EE 271
Introduction to Average and RMS Values
Spring 2004
This document defines the average and RMS value of a signal, which are measured by the digital
multimeter (DMM) and the digital oscilloscope. The differences between the way the oscilloscope
and the DMM measure the RMS value is also discussed.
Part I: Average Value
The average value (or D.C. component or the “mean” value) of a voltage (or current) is defined as
shown below.
Average Value = lim
T →∞
1
T
T
2
∫ v(t ) dt
−T
2
Some signals, called periodic signals, repeat every so often. Let's call the time between repetitions
T, then for periodic signals v (t ) = v(t + T ) for all t. An example of a periodic signal is
v(t ) = A cos(ωt ) + B which is graphed below. The value B is a constant called the dc value or the
offset value of the signal, and ω is the angular frequency in rad/sec. Often in practice, the
frequency is measured in Hz (cycles/sec or Hertz), where ω = 2πf and f is frequency in Hz. The
1 2π
period is related to the frequency as follows by T = =
.
f
ω
v(t)
B+A
B
B-A
T
Figure 1. v(t ) = A cos(ωt ) + B
T
t
For periodic signals, the average value (or the mean value) can be computed over just one period of
the signal, instead of over all time, as follows:
T
Average Value =
1
v(t) dt where T is the period of the signal.
T ∫0
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The digital multimeter (DMM) measures the average value of voltages and currents when it is
in the D.C. mode.
Example: Let v(t ) = A cos(ωt ) + B
T
1
1
Average Value = ∫ v(t) dt =
T 0
T
T
∫ ( A cos(ωt ) + B)dt = • • • = B
0
So if v(t ) = A cos(ωt ) + B and the DMM is set to D.C. mode, it will read the value of B.
Part II: RMS Value
The root-mean-square value (or RMS value) of a voltage (or current) is defined as shown below.
V RMS


1
=  lim
T →∞ T


1
2

2
∫−T v (t )dt 
2

T
2
For periodic signals, the RMS value can be computed over just one period of the signal, instead of
over all time, as follows:
1
V RMS
1T
2
=  ∫ v 2 (t )dt  where T is the period of the signal.
T 0

When the digital multimeter (DMM) is in A.C. mode, it first eliminates the D.C. component,
and then measures the RMS value.
Example: Let v(t ) = A cos(ωt ) + B
First the DMM set to AC mode eliminates the D.C. component: v AC (t ) = A cos(ωt )
1
1
2
1 T
2  1 T
A
2
VRMS =  ∫ v 2AC (t )dt  =  ∫ [A cos(ωt )] dt  dt = • • • =
2

T 0

T 0
So if v(t ) = A cos(ωt ) + B and the DMM is set to A.C. mode, it will read the RMS value to
be
A
.
2
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Part III: Digital Oscilloscope
The digital oscilloscope can also be used to measure the average and the RMS values of a signal.
However, it is important to note that the oscilloscope does not compute the RMS value in the same
way as the DMM if the oscilloscope is set to D.C. coupling. Unlike the RMS value measured
using the DMM set to AC mode, when the digital oscilloscope is set to D.C. coupling to
measure the RMS value of the signal, the oscilloscope does NOT eliminate the D.C.
component before computing the RMS value.
Example: Let v(t ) = A cos(ωt ) + B
If the oscilloscope is set to A.C. coupling, it eliminates the D.C. component of the signal
(same as the DMM): v AC (t ) = A cos(ωt )
1
1
1 T
2  1 T
2
A
2
VRMS =  ∫ v 2AC (t )dt  =  ∫ [A cos(ωt )] dt  dt = • • • =
2
T 0

T 0

So if the signal under consideration is v(t ) = A cos(ωt ) + B and the oscilloscope is set to
A
A.C. coupling, it will read the RMS value to be equal to
(same as the RMS reading
2
using the DMM in AC mode).
However, if the oscilloscope is set to D.C. coupling, since the oscilloscope does NOT
eliminate the D.C. component of the signal, the RMS value measured by the oscilloscope is
as follows:
1
1
1
1T
2  1 T
2
2
 A2
2
VRMS =  ∫ v 2 (t) dt  =  ∫ [A cos(ωt ) + B ] dt  dt = • • • = 
+ B 2 

 2
T 0

T 0

So if v(t ) = A cos(ωt ) + B and the oscilloscope is set to D.C. coupling, it will read an RMS
1
 A2
2
+ B 2  .
value of 
 2

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