RMS and Average
http://www.electronics-tutorials.ws/accircuits/rms-voltage.html
The RMS Voltage of an AC Waveform
In our tutorial about the AC Waveform we looked briefly at the RMS Voltage value of a sinusoidal
waveform and said that this RMS value gives the same heating effect as an equivalent DC power and in this
tutorial we will expand on this theory a little more by looking at RMS voltages and currents in more detail.
The term “RMS” stands for “Root-Mean-Squared”. Most books define this as the “amount of AC power that
produces the same heating effect as an equivalent DC power”, or something similar along these lines, but an RMS
value is more than just that. The RMS value is the square root of the mean (average) value of the squared function
of the instantaneous values. The symbols used for defining an RMS value are VRMS or IRMS.
The term RMS, ONLY refers to time-varying sinusoidal voltages, currents or complex waveforms were the
magnitude of the waveform changes over time and is not used in DC circuit analysis or calculations were the
magnitude is always constant. When used to compare the equivalent RMS voltage value of an alternating
sinusoidal waveform that supplies the same electrical power to a given load as an equivalent DC circuit, the RMS
value is called the “effective value” and is generally presented as: Veff or Ieff.
In other words, the effective value is an equivalent DC value which tells you how many volts or amps of DC that
a time-varying sinusoidal waveform is equal to in terms of its ability to produce the same power. For example,
the domestic mains supply in the United Kingdom is 240Vac. This value is assumed to indicate an effective value
of “240 Volts RMS”. This means then that the sinusoidal RMS voltage from the wall sockets of a UK home is
capable of producing the same average positive power as 240 volts of steady DC voltage as shown below.
RMS Voltage Equivalent
So how do we calculated the RMS Voltage of a sinusoidal waveform. The RMS voltage of a sinusoid or complex
waveform can be determined by two basic methods.

Graphical Method – which can be used to find the RMS value of any non-sinusoidal time-varying
waveform by drawing a number of mid-ordinates onto the waveform.

Analytical Method – is a mathematical procedure for finding the effective or RMS value of any periodic
voltage or current using calculus.
RMS Voltage Graphical Method
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Whilst the method of calculation is the same for both halves of an AC waveform, for this example we will consider
only the positive half cycle. The effective or RMS value of a waveform can be found with a reasonable amount
of accuracy by taking equally spaced instantaneous values along the waveform.
The positive half of the waveform is divided up into any number of “n” equal portions or mid-ordinates and the
more mid-ordinates that are drawn along the waveform, the more accurate will be the final result. The width of
each mid-ordinate will therefore be no degrees and the height of each mid-ordinate will be equal to the
instantaneous value of the waveform at that time along the x-axis of the waveform.
Graphical Method
Each mid-ordinate value of a waveform (the voltage waveform in this case) is multiplied by itself (squared) and
added to the next. This method gives us the “square” or Squared part of the RMS voltage expression. Next this
squared value is divided by the number of mid-ordinates used to give us the Mean part of the RMS voltage
expression, and in our simple example above the number of mid-ordinates used was twelve (12). Finally, the
square root of the previous result is found to give us the Root part of the RMS voltage.
Then we can define the term used to describe an RMS voltage (VRMS) as being “the square root of the mean of the
square of the mid-ordinates of the voltage waveform” and this is given as:
and for our simple example above, the RMS voltage will be calculated as:
So lets assume that an alternating voltage has a peak voltage (Vpk) of 20 volts and by taking 10 mid-ordinate values
is found to vary over one half cycle as follows:
Voltage
6.2V 11.8V 16.2V 19.0V 20.0V 19.0V 16.2V 11.8V 6.2V
0V
2
Angle
18o
36o
54o
72o
90o
108o
126o
144o
162o
180o
The RMS voltage is therefore calculated as:
Then the RMS Voltage value using the graphical method is given as: 14.14 Volts.
RMS Voltage Analytical Method
The graphical method above is a very good way of finding the effective or RMS voltage, (or current) of an
alternating waveform that is not symmetrical or sinusoidal in nature. In other words the waveform shape resembles
that of a complex waveform. However, when dealing with pure sinusoidal waveforms we can make life a little bit
easier for ourselves by using an analytical or mathematical way of finding the RMS value.
A periodic sinusoidal voltage is constant and can be defined as V(t) = Vm.cos(ωt) with a period of T. Then we can
calculate the root-mean-square (rms) value of a sinusoidal voltage (V(t)) as:
Integrating through with limits taken from 0 to 360 o or “T”, the period gives:
Dividing through further as ω = 2π/T, the complex equation above eventually reduces down too:
RMS Voltage Equation
Then the RMS voltage (VRMS) of a sinusoidal waveform is determined by multiplying the peak voltage value by
0.7071, which is the same as one divided by the square root of two ( 1/√2 ). The RMS voltage, which can also be
referred to as the effective value, depends on the magnitude of the waveform and is not a function of either the
waveforms frequency nor its phase angle.
