# [Pulse Wave] RMS Value and Average Value

Regarding the Pulse Wave: This article will explain the information below.

• How to obtain RMS Value and Average Value

## RMS Value and Average Value of Pulse Wave

The first illustration shows the root mean square (RMS) value and average value of a pulse 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).

## Waveform of a Pulse Wave

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

The pulse wave has different formulas for the red region $$\left(0 \leq t \lt T_1\right)$$ and the blue region $$\left(T_1 \leq t \lt T\right)$$.

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

Red Region

The region "$$0 \leq t \lt T_1$$" corresponds to the red region.

The red line is always $$V_M$$ during the time $$T_1$$.

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

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

Blue Region

The region "$$T_2 \leq t \lt T$$" corresponds to the blue region.

The blue rgsion 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 pulse wave can be represented by the following formula.

\begin{eqnarray}
v(t) = \begin{cases}
V_M & \left(0 \leq t \lt T_1\right) \\
\\
0 & \left(T_1 \leq t \lt T\right)
\end{cases}
\end{eqnarray}

## RMS Value of a Pulse Wave

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

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

The red region is $$0 \leq t \lt T_1$$", and the blue region is "$$T_1 \leq t \lt T$$".

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}{T} \left(\displaystyle \int_{0}^{T_1}v(t)^2dt+\displaystyle \int_{T_1}^{T}v(t)^2dt\right )}\\
\\
&=& \sqrt{\displaystyle\frac{1}{T} \left(\displaystyle \int_{0}^{T_1}V_M^2dt+\displaystyle \int_{T_1}^{T}0\right )}\\
\\
&=& \sqrt{\displaystyle\frac{1}{T} \displaystyle \int_{0}^{T_1}V_M^2dt}
\end{eqnarray}

$$V_M^2$$ is not a variable that changes with time $$t$$, but a constant. Therefore, it can be taken out of the integration. As a result, the following equation is obtained.

\begin{eqnarray}
V_{RMS} &=& \sqrt{\displaystyle\frac{1}{T_1} V_M^2\displaystyle \int_{0}^{T}1dt}\\
\\
&=& V_M\sqrt{\displaystyle\frac{1}{T_1} \displaystyle \int_{0}^{T}1dt}
\end{eqnarray}

When calculating the above equation, the following formula is obtained.

\begin{eqnarray}
V_{RMS} &=& V_M\sqrt{\displaystyle\frac{1}{T} \left[t\right]_{0}^{T_1}}\\
\\
&=& V_M\sqrt{\displaystyle\frac{1}{T} \left(T_1-0\right)}\\
\\
&=& \sqrt{\displaystyle\frac{T_1}{T}}V_M
\end{eqnarray}

Therefore, the root mean square (RMS) value $$V_{RMS}$$ of the pulse wave is as follows.

\begin{eqnarray}
V_{RMS} =\sqrt{\displaystyle\frac{T_1}{T}}V_M
\end{eqnarray}

## Average Value of a Pulse Wave

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

Formula to find the average value

\begin{eqnarray}
V_{AVE} &=& \displaystyle\frac{1}{T} \displaystyle \int_{0}^{T}|v(t)|dt
\end{eqnarray}

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

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 pulse wave, there is no negative region, so the formula for $$v(t)$$ and the formula for the absolute value $$|v(t)|$$ of $$v(t)$$ become the same.

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

\begin{eqnarray}
|v(t)| = \begin{cases}
V_M & \left(0 \leq t \lt T_1\right) \\
\\
0 & \left(T_1 \leq t \lt T\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}{T} \displaystyle \int_{0}^{T}|v(t)|dt\\
\\
&=& \displaystyle\frac{1}{T} \left(\displaystyle \int_{0}^{T_1}|v(t)|dt+\displaystyle \int_{T_1}^{T}|v(t)|dt\right )\\
\end{eqnarray}

Since the pulse wave is zero in the region $$T_1 \leq t \lt T$$, the above equation becomes:

\begin{eqnarray}
V_{AVE} &=& \displaystyle\frac{1}{T} \left(\displaystyle \int_{0}^{T_1}|v(t)|dt+\displaystyle \int_{T_1}^{T}0dt\right )\\
\\
&=&\displaystyle\frac{1}{T} \displaystyle \int_{0}^{T_1}|v(t)|dt\\
\end{eqnarray}

In the above equation, when substituting $$v(t)=V_M$$, the following formula is obtained.

\begin{eqnarray}
V_{AVE} &=& \displaystyle\frac{1}{T} \displaystyle \int_{0}^{T_1}V_Mdt\\
\\
&=& \displaystyle\frac{1}{T} V_M\displaystyle \int_{0}^{T_1}1dt\\
\\
&=& \displaystyle\frac{1}{T}V_M \left[t\right]_{0}^{T_1}\\
\\
&=& \displaystyle\frac{1}{T}V_M \left(T_1-0\right)\\
\\
&=&\displaystyle\frac{T_1}{T}V_M
\end{eqnarray}

Therefore, the average value $$V_{AVE}$$ of the pulse wave is as follows.

\begin{eqnarray}
V_{AVE} &=& \displaystyle\frac{T_1}{T}V_M
\end{eqnarray}

#### Summary

In this article, the following information on the "Pulse Wave" was explained.

• How to obtain RMS Value and Average Value

Thank you for reading.