Full Wave Rectifier all Formulas with Derivation

 

A full wave rectifier is a type of rectifier that converts a full waveform of AC voltage into DC voltage. For the conversion of AC voltage into DC voltage it uses two different types of circuit configurations i.e. Center Tapped Full Wave Rectifier and Full Wave Bridge Rectifier.

 

We have created dedicated articles for both types of full wave rectifiers, where we discussed their definitions, circuit diagrams, working principles, output waveforms and other important concepts.

 

In this article, we will cover all the formulas of full wave rectifiers with derivations. But, before moving to these concepts, we recommend you to first go through the basics of both types of full wave rectifiers, which will help you to better understand this article.

Full Wave Rectifier Formulas

 

To make it easy to understand all the formulas of full wave rectifiers, we consider a full wave rectifier using diodes for resistive load. The input and output waveforms of full wave rectifier are shown in the given figure. This output waveform is valid for both center tap and bridge rectifiers using diodes for resistive load.

By observing the above figure, we see that the rectified output waveform is not a pure DC signal, rather, it's pulsating in nature. Due to its unidirectional nature, it is referred to as a pulsating DC signal, even though its magnitude is not constant like a pure DC signal. Actually, the input and output waveform of voltage and current of rectifiers depend on the nature of load and rectifier circuit configuration. In order to evaluate the overall performance of the rectifier and load combinations, certain performance parameters related to their input and output must be calculated. 

Output Parameters of Full Wave Rectifier

Frequency of Output Voltage Waveform of Full Wave Rectifier

In the output waveform shown above, we see that from 0 to π radians, the output waveform follows the input, and from π to 2π radians, the output is the negative of the input. This output waveform repeats itself after π radians, which indicates that its fundamental time period is π radians which is half of the input AC waveform. Consequently, the fundamental frequency is 1/π rad/sec which is double that of the input waveform.

 

Average Value of the Output Voltage of Full Wave Rectifier

The formula for calculating the average value of a sinusoidal waveform is 

In the given formula,

• T is the fundamental time period of the output voltage waveform
• V_m is the peak value of output voltage waveform
• V_o is the average value of the output voltage waveform

After applying the given formula on the output voltage waveform, we get

RMS Value of the Output Voltage of Full Wave Rectifier

The formula for calculating the RMS value of a sinusoidal waveform is

where, 

• T represents the fundamental time period of the waveform,
• V_or represents the RMS value of the output waveform,
• V_m represents the peak value of the voltage waveform

After applying the given formula on the output waveform, we get

Full Wave Rectifier Fourier Series

As discussed above, the rectified output voltage waveform is not a pure DC waveform, it's pulsating in nature. This pulsating waveform contains both the DC component of voltage and AC component of voltage. This type of waveform can easily be resolved with the help of Fourier series.

Harmonics in Full Wave Rectifier

Harmonics on the DC side or Output side of the Full Wave Rectifier

The AC component present in the rectified output voltage waveform is referred to as ripples, and it is unwanted since our primary objective is to get pure DC signal. The measure of these unwanted ripples in the output waveform is called ripple factor. 

These unwanted AC voltage components present in the output waveform cause the overheating of loads and, consequently, a reduction in the performance of the rectifier. Therefore, filters are often used on the output side in order to remove these unwanted AC components present in the output waveform. We have created a dedicated article on how filters remove unwanted ripples from the output voltage waveform of the rectifiers.

Ripple Factor of Full Wave Rectifier

As discussed above, the ripple factor is the measure of unwanted ripple present in the output waveform. It is defined as the ratio of the RMS value AC component present in the output waveform to its average value.

The formula for RMS value of the AC voltage present in the output waveform or the ripple voltage formula of the full wave rectifier is

where,

• V_or  =  RMS value of the output voltage waveform
• V_o  =  Average value of the output voltage waveform
• V_ac  =  Ripple Voltage 

So, according to definition,

Equation that indicates by the blue box represents the general expression to calculate ripple factor for any rectifier circuit. This expression can also be written as

Where FF is the form factor. (discussed further in this article)

For Full Wave Diode Rectifier

The above given value of the ripple factor indicates that the RMS value of the AC component present in the output waveform is 0.483 times the DC component or average value of this waveform. This means the DC component present in the output waveform of the full wave rectifier is stronger than the AC component present.

Form Factor of Full Wave Rectifier

It is the ratio of RMS value of the output waveform to the average value of the output waveform.

Significance of Form Factor

• FF is the measure of the shape of the output voltage waveform.
• Generally, for rectified output waveform FF is greater than 1.
• As FF decreases and approaches unity then the smoothness of the waveform improves toward DC. As we know, for constant DC voltage, the RMS value of the output voltage V_or = the average value of the output voltage V_o.

 

PIV (Peak Inverse Voltage) of Full Wave Rectifier

As the name implies, it is the maximum voltage that appears across the switches used in the rectifier circuit during its blocking or off state. It is an important parameter in the design of rectifier circuits.

• For Center Tapped Rectifier, PIV of the switches used in the circuit  =  2V_m
• For Bridge Rectifier, PIV  =  V_m

Input Parameters of Full Wave Rectifier

Input Power Factor of Full Wave Rectifier

Input power factor is defined and ratio of active power supplied to the rectifier to the total volt amperes supplied to the rectifier. 

The input voltage taken from the power supply is generally sinusoidal, however, the ac input current is usually non sinusoidal due to the non-linear load and rectifier circuit combination. In such cases only the fundamental component of ac input current delivers the useful power. 

So, the active power supplied to the rectifier   =  V_sr. I_s1 cosɸ₁

where, ɸ₁ is the angle between supply voltage and the fundamental component of the current

Total volt amperes supplied to the rectifier  =  V_sr. I_sr

So,

In the above equation cosɸ₁ is the fundamental displacement factor or input displacement factor and ɸ₁ is the phase angle between the sinusoidal supply voltage and the fundamental component of the supply current I_s1.

Harmonics on Input side or AC side of the Full Wave Rectifier

As discussed above, the ac input current is usually non sinusoidal, and is made up of a fundamental component of current plus components of higher order frequencies. These higher frequency components are called harmonics in the input waveform. The measure of these harmonic content in the input current waveform is called Harmonic Factor also known as Total Harmonics Distortion.

 

Harmonic Factor or Total Harmonics Distortion of Full Wave Rectifier

The Harmonic Factor is defined as the ratio of the RMS value of the all the harmonic current components to the RMS value of the fundamental current component. 

 

The RMS value of the supply current is 

Current Distortion Factor of Full Wave Recitifier

It is defined as the ratio of RMS value of the fundamental component IS1 of the supply current to the RMS value of the input current. It is denoted by g.

Significance of Current Distortion Factor (g)

• Generally, for non-sinusoidal input waveform g is less than 1. 
• As g increases and approaches unity then the smoothness of the waveform improves towards sinusoidal.

Efficiency of Full Wave Rectifier 

Efficiency of a rectifier is the measure of ability of the rectifier to convert AC input power to DC output power. It is defined as the ratio DC output power of the rectifier to the AC input power to the rectifier.