Average Value

In this tutorial, we will look at calculating the average voltage value of a sinusoidal waveform using both the middle coordinate rule and the analytical rule.

The process used to find the Average Voltage of an alternative waveform is very similar to that used to find the RMS value, this time the difference is that the snapshot values are not squared and we cannot find the square root of the total average.

Regardless of whether the periodic waveform is a sine wave, square wave, or triangular waveform, its average voltage (or current) is defined as follows: "the portion of the area below the waveform by time".In other words, the average time that all instantaneous values along the timeline are a full period, ( T ).

For a periodic waveform, the area above the horizontal axis is positive and the area below the horizontal axis is negative.The result is that the average or average value of a symmetrical alternative quantity is therefore zero, because the area above the horizontal axis (positive half-loop) is the same as the area below the axis (negative half loop), thereby canceling each other out.This is because when we do the mathematics of the two fields, the negative field cancels out the positive area, which produces zero average voltage.

Then, the average value of a symmetrical variable quantity, such as a sinus wave, is the average value measured only in one half of a cycle, since as we mentioned earlier, the average value on a full cycle is zero, regardless.

Average value(voltage) or average current electric terms can be used for both AC waveforms and DC straightening calculations.The symbols used to represent an average value are defined as: VAV or IAV .

Average Voltage Graphics Method

Again, consider only the positive half loop from the previous RMS voltage tutorial.The average voltage of a waveform can be reconstructed graphically with reasonable accuracy by taking evenly spaced instantaneous values.

The positive half of the waveform is divided into any number of "n" equal parts or middle coordinates.

Average Voltage Graphics Method

Each medium coordinate value of the voltage waveform is added to the next one, and the total collected is divided by the number of medium coordinates used to give us the " Average Voltage " from V 1 to V 12.Then the average voltage (VAV) is the average sum of the middle coordinates of the voltage waveform and is given as follows:

and the average voltage for our simple example above is calculated as follows:

As before, let's re-assume that an alternative voltage of 20 volts changes over a half cycle as follows:

angle18 O36 O54 O72 o90 O108 o126 O144 O162 O180 O

Therefore, the average voltage value is calculated as follows:

Then, using the graphics method, the Average Voltage value for a half loop is given as follows: 12.64 Volts .

Average Voltage Analytical Method

As previously said, the average voltage of a periodic waveform, two halves of which are exactly similar, sinusoidal or non-sinusoidal, will be zero during a full cycle.But in the case of a non-symmetrical or complex wave, the average voltage (or current) should be mathematically taken throughout the entire periodic cycle.

The average value can be mathematically retrieved by taking the approximateity of the area below the curve to the distance or length of the base at various intervals, and this can be done using triangles or rectangles, as shown.

Zooming In

By approaching the areas of rectangles under the curve, we can get a rough idea of the actual area of each.All these fields can be summed up to find an average value.If an infinite number of smaller, thinner rectangles were used, the closer we got to 2/π, the more accurate the final result would be.

The area below the curve can be found with a variety of approach methods, such as the trapezial rule, the middle coordinate rule, or the Simpson rule.Then, using integral, the mathematical field below the positive half-cycle of the periodic wave defined as V (t) = Vp.cos(ωt) with a T period is given as follows:

Here: 0 and π are integral limits, as we determine the average value of the voltage for half a cycle.

The area below the curve is then finally given as Area = 2V P.Now that we know the area below the positive (or negative) half-cycle, we can easily determine the average value of the positive (or negative) region of a sinusoidal waveform by taking integral the sinusoidal amount for half a cycle and dividing it by half the period.

For example, the instantaneous tension of a sinusoid: v = Vp.sinε and the period of a sinusoid: if given as 2π, then:

Therefore, the standard equation for the Average Voltage of a sinus wave can be given as follows:

Average Voltage Equation

The average voltage (VAV)of a sinusoidal waveform is determined by multiplying the peak voltage value by a constant of 0.637, which is two divided pi (π). The average voltage, which can also be called the average value, depends on the size of the waveform and is not a function of the frequency or phase angle.

Therefore, this average value (voltage or current) of a sinusoidal waveform can also be shown as the equivalent DC value of the field and time.

Since the positive average field will be canceled by the negative average field ( V AVG – (-V AVG) ) in the sum of the two fields, the average value in a full cycle is zero, thus achieving zero average voltage in a full cycle.

According to our graphical example above, the peak voltage (V pk) is given as 20 Volts.Using the analytical method, the average voltage is therefore calculated as follows:

V AV = V pk x 0.637 = 20 x 0.637 = 12.74 volts

This is the same value as the chart method.

To find the peak value from a specific average voltage value, rearrange the formula and split it into constants. For example, if the synusoidal average value is 65 volts, what is the peak value Vpk?

V pk = V AV ÷ 0.637 = 65 ÷ 0.637 = 102 volts

Note that multiplying the peak or maximum value by a constant of 0.637 applies only to sinusoidal waveforms.

Average Voltage Summary

Then we can summarize as follows, when dealing with alternating voltages (or currents), the term Average value is usually taken during a full cycle, while the term Meydan(mean) is used for half of the periodic cycle.

The average value of a whole sinusoidal waveform over a full cycle is zero, since two halves cancel each other out, so the average value is taken over half a cycle. The average value of a sine wave voltage or current is 0.637 times its peak value (Vp or Ip). This mathematical relationship between average values applies to both AC current and AC voltage.

Sometimes it is necessary to calculate the value of direct voltage or current output from a rectifier or a pulse-type circuit such as a PWM motor circuit, because the voltage or current changes continuously, if not in reverse.Since there is no phase inversion, the average value is used, and the RMS (root-mean-square) value for this type of application is insignificant.

The main differences between an RMS Voltage and an Average Voltage are that the average value of a periodic wave is the average of all instantaneous areas taken under the curve during a given period of the waveform, and if there is a sinusoidal amount, this time is taken as half of the wave cycle. For convenience, a positive half loop is usually used.

The active value of the waveform or the root-mean-square (RMS) value is the effective heating value of the wave compared to a constant DC value, and is the square root of the average of the squares of instantaneous values taken during a full cycle.

For only a pure form of sinusoidal wave, both the average voltage and RMS voltage (or currents) can be easily calculated as follows:

Average value = 0.637 × maximum or peak value, Vpk

RMS value = 0.707 × maximum or peak value, Vpk

A final commentary on using the Average Voltage and RMS Voltage, both values can be used to represent the "Form Factor" of the sinusoidal alternate waveform. The form factor is defined as the shape of an AC waveform and is the division of the RMS voltage into the average voltage (form factor = rms value/average value).

The form factor for a sinusoidal or complex waveform is given as follows: ( π/(2√ 2 ) which is approximately equal to the constant 1.11.The form factor is a ratio and therefore there is no electrical unit.If the form factor of the sinusoidal waveform is known, the average voltage can be found using the RMS voltage value, and vice versa, since the average voltage is 0.9 times the RMS voltage value of a sinus wave.