RMS and Peak Value of a Sine Wave

Calculator and formulas for calculating the RMS or peak value of a sine wave

Sine RMS-Peak Voltage

Sine Wave

The voltage can be entered as RMS or peak value. The input of the peak value is preset.

V
Results
RMS voltage:
Peak voltage:

Sine Wave & Parameters

Sine wave
Parameters
\(\displaystyle U_s\) = Peak voltage [V]
\(\displaystyle U_{eff}\) = RMS voltage [V]
\(\displaystyle U_m\) = Mean voltage = 0V (sinusoidal)
Basic formulas
\[U_{eff} = \frac{U_s}{\sqrt{2}}\]
\[U_s = U_{eff} \cdot \sqrt{2}\]

Example calculations

Practical calculation examples

Example 1: Household voltage

Given: Ueff = 230V (mains voltage)

\[U_s = 230V \cdot \sqrt{2} = 230V \cdot 1{,}414 = 325{,}3V\]
Peak voltage in household mains
Example 2: Signal generator

Given: Us = 10V (peak voltage)

\[U_{eff} = \frac{10V}{\sqrt{2}} = \frac{10V}{1{,}414} = 7{,}07V\]
Typical test signal in electronics
Example 3: Audio signal

Given: Ueff = 1.41V (audio level)

\[U_s = 1{,}41V \cdot \sqrt{2} = 1{,}41V \cdot 1{,}414 = 2{,}0V\]
Typical audio signal amplitude
Ratios for sine wave
RMS ratio:
Ueff / Us: 1/√2 ≈ 0.707
Percent: ≈ 70.7%
Factor: 0.707
Peak value ratio:
Us / Ueff: √2 ≈ 1.414
Percent: ≈ 141.4%
Factor: 1.414

Formulas for Sine Wave

What is a sine wave?

The RMS value is defined as the DC value with the same thermal effect as the considered AC value. For a sinusoidal AC, it is the characteristic value of 1/√2 of the peak value.

Definition of RMS value

In a sine wave, the voltage oscillates harmonically between -Us and +Us. The mean value of the pure sine wave is always 0 volts. If the voltage is superimposed by a DC voltage, the mean value corresponds to the superimposed DC voltage.

Calculate RMS value
\[U_{eff} = \frac{U_s}{\sqrt{2}}\]

The RMS value is about 70.7% of the peak voltage.

Calculate peak value
\[U_s = U_{eff} \cdot \sqrt{2}\]

The peak value is about 141.4% of the RMS value.

Mathematical derivation

Calculation of the RMS value

For a sine wave u(t) = Us · sin(ωt) over a period T:

\[U_{eff} = \sqrt{\frac{1}{T} \int_0^T u^2(t) \, dt}\]
\[U_{eff} = \sqrt{\frac{1}{T} \int_0^T U_s^2 \sin^2(\omega t) \, dt}\]
\[U_{eff} = \frac{U_s}{\sqrt{2}}\]

Practical applications

Power engineering
  • Mains voltage (230V RMS)
  • Transformers
  • Generators
  • Motor control
Signal technology
  • Signal generators
  • Oscillators
  • Modulation carriers
  • Reference signals
Audio/RF technology
  • Audio amplifiers
  • Radio transmission
  • Antenna signals
  • Measuring instruments

Measurement aspects

Important measurement notes

Moving-coil meters can only measure the half-wave mean value, but due to the corresponding calibration of the scale, they display the RMS value. If a non-sinusoidal value is measured, incorrect readings are obtained.

True-RMS meters:
Measure correct RMS values
even for distorted signals
Mean value meters:
Only correctly calibrated
for pure sine signals
Oscilloscopes:
Show peak values
Conversion required

Spectral properties

Pure sine wave

An ideal sine voltage contains only a single frequency component:

\[u(t) = U_s \sin(\omega t + \phi)\]
Fundamental frequency: Contains only the carrier frequency
Harmonics: No harmonics in an ideal sine wave
THD: Total Harmonic Distortion = 0% (ideal)
Crest factor: Us/Ueff = √2 ≈ 1.414

Comparison with other waveforms

Crest factors of different signals
Peak-to-RMS ratio:
Sine wave: √2 ≈ 1.414
Triangle wave: √3 ≈ 1.732
Sawtooth wave: √3 ≈ 1.732
Square wave: 1.0
Practical significance:
Dimensioning: Peak voltage for insulation
Power dissipation: RMS value for heat calculation
Measuring range: Crest factor for headroom
Distortion: Deviation from ideal crest factor

Design notes

Important considerations
  • Insulation: Dimension voltage resistance according to peak value
  • Heating: Calculate power dissipation according to RMS value
  • Measurement accuracy: Use true-RMS meters for distorted signals
  • Overdrive: Consider crest factor for amplifiers
  • Power quality: Harmonics reduce ideal sine waveform
  • Safety: Design touch protection according to peak voltage

Is this page helpful?            
Thank you for your feedback!

Sorry about that

How can we improve it?


AC functions

Alternating voltage values  •  Alternating voltage and time  •  Frequency and wavelength  •  Alternating voltage value and angle  •  Frequency and periodic time  •  RMS value of a sinusoidal oscillation  •  RMS value of a sinusoidal oscillation with offset  •  RMS value of a sine pulse (half-wave rectification)  •  RMS value of a sine pulse (full-wave rectification)  •  RMS value of a square wave voltage  •  RMS value of a square pulse  •  RMS value of a triangle voltage  •  RMS value of a triangular pulse  •  RMS value of a sawtooth voltage  •  RMS value of a sawtooth pulse  •