amount of charge that leaks through the dielectric

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Physical Characteristics

Plate area – Capacitance is directly proportional to the plate area.

Plate separation – Capacitance is inversely proportional to the distance between the plates.

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Dielectric constant (or relative permittivity) – measure of a material’s ability to establish an electric field.

Capacitance is directly proportional to the dielectric constant.

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Capacitor types

Mica - very hard, heat resistant mineral

M ic aFo il

Fo ilM ic a

Fo il

Fo ilM ic a

Fo il

Mica capacitors are small with high working voltage. er (5)

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Ceramic disk

So ld e r

Le a d w ire so ld e re dto silve r e le c tro d e

C e ra m icd ie le c tric

D ip p e d p he no lic c o a ting

Silv e r e le c tro d e s d e p o site d o nto p a nd b o tto m o f c e ra m ic d isk

Ceramic disks are small non-polarized capacitors

They have relatively high capacitance due to high er (1200)

Capacitor types

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Plastic Film

Le a d w ire

Hig h -p urityfo il e le c tro d e s

Pla stic filmd ie le c tric

O u te r wra p o fp o lye ste r film

C a p a c ito r se c tio n(a lte rn a te strip s o ffilm d ie le c tric a ndfo il e le c tro d e s)

So ld e r c o a te d e n d

Plastic film capacitors are small and non-polarized.

They have relatively high capacitance due to larger plate area. er (depends on the film used)

Capacitor types

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Electrolytic (two types)

Symbol for any electrolytic capacitor

Aluminum electrolytic

+

_

Tantalum electrolytic

• Electrolytic capacitors have very high capacitance1. not as precise as other types and 2. tend to have more leakage current.

• Electrolytic capacitors are the only ones polarized.

Capacitor types

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Power Factor CorrectionDefibrillator Capacitor can deliver over 500 joules of energy

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Variable

• Variable capacitors:1. have small capacitance values and2. are adjusted manually3. called Trimmers, padders or tuning capacitors

because they are used to make very fine adjustments in a circuit.

• Varactor diode - A semiconductor that exhibits capacitive characteristic; it is adjusted with an electrical signal.

Capacitor types

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Capacitance meter

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Capacitor labeling

Small capacitors values are frequently stamped on them such as .001 or .01, which have implied units of microfarads.

+++

+

VT

TV

TT

47

MF

.022

Electrolytic capacitors have larger values, so are read as mF.

Usually stamped as mF, but some older ones may be shown as MF or MMF).

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Capacitors in series

1. Divide Voltage across them in proportion to their capacitance

2. Current is same at all points while charging; then goes to zero

tQ

I QnQQQT 21

CQ

V NS VVVV ...21

Q is charge and C is capacitance

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I

CQ

V

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Series capacitors

When connected in series, the total capacitance is smaller than the smallest one.

N

T

CCCC

C1

...1111

321

21

21

CCCC

CT

Capacitance for any number of capacitors in series

Capacitance for two capacitors in series

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Series capacitors

If a 0.001 mF capacitor is connected in series with an 800 pF capacitor, the total capacitance is 444 pF

0 .001 µ F 800 pF

C 1 C 2

1000pF

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Voltage in Series Capacitors

• Voltage depends on capacitance value• Smallest value capacitor will have the

largest voltage across it• Largest value capacitor will have the

smallest voltage across it.

CQ

V S

X

T

X VCC

V

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Parallel capacitors

1.When capacitors are connected in parallel, the total capacitance is the sum of the individual capacitors.

2.The voltage across capacitors in the same parallel branch is the same.

3.The total charge stored by each capacitor is proportional to its capacitance. QT=Q1+Q2+Q3+…+QN

tQ

I VC2 = VC1

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Parallel capacitors

The general equation for capacitors in parallel is

T 1 2 3 ... nC C C C C

1800 pF

If a 0.001 mF capacitor is connected in parallel with an 800 pF capacitor, the total capacitance is

0 .001 µ F 800 pF

C 1 C 2

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Capacitors in DC Circuits

A capacitor charges when connected to a DC voltage.When fully charged, there is no current flow.Except for leakage, a capacitor will remain charged for long periods.A capacitor appears as an “open” to a constant voltage.A capacitor appears as a “short” to an instantaneous change in voltage. When discharging the flow of current is in the opposite direction from the charging cycle.

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RC Time Constant:A fixed time interval that equals the product of the resistance and the capacitance in a series capacitor circuit.

e(farads)capacitancC

(ohms)resistanceR

ds)(tau)time(seconτ

RC

The RC time constant

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The RC time constant

When a capacitor is charged through a series resistor and dc source, the charging curve is exponential.