From the graphical example above, the peak voltage (Vpk) of the waveform was given as 20 Volts. By using the
analytical method just defined we can calculate the RMS voltage as being:
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VRMS = Vpk x 0.7071 = 20 x 0.7071 = 14.14V
Note that this value of 14.14 volts is the same value as for the previous graphical method. Then we can use either
the graphical method of mid-ordinates, or the analytical method of calculation to find the RMS voltage or current
values of a sinusoidal waveform. Note that multiplying the peak or maximum value by the constant 0.7071, ONLY
applies to sinusoidal waveforms. For non-sinusoidal waveforms the graphical method must be used.
RMS Voltage Summary
Then to summarise. When dealing with Alternating Voltages (or currents) we are faced with the problem of how
do we represent a voltage or signal magnitude. One easy way is to use the peak values for the waveform. Another
common method is to use the effective value which is also known by its more common expression of Root Mean
Square or simply the RMS value.
The root mean squared, RMS value of a sinusoid is not the same as the average of all the instantaneous values.
The ratio of the RMS value of voltage to the maximum value of voltage is the same as the ratio of the RMS value
of current to the maximum value of current. Most multi-meters, either voltmeters or ammeters, measure RMS
value assuming a pure sinusoidal waveform. For finding the RMS value of non-sinusoidal waveform a “True
RMS Multimeter” is required.
The RMS value of a sinusoidal waveform gives the same heating effect as a DC current of the same value. That
is if a direct current, I passes through a resistance of R ohms, the DC power consumed by the resistor as heat will
therefore be I2R watts. Then if an alternating current, i = Im.sinθ flows through the same resistance, the AC power
converted into heat will be: I2rms.R watts.
Then when dealing with alternating voltages and currents, they should be treated as RMS values unless otherwise
stated. Therefore an alternating current of 10 amperes will have the same heating effect as a direct current of 10
amperes and a maximum value of 14.14 amperes.
Having now determined the RMS value of an alternating voltage (or current) waveform, in the next tutorial we
will look at calculating the Average value, VAV of an alternating voltage and finally compare the two.
http://www.electronics-tutorials.ws/accircuits/average-voltage.html
The Average Voltage of a Sinusoid
Having looked at the RMS Voltage value of an alternating waveform in a previous tutorial, we can now
look at calculating another value using either the mid-ordinate rule or analytical rule to find a waveforms
“average” or mean voltage.
The process used to find the Average Voltage of an alternating waveform is very similar to that for finding its
RMS value, the difference this time is that the instantaneous values are not squared and we do not find the square
root of the summed mean.
The average voltage (or current) of a periodic waveform whether it is a sine wave, square wave or triangular
waveform is defined as: “the quotient of the area under the waveform with respect to time”. In other words, the
averaging of all the instantaneous values along time axis with time being one full period, (T).
For a periodic waveform, the area above the horizontal axis is positive while the area below the horizontal axis is
negative. The result is that the average or mean value of a symmetrical alternating quantity is zero because the
area above the horizontal axis (the positive half cycle) is the same as the area below the axis (the negative half
cycle) and cancel each other out in the sum of the two areas as a negative cancels a positive producing zero average
voltage.
Then the average or mean value of a symmetrical alternating quantity, such as a sine wave, is the average value
measured over only half a cycle since over a complete cycle the average value is zero regardless of the peak
amplitude.
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The electrical terms Average Voltage and Mean Voltage or or even average current, can be used in both an AC
and DC circuit analysis or calculations. The symbols used for representing an average value are defined as: VAV
or IAV.
Average Voltage Graphical Method
Again consider only the positive half cycle from the previous RMS voltage tutorial. The mean or average voltage
of a waveform can be found again with a reasonable amount of accuracy by taking equally spaced instantaneous
values.
The positive half of the waveform is divided up into any number of “n” equal portions or mid-ordinates. The
width of each mid-ordinate will therefore be no degrees (or t seconds) and the height of each mid-ordinate will be
equal to the instantaneous value of the waveform at that point along the x-axis of the waveform.
The Graphical Method
Each mid-ordinate value of the voltage waveform is added to the next and the summed total, V1 to V12 is divided
by the number of mid-ordinates used to give us the “Average Voltage”. Then the average voltage (VAV) is the
mean sum of mid-ordinates of the voltage waveform and is given as:
and for our simple example above, the average voltage is therefore calculated as:
So as before lets assume again that an alternating voltage of 20 volts peak varies over one half cycle as follows:
Voltage
6.2V 11.8V 16.2V 19.0V 20.0V 19.0V 16.2V 11.8V 6.2V
0V
5
Angle
18o
36o
54o
72o
90o
108o
126o
144o
162o
180o
The Average voltage value is therefore calculated as:
Then the Average Voltage value using the graphical method is given as: 12.64 Volts.
