The Study of Root Mean Square (RMS)
Value
Mechanical, Electrical, Electronics Engineering
INTRODUCTION So the RMS value y RMS is given by:
The root mean square value of a quantity is the
T T
square root of the mean value of the squared 1 1
values of the quantity taken over an interval. The
y RMS =
T ∫
y 2 (t )dt =
T ∫
A 2 sin2 ω tdt … (3)
0 0
RMS value of any function y = f (t ) over the range
t = a to t = b can be defined as: We know that cos 2θ = 1 − 2 sin2 θ , hence:
1
1
b
sin2 ω t = (1 − cos 2ω t ) … (4)
RMS value =
b− a∫y 2dt 2
a Substituting (4) into (3), we get, in sequence
One of the principal applications of RMS values is T
1 A2
with alternating currents and voltages. y RMS = ∫ (1 − cos 2ω t )dt
T 2
ROOT MEAN SQUARE (RMS) VALUE 0
The value of an AC voltage is continually T
A2
changing from zero up to the positive peak,
through zero to the negative peak and back to
y RMS =
2T ∫ (1 − cos 2ω t ) dt
0
zero again.
T
A2 sin 2ω t
y RMS = t − 2ω
2T 0
A2 sin 2ω T
y RMS = T −
2T 2ω
A 2 A 2 sin( 2ω T ) … (5)
y RMS = −
2 4ω T
As T → ∞ , the second (oscillatory) term in
equation (5) tends to zero. Hence, we obtain:
Figure-1: Difference between peak and RMS voltage A
y RMS =
Clearly, for most of the time it is less than the 2
peak voltage, so this is not a good measure of its 1
real effect. Instead we use the root mean square ⇒ RMS Value of y = × Peak Value of y
2
voltage ( VRMS ) which is 1 2 ≈ 0.7 of the peak
∴ y RMS = 0.7 × y peak
voltage ( Vpeak ):
The RMS value is the effective value of a varying
VRMS = 0.7 × Vpeak or Vpeak = 1.4 × VRMS voltage or current. It is the equivalent steady DC
Similar equations also apply to the current. (constant) value which gives the same effect. For
They are only true for sine waves (the most example, a lamp connected to a 6V RMS AC
common type of AC) because the factors (here supply will shine with the same brightness when
0.7 and 1.4) take different values for other connected to a steady 6V DC supply. However,
shapes. We can calculate these values as the lamp will be dimmer if connected to a 6V peak
follows: AC supply because the RMS value of this is only
Take a sine wave representing either current or 4.2V (it is equivalent to a steady 4.2V DC).
voltage with peak value A: What do AC meters show? Is it the RMS or
y (t ) = A sin ω t … (1) peak voltage?
AC voltmeters and ammeters show the RMS
where ω = 2π f (rads-1) and f = frequency (Hz). The value of the voltage or current. DC meters also
time average y (T ) over period T (seconds) of the show the RMS value when connected to varying
signal y (t ) is given by: DC provided that the DC is varying quickly; if the
T frequency is less than about 10Hz you will see the
1
y (T ) =
T ∫
y (t )dt … (2) meter reading fluctuating.
0
Value
Mechanical, Electrical, Electronics Engineering
INTRODUCTION So the RMS value y RMS is given by:
The root mean square value of a quantity is the
T T
square root of the mean value of the squared 1 1
values of the quantity taken over an interval. The
y RMS =
T ∫
y 2 (t )dt =
T ∫
A 2 sin2 ω tdt … (3)
0 0
RMS value of any function y = f (t ) over the range
t = a to t = b can be defined as: We know that cos 2θ = 1 − 2 sin2 θ , hence:
1
1
b
sin2 ω t = (1 − cos 2ω t ) … (4)
RMS value =
b− a∫y 2dt 2
a Substituting (4) into (3), we get, in sequence
One of the principal applications of RMS values is T
1 A2
with alternating currents and voltages. y RMS = ∫ (1 − cos 2ω t )dt
T 2
ROOT MEAN SQUARE (RMS) VALUE 0
The value of an AC voltage is continually T
A2
changing from zero up to the positive peak,
through zero to the negative peak and back to
y RMS =
2T ∫ (1 − cos 2ω t ) dt
0
zero again.
T
A2 sin 2ω t
y RMS = t − 2ω
2T 0
A2 sin 2ω T
y RMS = T −
2T 2ω
A 2 A 2 sin( 2ω T ) … (5)
y RMS = −
2 4ω T
As T → ∞ , the second (oscillatory) term in
equation (5) tends to zero. Hence, we obtain:
Figure-1: Difference between peak and RMS voltage A
y RMS =
Clearly, for most of the time it is less than the 2
peak voltage, so this is not a good measure of its 1
real effect. Instead we use the root mean square ⇒ RMS Value of y = × Peak Value of y
2
voltage ( VRMS ) which is 1 2 ≈ 0.7 of the peak
∴ y RMS = 0.7 × y peak
voltage ( Vpeak ):
The RMS value is the effective value of a varying
VRMS = 0.7 × Vpeak or Vpeak = 1.4 × VRMS voltage or current. It is the equivalent steady DC
Similar equations also apply to the current. (constant) value which gives the same effect. For
They are only true for sine waves (the most example, a lamp connected to a 6V RMS AC
common type of AC) because the factors (here supply will shine with the same brightness when
0.7 and 1.4) take different values for other connected to a steady 6V DC supply. However,
shapes. We can calculate these values as the lamp will be dimmer if connected to a 6V peak
follows: AC supply because the RMS value of this is only
Take a sine wave representing either current or 4.2V (it is equivalent to a steady 4.2V DC).
voltage with peak value A: What do AC meters show? Is it the RMS or
y (t ) = A sin ω t … (1) peak voltage?
AC voltmeters and ammeters show the RMS
where ω = 2π f (rads-1) and f = frequency (Hz). The value of the voltage or current. DC meters also
time average y (T ) over period T (seconds) of the show the RMS value when connected to varying
signal y (t ) is given by: DC provided that the DC is varying quickly; if the
T frequency is less than about 10Hz you will see the
1
y (T ) =
T ∫
y (t )dt … (2) meter reading fluctuating.
0
