Lec. (4)Chapter (2) AC- circuits

Capacitors and transient current

1

2.1 – Introduction

Capacitor displays its true characteristics only when a change in voltage is made in the network.

A capacitor is constructed of two parallel conducting plates separated by an insulator.

Capacitance is a measure of a capacitor’s ability to store charge on its plates.

Transient voltages and currents result when circuit is switched

2

2.2 – Capacitance

o Capacitance is a measure of a capacitor’s ability to store charge on its plates.

o A capacitor has a capacitance of 1 farad (F) if 1 coulomb (C) of charge is deposited on the plates by a potential difference of 1 volt across its plates.

o The farad is generally too large a measure of capacitance for most practical applications, so the microfarad (106 ) or picofarad (1012 ) is more commonly used. 3

A charged parallelplate capacitor.

Q = C V where C = eo A / d

for a parallel plate capacitor, where eo is the permittivity of

the insulating material (dielectric) between plates.

Recall that we used Gauss's Law to calculate the electric

field (E) between the plates of a charged capacitor:

E = s / eo where there is a vacuum between the plates.

Vab = E d, so E = Vab /d

The unit of capacitance is called the Farad (F).4

Fringing – At the edge of the capacitor plates the flux lines extend outside the common surface area of the plates.

2.2 – Capacitance

1. Fixed: mica, ceramic, electrolytic, tantalum and

polyester-film2. Variable Capacitors: The capacitance is changed by turning the shaft at one end

to vary the common area of the movable and fixed plates. The greater the common area the larger the capacitance.

2.3 Types of Capacitors:

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The potential energy stored in the system of positive charges that are separated from the negative charges is like a stretched spring that has potential energy associated with it.

Capacitors can store charge and ENERGY

DU = q DV

and the potential V increases as the charge is placed on the plates

2.4 ENERGY STORED IN A CAPACITOR

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(V = Q / C)

Since the V changes as the Q is increased, we have to integrate over all the little charges “dq” being added to a plate:

DU = q DV

QVu

V

QCCVu

2

1

,2

1 2

CVQC

Q

qdqC

C

dqqC

QVVdqu

Q

,2

1

,

2

0

8

U: is the energy stored in a capacitor

Energy density:

This is an important result because it tells us that empty space contains energy if there is an electric field (E) in the "empty"

space.

If we can get an electric field to travel (or propagate) we can send or

transmit energy and information through

empty space!!!

2

2

1Eu

9

The charges induced on the surface of the dielectric (insulator) reduce the electric field.

10

You slide a slab of dielectric between the plates of a parallel-plate capacitor. As you do this, the charges on the plates remain constant.

What effect does adding the dielectric have on the energy stored in the capacitor?

A. The stored energy increases.

B. The stored energy remains the same.

C. The stored energy decreases.

D. not enough information given to decide

Q1. 2

U α E2

11

2.5 Capacitors are in Series:

When capacitors are in series, the charge is the same on each

capacitor.321 VVVVt

C

QVCVQ

3

3

2

2

1

1

C

Q

C

Q

C

Q

C

Q

t

t

321 QQQQt

321

1111

CCCCt

2.5 Capacitors are in Parallel

When capacitors are in parallel , the total charge is the sum of that on each

capacitor.321 QQQQt CVQ

332211 VCVCVCVC tt

321 VVVVt

321 CCCCt

2.6 Capacitor charging and discharging

• Charging a capacitor that is discharged– When switch is closed, the current instantaneously

jumps to E/R– Exponentially decays to zero

• When switching, the capacitor looks like a short circuit• Voltage begins at zero and exponentially increases to E

volts• Capacitor voltage• cannot change instantaneously

2.5 – Initial Conditions

The voltage across a capacitor at the instant of the start of the charging phase is called the initial value. Once the voltage is applied the transient phase will commence until a leveling off occurs after five time constants called steady-state as shown in the figure.

RC Circuits

R

VC

++

--

a

b

RC-Circuit: Resistance R and capacitance C in series with a source of emf V.

Start charging capacitor. . .Applying KVL

R

VC

++

--

a

bi

q

C

IRC

qV

iR

RC Circuit: Charging Capacitor

Rearrange terms to place in differential form:

R

VC

++

--

a

bi

q

C

Multiply by C dt :

IRC

qV

dt

dqR

C

qV

IRC

qV

qt

qCV

dq

RC

dt

qCV

dq

RC

dt

00 )()(

RC Circuit: Charging Capacitor

qt

qCV

dq

RC

dt

00 )(

RC

t

q

CVeqCV

qCV

CV

RC

t

qCVCVRC

t

CVqCVRC

t

qCVRC

t

ln

)ln(ln

ln)ln(

)ln(0

)1( RC

t

eCVq

Instantaneous charge q on a charging capacitor:

RC Circuit: Charging Capacitor

At time t = 0: q = CV(1 - 1); q = 0

At time t = : q = CV(1 - 0); qmax = CV

The charge q rises from zero initially to its maximum value qmax = CVThe charge q rises from zero initially to its maximum value qmax = CV

)1( RC

t

eCVq

Example 2.1. What is the charge on a 4mF capacitor charged by 12V for a time t = RC?

Time, t

Qmax

q

Rise in Charge

Capacitor

t

0.63 Q

The time t = RC is known as the time constant.

R = 1400 W

V 4 mF

++

--

a

bi

e = 2.718

)1( RC

t

eCVq

)1( 1 eCVq CC

e

eVFq

3.30103.30

)1(1048

)1)(12)(104(

6

16

16

Example 2.1 (Cont.) What is the time constant t?

