Alternating Current
Simple Explanation
Alternating current (AC) periodically reverses direction, following a sine wave v=V0sin(Οt), unlike direct current (DC) which flows steadily in one direction. Since AC constantly changes, its "effective" voltage for delivering power is its RMS value, V_rms=V0/sqrt(2), not its peak.
Why Do We Need It?
Virtually all mains electricity (the power from a wall outlet) is AC β understanding its peak and RMS values is essential for safely and correctly working with real-world electrical power.
See It
A sine wave oscillating between a positive peak and a negative peak, repeating periodically, with the peak value marked
Formula
RMS Voltage of an AC Supply
V_rms = Vβ/β2
Since an alternating voltage constantly changes, its "effective" value for delivering power is not its peak β the RMS (root-mean-square) voltage is the steady (DC-equivalent) voltage that would deliver the same average power.
- V_rms
- β root-mean-square (effective) voltage, in volts (V)
- Vβ
- β peak (maximum) voltage of the AC waveform, in volts (V)
When to use it: Whenever converting between the peak voltage of an AC supply and the "effective" voltage that determines its power delivery (the value usually quoted for mains electricity).
Worked Example
Find the RMS voltage of an AC supply
An AC supply has a peak voltage of 170 V. Find its RMS voltage.
Why Does This Work?
Squaring a sine wave, averaging that square over a full cycle, then taking the square root, gives exactly V0/sqrt(2) β this "root of the mean of the square" process is precisely what makes the RMS value deliver the same average power as a steady DC voltage of that same value.
Real-Life Example
Why a "230 V" wall outlet is actually a moving target
A wall outlet rated "230 V" (common in much of the world) does not actually stay at 230 V for even an instant.
That 230 V is the RMS value β the actual instantaneous voltage constantly swings between a peak of about +325 V and a trough of about -325 V, sixty (or fifty) times each second.
Practice
An AC supply has a peak voltage of 311 V. Find its RMS voltage, rounded to the nearest whole volt.
MediumCommon mistake
Assuming the RMS voltage and the peak voltage are the same thing β the peak is always higher than the RMS value by a factor of sqrt(2), about 1.41 times larger.
Quick Review
- v = V0 sin(omega t) β AC constantly reverses direction.
- V_rms = V0 / sqrt(2) β the effective, power-delivering value.
- Mains electricity ratings quote RMS voltage, not peak.