Inductors and Capacitors

Introduction to time varying

circuits

Objectives

• To introduce inductors and capacitors as linear electric circuit

elements.

• To formulate the I-V relationship in inductors.

• To Calculate the energy stored in inductors.

• To formulate the I-V relationship in capacitors.

• To Calculate the energy stored in capacitors.

• To obtain series and parallel equivalents for inductors and

capacitors.

The inductor

• An inductor is a passive electrical device that stores energy in a magnetic

field, typically by combining the effects of many loops of electric current.

The inductance is measured in Henrys (H) (It is named after the American

scientist Joseph Henry).

L

+ – v

i

The inductor

• The voltage drop across the terminals is related to the current by

dt

diLv

The voltage across the terminals of an inductor is proportional to the time

rate of change of the current in the inductor.

If the current is constant (DC current) the inductor behaves as a short circuit.

constanti 0dt

di0v

Current can not change instantaneously in an inductor (it cannot change by a finite amount

in zero time)

dt

di v

I-V Relationship

• Integrate both sides dt

diLv

Ldivdt

t

t

ti

ti oo

vdtdiL)(

)(

t

to

o

vdttitiL )()(

)(1

)( o

t

ttivdt

Lti

o

• If to=0.

)0(1

)(0

ivdtL

tit

Power and Energy of an Inductor

vip

dt

diLiivP

)(

1o

t

ttivdt

Lvivp

o

dt

diLi

dt

dwp

Lididw

iw

idiLdw00

2

2

1Liw

Power in an inductor

Energy in an inductor

Example

• If 0i

A 10 5ttei

0t

0t

for

for

Sketch the current waveform.

At what instant of time is the current maximum.

Express the voltage across the terminals of the 100 mH inductor as a function of

time.

Sketch the voltage waveform.

Are the voltage and the current at a maximum at the same time.

At what instant of time does the voltage change polarity?

Is there ever an instantaneous change in voltage across the inductor? If so, at what

time?

Solution

b) At what instant of time is the current maximum.

tt teedt

di 55 )5(1010

0)5(1010 55 tt tee 005-10 t0|max dt

dist

5

1

a) Sketch the current waveform.

Solution

c) Express the voltage across the terminals of the 100 mH inductor

as a function of time.

V 5010101005010 535 tt etetLdt

diLv

d) Sketch the voltage waveform.

V 51 5tetv

0v 0t

0t

for

for

Note: Imax occurred when

vL= 0 at t = 0.2 Sec

plot i, v, p, and w versus time.

Example

a) Sketch the voltage as a

function of time.

b) Find the inductor current as

a function of time.

c) Sketch the current as a

function of time.

Ans:- a) The voltage as a function of time.

Example

b) Find the inductor current as a function of time.

)0(1

)(0

ivdtL

tit

)0(201

)(0

10 idtteL

tit

t

ttdtteInt

0

10

t

UdVInt0

tU dtedV t10

dtdU 10

10

teV

t

VdUUVInt0

t

tt

o

t

dtete

Int0

1010

10|

10

100

1

10

1010

tt eteInt

tt teeInt 1010 101100

1

tt teeti 1010

3101

10010100

20)(

A 1012)( 1010 tt teeti 0t

Example

A 1012)( 1010 tt teeti 0t

c) Sketch the current as a function of time.

Remarks

f) Sketch the p & W graphs, and comment why W is constant?

- Since both the source and the inductor is ideal, when the voltage

returns to zero, the energy is trapped inside the inductor and there is

no means of dissipating energy.

The capacitor

• A capacitor is a device that stores energy in the electric field created

between a pair of conductors on which equal but opposite electric

charges have been placed

• The capacitance is measured in Farads (F) (It is named after the English chemist Michael Faraday).

C

I-V Relationship

• The current drop across the terminals is related to the voltage by

dt

dvCi

• If the voltage is constant (DC voltage) the capacitor behaves as an open circuit.

i

+ – v

constantv 0dt

dv0i

• The voltage cannot change instantaneously across the terminals of a capacitor.

dt

dvi

The capacitor power and energy

dt

dvCvvip Power in a capacitor

Energy in a capacitor

dt

dvCv

dt

dwp

Cvdvdw

v

o

w

oCvdvdw 2

2

1Cvw

Cdvidt idtC

dv1

)(1

)( o

t

ttvidt

Ctv

o

Example

The voltage pulse is impressed across the terminals of a 0.5 µF capacitor:

)1(4

4

0

)(te

ttv

, t ≤ 0 s

, 0 ≤ t ≤ 1 s

, t ≥ 1 s

a) Derive the expressions for the capacitor current, power, and energy.

b) Sketch the voltage, current, power, and energy as functions of time.

c) Specify the interval of time when energy is being stored in the capacitor.

d) Specify the interval of time when energy is being delivered by the capacitor.

e) Evaluate the integrals

1

1

0

and pdtpdt

Example

a) Derive the expressions for the capacitor current, power, and

energy.

