fundamentals of engineering supplied-reference handbook - Ventech!

FOURIER SERIES
Every periodic function f(t) which has the period
T = 2 π/ ω 0 and has certain continuity conditions can be
represented by a series plus a constant
∞
0 0 0
n=
1
f (t)
1
0
T
2
T
∑ [ n n ]
f () t = a /2+ a cos( nω t) + b sin( nω t)
The above holds if f(t) has a continuous derivative f′(t) for
all t. It should be noted that the various sinusoids present in
the series are orthogonal on the interval 0 to T and as a
result the coefficients are given by
0
T
0
a = (1/ T) f( t) dt
T
0
a = (2 / T) f( t)cos( nω t) dt n=
1, 2, �
n
T
0
b = (2 / T) f( t)sin( nω t) dt n=
1, 2, �
n
∫
∫
∫
T
2
0
0
The constants an and bn are the Fourier coefficients of f(t)
for the interval 0 to T and the corresponding series is
called the Fourier series of f(t) over the same interval.
The integrals have the same value when evaluated over
any interval of length T.
If a Fourier series representing a periodic function is
truncated after term n = N the mean square value
V o
t
V o
2
F N of
the truncated series is given by the Parseval relation.
This relation says that the mean-square value is the sum
of the mean-square values of the Fourier components, or
N
2
2 2
( a / ) + ( 1/
2)
( a b )
2
N = 0 ∑
n=
1
F 2 n + n
and the RMS value is then defined to be the square root
of this quantity or FN.
Three useful and common Fourier series forms are defined
in terms of the following graphs (with ωo = 2π/T).
Given:
then
f
1
() ( ) ( ) n−1
2
t −1
( 4V
nπ)
cos ( n t)
= ∞
∑ o ω
n=
1
( n odd)
o
173
Given:
then
Given:
then
f
f
f
3
3
3
ELECTRICAL AND COMPUTER ENGINEERING (continued)
f (t)
2
T 0
f
f
2
2
() t
() t
2
T
2T
V o
τ ∞ sin(
nπτ
T )
n=
1 ( nπτ
T )
( nπτ
T ) jnωot
e
( nπτ
T )
Voτ
2Vo
= +
T T
V τ ∞
o sin
= ∑
T n=
−∞
∑ cos ω
t
( n t)
f (t) = "a train of impulses with weights A"
3
T 0 T 2T 3T
∞
() t = Aδ(
t − nT )
∑
n=
−∞
∞
() t = ( A T ) + ( 2A
T ) cos
( nω
t)
∞
() t = ( A T )∑
e
jnωot
n=
−∞
∑
n=
1
o
t
o