Average Voltage Analytical Method
As said previously, the average voltage of a periodic waveform whose two halves are exactly similar, either
sinusoidal or non-sinusoidal, will be zero over one complete cycle. Then the average value is obtained by adding
the instantaneous values of voltage over one half cycle only. But in the case of an non-symmetrical or complex
wave, the average voltage (or current) must be taken over the whole periodic cycle mathematically.
The average value can be taken mathematically by taking the approximation of the area under the curve at various
intervals to the distance or length of the base and this can be done using triangles or rectangles as shown.
Approximation of the Area
By approximating the areas of the rectangles under the curve, we can obtain a rough idea of the actual area of
each one. By adding together all these areas the average value can be found. If an infinite number of smaller
thinner rectangles were used, the more accurate would be the final result as it approaches 2/π.
The area under the curve can be found by various approximation methods such as the trapezoidal rule, the midordinate rule or Simpson’s rule. Then the mathematical area under the positive half cycle of the periodic wave
which is defined as V(t) = Vp.cos(ωt) with a period of T using integration is given as:
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Where: 0 and π are the limits of integration since we are determining the average value of voltage over one half a
cycle. Then the area below the curve is finally given as Area = 2VP. Since we now know the area under the positive
(or negative) half cycle, we can easily determine the average value of the positive (or negative) region of a
sinusoidal waveform by integrating the sinusoidal quantity over half a cycle and dividing by half the period.
For example, if the instantaneous voltage of a sinusoid is given as: v = Vp.sinθ and the period of a sinusoid is
given as: 2π, then:
Which is therefore given as the standard equation for the Average Voltage of a sine wave as:
Average Voltage Equation
Then the average voltage (VAV) of a sinusoidal waveform is determined by multiplying the peak voltage value by
the constant 0.637, which is two divided by pi (π). The average voltage, which can also be referred to as the mean
value, depends on the magnitude of the waveform and is not a function of either the frequency or the phase angle.
Referring to our graphical example above, the peak voltage, (Vpk) was given as 20 Volts. Using the analytical
method the average voltage is therefore calculated as:
VAV = Vpk x 0.637 = 20 x 0.637 = 12.74V
Which is the same value as for the graphical method. However, multiplying the peak or maximum value by the
constant 0.637 ONLY applies to sinusoidal waveforms.
Average Voltage Summary
Then to summarise. When dealing with alternating voltages (or currents), the term Average value is generally
taken over one complete cycle, whereas the term Mean value is used for one half of the periodic cycle.
The average value of a whole sinusoidal waveform over one complete cycle is zero as the two halves cancel each
other out, so the average value is taken over half a cycle. The average value of a sine wave of voltage or current
is 0.637 times the peak value, (Vp or Ip. This mathematical relationship between the average values applies to
both AC current and AC voltage.
Sometimes it is required to be able to calculate the value of the direct voltage or current output from a rectifier or
pulse type circuit such as a PWM motor circuit because the voltage or current, although not reversing, is changing
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continuously. Since there are no phase reversals the average value is used and the RMS (root-mean-square) value
is unimportant for this type of application.
The main differences between an RMS Voltage and an Average Voltage, is that the mean value of a periodic
wave is the average of all the instantaneous areas taken under the curve over a given period of the waveform, and
in the case of a sinusoidal quantity, this period is taken as one-half of the cycle of the wave. For convenience the
positive half cycle is generally used.
The effective value or root-mean-square (RMS) value of the waveform is the effective heating value of the wave
compared to a steady DC value and is the square root of the mean of the squares of the instantaneous values taken
over one complete cycle.
For a pure sinusoidal waveform ONLY, both the average voltage and the RMS voltage (or currents) can be easily
calculated as:
Average value = 0.637 × maximum or peak value, Vpk
RMS value = 0.707 × maximum or peak value, Vpk
One final comment about using Average Voltage and RMS Voltage. Both values can be used to represent the
“Form Factor” of a sinusoidal alternating waveform. Form factor is defined as being the shape of an AC
waveform and is the RMS voltage divided by the average voltage (form factor = rms value/average value).
So for a sinusoidal or complex waveform the form factor is given as: ( π/(2√2) ) which is approximately equal to
the constant, 1.11. Form factor is a ratio and therefore has no electrical units. If the form factor of a sinusoidal
waveform is known, then the average voltage can be found using the RMS voltage value and vice-versa.
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