RMS Value Of An Alternating Current: Formula, Derivations & Examples

RMS value of alternating current is equal to the amount of DC current which produces the same heating effect flowing through the same resistance during the same time. RMS Value is provided by Direct current flowing through a resistance. RMS value or Root Mean Square value is the amount of steady alternating current that is flowing through a given conductor for a given period of time. The formula for finding RMS value of AC is given by,

\(I_{rms} = \frac{I_0}{\sqrt{2}} = 0.707 I_0\)

Key Terms: RMS Value of Alternating Current, Resistance, Sine Current Wave, Sinusoidal, Non-sinusoidal, Alternating current, Dc current, Conductor

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What is an Alternating Current?

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Alternating current is described as a current whose magnitude varies over time and periodically reverses direction. The peak value of an alternating current is referred to as IoIo.

The general equation is given by,

II = I_o sin ωtI_o sin ωt

II = I_o cos ωtI_o cos ωt

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RMS Value of Alternating Current

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• The Root-Mean-Square of instantaneous current values is known as RMS.
• Alternating current is defined as the value of steady current that, when sustained across the same resistance for the same amount of time, generates the same amount of heat as A.C. current when maintained across the same resistance for the same amount of time. 

• AC has an RMS value that is higher than the average.
• The area covered in half-cycle can be used to calculate the RMS value of a sine current wave. This holds true for all waves, including sinusoidal, non-sinusoidal, symmetrical, and asymmetrical waves.
• The R.M.S. value of alternating current is also known as the effective value or virtual value. I_rms, I_eff, or I_v are the characters that represent it.

RMS Value of Alternating Current

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RMS Value of Alternating Voltage

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The RMS value of Alternating Voltage is:

• Alternating voltage's root mean square or R.M.S. value, is defined as the value of stable Potential Difference that would generate the same amount of heat in a given resistance in a given time, as A.C. voltage would when sustained across the same resistance for the same time.
• The effective value or virtual value of alternating voltage is also known as the R.M.S. value. V_rms, V_eff, or V_v are the symbols for it.

Formula for RMS Value of Alternating Current

RMS Value of AC is given by the formula,

\(I_{rms} = \frac{I_0}{\sqrt{2}} = 0.707 I_0\)

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Expression for RMS value of Alternating Voltage

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On the same lines, the RMS value of Alternating Voltage can be calculated. Alternating Voltage produces a modest amount of heat in a short period of time.

\(dt = \frac{(E_0 Sin \omega t)^2}{R} dt\)

We get E v 2T/R by integrating it from 0 to T and equating it with the heat produced by DC at the same time.

\(E_v = \frac{E_0}{\sqrt{2}}\)

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RMS Value of AC Derivation

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RMS Value of AC Derivation

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Things to Remember

• Alternating current is defined as a current whose amplitude changes over time and reverses direction on a regular basis.
• The RMS value of AC is higher than the average.
• The effective value or virtual value of alternating current is also known as the R.M.S. value. It is represented by the symbols Irms, Ieff, or Iv.
• The R.M.S. value refers to the effective or virtual value of alternating voltage. The RMS value is also denoted through these symbols Vrms, Veff, or Vv.
• The formula of RMS value of AC is: \(I_{rms} = \frac{I_0}{\sqrt{2}} = 0.707 I_0\)

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Previous Year Questions 

1. If the current in the primary of the transformer is 5A , and its efficiency is 90%, the output current would be….[JEE Main 2018]
2. A series LCR circuit is connected to an ac voltage source… [NEET 2020]
3. A transformer having efficiency of 90% is working on 200V and 3kW power… [NEET 2014]
4. An AC voltage is applied to a resistance R and an inductor L in series… [NEET 2011]
5. A current of 5A is flowing at 220V in the primary coil of a transformer… [KCET 2008]
6. A step-up transformer operates on a 230 V line and a load current of… [KCET 2018]
7. An electric heater rated 220V and 550W is connected to AC mains… [KCET 2009]
8. In R−L−C series circuit, the potential differences across each element is… [KCET 2013]
9. An inductor 20mH, a capacitor 50μF and a resistor 40Ω are connected in… [NEET 2016]
10. In the given circuit the reading of voltmeter V1 and V2 are 300 volts each… [NEET 2010]
11. Which of the following combinations should be selected for better tuning of… [NEET 2016]
12. In an A.C. circuit, V and I are given by V=150 sin(150t) volt and… [KCET 2011]
13. A transformer works on the principle of… [KCET 2011]
14. A voltage VPQ=V0cosωt (where V0 is a real amplitude) is applied between… [KEAM]
15. A transformer is used to light 100W−110V lamp from 220V mains… [KCET 2015]
16. In an L−C−R circuit, at resonance… [KCET 2015]
17. The average power dissipated in a pure inductor is… [KCET 2015]
18. In a series LCR circuit an alternating emf(v) and current (i) are given by… [KCET 2016]
19. A coil is self-inductance L is connected in series with a bulb… [NEET 2013]
20. A condenser of capacity C is charged to a potential difference of V1… [NEET 2010]