C

R

Iin itia l

t0(b ) C ha rg ing c urre nt

Vfina l

t0(a ) C a p a c ito r c ha rg ing vo lta g e

)0(1

fromchargingRC

t

Feous)(instantan eVv

t

FiFousinstantane eIIIi

)()(

t

FiFuousinstantane eVVVv

)()(

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When a capacitor is discharged through a resistor, the discharge curve is also an exponential. (Note that the current is negative.)

t

t

Iin itia l

0

( b ) D is c h a rg in g c u rre n t

V in itia l

0( a ) C a p a c i t o r d is c h a r g in g v o l t a g e

C

R

The RC time constant

)0(

fromgdischarginRC

t

ieVv

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Universal exponential curves

Specific values for current and voltage can be read from a universal curve. For an RC circuit, the time constant is

τ RC

100%

80%

60%

40%

20%

00 1 2 3 4 5

99%98%

95%

86%

63%

37%

14%

5% 2% 1%

Number of time constants

Per

cent

of f

inal

val

ue

Rising (charging) exponential

Falling (discharging) exponential

Any capacitor will charge in 5 time constants!!

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Universal exponential curves

τ RC100%

80%

60%

40%

20%

00 1 2 3 4 5

99%98%

95%

86%

63%

37%

14%

5% 2% 1%

Number of time constants

Per

cent

of f

inal

val

ue

How long will it take to charge the capacitor shown in the circuit below to 16 volts? Use the charging curve shown.

20v16 volts = 80% of final value80% (16 v) of charge occurs at ~ 1.6 time constantsτ = (10*103Ω)*(.001x10-6F) = 10 x10-6 sec = 10µs∴ Time to reach 16v = 1.6 x 10µs ~ 16 µsTime to reach 17.2 volts?

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Universal exponential curves

τ RC100%

80%

60%

40%

20%

00 1 2 3 4 5

99%98%

95%

86%

63%

37%

14%

5% 2% 1%

Number of time constants

Per

cent

of f

inal

val

ue

How long will it take to charge the capacitor shown in the circuit below to 16 volts? Use the charging curve shown.

20vTime to reach 17.2 volts?

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Capacitors in AC Circuits

When the source voltage is held at a constant amplitude and the frequency is:

1. increased the amplitude of the current increases which indicates a decrease in resistance to charging.

2. decreased the amplitude of the current decreases which indicates an increase in resistance to charging

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Capacitive Reactance

The opposition to ac by a capacitor is known as Capacitive reactance (XC). Unit is Ohms

Capacitive reactance is:1. inversely proportional to frequency2. inversely proportional to capacitance

1

2πCXfC

The capacitive reactance of a 0.047 mF capacitor when a frequency of 15 kHz is applied is 226 W

f – hertzC - farads

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Reactance for Series Capacitors

When capacitors are in series, the total capacitive reactance is the sum of the individual reactance's.

Assume three 0.033 mF capacitors are in series with a 2.5 kHz ac source. What is the total reactance?

5.79 kW

C( ) C1 C2 C3 Ctot nX X X X X

The reactance of each capacitor is

1 1

1.93 k2π 2π 2.5 kHz 0.033 μFCX

fC

C( ) C1 C2 C3

1.93 k 1.93 k 1.93 k

totX X X X

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Reactance for Parallel Capacitors

When capacitors are in parallel, the total reactance is the reciprocal of the sum of the reciprocals of the individual reactance's.

If the three 0.033 mF capacitors from the last example are placed in parallel with the 2.5 kHz ac source, what is the total reactance?

643 W

C( )

C1 C2 C3 C

11 1 1 1tot

n

X

X X X X

The reactance of each capacitor is 1.93 kW

21

21)(

CC

CCtotC

XX

XXX

C( )

C1 C2 C3

1 11 1 1 1 1 1

+ +1.93 k 1.93 k 1.93 k

totX

X X X

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Ohms Law for Capacitors

The capacitive reactance of a capacitor is the same as a resistor

CX

VI

Current and volts must be in the same units but it does matter which(i.e. rms, peak, p to p)

Calculate I

Irms= 2vrms/7957.7Ω = 0.25mA

1

2πCXfC

7957.7Ω

Xc same as resistance

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Capacitive Voltage Divider

Two capacitors in series are used as a capacitive voltage divider. The capacitors split the output voltage:

1. in proportion to their reactance and2. inversely proportional to their capacitance.

Vout

1000 pF

0.01 µFC2

C1

1.0 Vf = 33 kHz

SX

totX V

C

CV

StotC

CX V

X

XV

X

)(

XCX is the reactance of the capacitor CX

XC(tot) – total capacitive reactanceVX – voltage across capacitor CX

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What is the output voltage for the capacitive voltage divider?

