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Circuits, electronics and noise
RMS output of a half-wave rectifier
True RMS value of an unfiltered half-wave rectified sine, including its DC component, for an ideal diode and a resistive load.
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- ELN-12
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- v1
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- Circuits, electronics and noise
- Result
- Vrms [V]
The RMS of a half-wave rectified sine, over the whole input cycle, is Vp/2. The squared signal has an average of Vp²/4: half of the Vp²/2 of a full sine, because half the time the output is zero. This is the true RMS including the DC part, and is what a resistive load sees as heating.
True RMS value of an unfiltered half-wave rectified sine, including its DC component, for an ideal diode and a resistive load. It is larger than the average because the wave peaks.
Calculator
- Average value, Vp/π3.82 V
- RMS of the full sine, Vp/√28.485 V
- Form factor, RMS over average (π/2)1.571
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 12 V peak half-wave signal has an RMS of 6 V, so a resistive load dissipates 0.36 W per 100 Ω.
- The RMS (6 V) is larger than the average (3.82 V), and the ratio of 1.571 is the form factor of this waveform.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Vp Sine peak voltage | 0.5 to 500 V | 0.25 | 250 | 249.8 | +10% | +10 % |
Sweep of Sine peak voltage
| Sine peak voltage (V) | RMS output voltage (V) |
|---|---|
| 0.5 | 0.25 |
| 1.186 | 0.5928 |
| 2.812 | 1.406 |
| 6.668 | 3.334 |
| 15.81 | 7.906 |
| 37.49 | 18.75 |
| 88.91 | 44.46 |
| 210.8 | 105.4 |
| 500 | 250 |
Sweep one variable
Variables and how each one contributes
RMS output voltage
What the equation gives you. Everything else feeds it.
Sine peak voltage
The peak voltage of the sinusoidal source, in V. Scales the RMS value in direct proportion. Raising it by 10 % (to 13.2) moves the result by +10 %, from 6 to 6.6 V.
Worked example with the values above
- Square the peak: Vp² = 144.
- The mean square over a full cycle is Vp²/4 = 36 V².
- Vrms = √36 = 6 V.
Assumptions and limits
- Ideal diodes with zero forward drop for the rectifier formulas, and a resistive load.
- A sinusoidal supply and, for the ripple formula, a small ripple compared with the peak.
Common mistakes
- Using the peak value where an RMS or average value is needed.
- Forgetting that a full-wave rectifier recharges twice per supply cycle, so the ripple frequency is 2f.
- Ignoring the minimum Zener current, or its power rating.
Statistics and uncertainty
Rectifier formulas assume an ideal diode, so a real diode drop of about 0.7 V lowers the results for small peaks. Zener results depend on the supply, the Zener voltage tolerance and the load current, so use worst cases.
Where it is used
- Power supplies, battery chargers and meter circuits.
- Simple voltage references and regulators for sensors and microcontroller inputs.
Maths you need
- Average and RMS values: the mean and root-mean-square of a waveform
- Diode: conducts in one direction
- Capacitor discharge: ΔV = I Δt / C