# [Half-Wave Rectified Sine-Wave] RMS Value, Average Value, Form Factor, and Crest Factor

Regarding the Half-Wave Rectified Sine-Wave: This article will explain the information below.

• How to obtain RMS Value, Average Value, Form Factor, and Crest Factor

## RMS Value, Average Value, Form Factor, and Crest Factor of Half-Wave Rectified Sine-Wave

The first illustration shows the root mean square (RMS) value, average value, form factor, and crest factor of a half-wave rectified sine-wave (with a maximum value of $$V_M$$ and a period of $$T$$).

Now, let's explain how each of these values are calculated (we strive to include as many intermediate steps as possible).

Supplement

A half-wave rectified sine-wave is a waveform in which half of one cycle is zero in a sine wave. In the diagram above, the sine wave is zero in the area of "$$\displaystyle \frac{T}{2} \leq t \lt T$$".

Therefore, understanding first "how to calculate the RMS value, average value, form factor, and crest factor of a sine wave" can make it easier to comprehend "how to calculate the RMS value, average value, form factor, and crest factor of a half-wave rectified sine-wave".

For "how to calculate the RMS value, average value, form factor, and crest factor of a sine wave", a detailed explanation is provided in the article below.

## Waveform of a Half-Wave Rectified Sine-Wave

To find the root mean square (RMS) value and average value of a half-wave rectified sine-wave, we first need to express the half-wave rectified sine-wave as a formula.

The half-wave rectified sine-wave has different formulas for the red region $$\left(0 \leq t \lt \displaystyle \frac{T}{2}\right)$$ and the blue region $$\left(\displaystyle \frac{T}{2} \leq t \lt T\right)$$.

Let's calculate the formulas for the red region and blue region.

Red Region

The region "$$0 \leq t \lt \displaystyle \frac{T}{2}$$" corresponds to the red region.

The red region is part of the waveform of $$V_M\sin{{\omega}t}$$.

Therefore, the red region can be represented by the following formula.

\begin{eqnarray}
v(t)=V_M\sin{{\omega}t}
\end{eqnarray}

Blue Region

The region "$$\displaystyle \frac{T}{2} \leq t \lt T$$" corresponds to the blue region.

The blue region is zero.

Therefore, the blue region can be represented by the following formula.

\begin{eqnarray}
v(t)=0
\end{eqnarray}

By combining the red region and the blue region, the half-wave rectified sine-wave can be represented by the following formula.

\begin{eqnarray}
v(t) = \begin{cases}
V_M\sin{{\omega}t} & \left(0 \leq t \lt \displaystyle \frac{T}{2}\right) \\
\\
0 & \left(\displaystyle \frac{T}{2} \leq t \lt T\right)
\end{cases}
\end{eqnarray}

Next, we will calculate the RMS value, average value, form factor, and crest factor of the half-wave rectified sine-wave. However, using the above formula can make the calculation slightly complex.

Therefore, to simplify the calculation this time, we will convert the time axis (horizontal axis $$t$$) to the phase axis (horizontal axis $${{\omega}t}$$).

When the time axis (horizontal axis $$t$$) is converted to the phase axis (horizontal axis $${{\omega}t}$$), the half-wave rectified sine-wave becomes the following formula.

\begin{eqnarray}
v({\omega}t) = \begin{cases}
V_M\sin{{\omega}t} & \left(0 \leq {\omega}t \lt \pi\right) \\
\\
0 & \left(\pi \leq {\omega}t \lt 2\pi\right)
\end{cases}
\end{eqnarray}

Related article

The following article explains the "How to convert the time axis (horizontal axis $$t$$) to the phase axis (horizontal axis $${{\omega}t}$$)" in detail. If you are interested, please check it out from the link below.

## RMS Value of a Half-Wave Rectified Sine-Wave

The root mean square (RMS) value $$V_{RMS}$$ of a waveform $$v({{\omega}t})$$ is the square root of the mean of the square of $$v({{\omega}t})$$. It can be represented by the following formula:

\begin{eqnarray}
V_{RMS} &=& \sqrt{\displaystyle\frac{1}{2\pi} \displaystyle \int_{0}^{2\pi}v({{\omega}t})^2d({{\omega}t}})
\end{eqnarray}

The red region is $$0 \leq {\omega}t \lt \pi$$", and the blue region is "$$\pi \leq {\omega}t \lt 2\pi$$".

