228 Practical MATLAB® Applications for Engineers
R.3.10 The RMS or effective value of the waveform x(t) is given by
X
T
RMS xtdt
T
(^12)
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
RMSm Xm
2
0 707.
Because of its importance this relation is verifi ed as follows:
Let x(t) = Xm sin(ωt) then
X
T
RMS Xtdtm
T
(^12)
0
∫[sin()]
X
T
RMS Xtdtm
T
11
2
1
2
(^22)
0
cos( )
∫
X
T
RMS Xdtm tdt
T T
11
2
1
2
(^22)
0 0
∫∫ cos( )
Note that ∫
0
T
(1/2) cos(2ωt) dt = 0, then
X
T
Xdt
X
T
RMS m t
T
^11 m T
22
2
0
2
∫ * 0
X
XT
T
XX
RMSmmmXm
22
222
0 707
*
*.
If a voltage (or current) consists of a DC and AC components, such as
vt()A Vmsin( )t
then its VRMS value is given by
V
T
RMS AVm tdt
T
(^12)
0
∫ [sin()]
V
T
RMS AAVm tVm tdt
T
(^1222)
0
∫ [sin()sin()]
V
T
RMS A dt AV t dt V t dt
T
mm
T T
(^12)
0
22
∫∫∫ 0 0
sin() sin ( )