FOURIER SERIES CHAPTER 5. TOPIC: Fourier series definition Fourier coefficients The effect of...
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Transcript of FOURIER SERIES CHAPTER 5. TOPIC: Fourier series definition Fourier coefficients The effect of...
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.
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
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 coefficients
• Integral relationship to get Fourier coefficients
T
dttn0 0 0sin
T
o dttn0
0cos
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
THE EFFECT OF SYMMETRY ON FOURIER COEFFICIENTS
• Even symmetry
• Odd symmetry
• Half-wave symmetry
• Quarter-wave symmetry
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.
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
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(
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
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
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
Step 2: find response
• DC component: set n=0 atau ω=0• Time domain: inductor = short
circuit
capacitor = open circuit
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).
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
Solution (b):
• Voltage vi for first three harmonic:
oi
oi
oi
tsinV
tsinV
tsinV
048548
080380
0240240
5
3
1
Voltage vo for first three harmonic:
oo
o
ooo
ooo
..v
..v
..v
040891960048
028253160080
06816970700240
5
3
1