*

MT-144

NETWORK ANALYSIS

Mechatronics Engineering

(13)

Sinusoids and Phasors

Contents

IntroductionSinusoidsPhasorsPhasor Relationships for Circuit ElementsImpedance and AdmittanceKirchhoffs Laws in the Frequency DomainImpedance CombinationsApplications

Introduction

AC is more efficient and economical to transmit power over long distance.A sinusoid is a signal that has the form of the sine or cosine function.Circuits driven by sinusoidal current (ac) or voltage sources are called ac circuits.Why sinusoid is important in circuit analysis?Nature itself is characteristically sinusoidal.A sinusoidal signal is easy to generate and transmit.Easy to handle mathematically

Sinusoids

Sinusoids (Contd)

A period function is one that satisfies

f(t) = f(t+nT), for all t and for all integers n.

The period T is the number of seconds per cycleThe cyclic frequency f = 1/T is the number of cycles per second

Sinusoids (Contd)

Sinusoids (Contd)

To compare sinusoidsUse the trigonometric identitiesUse the graphical approach

The Graphical Approach

Phasors

Sinusoids are easily expressed by using phasorsA phasor is a complex number that represents the amplitude and the phase of a sinusoid.Phasors provide a simple means of analyzing linear circuits excited by sinusoidal sources.

Phasors (Contd)

Important Mathematical Properties

Phasor Representation

Phasor Representation (Contd)

Phasor Diagram

Sinusoid-Phasor Transformation

Phasor Relationships for Resistor

Time domain

Phasor domain

Phasor diagram

Phasor Relationships for Inductor

Time domain

Phasor domain

Phasor diagram

Phasor Relationships for Capacitor

Phasor diagram

Time domain

Phasor domain

Impedance and Admittance

Impedance and Admittance (Contd)

Impedance and Admittance (Contd)

Impedance and Admittance (Contd)

KVL and KCL in the Phasor Domain

Series-Connected Impedance

Parallel-Connected Impedance

Y- Transformations

Example 1

Example 2

Example 3

-Y transformation

Applications: Phase Shifters

Leading

output

Phase Shifters (Contd)

Lagging

output

Example

Applications: AC Bridges

AC Bridges (Contd)

Bridge for measuring L

Bridge for measuring C

Summary

Transformation between sinusoid and phasor is given as

Impedance Z for R, L, and C are given as

Basic circuit laws apply to ac circuits in the same manner as they do for dc circuits.

)

(

2

2

seconds.

every

itself

repeats

sinusoid

The

sinusoid

the

of

argument

the

)

(radians/s

frequency

angular

the

sinusoid

the

of

amplitude

the

where

sin

)

(

voltage

sinusoidal

he

Consider t

T:period

T

T

T

t

V

t

V

t

v

m

m

p

p

w

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sin(

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:

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(Hz)

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by

say that

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tan

where

)

cos(

sin

cos

1

2

2

A

B

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A

C

t

C

t

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t

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q

q

w

w

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)

1

.

53

cos(

5

sin

4

cos

3

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of

the

:

of

the

:

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form

l

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form

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form

r

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2

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z

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j

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x

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2

1

1

)

(

)

(

)

(

)

(

)

(

)

(

2

1

2

1

2

1

2

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2

1

2

1

2

1

2

1

2

1

2

1

2

1

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:

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Complex

:

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Square

:

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:

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:

tion

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:

on

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:

Addition

2

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2

2

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.

)

(

sinusoid

the

of

tion

representa

phasor

the

is

)

Re(

)

(

)

Re(

)

Re(

)

cos(

)

(

)

Im(

sin

)

Re(

cos

sin

cos

)

(

t

v

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e

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e

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t

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t

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e

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m

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m

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m

t

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m

m

j

j

j

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V

V

f

f

w

f

f

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f

f

w

w

f

f

w

f

f

f

=

=

=

=

=

+

=

=

=

+

=

+

Q

f

=

m

V

V

q

-

=

m

I

I

[

]

w

w

w

w

w

f

w

w

f

w

w

f

f

w

w

w

f

f

w

j

vdt

j

dt

dv

e

j

e

e

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e

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e

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t

V

t

V

dt

t

dv

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v

t

j

t

j

j

m

j

j

j

t

j

m

m

m

m

m

V

V

V

V

Similarly,

)

