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 Vs =Vm sinÏ‰t. 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 - Ï‰ = 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 Ï€ the output of the half
wave rectifier follows the input waveform and from Ï€ to 2Ï€ the
output of the half wave rectifier is zero. And the time period after which the
output waveform repeats itself is 2Ï€ means the fundamental
time period of the output waveform of half wave rectifier is 2Ï€.

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
2Ï€.

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 Ï€ and Ï€
to 2Ï€.

In the
above equation part (ii) is zero because from Ï€ to 2Ï€ 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 behaviour 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}
cosÉ¸_{1 }

where, É¸_{1}
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 cosÉ¸1 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 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.

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