PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

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

RMS

m 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

RMS



mmmXm

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 ( )