Engineering Circuit AnalysisEngineering Circuit Analysis

CH8 Fourier Circuit AnalysisCH8 Fourier Circuit Analysis

8.1 Fourier Series8.1 Fourier Series8.2 Use of Symmetry8.2 Use of Symmetry

Ch8 Fourier Circuit Analysis

8.1 Fourier Series8.1 Fourier Series

- Most of the functions of a circuit are periodic functions

- They can be decomposed into infinite number of sine and cosine functions that are harmonically related.

- A complete responds of a forcing function =

Partial response to each harmonics. erpositonsup

Harmonies: Give a cosine function

- : fundamental frequency ( is the fundamental wave form)

- Harmonics have frequencies

0

0

01

21:

2:

cos2

wfTT

wff

twtv

0w tv1

tnwatv nn 0cos

Amplitude of the nth harmonics(amplitude of the fundamental wave form)

,4,3,2, 0000 wwww

Freq. of the 1st harmonics (=fund. freq)

Freq. of the 2nd harmonics

Freq. of the nth harmonics

Freq. of the 3rd harmonics

Freq. of the 4th harmonics

Ch8 Fourier Circuit Analysis

8.1 Fourier Series8.1 Fourier Series

8.1 Fourier Series8.1 Fourier SeriesExample Fundamental: v1 = 2cosw0t

v3a = cos3w0t v3b = 1.5cos3w0t

v3c = sin3w0t

Ch8 Fourier Circuit Analysis

- Fourier series of a periodic function

Given a periodic function

can be represented by the infinite series as

( ) ( ) ( )Ttftftf +=:

tf

1000

0201

02010

sincos

2sinsin

2coscos

nnn tnwbtnwaa

twbtwb

twatwaatf

?

?

?0

n

n

b

a

a

Ch8 Fourier Circuit Analysis

8.1 Fourier Series8.1 Fourier Series

Example 12.1

3.01.0,0

1.01.0,5cos

t

ttVtV m

mVa 0

Ch8 Fourier Circuit Analysis

8.1 Fourier Series8.1 Fourier Series

Given a periodic function

It is knowing

21mVa

1n

12cos2

2

n

nVa mn

00b

It can be seen , we can evaluate 50

3

22

mVa 03a 15

24

mVa 05a 35

26

mVa

tV

tV

tV

tVV

tV

m

mmmm

30cos35

2

20cos15

210cos

3

25cos

2

-Review of some trigonometry integral observations

(a)

(c)

(d)

0sin0 0 dttnwT

0cos0 0 dttnwT

(b)

(e)

0sinsin2

1cossin

0 000 00 dttwnktwnkdttnwtkwTT

nkif

nkifT

dttwnktwnk

dttnwtkw

T

T

,0

,2

coscos2

1

sinsin

0 00

0 00

nkif

nkifT

dttwnktwnk

dttnwtkw

T

T

,0

,2

coscos2

1

cossin

0 00

0 00

Ch8 Fourier Circuit Analysis

8.1 Fourier Series8.1 Fourier Series

-Evaluations of nn baa ,,0

0a

T

nnn

TTdttnwbtnwadtadttf

01

000 00sincos

Based on (a) (b)

0sincos0

100

T

nnn dttnwbtnwa

Tdttf

Ta

00

1

0a ( is also called the DC component of ) tf

Ch8 Fourier Circuit Analysis

8.1 Fourier Series8.1 Fourier Series

na

Based on (b)

T

nn

T

nn

TT

dttkwtnwb

dttkwtnwadttkwatdtkwtf

01

00

01

000 000 0

cossin

coscoscoscos

T

n

n

T

nn

tdtnwtfT

a

aT

dttkwtnwa

0 0

01

00

cos2

2coscos

0cossin0

100

T

nn dttkwtnwb

0cos0 00 dttkwaT

Based on (c)

Based on (e)

Ch8 Fourier Circuit Analysis

8.1 Fourier Series8.1 Fourier Series

When k=n

nb

Based on (a)

T

nn

T

nn

TT

dttkwtnwb

dttkwtnwadttkwatdtkwtf

01

00

01

000 000 0

sinsin

sincossinsin

T

n

n

T

nn

tdtnwtfT

b

bT

dttkwtnwb

0 0

01

00

sin2

2sinsin

0sincos0

100

T

nn dttkwtnwa

0sin0 00 dttkwaT

Based on (c)

Based on (d)

Ch8 Fourier Circuit Analysis

8.1 Fourier Series8.1 Fourier Series

When k=n

nnnnn tnwbatnwbtnwa 022

00 cossincos

)(Hzf

22nnn bav

Harmonic

amplitude

7v

07 f06 f05 f04 f03 f02 f0f

5v6v4v3v2v1v

Phase spectrum

0f 02 f 03 f 04 f 05 f 06 f 07 f)(Hzf

n

n

nn a

b 1tan

Ch8 Fourier Circuit Analysis

8.1 Fourier Series8.1 Fourier Series

- Depending on the symmetry (odd or even), the Fourier series can be further simplified.

Even Symmetry

Observation: rotate the function curve along axis, the curve will overlap with the curve on the other half of .

Example :

Odd Symmetry

Observation: rotate the function curve along the axis, then along the axis, the curve will overlap with the curve on the other half .

Example :

tftf

tftf

wttf cos

tf tf

wttf sin

t tf

Ch8 Fourier Circuit Analysis

8.2 Use of Symmetry8.2 Use of Symmetry

Symmetry Algebra

(a) odd func. =odd func. × even func.

Example:

(b) even func. =odd func. × odd func.

Example:

(c) even func. =even func. × even func.

Example:

tωtωtω

cossin=2

2sin

tωtωtω

cos×cos=2

1+2cos

tωtωtω

sin×sin=2

2cos-1

Ch8 Fourier Circuit Analysis

8.2 Use of Symmetry8.2 Use of Symmetry

(d) even func. =const. +∑ even func. (No odd func.)

Example:

(e) odd func. =∑odd func.

Example:

tωtωtω 22 sin2-1=1-cos2=2cos

( ) φtωφtωφtω sincos+cossin=+sin

odd func. odd func.

Ch8 Fourier Circuit Analysis

8.2 Use of Symmetry8.2 Use of Symmetry

Apply the symmetry algebra to analyze the Fourier series.

If is an even function

If is an odd function

1

000 sincosn

nn tnwbtnwaatf

100 cos

0

nn

n

tnwaatf

b

10

0

sin

00

nn

n

tnwbtf

aa

tf

tf

Ch8 Fourier Circuit Analysis

8.2 Use of Symmetry8.2 Use of Symmetry

Half-wave symmetry f(t) = -f(t - ) or f(t) = -f(t + )2

T2

T

Ch8 Fourier Circuit Analysis

8.2 Use of Symmetry8.2 Use of Symmetry

evenisn

oddisntdtnwtfTb

evenisn

oddisntdtnwtfTa

T

n

T

n

0

sin4

0

cos4

0

0

0

0

Fourier series:

Ch8 Fourier Circuit Analysis

8.2 Use of Symmetry8.2 Use of Symmetry