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)

(

Re

)

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)

90

cos(

)

sin(

)

(

)

cos(

)

(

90

90

=

=

=

+

+

=

+

-

=

=

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=

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V

I

R

RI

t

RI

iR

v

I

t

I

i

m

m

m

m

=

=

+

=

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=

+

=

f

f

w

f

f

w

)

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law,

s

Ohm'

By

)

cos(

is

resistor

rough the

current th

the

If

I

V

I

L

j

t

LI

dt

di

L

v

I

t

I

i

m

m

m

w

f

w

w

f

f

w

=

+

+

=

=

=

+

=

)

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cos(

is

inductor

the

across

voltage

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)

cos(

is

inductor

rough the

current th

the

If

V

I

V

C

j

t

CV

dt

dv

C

i

V

t

V

v

m

m

m

w

f

w

w

f

f

w

=

+

+

=

=

=

+

=

)

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cos(

is

capacitor

rough the

current th

The

)

cos(

is

capacitor

the

across

voltage

the

If

1

1

1

Admittance

Impedance

Element

current

phasor

the

is

tage

phasor vol

the

is

re

whe

(S)

1

:

Admittance

,

)

(

:

Impedance

L

j

C

j

C

C

j

L

j

L

R

R

R

w

w

w

w

=

=

=

=

=

=

=

W

=

Y

Z

Y

Z

Y

Z

I

V

Z

Y

I

V

Z

C

j

w

1

=

Z

L

j

w

=

Z

=

w

w

0

-

=

+

=

+

=

voltage

leads

current

since

leading

or

capacitive

:

voltage

lags

current

since

lagging

or

inductive

:

then

positive,

is

If

negative

is

when

capacitive

positive

is

when

inductive

be

to

said

is

impedance

The

reactance

:

resistance

:

where

jX

R

jX

R

X

X

X

X

R

jX

R

Z

Z

Z

=

=

=

+

=

=

+

=

-

sin

cos

and

tan

where

1

2

2

q

q

q

q

Z

Z

Z

Z

Z

X

R

R

X

X

R

jX

R

+

-

=

+

=

+

-

=

-

-

+

=

+

=

+

+

=

=

2

2

2

2

2

2

1

1

e

susceptanc

:

e

conductanc

:

where

1

X

R

X

B

X

R

R

G

X

R

jX

R

jX

R

jX

R

jX

R

jX

R

jB

G

B

G

jB

G

Z

Y

0

)

Re(

)

Re(

)

Re(

as

written

be

can

This

0

)

cos(

)

cos(

)

cos(

form.

cosine

in

written

be

may

ge

each volta

state,

steady

sinusoidal

In the

0

loop.

closed

a

around

voltages

the

be

,

,...,

,

let

KVL,

For

2

1

2

1

2

2

1

1

2

1

2

1

=

+

+

+

=

+

+

+

+

+

+

=

+

+

+

t

j

j

mn

t

j

j

m

t

j

j

m

n

mn

m

m

n

n

e

e

V

e

e

V

e

e

V

t

V

t

V

t

V

v

v

v

v

v

v

n

w

q

w

q

w

q

q

w

q

w

q

w

(

)

[

]

0

phasor.

for

holds

KCL

manner,

similar

a

In

!

!

phasor!

for

holds

KVL

0

,

any

for

0

Since

0

Re

then

,

Let

0

Re

2

1

2

1

2

1

2

1

2

1

=

+

+

+

=

+

+

+

=

+

+

+

=

=

+

+

+

n

n

t

j

t

j

n

j

mk

K

t

j

j

mn

j

m

j

m

t

e

e

e

V

e

e

V

e

V

e

V

k

n

I

I

I

V

V

V

V

V

V

V

w

w

q

w

q

q

q

)

(

gives

KVL

Applying

2

1

2

1

n

n

Z

Z

Z

I

V

V

V

V

+

+

+

=

+

+

+

=

V

Z

Z

V

Z

V

I

Z

Z

Z

I

V

Z

eq

k

k

eq

n

eq

=

=

+

+

+

=

=

,

2

1

)

