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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
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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
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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
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8.1 Fourier Series8.1 Fourier SeriesExample Fundamental: v1 = 2cosw0t
v3a = cos3w0t v3b = 1.5cos3w0t
v3c = sin3w0t
Ch8 Fourier Circuit Analysis
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- 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
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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
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-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
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-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
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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
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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
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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
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- 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
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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
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(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
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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
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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
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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