228
Practical MATLAB
® Applications for Engineers
R.3.10 The RMS or effective value of the waveform x(t) is given by
X
T
x t dt
RMS
T
ϭ
1 2
0
( )
∫
R.3.11 If x(t) is a sinusoidal wave, then the aforementioned relation can be evaluated and
the result is given by
X
X
X
RMS
m
m
ϭ
ϭ
2
0 707
.
Because of its importance this relation is verifi ed as follows:
Let x(t) = X m sin(ωt) then
X
T
X
t d t
RMS
m
T
ϭ
1
2
0
[ sin( )]
∫
X
T
X
t d t
RMS
m
T
ϭ
Ϫ
1
1
2
1
2
2
2
0
cos(
)

 

 
∫
X
T
X dt
t dt
RMS
m
T
T
ϭ
Ϫ
1 1
2
1
2
2
2
0
0
cos(
)
∫
∫








Note that ∫
0
T
(1/2) cos(2ωt) dt = 0, then
X
T
X dt
X
T
t
RMS
m
T
m
T
ϭ
ϭ
1 1
2
2
2
0
2
0
∫
*
X
X T
T
X
X
X
RMS
m
m
m
m
ϭ
ϭ
ϭ
ϭ
2
2
2
2
2
0 707
*
* .
If a voltage (or current) consists of a DC and AC components, such as
v t
A V
t
m
( )
sin( )
ϭ ϩ
then its V RMS value is given by
V
T
A V
t dt
RMS
m
T
ϭ
1
2
0
[
s i n ( ) ]
ϩ
∫
V
T
A
AV
t V
t dt
RMS
m
m
T
ϭ
1
2
2
2
0
[
s i n ( )
s i n( ) ]
ϩ
ϩ
∫
V
T
A dt
AV
t dt
V
t dt
RMS
T
m
m
T
T
ϭ
1
2
0
2
2
0
0
∫
∫
∫

 

 
ϩ
ϩ
sin( )
sin ( )
CRC_47760_CH003.indd 228
CRC_47760_CH003.indd 228
7/23/2008 1:27:29 PM
7/23/2008 1:27:29 PM