In this short piece of article, we will derive an expression for the root mean square value of alternating current for a full cycle, so let’s get started…

## Root mean square (RMS) or virtual or effective value of AC

It is defined as the value of a direct current which produces the same heating effect in the given resistor as is produced by the given alternating current when passed for the same time. It is denoted as $I_{rms},\; I_v\;\text{or}\;I_{eff}$.

## RMS value of alternating current for a full cycle

Suppose an alternating current $I=I_0\sin\omega t$ is passed through a circuit of resistance $R$. Then, the amount of heat produced in small time $dt$ will be

$$ d H=I^{2} R d t $$ If $T$ is the time period of AC then heat produced in one complete cycle will be $$ H=\int_{0}^{T} I^{2} R d t $$ Let $I_{\text {eff }}$ be the effective value of AC Then heat produced in time $T$ must be

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$$ \therefore \quad I_{e f f}^{2} R T=\int_{0}^{T} I^{2} R d t \text { or } I_{e f f}^{2}=\frac{1}{T} \int_{0}^{T} I^{2} d t $$ |

But, $\displaystyle{\frac{1}{T} \int_{0}^{T} I^{2} d t}$ is the mean of the squares of the instantaneous values of AC over one complete cycle, hence the effective or virtual value of AC equals its root mean square value, i.e., $$ I_{e f f}=I_{r m s}=\sqrt{\frac{1}{T} \int_{0}^{T} I^{2} d t} $$ Now

$$ \begin{aligned} \int_{0}^{T} I^{2} d t &=\int_{0}^{T} I_{0}^{2} \sin ^{2} \omega t d t \\ &=I_{0}{ }^{2} \int_{0}^{T} \frac{1-\cos 2 \omega t}{2} d t \\ &=\frac{I_{0}{ }^{2}}{2}\left[t-\frac{\sin 2 \omega t}{2 \omega}\right]_{0}^{T} \\ &=\frac{I_{0}^{2}}{2}\left[(T-0)-\frac{1}{2 \omega} \mid \sin \frac{4 \pi}{T} t\mid_{0}^{T}\right] \\ &=\frac{I_{0}{ }^{2}}{2}\left[T-\frac{1}{2 \omega}(\sin 4 \pi-\sin 0)\right] \\ &=\frac{I_{0}{ }^{2}}{2}[\mathrm{~T}-0]=\frac{I_{0}{ }^{2} T}{2} \\ \therefore \quad I_{e f f} \text { or } I_{r m s} &=\sqrt{\frac{1}{T} \cdot \frac{I_{0}{ }^{2} T}{2}} \\ \text { or } \quad I_{e f f} \text { or } I_{r m s} &=\frac{1}{\sqrt{2}} I_{0}=0.707 I_{0} \end{aligned} $$ |

Thus the effective or RMS value of an AC is $\frac{1}{\sqrt{2}}$ times its peak value.

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## Frequently Asked Questions – FAQs

##### What is the value of RMS value?

The RMS value of a sinusoidal current (or any time-varying current) is **defined as the value of a direct current which produces the same heating effect in the given resistor as is produced by the given alternating current when passed for the same time**.

##### What is the peak value of AC?

The peak value of the alternating current is √2 times the root mean square value of the alternating current. i.e $$I_0=\sqrt{2}I_{rms}$$

##### What is the RMS value of the AC formula?

the effective or RMS value of an AC is $\frac{1}{\sqrt{2}}$ times its peak value. i.e $$I_{r m s} =\frac{1}{\sqrt{2}} I_{0}=0.707 I_{0}$$

##### What is the mean value of AC?

The mean value of an alternating current is **the total charge flown for one complete cycle divided by the time taken to complete the cycle** i.e. time period T. The magnitude of an alternating current changes from time to time and its direction also reverses after every half cycle.

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