Half Wave Rectifier Formulas with Derivation

 

A half wave rectifier is a type of rectifier in electronics that converts half of the AC voltage waveform into DC voltage. 

 

Half wave rectifier circuits are the simplest and cost-efficient circuit among all the rectifier circuits in electronics because they use only one switch or electronic device to convert AC voltage into DC voltage. We have already created a separate article about the circuit diagram and working of half wave rectifier. If you want to read this article then visit Rectifier:  Definition, Working and Types.

 

In this article we will discuss in detail about all the formulas of half wave rectifiers with derivation. 

 

 

Half Wave Rectifier Formula

 

To easily understand half wave rectifier all formulas, let us take the example of a half wave rectifier using a diode with pure resistive load. 

Consider V_s = V_m sinωt is the AC voltage applied to the half wave rectifier

 

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 know that the prime objective of the rectifiers is to convert AC voltage into DC voltage but they do not convert it into pure DC instead of doing that they convert it into pulsating waveform. (discussed this in detail in Rectifier)

 

Similarly, a half wave rectifier converts half waveform of AC voltage into pulsating voltage waveform. A pulsating waveform is a waveform that has constant direction like a DC but its magnitude changes periodically. 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 repeat itself is 2π means the fundamental time period of the output waveform of half wave rectifier is 2π.

 

As the output waveform of the half wave rectifier is not constant like DC. Thus, to analyze the effect of this waveform first of all we have to calculate the Average and the RMS value of the output waveform of half wave rectifier.

Average Value of Half Wave Rectifier

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

In the given formula, T is the fundamental time period of the output waveform of half wave rectifier and V_m is the peak value of the output 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 average voltage of half wave rectifier

Half Wave Rectifier RMS Value

The formula for calculating the rms value of any 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. o 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 DC, it is pulsating in nature. This output pulsating waveform is made up of a constant DC voltage plus harmonic components of the voltage.

So, the RMS Value of this Voltage Waveform is

Second part of the above equation represents the AC component present in the output of half wave rectifier.

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

The output voltage waveform of the wave rectifier is combination of DC voltage and AC voltage. This type of waveform can be easily resolved with help of Fourier series as follows 

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, ripple factor of half wave rectifier is the measure of unwanted AC components that are 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. However, this 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.

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.

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.

If we analyze the expression for input power factor then we will find,

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

Harmonic Factor 

Above we discussed that the input current is usually non sinusoidal, the current is made up of fundamental component of current plus current components of higher frequencies.

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 

Half Wave Rectifier Efficiency

Efficiency of half wave rectifier is defined as the ratio of DC output power to the AC output 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.