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What is RMS value?

The size of an AC voltage or current expressed as the DC value that would dissipate the same average power in the same resistance. It is the square root of the mean of the square (root mean square); for a sine wave it is 1/√2 ≈ 0.707 times the peak value. Household AC of 100 V is an RMS value, with a peak of about 141 V. AC power calculations and phasor magnitudes normally use RMS values.

Read the beginner explainer: AC vs DC Explained: Impedance, Phase and Power Factor

Japanese name: 実効値

RMS value appears in 10 practice questions across 4 topics, including Circuit and electromagnetism basics, Circuit Analysis: AC, Linear Circuits and Transients, Power Electronics Circuits.

Parallel RLC circuit driven by a sinusoidal AC source V I IR IL IC R L C Y = 1/R + j(ωC − 1/(ωL))
Parallel RLC circuit driven by a sinusoidal AC source · View in diagrams

Field: Circuit Analysis: AC, Linear Circuits and Transients

Try it: questions about RMS value

Which is closest to the peak value (amplitude) of a sinusoidal AC voltage with an RMS value of 100 V?

  1. AAbout 141 V
  2. B100 V
  3. CAbout 71 V
Show answer and explanation

Answer: A. About 141 V

For a sine wave the RMS value is 1/√2 times the peak, so the peak is √2 times the RMS value: 100 V × √2 ≈ 141 V (141.4 V). The RMS value is the DC voltage that would dissipate the same power in the same resistor; AC voltmeter readings and nominal mains voltages (such as 100 V) are RMS values. The peak-to-peak value is about 283 V, so component voltage ratings must be judged against the peak, not the RMS value.

Which of the following expresses the sinusoidal AC voltage v(t) = 100√2 sin(ωt + π/6) [V] as a phasor (polar form) whose magnitude is the RMS value?

  1. A100∠60° V
  2. B100∠30° V
  3. C141∠30° V
Show answer and explanation

Answer: B. 100∠30° V

Method 1 (from the definition): the magnitude of the phasor is the RMS value, the peak value 100√2 V divided by √2, which is 100 V. The angle is the initial phase π/6 rad = 30°, so the phasor is 100∠30° V. Method 2 (check with complex numbers): 100∠30° = 100(cos 30° + j sin 30°) = 86.6 + j50 V; multiplying by √2 and e^(jωt) and taking the imaginary part gives back the original v(t). In circuit calculations it is usual to use the RMS value, not the peak value, as the magnitude of a phasor.

RMS value in the explanations

Questions that mention RMS value (10)

  1. Which is closest to the peak value (amplitude) of a sinusoidal AC voltage with an RMS value of 100 V?
  2. Which of the following expresses the sinusoidal AC voltage v(t) = 100√2 sin(ωt + π/6) [V] as a phasor (polar form) whose magnitude is the RMS value?
  3. In the parallel RLC circuit in the figure, the RMS currents in the resistor R, the coil L and the capacitor C are IR = 4 A, IL = 5 A and IC = 2 A. Which of the following is the RMS value of the current I drawn from the source?
  4. In the series RLC circuit in the figure, R = 10 Ω, ωL = 30 Ω, 1/(ωC) = 20 Ω and the source voltage is V = 100 V RMS. Which pair correctly gives the RMS value of the current I and its phase relative to the source voltage?
  5. A three-phase inverter on a 400 V DC supply is operated with sinusoidal PWM at a modulation index of 1. Which of the following is closest to the RMS value of the fundamental of the output line-to-line voltage?
  6. What is the frequency of a sinusoidal AC voltage whose instantaneous value is v = 10 sin(100πt) [V]? (t in seconds)
  7. When a sinusoidal AC voltage of 100 V RMS is applied to a 50 Ω resistor, what is the average power dissipated?
  8. What is the maximum quantization error of an ideal ADC? (LSB: the voltage corresponding to one least significant bit)
  9. Which is the approximate formula for the signal-to-noise ratio (SNR) of an ideal N-bit ADC, set by quantization noise? (For a full-scale sine-wave input)
  10. A sinusoidal AC voltage of 100 V RMS is full-wave rectified with an ideal diode bridge. Without a smoothing capacitor, which of the following is closest to the average output voltage?

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Related terms

Phasor Inverter Sinusoidal PWM Three-phase AC Quantization error Thermal noise (Johnson noise) Jitter ENOB (effective number of bits) Smoothing capacitor Rectifier circuit

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