Sine Pulse RMS Value (Full-Wave Rectification)

Calculator and formulas for calculating the RMS and mean value of sine pulses

Sine Pulse Calculator (Full-Wave)

Full-Wave Rectification

This function calculates the RMS and mean value of a sine pulse from a full-wave rectification. Both half-waves are rectified.

V
Results
RMS voltage:
Mean voltage:

Full-Wave Rectification

Sine pulse after full-wave rectification

Sine pulse after full-wave rectification - both half-waves rectified

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

Example calculations

Practical calculation examples

Example 1: Standard full-wave rectification

Given: Us = 10V (peak voltage of the sine wave)

\[U_{eff} = \frac{10V}{\sqrt{2}} = \frac{10V}{1{,}414} = 7{,}07V\]
\[U_m = \frac{2 \cdot 10V}{\pi} = \frac{20V}{3{,}14159} = 6{,}37V\]
Optimal energy utilization with 70.7% RMS value
Example 2: Rectified mains voltage

Given: Us = 325V (peak of 230V mains voltage)

\[U_{eff} = \frac{325V}{\sqrt{2}} = 230V\]
\[U_m = \frac{2 \cdot 325V}{\pi} = 207V\]
Full utilization of mains voltage
Example 3: Low voltage power supply

Given: Us = 17V (12V transformer with √2 factor)

\[U_{eff} = \frac{17V}{\sqrt{2}} = 12{,}0V\]
\[U_m = \frac{2 \cdot 17V}{\pi} = 10{,}8V\]
Typical 12V power supply after rectification
Ratios for full-wave rectification
RMS ratio:
Ueff / Us: 1/√2 ≈ 0.707
Percent: ≈ 70.7%
Factor: 0.707
Mean value ratio:
Um / Us: 2/π ≈ 0.637
Percent: ≈ 63.7%
Factor: 0.637

Theory of Full-Wave Rectification

What is full-wave rectification?

In full-wave rectification, both the positive and negative parts of the sinusoidal AC voltage are used, so the entire wave is converted into positive voltages. This leads to significantly better energy utilization compared to half-wave rectification.

RMS value after full-wave rectification

The RMS value is defined as the DC value with the same thermal effect as the considered AC value. The RMS voltage after full-wave rectification is the square root of the mean square value over the entire period. Since both half-waves are used, the RMS value remains the same as the original sine wave.

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

The RMS value is the same as for a normal sine wave (≈ 70.7%).

Mean value after full-wave rectification

The mean voltage (average voltage) after full-wave rectification is calculated from the mean value of the entire positive and negative half-waves. Since both half-waves are rectified, the mean value is twice as high as for half-wave rectification.

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

The mean value is about 63.7% of the peak voltage.

Mathematical derivation

Calculation of the RMS value

For a full-wave rectified sine over a full period T:

\[U_{eff} = \sqrt{\frac{1}{T} \int_0^T u^2(t) \, dt}\]
For 0 ≤ t ≤ T/2: u(t) = Us sin(ωt) (positive half-wave)
For T/2 < t ≤ T: u(t) = Us |sin(ωt)| (negative half-wave rectified)
\[U_{eff} = \sqrt{\frac{1}{T} \int_0^T U_s^2 \sin^2(\omega t) \, dt} = \frac{U_s}{\sqrt{2}}\]
Calculation of the mean value

The mean value is calculated over a full period:

\[U_m = \frac{1}{T} \int_0^T |u(t)| \, dt\]
\[U_m = \frac{2}{T} \int_0^{T/2} U_s \sin(\omega t) \, dt = \frac{2 \cdot U_s}{\pi}\]

Practical applications

Power supplies
  • Standard rectifiers
  • High efficiency
  • Low ripple voltage
  • Good transformer utilization
Industrial applications
  • DC motor drives
  • Battery chargers
  • Electroplating systems
  • Welding equipment
Electronics
  • Amplifier power supplies
  • LED drivers
  • Voltage regulator inputs
  • Laboratory power supplies

Comparison of rectification types

Rectification comparison
No rectification:
Ueff = Us/√2 ≈ 0.707
Um = 0V
AC
Half-wave rectification:
Ueff = Us/2 = 0.5
Um = Us/π ≈ 0.318
50% efficiency
Full-wave rectification:
Ueff = Us/√2 ≈ 0.707
Um = 2Us/π ≈ 0.637
100% efficiency

Advantages of full-wave rectification

Electrical advantages:
  • Double efficiency: 100% vs. 50% for half-wave
  • Lower ripple: 100Hz vs. 50Hz fundamental frequency
  • Better utilization: Transformer and diodes
  • Higher voltage: Double mean value
Practical advantages:
  • Smaller filters: Less capacitor capacity required
  • Better regulation: More even DC voltage
  • Less heating: Lower losses
  • Standard solution: In most power supplies

Design notes

Practical considerations
  • Circuit types: Bridge rectifier or center-tap circuit
  • Diode voltage rating: At least 1.4 × Us (bridge) or 2.8 × Us (center-tap)
  • Current load: More even distribution on the diodes
  • Transformer design: Better utilization of the iron core
  • Filter design: Capacitor for 100Hz (i.e. 2 × fmains) dimensioning
  • Cost efficiency: Higher initial investment, but better overall efficiency

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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  •