In the formula for finding the absolute value, if we separate the red region and the blue region, we get the following formula.

\begin{eqnarray}
V_{RMS} &=& \sqrt{\displaystyle\frac{1}{2\pi} \left(\displaystyle \int_{0}^{\pi}v({{\omega}t})^2d({\omega}t)+\displaystyle \int_{\pi}^{2\pi}v({{\omega}t})^2d({\omega}t)\right )}\\
\\
&=& \sqrt{\displaystyle\frac{1}{2\pi} \left(\displaystyle \int_{0}^{\pi}(V_M\sin{{\omega}t})^2d({\omega}t)+\displaystyle \int_{\pi}^{2\pi}0^2d({\omega}t)\right )}\\
\\
&=& \sqrt{\displaystyle\frac{1}{2\pi}\displaystyle \int_{0}^{\pi}(V_M\sin{{\omega}t})^2d({\omega}t)}
\end{eqnarray}

The formula for the red region is $$V_M\sin{{\omega}t}$$, and the formula for the blue region is $$-V_M\sin{{\omega}t}$$, but in the calculation of the effective value, we square it, so we can combine the above formulas. If we combine them, we get the following formula.

\begin{eqnarray}
V_{RMS} &=& \sqrt{\displaystyle\frac{1}{\pi} \displaystyle \int_{0}^{2\pi}(V_M\sin{{\omega}t})^2d({\omega}t)}
\end{eqnarray}

In the above formula, if you let

\begin{eqnarray}
X &=& \displaystyle \int_{0}^{\pi}(V_M\sin{{\omega}t})^2 d({\omega}t)
\end{eqnarray}

then the root mean square (RMS) value $$V_{RMS}$$ of a waveform $$v({{\omega}t})$$ can be represented by the following formula:

\begin{eqnarray}
V_{RMS} &=& \sqrt{\displaystyle\frac{1}{2\pi} X}
\end{eqnarray}

Next, we calculate the value of $$X$$.

\begin{eqnarray}
X &=& \displaystyle \int_{0}^{\pi}v({{\omega}t})^2 d({\omega}t)\\
\\
&=& \displaystyle \int_{0}^{\pi}{V_M}^2{\sin}^2{\omega}t d({\omega}t)\\
\\
&=& {V_M}^2\displaystyle \int_{0}^{\pi}{\sin}^2{\omega}t d({\omega}t)\\
\\
&=& {V_M}^2\displaystyle \int_{0}^{\pi}\displaystyle \frac{1-{\cos2{\omega}t}}{2} d({\omega}t)\\
\\
&=& {V_M}^2 \left[\displaystyle\frac{1}{2}{{\omega}t}-\frac{1}{4}{\sin2{\omega}t} \right]_{0}^{\pi}\\
\\
&=& {V_M}^2 \left( \displaystyle\frac{1}{2}×{\pi}-\frac{1}{4}{\sin{2\pi}}\right)\\
\\
&=& \displaystyle\frac{{V_M}^2 {\pi}}{2}
\end{eqnarray}

Therefore, the root mean square (RMS) value $$V_{RMS}$$ of a waveform $$v({{\omega}t})$$ becomes:

\begin{eqnarray}
V_{RMS} &=&\sqrt{\displaystyle\frac{1}{2\pi} X}\\
\\
&=&\sqrt{\displaystyle\frac{1}{2\pi} \displaystyle\frac{{V_M}^2 {\pi}}{2}}\\
\\
&=&\displaystyle\frac{1}{2}V_M
\end{eqnarray}

## Average Value of a Half-Wave Rectified Sine-Wave

The average value $$V_{AVE}$$ of a waveform $$v({{\omega}t})$$ is the average value of the absolute value $$|v({{\omega}t})|$$ of $$v({{\omega}t})$$ and can be represented by the following formula:

Formula to find the average value

\begin{eqnarray}
V_{AVE} &=& \displaystyle\frac{1}{2\pi} \displaystyle \int_{0}^{2\pi}|v({{\omega}t})|d({{\omega}t})
\end{eqnarray}

This formula for calculating the average value uses the absolute value $$|v({{\omega}t})|$$ of $$v({{\omega}t})$$.

Therefore, if there is a negative region in the waveform, it needs to be converted so that the negative region becomes positive.

