FOURIER SERIES CHAPTER 5. TOPIC: Fourier series definition Fourier coefficients The effect of symmetry on Fourier series coefficients Alternative trigonometric.

FOURIER SERIES

CHAPTER 5

TOPIC:• Fourier series definition• Fourier coefficients• The effect of symmetry on Fourier series

coefficients• Alternative trigonometric form of Fourier series• Example of Fourier series analysis for RL and RC

circuit• Average power calculation of periodic function• rms value of periodic function• Exponential form of Fourier series• Amplitude and phase spectrum

FOURIER SERIES DEFINITION

• The Fourier Series of a periodic function f(t) is a representation that resolves f(t) into a DC component and an AC component comprising an infinite series of harmonic sinusoids.

FOURIER SERIES

• Periodic function

)()( nTtftf

0T 2 T

Vm

m as a , t

f ( t)

trigonometric form of Fourier series

tnbtnaatf onn

onv sincos)(1

Fourier coefficients

Harmonic frequency

DC AC

Condition of convergent a Fourier series (Dirichlet conditions):1. F(t) adalah single-valued

2. F(t) has a finite number of finite discontinuities in any one period

3. F(t) has a finite number of maxima and minima in any one period

4. The intergral

Tt

t

o

0

dt)t(f





• Integral relationship to get Fourier coefficients

T

dttn0 0 0sin

T

o dttn0

0cos

T

oo dttmtn0

0cossin

)(0sinsin0

nmdttmtnT

oo

)(0coscos0

nmdttmtnT

oo

T

o

Tdttn

0

2

2sin

T

o

Tdttn

0

2

2cos

av coefficient

T

v dttfT

a0

)(1

an coefficient

T

on dttncos)t(fT

a0

2

bn coefficient

T

on dttntfT

b0

sin)(2

TOPIC:• Fourier series definition• Fourier coefficients• The effect of symmetry on Fourier series coefficients• Alternative trigonometric form of Fourier series• Example of Fourier series analysis for RL and RC


THE EFFECT OF SYMMETRY ON FOURIER COEFFICIENTS

• Even symmetry

• Odd symmetry

• Half-wave symmetry

• Quarter-wave symmetry

Even Symmetry

• A function is define as even if

)()( tftf

Even function example


0

0cos)(4

)(2

2/

0

2/

0

n

T

on

T

v

b

dttntfT

a

dttfT

a

Odd Symmetry

• A function is define as odd if

)()( tftf

Odd function example

Odd function characteristic

2/

2/0)(

T

T o dttf


2/

0sin)(

4

0

0

T

on

n

v

dttntfT

b

a

a

Half-wave symmetry

• half-wave function:

2)(

Ttftf

half-wave function

Fourier coefficients for half wave function:

evennutk

oddnutkdttntfTb

evennutk

oddnutkdttntfTa

a

T

on

T

on

v

0

sin)(4

0

cos)(4

0

2/

0

2/

0

Quarter-wave symmetry

• A periodic function that has half-wave symmetry and, in addition, symmetry about the mid-point of the positive and negative half-cycles.

Example of quarter-wave symmetry function

Even quarter-wave symmetry

nnilaisemuautkb

evennutk

oddnutkdttncos)t(fTa

a

n

/T

on

v

0

0

8

0

4

0

Odd quarter-wave symmetry

evennutk

oddnutkdttnsin)t(fTb

nnilaisemuautka

a

/T

on

n

v

0

8

0

0

4

0




ALTERNATIVE TRIGONOMETRIC FORM OF THE FOURIER SERIES

• Fourier series

tnbtnaatf onn

onv sincos)(1

• Alternative form

)cos()(1

nn

onv tnAatf

• Trigonometric identity

tnA

tnAa

tnAa

onn

nonnv

nn

onv

sin)cos(

cos)cos(

)cos(

1

1

• Fourier series

sinsincoscos)cos(


n

nn

nnn

a

b

baA

1

22

tan

nnnn jbaA

Example 1

• Obtain the Fourier series for the waveform below:

Solution:

• Fourier series:

tnbtnaatf onn

onv sincos)(1

Waveform equation:

