RMS and Mean Value of a Sine Wave with Offset

Calculator and formulas for calculating the RMS and mean value of a sine wave with offset

Sine Wave with Offset Calculator

Sine Wave with Offset

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

V
V
Results
RMS voltage (total):
Offset voltage (DC):

Sine Wave with Offset & Parameters

Sine wave with offset
Parameters
\(\displaystyle U_s\) = Peak voltage [V]
\(\displaystyle U_{eff}\) = RMS voltage [V]
\(\displaystyle U_0\) = Offset (DC) [V]
\(\displaystyle U_m\) = Mean value [V]
Basic formulas
\[U_{eff} = \sqrt{U_0^2 + \frac{U_s^2}{2}}\]
\[U_m = U_0\]
\[U_s = U_{eff} \cdot \sqrt{2}\] (if U_0 = 0)

Example calculations

Practical calculation examples

Example 1: Sine wave with offset

Given: Us = 10V, U0 = 5V

\[U_{eff} = \sqrt{5^2 + \frac{10^2}{2}} = \sqrt{25 + 50} = \sqrt{75} = 8.66V\]
\[U_m = 5V\]
RMS and mean value with DC offset
Example 2: Pure sine wave (no offset)

Given: Us = 10V, U0 = 0V

\[U_{eff} = \sqrt{0^2 + \frac{10^2}{2}} = \sqrt{50} = 7.07V\]
\[U_m = 0V\]
Standard sine wave without offset
Example 3: Only DC voltage

Given: Us = 0V, U0 = 5V

\[U_{eff} = \sqrt{5^2 + 0} = 5V\]
\[U_m = 5V\]
Only DC voltage, no AC component
Ratios for sine wave with offset
RMS value:
Ueff: \(\sqrt{U_0^2 + U_s^2/2}\)
Mean value:
Um: U0

Formulas for Sine Wave with Offset

What is a sine wave with offset?

A sine wave with offset is a sinusoidal voltage that is superimposed by a DC voltage. The offset shifts the entire waveform up or down. The RMS value and mean value are calculated as follows.

Definition of RMS and mean value

The RMS value is defined as the DC value with the same thermal effect as the considered AC value. The mean value is the average value over a period. For a sine wave with offset, the mean value corresponds to the offset.

Calculate RMS value
\[U_{eff} = \sqrt{U_0^2 + \frac{U_s^2}{2}}\]

The RMS value is always greater than or equal to the offset.

Calculate mean value
\[U_m = U_0\]

The mean value is equal to the offset (DC component).

Mathematical derivation

Calculation of RMS and mean value

For a sine wave with offset: u(t) = U0 + Us · sin(ωt)

\[U_{eff} = \sqrt{\frac{1}{T} \int_0^T (U_0 + U_s \sin(\omega t))^2 \, dt}\]
\[U_{eff} = \sqrt{U_0^2 + \frac{U_s^2}{2}}\]
\[U_m = U_0\]

Practical applications

Power supplies
  • Rectifiers with residual ripple
  • DC offset in amplifiers
  • Signal biasing
  • Measurement technology
Signal technology
  • Audio signals with DC offset
  • Sensor signals
  • Modulation with offset
  • Reference signals
Measurement technology
  • Oscilloscope measurements
  • True-RMS meters
  • Mean value meters
  • Data acquisition

Measurement aspects

Important measurement notes

When measuring signals with offset, it is important to distinguish between RMS value and mean value. True-RMS meters measure the correct RMS value, while mean value meters only display the mean value.

True-RMS meters:
Measure correct RMS values
even for signals with offset
Mean value meters:
Only display the mean value
not the RMS value
Oscilloscopes:
Show both RMS and mean value
depending on settings

Design notes

Important considerations
  • Offset: Can affect the operation of amplifiers and circuits
  • Measurement: Use the correct instrument for RMS or mean value
  • Signal quality: Offset can distort signal processing
  • Safety: High offset can be dangerous in power circuits

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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  •