Question

The value of form factor in case of half wave rectifier is:
$\begin{aligned} & A.\,\,\,1.11 \\\ & B.\,\,\,1.57 \\\ & C.\,\,\,1.27 \\\ & D.\,\,\,0.48 \\\ \end{aligned}$

Answer 1

To get the form factor for a half wave rectifier, first we need to get the RMS value of the current and the average value of the current, as form factor depends on this. Mainly we need to find the average value of the current, as the RMS value of the current is dependent on the average value of a current for a half-wave rectifier.

Formula used:
$\begin{aligned} & \text{Form Factor of a Half-Wave Rectifier = }\dfrac{{{I}_{RMS}}}{{{I}_{AVG}}} \\\ & {{I}_{AVG}}=\dfrac{1}{2\pi }\int\limits_{0}^{\pi }{i.d\theta } \\\ \end{aligned}$

Complete solution:
We know that,
$\text{Form Factor of a Half-Wave Rectifier = }\dfrac{{{I}_{RMS}}}{{{I}_{AVG}}}$
Now,
Average value of the current is,
${{I}_{AVG}}=\dfrac{1}{2\pi }\int\limits_{0}^{\pi }{i.d\theta }$
$\begin{aligned} & \Rightarrow {{I}_{AVG}}=\dfrac{1}{2\pi }\int\limits_{0}^{\pi }{{{I}_{\max }}\sin \theta .d\theta } \\\ & \therefore {{I}_{AVG}}=\dfrac{{{I}_{\max }}}{\pi } \\\ \end{aligned}$

And, For a half-wave rectifier, the RMS current (${{I}_{RMS}}$) is equal to the average current (${{I}_{AVG}}$) multiply by $\dfrac{\pi }{2}$.

Then, the RMS value of current for a half-wave rectifier is:
$\begin{aligned} & {{I}_{RMS}}={{I}_{AVG}}\times \dfrac{\pi }{2} \\\ & \Rightarrow {{I}_{RMS}}=\dfrac{{{I}_{\max }}}{\pi }\times \dfrac{\pi }{2} \\\ & \therefore {{I}_{RMS}}=\dfrac{{{I}_{\max }}}{2} \\\ \end{aligned}$

Hence,
$\begin{aligned} & \text{Form Factor of a Half-Wave Rectifier = }\dfrac{{{I}_{RMS}}}{{{I}_{AVG}}} \\\ & \Rightarrow F.F.=\dfrac{\dfrac{{{I}_{\max }}}{2}}{\dfrac{{{I}_{\max }}}{\pi }} \\\ & \Rightarrow F.F.=\dfrac{\pi }{2} \\\ & \therefore F.F.\approx 1.57 \\\ \end{aligned}$

Therefore, the correct answer is Option (B).

Additional Information:
A rectifier is an electrical device that converts alternating current (AC), which periodically reverses direction, to direct current (DC), which flows in only one direction. The reverse operation is performed by the inverter.
Rectifiers have many uses, but are often found serving as components of DC power supplies and high-voltage direct current power transmission systems. Rectification may serve in roles other than to generate direct current for use as a source of power. As noted, detectors of radio signals serve as rectifiers. In gas heating systems flame rectification is used to detect the presence of a flame.

Note:
Rectifier is an important topic in physics and there are many questions related to this topic in the paper. To get the solution for this question, we should remember the formula for half-wave rectifier, average value of the current and how the rms value is related to average value. Once all the value is known, then getting the form factor becomes easy.