)(

210

101)(

Ttf

t

ttf

av coefficient

2

1

2

1

012

1

)(1

1

0

1

0

2

1

0

t

dtdt

dttfT

aT

v

an coefficient

01

1

012

2

2

1

0

1

0

2

1

0

nsinn

tnsinn

dttncosdttncos

dttncos)t(fT

aT

on

bn coefficient

evenn

oddnn

n

nn

tnn

dttndttn

dttntfT

b

n

T

on

0

2)1(1

1

)1(cos1

cos1

sin0sin12

2

sin)(2

1

0

1

0

2

1

0

Fit in the coefficients into Fourier series equation:

tnbtnaatf onn

onv sincos)(1

2

1va

0na

evenn

oddnnbn0

2

...5sin5

23sin

3

2sin

2

2

1)( ttttf

12sin12

2

1)(

1

kntnn

tfk

By using n=integer….




Steps for applying Fourier series:

• Express the excitation as a Fourier Series

• Find the response of each term in Fourier Series

• Add the individual response using the superposition principle

Periodic voltage source:

)tcos(V o 11

)tncos(V non

)tcos(V o 22 2

Step 1: Fourier expansion

1

)cos()(n

nono tnVVtv

nnn VV

Step 2: find response

• DC component: set n=0 atau ω=0• Time domain: inductor = short

circuit

capacitor = open circuit

Steady state response (DC+AC)

22 V

11 V

nnV

)(Z 0

)n(Z o)(Z o2

)(Z o

(d)

Step 3: superposition principle

1

21

)cos(

.....)()()()(

nnono

o

tnII

titititi

example:

5

H2

)(tvo)(tvs

Question:

• If

Obtain the response of vo(t) for the

circuit using ωo=π.

12sin12

2

1)(

1

kntnn

tvk

s

Solution:

• Using voltage divider:

ssn

no V

nj

njVLjR

LjV

25

2

• DC component (n=0 @ ω=0)

02

1 os VV

• nth harmonic

os nV 90

2

Response of vo:

22

1

122

425

524

902

52425

902

n

/ntan

n/ntann

nV o

o

o

In time domain:

12

5

2tancos

425

4 1

122

knuntuk

ntn

nv

ko

Example of symmetry effect on Fourier coefficients (past year):

Satu voltan berkala segiempat, vi (t) ( Rajah (b))

digunakan ke atas litar seperti yang ditunjukkan

pada Rajah (a). Jika Vm = 60π V dan tempoh,

T = 2π s,

a) Dapatkan persamaan Siri Fourier untuk vi (t).

b) Dapatkan tiga sebutan pertama bukan sifar bagi Siri Fourier untuk vo (t).

mV

mV

T2T0

iV

t

Rajah (a)

Rajah (b)

Solution (a):

• Response is the Odd Quarter-wave symmetry…

evennutk

oddnutkdttnsin)t(fTb

nnilaisemuautka

a

/T

on

n

v

0

8

0

0

4

0

Equation of vi (t) for 0<t< T/4:

mi Vtv )(Harmonic frequency:

12

22

2

T

fo

bn coefficient:

n

n

ntcos)(

dttnsinVT

b

/

/

omn

240

602

8

8

2

0

2

0

Fourier series for vi(t):

...,,n

i ntsinn

)t(v531

1240

Solution (b):

• Voltage vi for first three harmonic:

oi

oi

oi

tsinV

tsinV

tsinV

048548

080380

0240240

5

3

1

Circuit transfer function:

21

1

1

1)(

RCjH

Transfer function for first three harmonic:

1960251

1

316091

1

707011

1

5

3

1

.H

.H

.H

Voltage vo for first three harmonic:

oo

o

ooo

ooo

..v

..v

..v

040891960048

028253160080

06816970700240

5

3

1

First three nonzero term:

.......tsin.

tsin.tsin.)t(vo54089

3282568169

FOURIER SERIES CHAPTER 5. TOPIC: Fourier series definition Fourier coefficients The effect of symmetry on Fourier series coefficients Alternative trigonometric.

Documents