The idea of

The average power absorbed by the resistor in the ac circuit is

while the power absorbed by the resistor in the dc circuit is

Equating the expressions in Equations.(1) and (2) and solving for

The effective value of the voltage is found in the same way as current; that is

This indicates that the effective value is the (square)

For any periodic function

and the square root (√) of that mean. The rms value of a constant is the constant itself. For the sinusoid

Similarly, for

Keep in mind that Equations.(8) and (9) are only valid for sinusoidal signals.

Before moving on, remember all the equations we had in Instantaneous Power and Average Power Formula.

The average power can be written in terms of the rms values

Similarly, the average power absorbed by a resistor R can be written as

When a sinusoidal voltage or current is specified, it is often in terms of its maximum (or peak) value or its rms value, since its average value is zero. The power industries specify phasor magnitudes in terms of their rms values rather than peak values. For instance, the 110 V available at every household is the rms value of the voltage from the power company.

It is convenient in power analysis to express voltage and current in their rms values. Also, analog voltmeters and ammeters are designed to read directly the rms value of voltage and current, respectively.

The period of the waveform is

The rms value is

The power absorbed by a 2 Ω resistor is

The period of the voltage waveform is T = 2π, and

The rms value is obtained as

But sin

The average power absorbed is

*effective value*arises from the need to measure the effectiveness of a voltage or current source in delivering power to a resistive load.Theeffective valueof a periodic current is the dc current that delivers the same average power to a resistor as the periodic current.

*Make sure to read what is ac circuit first.*## How to Calculate Root Mean Square Value

In Figure.(1), the circuit in (a) is ac while the circuit in (b) is dc. Our objective is to find*I*that will transfer the same power to resistor R as the sinusoid_{eff}*i*.Figure 1. Finding the effective current: (a) ac circuit, (b) dc circuit |

(1) |

(2) |

*I*, we obtain_{eff}(3) |

(4) |

*root*of the*mean*(or average) of the*square*of the periodic signal. Thus, the effective value is often known as the*root-mean square*value, or*rms*value for short; and we write(5) |

*x(t)*in general, the rms value is given by(6) |

TheEquation.(6) states that to find the rms value ofeffective valueof a periodic signal is its root mean square (rms) value.

*x(t)*, we first find its*square x*and then find the mean of that, or^{2}(7) |

*i(t)*=*I*cos ωt, the effective or rms value is_{m}(8) |

*v(t)*=*V*cos ωt,_{m}(9) |

Before moving on, remember all the equations we had in Instantaneous Power and Average Power Formula.

The average power can be written in terms of the rms values

(10) |

(11) |

It is convenient in power analysis to express voltage and current in their rms values. Also, analog voltmeters and ammeters are designed to read directly the rms value of voltage and current, respectively.

## Root Mean Square Value Examples

For better understanding let us review examples below :**1. Determine rms value of the current waveform in Figure.(2). If the current is passed through a 2 Ω resistor, find the average power absorbed by the resistor.**Figure 2 |

*Solution :*The period of the waveform is

*T*= 4. Over a period, we can write the current waveform asThe rms value is

The power absorbed by a 2 Ω resistor is

**2. The waveform is shown in Figure.(3) is a half-wave rectified sine wave. Find the rms value and the amount of average power dissipated in a 10 Ω resistor.**Figure 3 |

*Solution :*The period of the voltage waveform is T = 2π, and

The rms value is obtained as

But sin

^{2}*t*= ½(1 - cos 2*t*). Hence,The average power absorbed is

Have you understood what is root mean square? Don't forget to share and subscribe! Happy learning!

*Reference: Fundamentals of electric circuits by Charles K. Alexander and Matthew N. O. Sadiku*

Untuk Bahasa Indonesia baca Rumus Nilai Efektif Atau RMS.

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