1

1

1

(

gives

KCL

Applying

2

1

2

1

n

n

Z

Z

Z

V

I

I

I

I

+

+

+

=

+

+

+

=

I

Y

Y

I

Y

I

V

Y

Y

Y

V

I

Y

eq

k

k

eq

n

eq

=

=

+

+

+

=

=

,

2

1

3

1

3

3

2

2

1

2

1

3

3

2

2

1

1

1

3

3

2

2

1

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

+

+

=

+

+

=

+

+

=

-

c

b

a

ion:

Convers

Y

c

b

a

b

a

c

b

a

a

c

c

b

a

c

b

ion:

Y Convers

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

+

+

=

+

+

=

+

+

=

-

3

2

1

(

)

(

)

(

)

(

)

(

)

(

)

07

.

11

22

.

3

8

11

10

8

2

3

10

10

8

||

2

3

10

2

.

0

8

||

10

1

3

2

1

||

H

2

.

0

8

F

10

3

F

2

in

j

j

j

j

j

j

j

j

j

m

j

m

j

m

m

-

=

+

+

-

+

-

=

+

-

+

-

=

+

+

+

=

+

=

+

W

+

W

w

w

w

Z

Z

Z

Z

rad/s.

50

for

Find

in

=

w

Z

(t).

Find

o

v

[

]

(

)

(

)

(

)

)

96

.

15

4

cos(

15

.

17

)

(

96

.

15

15

.

17

15

20

96

.

30

8575

.

0

15

20

100

60

100

25

||

20

60

25

||

20

4

,

15

20

)

15

4

cos(

20

+

=

=

-

=

-

+

=

-

+

-

=

=

-

=

-

=

t

t

v

j

j

j

j

j

j

t

v

Sol:

o

s

o

s

s

V

V

V

w

.

Find

I

(

)

(

)

-

=

=

=

=

+

=

+

+

-

+

+

=

-

=

-

=

=

=

+

=

+

-

+

-

=

204

.

4

666

.

3

204

.

4

64

.

13

0

50

204

.

4

64

.

13

1

6

.

13

8

6

||

3

12

2

.

3

6

.

1

10

)

4

2

(

8

2

.

3

10

)

8

(

4

8

.

0

6

.

1

8

4

2

4

)

4

2

(

4

Z

V

I

Z

Z

Z

Z

Z

Z

Z

j

j

j

j

j

j

j

j

j

j

j

j

Sol:

cn

bn

an

cn

bn

an

i

i

i

o

RC

C

R

RC

RC

j

RC

j

C

j

R

R

V

V

V

V

+

=

+

=

+

=

-

w

w

w

w

w

w

1

tan

1

1

1

1

2

2

2

(

)

i

i

i

o

RC

C

R

RC

j

C

j

R

C

j

V

V

V

V

-

+

=

+

=

+

=

-

w

w

w

w

w

1

2

2

2

tan

1

1

1

1

1

1

=

=

=

-

=

=

-

-

=

-

=

-

=

-

-

=

-

=

90

3

1

45

3

2

45

2

2

45

2

2

20

20

20

45

3

2

24

12

4

12

20

4

12

20

40

)

20

20

(

20

)

20

20

(

||

20

1

1

1

i

o

i

i

i

j

j

j

j

j

j

j

j

Sol:

V

V

V

V

V

V

V

Z

Z

V

Z

leading..

90

of

phase

a

provide

to

circuit

an

Design

RC

2

1

3

1

3

2

3

2

1

2

3

2

2

1

2

1

2

1

:

condition

Balanced

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

V

Z

Z

Z

V

V

Z

Z

Z

V

V

V

=

=

+

=

+

+

=

=

+

=

=

x

x

x

x

s

x

x

s

s

x

L

R

R

L

2

1

=

s

x

C

R

R

C

2

1

=

C

j

L

j

R

C

L

R

w

w

1

,

,

=

=

=

Z

Z

Z

f

f

w

=

+

=

m

m

V

t

V

t

v

V

)

cos(

)

(