Fourier series , electrical application,Md nazmul islam ,

1. Welcome to Presentation
Fourier Series and it’s Applications
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2. A Fourier series is an expansion of a periodic function
in terms of an infinite sum of sines and cosines.
The general Fourier series of a periodic function f(x) in the interval
[-L L] is define as
𝑓 𝑥 =
1
2
𝑎0 +
𝑛=1
∞
𝑎 𝑛 cos
𝑛𝜋𝑥
𝐿
+ 𝑏 𝑛 sin
𝑛𝜋𝑥
𝐿
Where, 𝑎0 =
1
𝐿 −𝐿
𝐿
𝑓 𝑥 𝑑𝑥
𝑎n =
1
𝐿
−𝐿
𝐿
𝑓 𝑥 𝑠𝑖𝑛
𝑛𝜋𝑥
𝐿
𝑑𝑥
𝑏n =
1
𝐿
−𝐿
𝐿
𝑓 𝑥 𝑐𝑜𝑠
𝑛𝜋𝑥
𝐿
𝑑𝑥
What is Fourier series ?
Joseph Fourier
Heat transfers in a metal plate
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3. 𝐿ⅇ𝑡 𝑓 𝑥 =
1
2
𝑎0 + 𝑎1cosx + 𝑎2cos2x + 𝑎3cos3x + ….. + 𝑎 𝑛cosnx + 𝑏1sin x + 𝑏2sin2x
+ 𝑏3sin3x + …. +𝑏 𝑛sinnx ... … … … … … … … … … … … (1)
𝐻𝑒𝑛𝑐𝑒, 𝑎0 =
1
𝜋
0
2𝜋
𝑓 𝑥 𝑑𝑥
=
1
𝜋
0
2𝜋
𝑥 𝑑𝑥
=
1
𝜋
𝑥2
2 0
2𝜋
Example : Find the Fourier series representing f(x)= x, 0<x<2π and
sketch its graph from x =-4π to x = 4π
Solution:
= 2𝜋
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4. 𝑎 𝑛 =
1
𝜋 0
2𝜋
x cos nx d𝑥
=
𝑥 sin 𝑛𝑥
𝑛
+
cos 𝑛𝑥
𝑛2
0
2𝜋
= 0 +
cos 2n𝜋
𝑛2 − 0 −
cos 00
𝑛2
= 1 − 1
= 0
And ,
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5. 𝑏 𝑛 =
1
𝜋 0
2𝜋
x sin nx d𝑥
=
1
𝜋
−
𝑥 cos 𝑛𝑥
𝑛
+
sin 𝑛𝑥
𝑛2
0
2𝜋
=
1
𝜋
( −
2𝜋 cos 2n𝜋
𝑛
+ 0)
=−
2
𝑛
Also,
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6. Putting the values of 𝑎0 , 𝑎 𝑛 , 𝑏 𝑛 in (1) , we get
x = π -2 [ sin x +
1
2
𝑠𝑖𝑛2𝑥 +
1
3
𝑠𝑖𝑛3𝑥 + … ]
0 2π 4π-2π-4π
f(x)
The graph of this function is
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7. Application
of Fourier
Series
Expanding
periodic
function
outside of
closed interval
Noise
cancellation
Expansion of
oscillating
function
Wave
simplification
Heat
distribution
mapping
Signal
Processing
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8. Electrical Circuits and Fourier Series
An equivalent circuit.
Each voltage source
is a term of the
Fourier series of vs(f).
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9. Signal Processing &The Fourier
Series
Sinusoidal Inputs
Sinusoidal Inputs
Non-sinusoidal Inputs
How to Lighted the bulb
at non-sinusoidal input ?
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10. CONTACT ME : NAZMULISLAM1581@GMAIL.COM

11. Thanks Everybody
“ Thⅇ Important thing about a problⅇm is not its solution , but
thⅇ strⅇngth wⅇ gain in finding thⅇ solution ” - Anonymous
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