Fourier Series Representation

Fourier Series Representation

Fourier series Fourier series is a technique for expressing a

periodic functions in terms of sinusoids. Once the source function is expressed in terms of sinusoids, we can apply the phasor method to analyze circuits.

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Trigonometric Fourier Series  According to Fourier theorem, any practical periodic function of

frequency ωo can be expressed as an infinite sum of sine or cosine functions that are integral multiples of ωo . Thus g(t) can be expressed as

g (t ) = a0 + a1 cosω 0t + a2 cos2ω 0t +  + an cosnω 0t +  + b1 cosω 0t + b2 cos2ω 0t +  + bn cosnω 0t +  ∞

g (t ) = a 0 + ∑ (a n cos nω 0 t + bn sin nω 0 t ) n =1

(t 0 < t < t 0 + T )

This Equation is the trigonometric Fourier series representation of g(t) over an interval (t0, t0 +To). Where ωo = 2 π fo= 2 π / To is called fundamental frequency in rad / sec. 3

Trigonometric Fourier Series A major task in Fourier series is the determination of

the Fourier coefficients ao,,an & bn..

an

∫ =

( t 0 +T )

t0

∫

g (t ) cos nω 0 t dt

( t 0 +T )

t0

( t 0 +T )

bn

∫ =

t0

cos 2 nω 0 t dt

g (t ) sin nω0 t dt

( t 0 +T )

∫

t0

sin 2 nω0 t dt

If we put n=0 in above Eq, we get

1 a0 = T

∫

( t 0 +T )

t0

g (t ) dt 4

Trigonometric Fourier Series We also have

∫

( t 0 +T )

t0

cos nω 0t dt = ∫ 2

( t 0 +T )

t0

T sin nω 0t dt = 2 2

2 ( t 0 +T ) an = ∫ g (t ) cos nω 0 t dt t T 0

2 ( t 0 +T ) bn = ∫ g (t ) sin nω 0 t dt T t0

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Trigonometric Fourier Series Example ƒ(t)

A

We shall now expand a

function ƒ(t) shown in figure2.1(a,b,c). Using trigonometric Fourier series over the interval (0,1). It is evident that

0

1

ƒ1(t) A

0

ƒ(t) =At, (0
t

1

t

ƒ2(t) A

Time interval T = 1 0

1

Figure 2.1(a,b,c)

t

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Trigonometric Fourier since 2π ω = = 2π Series Example T o

thus ∞

f (t ) = ao + ∑ (an cos nωo t + bn sin nωo t )

→11

n =1

f (t ) = ao + a1 cos 2πt + a2 cos 4πt + ......... + an cos 2nπt + ......... + b1 sin 2πt + b2 sin 4πt + ..... + bn sin 2nπt + ..... 1 ao = T

t 2  1 A ∫0 Atdt = A 2  0 = 2 1

→12

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Trigonometric Fourier 2 a = ∫ A cos n( 2π ) tdt Series T Example 1

n

0

2 A  t sin 2πnt 1 1 sin 2πnt  an = − ∫ 1. dt   T  2πn 0 0 2πn   t sin 2πnt cos 2πnt  1 an = 2 A + 2  2 π n ( 2πn )  0  1 A an = [ cos 2πnt + 2πnt sin 2πnt ] 2 2 2π n 0 A an = [ (1 − 1) + (1.0 − 0.1) ] 2 2 2π n

a n =0

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Trigonometric Fourier Series Example

2 1 2 A  ( − cos 2πnt ) 1 1 ( − cos 2πnt )  bn = ∫ At sin 2πntdt = t −∫ dt   0 T 0 1  2πn 2πn 0 

 ( − t cos 2πnt ) ( sin 2πnt )  1 2A bn = 2 A − =− 2  2πn ( 2πn )  0 2πn 

A bn = − πn

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Trigonometric Fourier Series Example Hence f(t) can be expressed as

A ∞ f (t ) = + ∑ bn sin 2π nt 2 n =1  A − A f (t ) = + ∑  nπ 2 n =1 ∞

  sin 2π nt 

1 1 ∞ 1  f (t ) = A − ∑ sin 2π nt   2 π n =1 n 

(0
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DIRICHLET CONDITIONS A function that can be represented by a Fourier series must meet certain requirements, because the infinite series may or may not converge. These conditions on g(t) to yield a convergent Fourier series are as follows: g(t) is single valued every where. g(t) has a finite number of finite discontinuities in any one period. g(t) has a finite number of maxima and minima in any one period. The integral ∫ |g(t)| dt < ∞ for any t0 These conditions are called Dirichlet conditions. 11