Root-Mean-Square Voltages
When a time-varying voltage V(t) is applied to a resistor, the power dissipated in it,
V2/R is also time-dependent. The average power is the energy consumed in a cycle
over the time of one period. If we were to use a steady d.c. voltage to give the
same average power, then this “d.c. equivalent steady voltage” is called the root
mean square voltage of the a.c.
I. Complete sinusoidal waveform:
A sinusoidal a.c. is applied to a resistor
V = V0 sin t
V
2
The instantaneous power dissipated in R is P =
V 2 V0

sin2t
R
R
0
T
The average power ,
P
2
0
1 T
1V
Pdt

T 0
T R

T
0
sin 2ω t 
2
0
1V

2 R
(
V0
2
R
)2
………..(1)
The power of a steady d.c. voltage Vdc is
Vdc2
P
R
………(2)
If (2) is exactly equal to (1), then
V
0 2
Vdc2 ( 2 )

R
R
This implies
Vdc 
.
V0
2
This “d.c. equivalent steady voltage” Vdc is called the root-mean-square voltage
(Vrms) of the a.c. because it is calculated from the procedures “root”, “mean”
and “square”, but in the reverse order.
1. Squaring the time-depending a.c. voltage.
2. Taking the mean of the squared V over a cycle.
3. Taking the square root of the mean.
Vrms  V 2
2
Vrms
 I 2rms R  Vrms I rms
In a pure R, P 
R
II. Half sinusoidal waveform:
V
V0
Vrms 
V0
2
III. Square waveform:
V
Va
-Vb
T
2
t
T
Vrms =
Va2  Vb2
2