Periodic currents and voltages deliver an average power to resistive loads. The amount of power that is delivered depends on the characteristics of the particular waveform. A method of comparing the power delivered by different waveforms is therefore very useful. One such method is the use of RMS or effective values for periodic currents or voltages.

Thus if I_{rms} is the RMS value of i, we may write

$P=RI_{rms}^{2}=\frac{1}{T}\int\limits_{0}^{T}{R{{i}^{2}}}dt$

From which the RMS current is

$\begin{matrix} {{I}_{rms}}=\sqrt{\frac{1}{T}\int\limits_{0}^{T}{{{i}^{2}}dt}} & \cdots & (1) \\\end{matrix}$

In a similar manner, it is easily shown that the rms voltage is

${{V}_{rms}}=\sqrt{\frac{1}{T}\int\limits_{0}^{T}{{{v}^{2}}dt}}$

The term rms is an abbreviation for root-mean-square. Inspecting (1), we see that we are indeed taking the square root of the average, or mean, value of the square of the current.

From the definition, the RMS value of a constant (dc) is simply the constant itself. The dc case is a special case (ω=0) of the most important type of waveform, the sinusoidal current or voltage.

Suppose we now consider a sinusoidal current $i={{I}_{m}}\cos (\omega t+\phi )$ . Then, from (1) and **Table 1**, we find

${{I}_{rms}}=\sqrt{\frac{\omega I_{m}^{2}}{2\pi }\int\limits_{0}^{{}^{2\pi }/{}_{\omega }}{{{\cos }^{2}}(\omega t+\phi )}}=\frac{{{I}_{m}}}{\sqrt{2}}$

$f(t)$ | $\int\limits_{0}^{{}^{2\pi }/{}_{\omega }}{f(t)dt,\text{ }\omega \ne \text{0}}$ |
---|---|

$1.\text{ sin(}\omega \text{t+}\alpha \text{),cos(}\omega \text{t+}\alpha \text{)}$ | $0$ |

$2.\text{ sin(n}\omega \text{t+}\alpha \text{),cos(n}\omega \text{t+}\alpha \text{)*}$ | $0$ |

$3.\text{ si}{{\text{n}}^{\text{2}}}\text{(}\omega \text{t+}\alpha \text{),co}{{\text{s}}^{\text{2}}}\text{(}\omega \text{t+}\alpha \text{)}$ | ${}^{\pi }/{}_{\omega }$ |

$4.\text{ sin(m}\omega \text{t+}\alpha \text{)cos(n}\omega \text{t+}\alpha \text{)*}$ | $0$ |

$5.\text{ cos(m}\omega \text{t+}\alpha \text{)cos(n}\omega \text{t+}\beta \text{)*}$ | $\left\{ \begin{matrix} \begin{matrix} 0, & m\ne n \\ \end{matrix} \\ \begin{matrix} \frac{\pi \cos (\alpha -\beta )}{\omega }, & m=n \\ \end{matrix} \\ \end{matrix} \right.$ |

Table 1: Integrals of Sinusoidal Functions and Their Products

Thus a sinusoidal current having an amplitude I_{m} delivers the same average power to a resistance R as does a dc current which is equal to$\frac{{{I}_{m}}}{\sqrt{2}}$. We also see that RMS current is independent of the frequency ω or the phase ϕ of the current i. similarly, in the case of a sinusoidal voltage, we find that

${{V}_{rms}}=\frac{{{V}_{m}}}{\sqrt{2}}$

Substituting these values into the following power relations,

\[\begin{align} & P=\frac{{{V}_{m}}{{I}_{m}}}{2}\cos \theta \\ & and \\ & P=\frac{1}{2}I_{m}^{2}\text{ }\operatorname{Re}(Z) \\\end{align}\]

We get,

$\begin{matrix} P={{V}_{rms}}{{I}_{rms}}\cos \theta & \cdots & (2) \\\end{matrix}$

And

$\begin{matrix} P=I_{rms}^{2}\operatorname{Re}(Z) & \cdots & (3) \\\end{matrix}$

In practice, RMS values are usually used in the fields of power generation and distribution. For instance, the nominal 115-V AC power which is commonly used for household appliances is an RMS value. This the power supplied to our homes is provided by a 60 Hz voltage having a maximum value of $115\sqrt{2}\approx 163V$ . On the other hand, maximum values are more commonly used in electronics and communications.