Power Electronics- Lecture 4

Yongsug Suh

Environmental-friendly Energy Conversion Laboratory, CBNU

ContentsFourier series analysis

Example ⇒ square waveform, triangular waveform

Symmetry in Fourier series analysis

Lecture 4 – Fourier Series Analysis

Environmental-friendly Energy Conversion Laboratory, CBNU

Fourier Analysis of Repetitive Waveform

∫

∫

∫

−

−

==

==

=

π

π

π

π

π

π

..1,2,3,4...h 1

..1,2,3,4...h 1

T1F 0

)()sin()(

)()cos()(

)(

wtdhwttfb

wtdhwttfa

dttf

h

h

T

Fourier analysis of repetitive waveforms

⇒ w; repeating frequency of

∑∞

=

++=1h

0F )]sin()cos([ hwtbhwta f(t) hh

)T2( π

= f(t)

Environmental-friendly Energy Conversion Laboratory, CBNU

2DT

⋅⋅⋅⋅⋅+⋅⋅+⋅+⋅=∴

==

===

⋅=

∫

∫∫

−

−

)2cos()2sin(212)cos()sin(2)(

0)()sin(

)sin(2)()cos(2)()cos(0

wtDAwtDADAtf

wtdhwtAb

hDh

AwtdhwtAwtdhwtAa

AD

D

Dh

DD

Dh

ππ

ππ

π

ππππ

π

π

ππ

π

1

1

F 0

1=h

Example– Fourier Series of Square Waveform

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Simplified Harmonic Terms & Total RMS Value

∑∞

=

−

⎟⎠⎞

⎜⎝⎛+=

−=++=

+=

1

2,2

0

1

2)(

)(tan],cos[

)sin()cos(

h

pkh

h

hhhhh

hhh

FFtf

abhwtba

hwtbhwta

)()(

F

22 φφ

Total RMS of

pkhF ,

Cosine and sine terms of same frequency can be merged into a single frequency term

Total RMS of f(t) becomes

Environmental-friendly Energy Conversion Laboratory, CBNU

2T

[ ]

⋅⋅⋅⋅−−−−=∴

=>=⎪⎩

⎪⎨⎧

=

=−

=−=

=

wtAwtAwtAAtf

bhA

hhAa

AF

hh

5cos5

43cos3

4cos42

)(

04

1)cos(2

2

22222

2222

0

πππ

πππ

function even evenh; 0

oddh ;

Example– Fourier Series of Triangular Waveform

Environmental-friendly Energy Conversion Laboratory, CBNU

Symmetry in Fourier series

==

h

h

ab

Even function

Odd function

Half-wave even terms = 0

Even + Half-wave 0 + even term=0

Odd + Half-wave 0 + even term=0

0 0

====

h

h

a-f(-t)f(t)bf(-t) f(t)

)T-f(tf(t)2

+=

Symmetry in the repetitive waveform can greatly simplify the Fourier series calculation

MISSION

The Laboratory performs the research and educational activity in the area of electrical system that can be applied in environmental-friendly & sustainable energy conversion process.