An alternating quantity is one whose magnitude changes continuously and the direction changes periodically. In other words, a quantity such as voltage or current which repeats its variations (or wave shape) after a certain time interval (called time period) is called an alternating quantity. In electrical engineering, we deal with two major alternating quantities namely – alternating voltage and alternating current.
In this article, we will discuss the average value of an alternating quantity
(i.e. alternating current and alternating voltage).
What is the Average Value of Alternating Quantity?
The average value of an alternating quantity is defined as
under-
The average or mean of all the instantaneous values of the
alternating quantity over one cycle is called the average value of the ac quantity.
The average value is also called the mean value. It is denoted by a capital letter with a subscript ‘av’ such as Iav for current, and Vav
for voltage.
Therefore, the average value of an ac quantity can be
expressed as,
In other words, the average value of an ac quantity can also be defined as the total
area under the curve of the waveform for a certain time period (T) divided by the time period, i.e.
In actual practice, the alternating
quantities may have the following two types of waveforms-
- Symmetrical Waveform
- Unsymmetrical Waveform
For an alternating quantity having a symmetrical waveform, the positive half
cycle is exactly identical to the negative half cycle. Therefore, the average value of a symmetrical wave ac quantity
over a complete cycle is zero. For this reason, for symmetrical wave ac
quantities, the average value is computed for only half a cycle.
However, for an alternating quantity
having an unsymmetrical waveform, the
positive half-cycle and the negative half-cycle are not identical. Thus, the
average value of unsymmetrical wave ac quantity over a complete is not zero and
hence computed for the complete cycle.
There are following two methods for
determining the average value of an ac quantity-
- Graphical Method (Mid-Ordinate Method)
- Analytical Method
Determination of Average Value by Mid-Ordinate Method
The mid-ordinate
method of computing the average value of an alternating quantity is usually
used when the mathematical equation of an ac waveform is not known. This method
is also called the graphical method
because we use the waveform of the ac quantity to determine the average value.
Consider a waveform of a sinusoidal
alternating quantity where the positive half cycle is exactly identical to the
negative half cycle as shown in figure-1.
Now, in order to determine the average value of the quantity, we divide the time base t of half cycle into n equal parts of t/n duration. In this method, the larger the value of n, the better the approximation of the average value. Suppose x1, x2, x3,…, xn are the mid-ordinate values in the successive time intervals.
Therefore, the average value of the ac
wave is,
`\"Average value"=(x_1+x_2+x_3+⋯+x_n)/n`
`\⟹"Avg value"=(x_1 (t/n)+x_2 (t/n)+⋯+x_n (t/n))/t`
`\∴"Avg value"=("Area of the half wave")/("Time Base")`
Determination of Average Value by Analytical Method
The analytical
method of determining the average value of an alternating quantity is
generally used when the mathematical equation of the ac waveform is known.
In order to understand the process of
finding the average value by the analytical method, consider a sinusoidal ac quantity
whose waveform is shown in figure-2.
This ac waveform is expressed by the following equation,
Where Xm is the maximum value of the alternating quantity.
Consider an elementary strip of small
thickness dθ
as shown in figure-2. Let the ordinate of this strip is x(t). Then, the area of
the strip is given by,
Since the sinusoidal waveform is a
symmetrical wave, we consider only half a cycle for the computation of the
average value.
Thus, the total area of the half cycle can
be determined by,
`\∫dA=∫_0^Ï€ x dθ`
`\⟹A=∫_0^Ï€ X_m sinθ dθ`
Now, from the definition of the average
value, we have,
`\"Average value"=("Area of half cycle")/("Duration of half cycle")`
`\∴X_(av)=(∫_0^Ï€ X_m sinθ dθ)/(Ï€-0)`
`\⟹X_(av)=X_m/Ï€ [-cosθ ]_0^Ï€`
`\⟹X_(av)=X_m/Ï€ [-cosÏ€+cos0 ]`
`\⟹X_(av)=X_m/Ï€ [-(-1)+1]`
`\∴X_(av)=(2X_m)/Ï€=0.637X_m`
Therefore, the average value of the sinusoidal alternating current is given by,
And, the average value of the sinusoidal alternating voltage is given by,
Note
– In general, the average value of an ac quantity x(t) is given by,
Conclusion
Therefore, the average value of an
alternating quantity is the mean value of the quantity calculated over a
complete cycle. In practice, the most widely used alternating quantity is the
sinusoidal waveform alternating quantity. The average value of a sinusoidal
alternating quantity is always computed over half cycle and is equal to 0.637
times the maximum value of the quantity. The average value of sinusoidal ac
quantity over a complete cycle is zero because this has exactly identical
positive and negative half cycles. There are two methods for determining the
average value of an alternating quantity namely the graphical method and the analytical
method. The graphical method is used when the equation of the quantity is not
known, while the analytical method is used when the equation of the quantity is
known.
Numerical
Example – A voltage wave is represented by,
Determine the average value of the given
voltage.
Solution
– From the given equation, it is clear that the voltage has a sinusoidal waveform
whose maximum value is
The average value is given by,
`\V_(av)=(2V_m)/Ï€=0.637V_m`
`\⟹V_(av)=0.637×240`
`\∴V_(av)=152.88" V"`
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