Higher Engineering Mathematics, Sixth Edition

614 Higher Engineering Mathematics

x

f(x)

f(x)

P 1

0

(a)

(b)

(c)

4

24

2  2 /2

2 

/2 



x

f(x) f(x)

4/3 sin 3x

2  2 /2 0

2 

/2 



x

f(x) f(x)

4/5 sin 5x

P 2

P 2

P 1

P 3

0

2 /2

2 

2 

/2 



Figure 66.4

Whenx=

π

2

, f(x)= 1 ,

sinx=sin

π

2

= 1 ,

sin3x=sin

3 π

2

=− 1 ,

sin5x=sin

5 π

2

= 1 ,and so on.

Hence 1=

4

π

[

1 +

1

3

(− 1 )+

1

5

( 1 )+

1

7

(− 1 )+···

]

i.e.

π

4

= 1 −

1

3

+

1

5

−

1

7

+···

Problem 4. Determine the Fourier series for

the full wave rectified sine wavei=5sin

θ

2

shown

in Fig. 66.5.

0

5

22  2  4 

i i 5 5 sin /2



Figure 66.5

i=5sin

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 continuous

between 0 and 2π.

a 0 =

1

2 π

∫ 2 π

0

f(θ )dθ=

1

2 π

∫ 2 π

0

5sin

θ

2

dθ

=

5

2 π

[

−2cos

θ

2

] 2 π

0

=

5

π

[(

−cos

2 π

2

)

−(−cos0)

]

=

5

π

[( 1 )−(− 1 )]=

10

π

an=

1

π

∫ 2 π

0

5sin

θ

2

cosnθdθ

=

5

π

∫ 2 π

0

1

2

{

sin

(

θ

2

+nθ

)

+sin

(

θ

2

−nθ

)}

dθ

(see Chapter 40,page 401)

=

5

2 π

[

−cos

[

θ

( 1

2 +n

)]

( 1

2 +n

)

−

cos

[

θ

( 1

2 −n

)]

( 1

2 −n

)

] 2 π

0