Time, t

Qmax

q

Rise in Charge

Capacitor

t

0.63 Q

The time t = RC is known as the time constant.

R = 1400 W

V 4 mF

++

--

a

bi

In one time constant (5.60 ms in this example), the charge rises to 63% of its maximum value (CV).

t = (1400 W)(4 mF)

t = 5.60 mst = 5.60 ms

RC Circuit: Decay of Current

As charge q rises, the current i will decay. Current decay as a capacitor

is charged:

)1( RC

t

eCVq

RC

t

RC

t

eRC

CV

CVeCVdt

d

dt

dqi

)(

RC

t

eR

Vi

when t = 0

when t = i=0

R

Vi max

Current Decay

Time, t

Ii

Current Decay

Capacitor

t

0.37 I

R

VC

++--

a

bi

q

C

RC

t

eR

Vi

when t = 0

when t = i=0

R

Vi max

Transients in Capacitive Networks: Charging Phase

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• The placement of charge on the plates of a capacitor does not occur instantaneously.

• Transient Period – A period of time where the voltage or current changes from one steady-state level to another.

• The current ( ic ) through a capacitive network is essentially zero after five time constants of the capacitor charging phase.

Steady State Conditions

Circuit is at steady state When voltage and current reach their final

values and stop changing Capacitor has voltage across it, but no

current flows through the circuit Capacitor looks like an open circuit

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Example 2.2. What is the current i after one time constant ( = t RC)? Given R=1400W and C=4mF.

RC

t

eR

Vi

The time t = RC is known as the time constant.i.e t = t = RC

max

1

37.0 ii

eR

Vi

Charge and Current During the Charging of a Capacitor.

Time, t

Qmax

q

Rise in Charge

Capacitor

t

0.63 I

Time, t

Ii

Current Decay

Capacitor

t

0.37 I

In a time t of one time constant, the charge q rises to 63% of its maximum, while the current i decays to 37% of its maximum value.

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Capacitor Discharging

Assume capacitor has E volts across it when it begins to discharge

Current will instantly jump to –E/R Both voltage and current will decay

exponentially to zero

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RC Circuit: Discharge

R

VC

++

--

a

b

After C is fully charged, we turn switch to b, allowing it to discharge.Discharging capacitor. . . loop rule gives:

; q

iR iRC

E

R

VC

++

--

a

bi

q

C

Negative because of decreasing I.

Discharging From q0 to q:

; dq

q RCi q RCdt

Instantaneous charge q on discharging capacitor:

R

VC

++

--

a

bi

q

C

RC

t

q

qRC

tq

RC

dt

q

dq

q

q

tq

q

ln

ln

0

RC

t

eqq

Discharging Capacitor

R

VC

++

--

a

bi

q

CNote qo = CV and the instantaneous current is: dq/dt.

Current i for a discharging capacitor.

RC

t

eqq

RC

t

RC

t

eRC

CV

CVedt

d

dt

dqi

)(

RC

t

eR

Vi

Prob. 2.3 How many time constants are needed for a capacitor to reach 99% of final charge?

4.61 time constants

4.61 time constants

01.0ln

01.0

199.0

1

)1(

max

max

RC

t

e

e

eq

q

eqq

RC

t

RC

t

RC

t

RC

t

)1( RC

t

eCVq

61.4RC

t

Prob. 2.4. Find time constant, qmax, and time to reach a charge of 16 mC if V = 12 V and C = 4 mF.

R

V1.8 mF

++

--

a

bi

1.4 MW

C12 V

FCVVFq

tq

41216

??? max

S

F

RC

52.2

)108.1)(104.1( 66

)1(maxRC

t

eqq

C

VF

CVq

6.21

)12)(1016( 6

max

))1/(1ln(

)1/(1

1

1

)1(

max

max

max

max

max

q

q

RC

t

q

qe

q

qe

eq

q

eqq

RC

t

RC

t

RC

t

RC

t

Prob. 2.4. continued

St

t

C

Ct

q

q

RC

t

4.3

)259.0/1ln(52.2

))6.21

161/(1ln(

52.2

))1/(1ln(max

Time to reach 16 mC: t=3.4S

2.12 – Energy Stored by a Capacitor

o The ideal capacitor does not dissipate any energy supplied to it. It stores the energy in the form of an electric field between the conducting surfaces.

o The power curve can be obtained by finding the product of the voltage and current at selected instants of time and connecting the points obtained.

o WC is the area under the curve.

2.14 – Applications

Capacitors find applications in:o Electronic flash lamps for cameraso Line conditionerso Timing circuitso Electronic power supplies

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37

An RC Timing Application

RC circuits Used to create delays for alarm, motor control,

and timing applications Alarm unit shown contains a threshold

detector When input to this detector exceeds a preset

value, the alarm is turned on

38

An RC Timing Application

Pulses have a rise and fall time Because they do

not rise and fall instantaneously

Rise and fall times are measured between the 10% and 90% points

39

The Effect of Pulse Width Width of pulse relative to a circuit’s time constant

Determines how it is affected by an RC circuit If pulse width >> 5

Capacitor charges and discharges fully With the output taken across the resistor, this is a

differentiator circuit

If pulse width = 5 Capacitor fully charges and discharges during each

pulse If the pulse width << 5

Capacitor cannot fully charge and discharge This is an integrator circuit

40

Simple Wave-shaping Circuits

Circuit (a) provides approximate integration if 5 >>T

Circuit (b) provides approximate differentiation if T >> 5

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Capacitive Loading(stray capacitance):

Stray Capacitance Occurs when conductors are

separated by insulating material Leads to stray capacitance In high-speed circuits this can

cause problems