Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

6.6.1 
rmS Value of a composite Waveform
We will employ the superposition principle for average power to arrive at an expression for the rms 
value (i.e., the effective value) of a periodic waveform that is the sum of many sinusoidal periodic 
waveforms of different frequencies. 
Let v(t) 
=
v
1
(t) 
+
v
2
(t) 
+ 
… 
+
v
n
(t) be a composite waveform comprising n distinct frequency 
sinusoidal waveforms. Notice the word distinct used in the last sentence. No two components in the 
sum can have same frequency value. If there are two sources of same frequency, they have to be 
combined and treated as a single term.
Now imagine that we apply this signal as a voltage across 1
W
resistor. The average power delivered 
to the resistor is found by applying power superposition principle as the sum of average power 
delivered to the same resistor when each source is acting alone. But that will be nothing but the square 
of rms value of each component waveform. The rms value of v(t) is obtained by finding the square root 
of average power delivered to the 1
W
resistor. Therefore,
V
V
V
V
V
V
V
V
rms
1rms
2rms
n rms
rms
1rms
2rms
n rms
2
2
2
2
2
2
2
=
+
+ +
∴
=
+
+ +
. 
(6.6-2)
The effective value of a waveform that is the sum of many sinusoidal waveforms with 
distinct frequencies (including zero frequency) is obtained by taking the square root of 
sum of squares of rms values of individual components.
example: 6.6-1
The voltage applied across an electrical load circuit is v(t) 
=
10 
+
50 sin100
p
 t 
+
30 sin (150
p
 t – 30
°
) 
V and the current delivered to the load circuit is seen to be i(t) 
=
2.5 
+
5 sin (100
p
 t –45
°
)
+
1.5 sin 
(150
p
 t – 80
°
) A. (i) Find the average power delivered to the load circuit, rms value of applied voltage 
and load current. (ii) What is the period over which averaging was carried out in the last step?
Solution
(i) All the three components in voltage waveform have distinct frequencies. Similarly, all the three 
components in current waveform too have distinct frequencies.
Therefore, average power delivered by applying power superposition principle is obtained 
as
P
=
×
+
×
− − ° +
×
− ° − − °
=
+
10 2 5
50
2
5
2
0
45
30
2
1 5
2
30
80
25 1
.
cos(
(
))
.
cos(
(
))
225
45
22 5
50
127 84
cos
. cos
.
° +
° =
W
The rms values of voltage and current waveforms are obtained by applying Eqn. 6.6-2.
V
rms
V
=
+ 



+ 



=
10
50
2
30
2
42 43
2
2
2
.

---

The Power Superposition Principle 
6.33
I
rms
A
=
+ 



+ 



=
2 5
5
2
1 5
2
4 46
2
2
2
.
.
.
(ii) 
w
p
w
p
w
w
p
p
1
2
1
2
1
2
100
150
100
150
2
3
=
=
=
= =
and
k
k
The lowest values possible for k
1
and k
2
are 2 and 3 respectively. 
Therefore, T 
=
k
1
1
2
2 2
100
0 04
p
w
p
p
=
×
=
.
s
Therefore, the period over which averaging was carried out in calculating average power 
and rms values is 40 ms.