RMS value of a triangular voltage

Calculator and formulas for calculating the RMS value of a triangular voltage

Triangular Voltage Calculator

Symmetrical triangular voltage

This function calculates the RMS value of a symmetrical triangular voltage. The mean value is always 0 volts for a symmetrical voltage.

Peak voltage U_s

Decimal places

Result

RMS voltage:

Triangular voltage & parameters

Parameters

\(\displaystyle U_S\) = Peak voltage [V]

\(\displaystyle U_{eff}\) = RMS voltage [V]

\(\displaystyle U_m\) = Mean voltage = 0V (symmetrical)

Basic formula

\[U_{eff} = \frac{U_S}{\sqrt{3}}\]

The RMS value is about 57.7% of the peak voltage.

Example calculations

Practical calculation examples

Example 1: Standard triangular voltage

Given: U_s = 10V

\[U_{eff} = \frac{10V}{\sqrt{3}} = \frac{10V}{1{,}732} = 5{,}77V\]

The RMS value is about 57.7% of the peak value

Example 2: Audio signal generator

Given: U_s = 5V (typical test signal)

\[U_{eff} = \frac{5V}{\sqrt{3}} = 2{,}89V\]

Used in measurement technology and signal generation

Example 3: Power electronics

Given: U_s = 325V (mains voltage peak value)

\[U_{eff} = \frac{325V}{\sqrt{3}} = 187{,}6V\]

Application in triangle PWM and switching regulators

Ratios for triangular voltage

RMS ratio:

U_eff / U_s: 1/√3 ≈ 0.577

Percent: ≈ 57.7%

Factor: 0.577

Mean value:

U_m: 0V (symmetrical)

Positive half-wave: Linear rising

Negative half-wave: Linear falling

Formula for triangular voltage

What is a triangular voltage?

The RMS value of a triangular voltage (also called a triangular wave) can be easily calculated, as it has a regular symmetrical shape and the RMS value can be calculated directly from the peak voltage U_S.

Definition of RMS value

The RMS value is defined as the DC value with the same thermal effect as the considered AC value. For symmetrical triangular voltages, it is calculated using the following formula:

RMS value formula

\[U_{eff} = \frac{U_S}{\sqrt{3}}\]

The mean value of the voltage is always 0 volts for symmetrical triangular voltages.

Properties of the triangular wave

The triangular wave has a linear rise (during the positive half-wave) and a linear fall (during the negative half-wave). The RMS value is calculated from the quadratic mean value over the entire period of the triangular voltage. The RMS value is therefore less than the peak value, and the factor 1/√3 describes the ratio between peak value and RMS value.

Mathematical derivation

For a symmetrical triangular wave with period T:

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

\[U_{eff} = \frac{U_S}{\sqrt{3}}\]

Comparison with other signals

Sine voltage: U_eff = U_s/√2 ≈ 0.707

Triangular voltage: U_eff = U_s/√3 ≈ 0.577

Square voltage: U_eff = U_s = 1.0

Sawtooth voltage: U_eff = U_s/√3 ≈ 0.577

Practical applications

Signal generation

Function generators
VCO (Voltage Controlled Oscillator)
Test signals
Modulation methods

Power electronics

PWM control
Triangle comparators
Switching regulators
Inverters

Measurement technology

Calibration signals
Sweep generators
Integrator test
Linearity measurement

Spectral properties

Harmonic components

A triangular voltage contains only odd harmonics with decreasing amplitude:

\[u(t) = \frac{8U_S}{\pi^2} \sum_{n=1,3,5...}^{\infty} \frac{(-1)^{(n-1)/2}}{n^2} \sin(n\omega t)\]

Fundamental: 8U_S/(π²) ≈ 0.811 · U_S

3rd harmonic: -8U_S/(9π²) ≈ -0.090 · U_S

5th harmonic: 8U_S/(25π²) ≈ 0.032 · U_S

Design notes

Important properties

Linearity: Linear slopes facilitate mathematical treatment
Symmetry: Mean value is always zero for symmetrical signals
Harmonics: Faster decay of harmonics than with square voltage
Integration: Triangle → parabola; differentiation: triangle → square
Filter behavior: Better HF properties than square voltage
EMC: Lower interference emission due to less steep slopes

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