Describing a sine wave
A sinusoidal voltage is fixed by three numbers: its peak Vp, its frequency f and its phase φ. The period is T = 1/f and the angular frequency ω = 2πf.
v(t) = Vp·sin(ωt + φ) T = 1/f, ω = 2πf Vpp = 2·Vp
RMS: root of the mean of the square
The average of a sine wave over a period is zero, so the average is useless for power. RMS (root-mean-square) asks a better question: what DC voltage would heat a resistor equally? Square the waveform so every part is positive, take the mean over one period, then the square root.
For a sine, the mean of sin² is exactly ½, so Vrms = Vp/√2. That is why Thai mains, quoted as 220 V rms, actually peaks at 220·√2 ≈ 311 V.
Vrms = √( (1/T)·∫ v² dt ) sine: Vrms = Vp / √2 ≈ 0.707·Vp
Square and triangle waves break the √2 rule
The factor √2 comes from the shape of a sine. A square wave sits at ±Vp all the time, so v² is Vp² everywhere and Vrms = Vp. A triangle wave spends more time near zero, so Vrms = Vp/√3. The ratio Vp/Vrms is called the crest factor.
- Sine: Vrms = 0.707·Vp, crest factor 1.414
- Square: Vrms = Vp, crest factor 1
- Triangle: Vrms = 0.577·Vp, crest factor 1.732
The phasor: a rotating arrow
Picture an arrow of length Vp rotating anticlockwise at ω. Its height at any instant is Vp·sin(ωt + φ): the sine wave. The phase φ is just where the arrow starts. Phasors turn calculus on sines into geometry, which is how the RLC and three-phase sections work.
Worked example (the simulator’s default values)
- Sine wave, Vp = 311 V, f = 50 Hz.
- T = 1/50 = 20 ms; ω = 2π·50 = 314.2 rad/s; Vpp = 622 V.
- Vrms = 311/√2 = 220 V: the number on the electricity bill.
- Switch to a square wave of the same peak: Vrms = 311 V, so the same peak delivers twice the heating power (Vrms² is twice as big).
Common mistakes
- Using Vrms = Vp/√2 for every waveform. It is true only for sines.
- Confusing peak and peak-to-peak: an oscilloscope’s Vpp is twice the peak.
- Treating the average of a sine as its “effective” value. The average over a period is zero; RMS is the effective value.
Common questions
How do you convert peak voltage to RMS?
For a sine wave divide by √2: Vrms = Vp/√2 ≈ 0.707·Vp. For other shapes use the definition: square, average over a period, square root.
Why is 220 V mains 311 V peak?
220 V is the RMS value. The peak of a sine is √2 times its RMS, and 220·√2 ≈ 311 V.
What is RMS used for?
RMS is the equivalent DC value for heating: a resistor dissipates P = Vrms²/R on AC, exactly as it would on that much DC. Ratings of mains, meters and loads are all RMS.