# Average value of full wave and half wave rectifier

**Because Average value**

The arithmetic average of all the instantaneous values of a signal is called as its average value.

* Average value = area under curve / base*

#### Average value of a sine wave

The average of a sinusoidal wave form can be calculated as,

*Average value = Area of a unit cycle/ base length of a unit cycle*

Derivation to mathematically find the resultant average value for a unit cycle of a sine wave,

V = V_{m}sinωt, V_{m} – Maximum Voltage or peak voltage, V – Instantaneous voltage.

The average value of a function f(x) on an interval [a, b] = (1/b-a) _{a} ∫^{b} f(x) dx.

The area under the curve is the Integral of function f(x) in the interval from a to b. And the base length is the difference between the limits b and a.

For a unit cycle of a sine wave, the area of the region has obtained by integrating the sine wave equation and the base length from the difference of limits 0 and 2π.

Hence the average voltage, Vavg = V_{m}/2π _{0}∫^{2π} sinωt dωt | V_{m} is a constant value.

= V_{m}/2π ( _{0 }∫^{π} sinωt dωt + _{π} ∫^{2π} sinωt dωt ) = V_{m}/2π [ – cosωt]_{0}^{π }+ V_{m}/2π [ – cosωt]_{0}^{π }.

= V_{m}/2π [- cosπ + cos0] + V_{m}/2π [- cos2π + cosπ]

Therefore, Vavg = V_{m}/2π [1+1] + V_{m}/2π [-1-1] = 2V_{m}/2π – 2V_{m}/2π = 0

The average value of a sinusoidal alternating quantity for a complete cycle will be equal to zero. Because, the positive and negative half cycle is equal in magnitude and thus the total value cancels out on summation.

##### Average value of Half wave rectifier

Negative half cycles are absent in the output wave form of a half wave rectifier. So, in order to find the average value of the rectifier, the area under the positive half cycle has divided by the total base length.

The area under the positive half cycle is the integral of sinusoidal wave equation from the limits 0 to π. The total base length is the difference of limits of a complete cycle (2π – 0 = 2π), which includes the base length of both the positive and negative cycles.

The average output voltage of a half wave rectifier can be derived as,

A**verage voltage,** V_{DC} = V_{m}/2π _{0}∫^{π} sinωt dωt

= V_{m}/2π [ – cosωt]_{0}^{π} = V_{m}/2π [- cosπ + cos0]

= V_{m}/2π [1+1] = 2V_{m}/2π_{ }= V_{m}/π

The average voltage equation for a half wave rectifier is V_{DC} = V_{m}/π.

##### Average value of Full wave rectifier

In a full wave rectifier, the negative polarity of the wave will be converted to positive polarity. So the average value can be found by taking the average of one positive half cycle.

Derivation for average voltage of a full wave rectifier,

The **average voltage,** V_{DC} = V_{m}/π _{0}∫^{π} sinωt dωt

= V_{m}/π [ – cosωt]_{0}^{π} = V_{m}/π [- cosπ + cos0]

= V_{m}/π [1+1] = 2V_{m}/π_{ }

Average voltage equation for a full wave rectifier is V_{DC} = 2V_{m}/π.

So during calculations, the average voltage can be obtained by substituting the value of maximum voltage in the equation for V_{DC}.