Half Wave Rectifier Formulas with Derivation

 

A half wave rectifier is a type of rectifier that converts half of the AC voltage waveform into DC voltage, and the process of conversion is known as half wave rectification.

 

Half wave rectifier circuits are the simplest and cost-efficient circuit among all the rectifier circuits because they use only one switch or semiconductor device to convert AC voltage into DC voltage. 

 

In this article, we will discuss all the formulas of half wave rectifiers with derivations. But, before moving on to the main concepts, it is recommended that first understand the basics, circuit diagram and working of half wave rectifiers. This will help you to understand the derivation and formulas of half wave rectifiers easily.

We have already created an article dedicated to all the basic concepts of half wave rectifiers. If you want to read this article then visit: Half Wave Rectifiers.

Half Wave Rectifier Formula

 

To easily understand all the formulas of a half wave rectifier, let us take the example of a half wave rectifier using a diode with pure resistive load. The reason behind using diode is its un-controlling nature, means, there is no such extra parameter through which we can control its conduction. And the reason behind taking resistive load is the linear relationship between the voltage and current.

 

Consider the circuit given below

Let us say, the AC voltage applied to the given circuit is V_s =V_m sinωt. We can draw the waveform for the given expression shown below.

Where,

 

• V_s      =         Instantaneous Value of the applied voltage waveform
• V_m     =         Peak Value of the applied voltage waveform
• ω       =         Angular frequency
• t         =         time
• V_sr      =        RMS Value of the applied voltage waveform

As we discussed in the working of half wave rectifiers, during the positive half of the AC supply the diode is forward biased and it conducts the current. In this conduction the voltage across the load is similar to the input voltage. 

And during the negative half of the AC supply, the diode is reverse biased and it does not conduct the current. In this condition the voltage across the load is zero.

The input and output waveforms for the half wave rectifier is shown in the given figure.

In the above figure we see that from 0 to π the output of the half wave rectifier follows the input waveform and from π to 2π the output of the half wave rectifier is zero. And the time period after which the output waveform repeats itself is 2π means the fundamental time period of the output waveform of half wave rectifier is 2π.

If we carefully observe the output waveform of the given rectifier circuit, we find that this waveform is not a pure DC signal, it's a pulsating unidirectional waveform. Due to the unidirectional nature of this waveform, it is referred to as a pulsating DC signal, although the magnitude of this waveform is not constant like a pure DC signal. Thus, to analyze the effect of this waveform, we first need to calculate its average and RMS values.

Average Value of Half 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 waveform and V_m is the peak value of the waveform.

Above we have discussed that the fundamental time period of the output waveform of the half wave rectifier is 2π. 

After applying the given formula on the output waveform of half wave rectifier then we will get

Half Wave Rectifier RMS Value

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,
• Vm represents the peak value of the voltage waveform

 

After applying the given formula on the output waveform of half wave rectifier, we will get the rms value of half wave rectifier.

For ease of the calculation, we can split above equation into two parts i.e. 0 to π and π to 2π.

In the above equation part (ii) is zero because from π to 2π the output of the half wave rectifier is zero.

Half Wave Rectifier Fourier Series

As we have discussed above, the rectified output voltage is not a constant waveform like a pure DC signal, it’s pulsating in nature. This waveform is made up of the combination of both DC voltage and AC voltage. This type of waveform can easily be resolved with the help of Fourier series as follows.

Harmonics Analysis of Half Wave Rectifier

Harmonics on DC side or output side of the Half Wave Rectifier

The AC component present in the rectified output voltage waveform contains fundamental plus higher order harmonics. Therefore, this output waveform is made up of DC component plus fundamental component plus higher order harmonics component of voltage. The presence of harmonic components causes the reduction in performance of the rectifier and overheating of loads.

The RMS Value of this Voltage Waveform is

The second part of the above equation represents the AC component present in the output voltage waveform of the half wave rectifier. This is the unwanted part of the rectified output waveform, as the prime objective is to get pure DC signal.

(V_ac)²   =      (V_or)²   -   (V_o)²       =          (V_o1)²   +  (V_o2)²   +   (V_o3)² …...………..

Ripple Factor of Half Wave Rectifier

Ripple Factor of any waveform is defined as the ratio of the RMS value of the AC component present in the waveform to the average value of the waveform. 

It is the measure of the fluctuating component or AC component present in the waveform.

In a similar context, the Ripple factor of half wave rectifier is the measure of the unwanted AC component present in the output waveform of the Half wave Rectifier and it is defined as the ratio of the RMS value of the AC component present in the output waveform to its average value.

Above we have discussed, the expression for RMS value of the AC voltage present in the output waveform of half wave rectifier is 

It is also known as the ripple voltage present in the rectified output voltage waveform.

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 Half Wave Rectifier

If we carefully observe the ripple factor of the half wave rectifier, then we find that the RMS value of the AC component present in the output waveform is 1.21 times the DC component or average value of this waveform. 

This indicates that the AC component present in the output waveform of the half wave rectifier is stronger than the DC component present in it and it is not desired.

Form Factor of Half Wave Rectifier

Form Factor of any waveform is defined as the ratio of RMS value of the waveform to the average value of the waveform.

Harmonics on AC side or input side of the Half Wave Rectifier

As we discussed above, the rectified output voltage contains the DC component plus harmonics component of voltage. A rectifier also injects harmonics on the AC side or input side due to the non-linear behavior of switches used in rectifier circuit and load. Actually, the input and output voltage and current waveform depends upon the combination of load and rectifier configuration.

The input voltage is usually sinusoidal, whereas the input current is non sinusoidal due to the reason mentioned above. The input current is made up of fundamental component of current plus current components of higher frequencies. In such waves the measure of harmonic content is known as the harmonic factor. 

Harmonic Factor or Total Harmonic Distortion in Half Wave Rectifier

Harmonic factor is the measure of harmonic content in the input side of the rectifier, it is also known as Total Harmonic Distortion. It is defined as the ratio of the RMS value of all the harmonic components to the fundamental component of the input current.

The RMS value of the supply current is

Current Distortion Factor of half Wave Rectifier

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

Input Power Factor of Half 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 voltage applied to the rectifier is generally sinusoidal however, the average input current is usually non sinusoidal. In that case, only the fundamental component of AC with current takes part in supplying the active 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 rectifier  =  V_sr. I_sr

So,

In the above equation cosɸ1 is the fundamental displacement factor or input displacement factor.

If we carefully observe the input power factor expression, we find that the Input power factor is equals to the product of Current Distortion Factor (CDF) and the Fundamental Displacement Factor (FDF).

Input power factor = Current Distortion Factor (CDF)  *  Fundamental Displacement Factor (FDF)

Half Wave Rectifier Efficiency

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

Above driven efficiency i.e. 40.5% is the maximum efficiency of half wave rectifier. It is derived by taking certain assumption. It can be varied in practical conditions.