Higher Engineering Mathematics

660 FOURIER SERIES

−π −π/ 2 0 π/ 2 π

−π

− 4

π

4

f(x)

P 1

f(x)

x

−π −π/ 2 0 π/ 2 π x

P 2

P 1

f(x) f(x)

π

−π

4 /3 sin 3x

f(x)

π

−π/ 2

−π^0 π/^2 π x

4 /5 sin 5x

P 2

P 3

f(x)

(c)

(b)

(a)

−π

Figure 69.4

Problem 4. Determine the Fourier series for

the full wave rectified sine wave i=5 sin

θ

2

shown in Fig. 69.5.

i = 5 sin /2 θ

5

− 2 π 0 2 π 4 π θ

i

Figure 69.5

i=5 sin

θ

2

is a periodic function of period 2π.

Thus

i=f(θ)=a 0 +

∑∞

n= 1

(ancosnθ+bnsinnθ)

In this case it is better to take the range 0 to 2π

instead of−πto+πsince the waveform is continu-

ous between 0 and 2π.

a 0 =

1

2 π

∫ 2 π

0

f(θ)dθ=

1

2 π

∫ 2 π

0

5 sin

θ

2

dθ

=

5

2 π

[

−2 cos

θ

2

] 2 π

0

=

5

π

[(

−cos

2 π

2

)

−(−cos 0)

]

=

5

π

[(1)−(−1)]=

10

π

an=

1

π

∫ 2 π

0

5 sin

θ

2

cosnθdθ

=

5

π

∫ 2 π

0

1

2

{

sin

(

θ

2

+nθ

)

+sin

(

θ

2

−nθ

)}

dθ

(see Chapter 40, page 400)

=

5

2 π

[

−cos

[

θ

( 1

2 +n

)]

( 1

2 +n

)

−

cos

[

θ

( 1

2 −n

)]

( 1

2 −n

)

] 2 π

0