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} sinÏ‰t 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

- Ï‰ = 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 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 repeat itself is 2Ï€ means the fundamental
time period of the output waveform of half wave rectifier is 2Ï€.

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 2Ï€.

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 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 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 AC component present in the waveform to
the average value of the waveform. It is the measure of the fluctuating
components in the given 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 RMS value AC component present in the output waveform of half wave rectifier is

In above expression Vac represents the half wave rectifier ripple voltage formula

So, according to definition
ripple factor

Equation that indicates by the blue box represents the general expression to calculate ripple factor for any rectifier circuit.

**For Half Wave Rectifier**

**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} 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.**

**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.

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