Alternating Current Analysis
227
where j = √
___
−1 (Euler’s identity) and
e
n
n
ϭ ϩ ϩ ϩ
1
2
3
2
3
!
!
, ,
!
(Taylor s series)
’
cos( )
!
!
!
ϭ
ϩ Ϫ
1
2
4
6
2
4
6
−
sin( )
!
!
!
ϭ Ϫ ϩ Ϫ
3
5
7
3
5
7
Recall that cos(θ) = real[e
jθ
] (from Chapter 5 of the book titled Practical MATLAB
®
Basics for Engineers), sin(θ) = imaginary [e
−jθ ], and ͉e
jθ
͉ = √
_____________
co s
2 θ + si n
2
θ = 1.
R.3.6 The average value of a given signal x(t) is given by
X
T
x t dt
AVG
T
ϭ
1
0
( )
∫
assuming that x(t) is a periodic wave with period T.
R.3.7 A DC voltmeter connected across an AC drop will indicate its average value.
R.3.8 The average value of a sinusoidal wave with period T is given by
X
T
X
t d t
AVG
m
T
ϭ
1
0
sin( )
∫
which is also equal to
X
X
t d t
AVG
m
ϭ
ϭ
2
0
0
2
sin( ) ( )
π
∫
R.3.9 The average value of a sinusoidal wave over 1/2 period (T) is given by
X
T
X
t d t
AVG
m
T
ϭ ր
ր
1
2 0
2
sin( )
∫
or by
X
X
t d t
AVG
m
ϭ
sin( ) ( )
0
π
∫
which is equal to
X
X
AVG
m
ϭ
2
0 64
X m ϭ .
CRC_47760_CH003.indd 227
CRC_47760_CH003.indd 227
7/23/2008 1:27:29 PM
7/23/2008 1:27:29 PM