In the case of a half-wave rectified sine-wave, there is no negative region, so the formula for $$v({{\omega}t})$$ and the formula for the absolute value $$|v({{\omega}t})|$$ of $$v({{\omega}t})$$ become the same.

Therefore, the absolute value $$|v({{\omega}t})|$$ of $$v({{\omega}t})$$ can be represented by the following formula.

\begin{eqnarray}
|v({{\omega}t})| = \begin{cases}
V_M\sin{{\omega}t} & \left(0 \leq {\omega}t \lt \pi\right) \\
\\
0 & \left(\pi \leq {\omega}t \lt 2\pi\right)
\end{cases}
\end{eqnarray}

In the formula to find the average value, if we separate each term, we get the following formula.

\begin{eqnarray}
V_{AVE} &=& \displaystyle\frac{1}{2\pi} \displaystyle \int_{0}^{2\pi}|v({\omega}t)|d({{\omega}t})\\
\\
&=& \displaystyle\frac{1}{2\pi}\left(\displaystyle \int_{0}^{\pi}|v({\omega}t)|d({{\omega}t})+ \int_{\pi}^{2\pi}|v({\omega}t)|d({{\omega}t}) \right)\\
\end{eqnarray}

Since the half-wave rectified sine-wave is zero in the region $$\pi \leq {\omega}t \lt 2\pi$$, the above equation becomes:

\begin{eqnarray}
V_{AVE} &=& \displaystyle\frac{1}{2\pi}\displaystyle \int_{0}^{\pi}|v({\omega}t)|d({{\omega}t})
\end{eqnarray}

If we calculate the above formula, the average value $$V_{AVE}$$ of the half-wave rectified sine-wave becomes:

\begin{eqnarray}
V_{AVE} &=& \displaystyle\frac{1}{2\pi}\displaystyle \int_{0}^{\pi}V_M\sin{{\omega}t}d({{\omega}t})\\
\\
&=& \displaystyle\frac{1}{2\pi}V_M\left[-\cos{\omega}t \right]_{0}^{\pi}\\
\\
&=& \displaystyle\frac{V_M}{2\pi}\left(-\cos\pi + \cos0 \right)\\
\\
&=& \displaystyle\frac{1}{\pi}V_M
\end{eqnarray}

## Form Factor of a Half-Wave Rectified Sine-Wave

The form factor can be represented by the following formula:

\begin{eqnarray}
\mbox{Form Factor} &=& \displaystyle\frac{\mbox{Root Mean Square Value}~~V_{RMS}}{\mbox{Average Value}~~V_{AVE}}
\end{eqnarray}

Since we've already calculated the root mean square (RMS) value $$V_{RMS}$$ and average value $$V_{AVE}$$ of the sine wave, we can substitute these values into the formula to calculate the form factor of the half-wave rectified sine-wave.

\begin{eqnarray}
\mbox{Form Factor} = \displaystyle\frac{\displaystyle\frac{1}{2}V_M}{\displaystyle\frac{1}{\pi}V_M} = \displaystyle\frac{\pi}{2}
\end{eqnarray}

## Maximum Value of a Half-Wave Rectified Sine-Wave

As can be seen from the waveform, the maximum value $$V_{PEAK}$$​ of a half-wave rectified sine-wave is represented by the following value:

\begin{eqnarray}
\mbox{Maximum Value}~~V_{PEAK}=V_M
\end{eqnarray}

## Crest Factor of a Half-Wave Rectified Sine-Wave

The Crest factor can be represented by the following formula:

\begin{eqnarray}
\mbox{Crest Factor} &=& \displaystyle\frac{\mbox{Peak Value}~~V_{PEAK}}{\mbox{Root Mean Square Value}~~V_{RMS}}
\end{eqnarray}

Since we've already calculated the root mean square (RMS) value $$V_{RMS}$$ and peak value $$V_{PEAK}$$ of the sine wave, we can substitute these values into the formula to calculate the peak factor (crest factor) of the half-wave rectified sine-wave.

The Crest factor of the half-wave rectified sine-wave is represented by the following value:

\begin{eqnarray}
\mbox{Crest Factor} = \displaystyle\frac{V_M}{\displaystyle\frac{1}{2}V_M} = 2
\end{eqnarray}

#### Summary

In this article, the following information on the "Half-Wave Rectified Sine-Wave" was explained.

• How to obtain RMS Value, Average Value, Form Factor, and Crest Factor