11

1 14.82 k

2π 2π 33 kHz 1000 pFCXfC

22

1 1482

2π 2π 33 kHz 0.01 μFCXfC

Vout

1000 pF

0.01 µFC2

C1

1.0 Vf = 33 kHz

2

( )

482 1.0 V =

5.30 kC

out sC tot

XV V

X

( ) 1 2

4.82 k 482 5.30 k

C tot C CX X X

91 mV

Capacitive Voltage Divider

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Instead of using a ratio of reactances in the capacitor voltage divider equation, you can use a ratio of the total series capacitance to the output capacitance (multiplied by the input voltage). The result is the same. For the problem presented in the last slide,

Vout

1 2( )

1 2

1000 pF 0.01 μF909 pF

1000 pF 0.01 μFtot

C CC

C C

1000 pF

0.01 µFC2

C1

1.0 Vf = 33 kHz

( )

2

909 pF1.0 V =

0.01 μFtot

out s

CV V

C

91 mV

Capacitive Voltage Divider

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Capacitive phase shift

When a sine wave is applied to a capacitor, there is a phase shift between voltage and current such that current always leads the voltage by 90o.

VC0

I 0

90 o

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Power in a capacitor

Energy is stored by the capacitor during a portion of the ac cycle and returned to the source during another portion of the cycle.

Power is maximum where the V and I curves cross

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1. Voltage and current are always 90o out of phase. 2. No true power is dissipated by a capacitor, because

stored energy is returned to the circuit.3. The rate at which a capacitor stores or returns

energy is called reactive power. 4. The unit for reactive power is the VAR (volt-

ampere reactive).

Crms

C

rms

rmsrms

XIX

V

IV

2r

2

r

r

P

P

P

Power in a capacitor

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Rectifiers

Convert a sinusoidal wave (ac) to a pulsing dc wave

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Power supply filtering

Pulsing voltage and current are not usually desirable for operating equipment

A capacitor adds a smoothing effect to the wave by discharging on the down slope of the curve.

Rectifier

60 Hz acC Load

resistance

The filter smoothes the pulsating dc from the rectifier.

AC Circuits

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Selected Key Terms

Capacitor

Dielectric

Farad

RC time constant

An electrical device consisting of two conductive plates separated by an insulating material and possessing the property of capacitance.

The insulating material between the conductive plates of a capacitor.

The unit of capacitance.

A fixed time interval set by the R and C values, that determine the time response of a series RC circuit. It equals the product of the resistance and the capacitance.

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Capacitive reactance

Instantaneous power (p)

True power (Ptrue)

Reactive power (Pr )

VAR (volt-ampere

reactive)

The value of power in a circuit at a given instant of time.

The power that is dissipated in a circuit usually in the form of heat.

The opposition of a capacitor to sinusoidal current. The unit is the ohm.

Selected Key Terms

The rate at which energy is alternately stored and returned to the source by a capacitor. The unit is the VAR.

The unit of reactive power.

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Quiz

1. The capacitance of a capacitor will be larger if

a. the spacing between the plates is increased.

b. air replaces oil as the dielectric.

c. the area of the plates is increased.

d. all of the above.

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Quiz

2. The major advantage of a mica capacitor over other types is

a. they have the largest available capacitances.

b. their voltage rating is very high

c. they are polarized.

d. all of the above.

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Quiz

3. Electrolytic capacitors are useful in applications where

a. a precise value of capacitance is required.

b. low leakage current is required.

c. large capacitance is required.

d. all of the above.

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Quiz

4. If a 0.015 mF capacitor is in series with a 6800 pF capacitor, the total capacitance is

a. 1568 pF.

b. 4678 pF.

c. 6815 pF.

d. 0.022 mF.

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Quiz

5. Two capacitors that are initially uncharged are connected in series with a dc source. Compared to the larger capacitor, the smaller capacitor will have

a. the same charge.

b. more charge.

c. less voltage.

d. the same voltage.

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Quiz

6. When a capacitor is connected through a resistor to a dc voltage source, the charge on the capacitor will reach 50% of its final charge in

a. less than one time constant.

b. exactly one time constant.

c. greater than one time constant.

d. answer depends on the amount of voltage.

100%

80%

60%

40%

20%

00 1 2 3 4 5

99%98%

95%

86%

63%

37%

14%

5% 2% 1%

Number of time constants

Per

cent

of f

inal

val

ue

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Quiz

8. The capacitive reactance of a 100 mF capacitor to 60 Hz is

a. 6.14 k .W

b. 265 .W

c. 37.7 .W

d. 26.5 W

AC Circuits

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