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6.5.1 
rmS Value of Sinusoidal Waveforms
Let v(t) 
=
V
m
sin
w
t V. We find its rms value as,
V
V
t dt
V
t dt
V
V
rms
m
m
m
=
=
−
=
=
∫
∫
2
2
1
2
1
2
1
2
2
0
2
2
2
0
2
2
p
w
w
p
w
w
p
w
p
w
sin
[
cos
]
m
m
2
.
Thus, a sinusoidal voltage is only as effective as a DC voltage that is 70.7% of its peak value as far 
as its heating capability is concerned.
Let v(t) 
=
V
m
sin
w
t V and i(t) 
=
I
m
sin(
w
t 
+
q
) A be the voltage across and current through an 
electrical element as per passive sign convention, respectively. Then we know that the average power 
delivered to the element is given by 0.5V
m
I
m
cos
q
W. Now we can express this power equation in terms 
of rms values of voltage and current.
P V I
=
rms rms
W
cos
q
(6.5-2)
Two other measures are useful in the context of periodic waveforms. They are cycle average and 
cycle average of absolute value of the waveform. We denote them by V
cav
and V
caav
, respectively.
V
T
v t dt
V
T
v t dt
T
T
cav
caav
and
=
=
∫
∫
1
1
0
0
( )
| ( ) |
(6.5-3)
Note that both in the definitions of rms value in Eqn. 6.5-1 and average values in Eqn. 6.5-3, we 
are free to carry out the integration over any interval of width equal to the period of the waveform. It 
does not have to be between 0 to T.
A pure alternating waveform will have a cycle average of zero.

---

Effective Value (RMS Value) of Periodic Waveforms 
6.27
The cycle average of absolute value of a pure alternating waveform that has identical positive 
half-cycle width and negative half-cycle width will be same as the average of positive half-cycle over 
0.5T. Hence, the cycle average of absolute value is also called half-cycle average in the case of such 
a waveform. A sinusoidal waveform is one such waveform. The V
caav
value for a sinusoidal waveform 
v(t) 
=
V
m
sin
w
t is obtained as follows:
V
T
v t dt
V
t dt
V
t
V
T
caav
m
m
m
=
=
=
=
∫
∫
2
2
0
0 5
0
0
( )
sin
cos
.
w
p
w
p
w
p
p
w
p
w
The ratio between the rms value and the half-cycle average value of a periodic waveform with 
equal half-cycle widths and zero cycle average (i.e., a pure alternating waveform) is defined as its form 
factor. Therefore, the form factor of a pure sinusoidal waveform 
=
p
2 2
1 11
≈
.
.
The ratio between peak value and rms value of an alternating waveform with equal half-cycle 
widths and zero cycle average is defined as its crest factor. The crest factor of a pure sinusoidal 
waveform is 2.