8/12/2019 ECE 1311 Ch6

1/31

ECE 1311

Chapter 6 Capacitors and Inductors

1

8/12/2019 ECE 1311 Ch6

2/31

Outlines

2

CapacitorsSeries and parallel capacitors

Inductors

Series and parallel inductors

8/12/2019 ECE 1311 Ch6

3/31

Introduction

3

Capacitors and inductorsPassive elements in a linear circuit.Both elements store energy.Circuit analysis techniques covered in Ch. 3 and Ch. 4 are

applicable to circuits with capacitors and inductors.

8/12/2019 ECE 1311 Ch6

4/31

Capacitors

4

Consist of two conducting platesseparated by an insulator.

If it is connected to a voltage source,the source deposits:

+ve charge (q) on one plate.-ve charge ( q) on another plate.

The Mc-Graw-Hill Companies

A capacitor with voltage v

8/12/2019 ECE 1311 Ch6

5/31

Capacitors

5

Capacitors store electric charge.

The amount of charge (q) stored is directly proportional to theapplied voltage ( v) so that:

Where C is the capacitance of the capacitor and is measured in farad (F).

Previously:

Current-voltage relationship:

.

Cvq

dt

Cdv

dt

dCvi

Cvqand dt dq

i

dt Cdv

i

The Mc-Graw-Hill Companies

Common symbolfor capacitors

8/12/2019 ECE 1311 Ch6

6/31

8/12/2019 ECE 1311 Ch6

7/31

Capacitors

7

Energy stored:

vq

vwhereC q

wor Cvw

vce

Cvt CvvdvC w

dt dt dv

vC pdt w

t v

v

t t

221

0)(sin

)(21)(

21

22

22)(

)(

8/12/2019 ECE 1311 Ch6

8/31

Summary - Capacitors

8

Charge (q)

Current (i)

Voltage (v)

Power (p)

Energy (w)

Cvq

dt dv

C i

t

t

t vidt C

v0

)(1 0

dt dv

Cv p

2

21

Cvw

8/12/2019 ECE 1311 Ch6

9/31

Capacitors

9

Property 1:A capacitor is an open circuit to dc.

If a voltage across a capacitor is constant, the current through it is 0.

Property 2:The voltage on a capacitor cannot change abruptly.

Instantaneous change of voltage is NOT ALLOWED but,Current may change instantaneously.

.

Allowed Not allowed

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

10/31

Capacitors

10

Property 3:An ideal capacitor does not dissipate energy.It takes power from the circuit and stores it.It returns previously stored energy when delivering power to thecircuit.

8/12/2019 ECE 1311 Ch6

11/31

Example 1

11

What is the voltage across 3F capacitor if the chargeon one plate is 0.12 mC?How much energy is stored?

Answers:40 V2.4 mJ

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

12/31

Example 2

12

If a 10 F capacitor is connected to a voltage sourcewith v(t)=50 sin 2000t ; find the current through thecapacitor.

Answer:cos 2000t A

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

13/31

Example 3

13

An initially uncharged 1 mF capacitor has the currentas below.Calculate the voltage across it at t=2 ms and at t=5ms.

Answers:100 mV400 mV

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

14/31

Example 4

14

Under dc conditions, find the energy stored in thecapacitors in figure below.

Answers:810 J135 J

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

15/31

Parallel Capacitors

15

N eq C C C C C ...321

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

16/31

Series Capacitors

16

N eq C C C C C 1

...1111

321

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

17/31

Example 5

17

Determine the equivalent capacitance seen at theterminals of the circuit.

Answer:40 F

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

18/31

Example 6

18

Find the voltages across each of the capacitors.

The Mc-Graw-Hill Companies

Answers:v 1=30 Vv 2=30 Vv 3=10 Vv 4=20 V

8/12/2019 ECE 1311 Ch6

19/31

Inductors

19

It is a passive element and it storesenergy.

Voltage-current relationship:The voltage across an inductor is directlyproportional to the time rate of change ofthe current.

Where L is the inductance of the inductor andis measured in henry (H).

dt di

Lv

Common symbolfor inductors

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

20/31

Inductors

20

Current-voltage relationship:

Where i(t0 ) is the total current for -

8/12/2019 ECE 1311 Ch6

21/31

Inductors

21

Energy stored:

2

22

21

0)(sin

)(21)(

21

Liw

ice

Lit Liidi Lw

idt dt di

L pdt w

t

t t

8/12/2019 ECE 1311 Ch6

22/31

Summary - Inductors

22

Voltage (v)

Current (i)

Power (p)

Energy (w)

dt di

Lv

)()(1

0

0

t idt t v L

it

t

idt di

L p

2

21 Liw

8/12/2019 ECE 1311 Ch6

23/31

Inductors

23

Property 1:An inductor acts like a short circuit at dc.Voltage across an inductor is 0 when current is constant.

Property 2:Instantaneous change of current is NOT ALLOWED but,Voltage may change instantaneously.

.

Allowed Not allowed

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

24/31

Inductors

24

Property 3:An ideal inductor does not dissipate but stores energy.It takes power from the circuit and stores it.It returns previously stored energy when delivering power to thecircuit.

8/12/2019 ECE 1311 Ch6

25/31

Example 7

25

If the current through a 1 mH inductor is i(t)=20 cos100t mA, find the terminal voltage and the energystored.

The Mc-Graw-Hill Companies

Answers:-2sin 100t mV200 cos 2(100t)

8/12/2019 ECE 1311 Ch6

26/31

Example 8

26

The terminal voltage of 2H inductor is v=10(1-t) v.Find the current flowing through it at t=4 s and energy storedin it within 0

8/12/2019 ECE 1311 Ch6

27/31

Example 9

27

Determine V C , i L and the energy stored in thecapacitor and inductor in the circuit under dcconditions.

The Mc-Graw-Hill Companies

Answers:6 V3 A72 J27 J

8/12/2019 ECE 1311 Ch6

28/31

Series Inductors

28

N eq L L L L L ...321

The Mc-Graw-Hill Companies

8/12/2019 ECE 1311 Ch6

29/31

Parallel Inductors

29 The Mc-Graw-Hill Companies

N eq L L L L L1

...1111

321

8/12/2019 ECE 1311 Ch6

30/31

Example 10

30

Calculate equivalent inductance for the inductiveladder network.

The Mc-Graw-Hill Companies

Answer:25 mH

8/12/2019 ECE 1311 Ch6

31/31

Example 11

31

In the circuit; i 1(t)=0.6e -2t A. If i(0)=1.4 A; find

i 2(0),i 2 (t) and i(t)v(t), v 1 (t) and v 2 (t)

.

h ll