μA 2

μA 2

0

105.04

105.04

105.00

)()1(6)1(

6

6

tt eedt

dvCti

, t ≤ 0 s

, 0 ≤ t ≤ 1 s

, t ≥ 1 s

W 8

W 8

0

)()1(2

te

tivtp

, t ≤ 0 s

, 0 ≤ t ≤ 1 s

, t ≥ 1 s

μJ 4

μJ 4

0

8

8

0

2

1

)1(2

2

)1(2

22

tt e

t

Ce

CtCvw

, t ≤ 0 s

, 0 ≤ t ≤ 1 s

, t ≥ 1 s

Solution

b)Sketch the voltage, current, power, and energy as functions of

time.

Solution

c) Specify the interval of time when energy is being stored in the capacitor.

- (0 < t < 1 µs)

d) Specify the interval of time when energy is being delivered by the capacitor.

- (t > 1 µs)

e) Evaluate the integrals

1

1

0

and pdtpdt

1

0

1

0

48 Jtdtpdt

1

)1(2

1

μJ 48 dtepdt t

Energy Stored

Energy Extracted

Example

An uncharged 0.2 µF capacitor is driven by a triangle angular current pulse.

The current pulse is described by

a) Derive the expressions for the capacitor voltage, power, and energy.

b) Sketch the current, voltage, power, and energy as functions of time.

c) Why does a voltage remain on the capacitor after the current returns to zero.

0

A 50002.0

A 5000

0

)(t

tti

, t ≤ 0 s

, 0 ≤ t ≤ 20 µs

, t ≥ 40 µs

, 20 ≤ t ≤ 40 µs

Solution

Derive the expressions for the capacitor voltage, power, and energy.

0 ≤ t ≤ 20 µs

20 ≤ t ≤ 40 µs

V 105.1250002.0

10

2.0

1 2920

0

20

0ttdtidtv

ss

W105.62 312tvip

J 10625.152

1 4122 tCvw

V )10105.1210( 296 ttv

W)2105.2105.7105.62( 529312 tttvip

J )10210125.0105.210625.15(2

1 526394122 ttttCvw

Solution

t ≥ 40 µs V 10v 0 vip J 102

1 2 Cvw

Sketch the current, voltage, power, and energy as functions of time.

Series-parallel Combinations

dt

diLv 11

+ – + – + –

v + –

v1 v2 v3

i

dt

diLv 22

dt

diLv 33

dt

diLLLvvvv )( 321321

neq LLLLL 321

Series-Parallel Combinations

+ –

v

i1 i2 i3

i

i1(to) i2(to) i3(to)

)(1

)( 1

1

1 o

t

ttivdt

Lti

o

)(1

)( 2

2

2 o

t

ttivdt

Lti

o

)(1

)( 3

3

3 o

t

ttivdt

Lti

o

)()()(111

321

321

321 ooo

t

ttititivdt

LLLiiii

o

)(1

o

t

teq

tivdtL

io

321

1111

LLLLeq

neq LLLLL

11111

321

Series-Parallel Combinations

Series-Parallel Combinations

Example

Find Lab? a

b

5 H 14 H

8 H 10 H

15 H 60 H

80 H

30 H 20 H H 1230//201 eqL

H 208122 eqL

H 1680//203 eqL

H 3016144 eqL

H 2060//305 eqL

H 3020106 eqL

H 1015//307 eqL

H 15510 abL

Example

Find Cab?

Ans. :

8 µF // 16 µF 24 µF

6 µF series 4 µF

2.4 µF // 1.6 µF 4 µF

12 µF series 4 µF 3 µF

5 µF // 3 µF 8 µF

24 µF series 8 µF 6 µF

FFF

FF

4.2

64

46

D.C. Conditions

Example

In the circuit shown, find the capacitor voltage VC, the inductor current IL, and

the energy stored in the capacitor and under dc conditions.