RMS value of a rectangular pulse voltage

Calculator and formulas for calculating the RMS value of a rectangular pulse voltage

Rectangular Pulse Calculator

Rectangular Pulse Voltage

Enter the values for pulse duration (ti), period duration (T), and the peak voltage Us of the pulse.

ms
ms
V
Results
RMS voltage:
Mean voltage:

Rectangular pulse & parameters

Rectangular pulse
Parameters
\(\displaystyle U_s\) = Peak voltage [V]
\(\displaystyle U_{eff}\) = RMS voltage [V]
\(\displaystyle U_m\) = Mean voltage [V]
\(\displaystyle T\) = Period duration [ms]
\(\displaystyle t_i\) = Pulse duration [ms]
Basic formulas
\[U_{eff} = \sqrt{\frac{t_i}{T}} \cdot U_s\]
\[U_m = \frac{U_s \cdot t_i}{T}\]

Example calculations

Practical calculation examples

Example 1: 50% duty cycle

Given: Us = 10V, ti = 50ms, T = 100ms

\[U_{eff} = 10V \cdot \sqrt{\frac{50}{100}} = 10V \cdot \sqrt{0{,}5} = 10V \cdot 0{,}707 = 7{,}07V\]
\[U_m = \frac{10V \cdot 50ms}{100ms} = 5{,}0V\]
50% duty cycle - common in digital technology
Example 2: 20% duty cycle

Given: Us = 10V, ti = 20ms, T = 100ms

\[U_{eff} = 10V \cdot \sqrt{\frac{20}{100}} = 10V \cdot \sqrt{0{,}2} = 10V \cdot 0{,}447 = 4{,}47V\]
\[U_m = \frac{10V \cdot 20ms}{100ms} = 2{,}0V\]
Low duty cycle - typical for PWM control
Example 3: PWM signal

Given: Us = 5V, ti = 3ms, T = 10ms (30% PWM)

\[U_{eff} = 5V \cdot \sqrt{\frac{3}{10}} = 5V \cdot \sqrt{0{,}3} = 5V \cdot 0{,}548 = 2{,}74V\]
\[U_m = \frac{5V \cdot 3ms}{10ms} = 1{,}5V\]
Typical PWM signal in motor control
Duty cycle effects
RMS ratio:
Ueff / Us: √(ti/T)
At 50%: 0.707
At 25%: 0.5
At 10%: 0.316
Mean value ratio:
Um / Us: ti/T
At 50%: 0.5
At 25%: 0.25
At 10%: 0.1

Formulas for rectangular pulse

What is a rectangular pulse voltage?

The RMS value of a rectangular pulse voltage depends on how the voltage changes over time. A rectangular 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 pulse length can vary.

Definition of RMS value

The RMS value is defined as the DC value with the same thermal effect as the considered AC value. Suppose the voltage has a positive value Us for part of the time and is 0V for the rest of the time, then the RMS value of the rectangular pulse voltage is calculated as:

RMS value
\[U_{eff} = \sqrt{\frac{t_i}{T}} \cdot U_s\]

Depends on the duty cycle ti/T.

Mean value
\[U_m = \frac{U_s \cdot t_i}{T}\]

Proportional to the duty cycle.

Mathematical derivation

Calculation

For a rectangular pulse over a period T:

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

Practical applications

Digital technology
  • Clock signals
  • PWM control
  • Logic levels
  • Trigger signals
Power electronics
  • Switching regulators
  • Motor control
  • LED dimmers
  • DC-DC converters
Measurement technology
  • Oscilloscopes
  • Signal generators
  • Pulse width analysis
  • Duty cycle measurement

Duty cycle and properties

Duty cycle

The duty cycle D defines the ratio of pulse duration to period duration:

\[D = \frac{t_i}{T} \quad \text{(in %: } D_{\%} = \frac{t_i}{T} \cdot 100\% \text{)}\]
D = 10%:
Ueff = 0.316 · Us
Um = 0.1 · Us
D = 25%:
Ueff = 0.5 · Us
Um = 0.25 · Us
D = 50%:
Ueff = 0.707 · Us
Um = 0.5 · Us
D = 90%:
Ueff = 0.949 · Us
Um = 0.9 · Us

Spectral properties

Harmonics in rectangular pulses

Rectangular pulses contain all harmonics with decreasing amplitude:

\[u(t) = \frac{U_s \cdot D}{1} + \frac{2U_s}{\pi} \sum_{n=1}^{\infty} \frac{\sin(n\pi D)}{n} \cos(n\omega t)\]
DC component: Us · D (= mean value)
Fundamental: 2Ussin(πD)/π
Harmonics: Amplitude ∝ sin(nπD)/n

Design notes

Practical considerations
  • Power dissipation: P = Ueff²/R - important for dimensioning
  • EMC: Rectangular pulses generate broadband interference
  • Filtering: Lower duty cycles require stronger filters
  • Switching times: Real pulses have finite rise/fall times
  • Measurement technology: Instrument bandwidth must be sufficient
  • Cooling: High duty cycles increase thermal load

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