RMS value of a square wave voltage

Calculator and formulas for calculating the RMS value of a square wave voltage

Square Wave Calculator

Square wave (50% duty cycle)

It is assumed that the pulse duration (ti) and the pause (tp) have the same length (50% duty cycle).

V
Results
RMS voltage:
Mean voltage:

Square wave & parameters

Square wave
Parameters
\(\displaystyle U_s\) = Peak voltage [V]
\(\displaystyle U_{eff}\) = RMS voltage [V]
\(\displaystyle U_m\) = Mean voltage [V]
\(\displaystyle t_i\) = Pulse duration (50% of period)
\(\displaystyle t_p\) = Pause duration (50% of period)
Formulas (50% duty cycle)
\[U_{eff} = \frac{U_s}{\sqrt{2}}\]
\[U_m = \frac{U_s}{2}\]

Example calculations

Practical calculation examples

Example 1: Standard square wave

Given: Us = 10V (50% duty cycle)

\[U_{eff} = \frac{10V}{\sqrt{2}} = \frac{10V}{1{,}414} = 7{,}07V\]
\[U_m = \frac{10V}{2} = 5{,}0V\]
Typical square wave with equal pulse and pause times
Example 2: Digital logic voltage

Given: Us = 5V (TTL level, 50% duty cycle)

\[U_{eff} = \frac{5V}{\sqrt{2}} = 3{,}54V\]
\[U_m = \frac{5V}{2} = 2{,}5V\]
Common in digital technology for clock signals
Example 3: Symmetrical AC square wave

Given: Us = 10V (±5V square AC voltage)

\[U_{eff} = U_s = 10V\] (for symmetrical AC square wave)
\[U_m = 0V\] (mean value is zero)
For ±Us we have Ueff = Us
Comparison of square wave types
Square pulse voltage (0V to Us):
Ueff: Us/√2 ≈ 0.707 · Us
Um: Us/2 = 0.5 · Us
Application: Digital technology, PWM
Square AC voltage (-Us to +Us):
Ueff: Us = 1.0 · Us
Um: 0V (symmetrical)
Application: Inverters

Formulas for square wave

What is a square wave voltage?

The RMS value of a square pulse voltage depends on how the voltage changes over time. A square pulse voltage usually has a fixed peak value Us and alternates between two voltage values over time, for example Us and 0 V or Us and -Us.

The calculator above computes the variant with Us and 0 V. The time at Us and 0 V (ti and tp) is 50% each. For a calculator with variable pulse length, see here.

Definition of RMS value

The RMS value is defined as the DC value with the same thermal effect as the considered AC value.

Square pulse voltage (0V to Us)

Square pulse voltage

The RMS value for square waves with equal pulse duration (ti) and pause (tp) is:

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

About 70.7% of the peak voltage.

Mean value
\[U_m = \frac{U_s}{2}\]

Half the peak voltage.

Square AC voltage (-Us to +Us)

Square AC voltage

For a symmetrical square AC voltage, the RMS value is equal to the peak value:

Symmetrical AC voltage
\[U_{eff} = U_s\]

Since the voltage in both cases has the same magnitude (but different sign), the RMS value remains equal to the amplitude of the square wave.

The mean value for a symmetrical square AC voltage is always 0 volts.

Mathematical derivation

Calculation of RMS value

For a square wave with 50% duty cycle over a period T:

\[U_{eff} = \sqrt{\frac{1}{T} \int_0^T u^2(t) \, dt}\]
For 0 ≤ t ≤ T/2: u(t) = Us
For T/2 < t ≤ T: u(t) = 0
\[U_{eff} = \sqrt{\frac{1}{T} \int_0^{T/2} U_s^2 \, dt} = U_s \sqrt{\frac{1}{2}} = \frac{U_s}{\sqrt{2}}\]

Practical applications

Digital technology
  • TTL/CMOS logic levels
  • Clock signals (50% duty cycle)
  • Data transmission
  • Microcontroller outputs
Power electronics
  • Inverters
  • DC-AC converters
  • H-bridges
  • Motor control
Signal processing
  • Modulation methods
  • Reference signals
  • Test signals
  • Calibration

Spectral properties

Harmonics in square waves

Square waves contain all odd harmonics:

\[u(t) = \frac{4U_s}{\pi} \sum_{n=1,3,5...}^{\infty} \frac{1}{n} \sin(n\omega t)\]
Fundamental: 4Us/π ≈ 1.273 · Us
3rd harmonic: 4Us/(3π) ≈ 0.424 · Us
5th harmonic: 4Us/(5π) ≈ 0.255 · Us

Design notes

Practical considerations
  • Power dissipation: P = Ueff²/R - reduced at 50% duty cycle
  • EMC: Square waves generate many harmonics
  • Filtering: Low-pass filters required for harmonic suppression
  • Switching times: Real square waves have finite edge times
  • Coupling: Transformers suitable for AC transmission
  • Measurement technology: True-RMS meters for correct RMS measurement

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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  •