Sinusoidal AC and rms values

Write the source as v(t) = V₀ sin(ωt + φ). The peak value V₀ sets the height in volts, the angular frequency ω sets how fast the phase advances in rad s⁻¹, and the phase constant φ says where in the cycle the clock was started. Only ω appears in the circuit equations, but instruments report the frequency f in hertz, so convert: ω = 2πf and T = 1/f. European mains at 50.0 Hz gives T = 20.0 ms and ω = 314 rad s⁻¹. Phase is read the same way in either currency. Two sinusoids of the same frequency keep a fixed phase difference, and 30° of it corresponds to (30/360) × 20.0 ms = 1.67 ms of lead or lag at 50 Hz. A phase difference between waves of different frequencies is not constant, so the idea only earns its keep at a single frequency.

Integrate a full period of the source: ∫₀ᵀ V₀ sin(ωt + φ) dt = 0, because the negative half exactly undoes the positive half. That is not a technicality. A moving-coil meter, which responds to the mean, reads zero across a live socket, and no average of v itself can appear in a rating that must predict heating. The rectified mean ⟨|v|⟩ = 2V₀/π = 0.637 V₀ is at least non-zero, but it is still the wrong quantity: power in a resistor goes as v², not as |v|, so an average taken before squaring throws away the very nonlinearity that decides the heating. Whatever average we choose has to be taken after the square.

Square the source: v² = V₀² sin²(ωt + φ). Use the identity sin²θ = ½(1 − cos 2θ) to get v² = (V₀²/2)[1 − cos(2ωt + 2φ)]. The cosine term is itself a sinusoid, of twice the frequency, so over a whole number of periods it averages to zero and only the constant survives: ⟨v²⟩ = V₀²/2. Take the root and Vᵣₘₛ = V₀/√2 ≈ 0.707 V₀. The same result follows without integration: sin²θ + cos²θ = 1 everywhere, and over a full cycle the two averages must be equal, so each is ½. Notice what the algebra also delivered — the square of a sinusoid oscillates at 2ω about its mean, and that doubled frequency is about to matter.

For a resistor, p(t) = v²/R = (V₀²/2R)[1 − cos(2ωt + 2φ)]. Averaging gives P_avg = V₀²/2R = Vᵣₘₛ²/R, and with Ohm's law P_avg = Iᵣₘₛ² R = Vᵣₘₛ Iᵣₘₛ. That is the whole point of the definition: every DC power formula survives if rms values are inserted, so a 230 V rms supply into a 53 Ω heater delivers 998 W exactly as a 230 V battery would. What does not survive is steadiness. The instantaneous power swings from 0 to 2P_avg at 2f — 100 Hz on a 50 Hz supply — which is why filament lamps and cheap LEDs flicker at 100 Hz, why a single-phase motor's torque pulses, and why a full-wave rectifier's smoothing capacitor is sized against 100 Hz ripple rather than 50 Hz.

Repeat the recipe on other repeating waveforms and the factor changes. A symmetric square wave spends all its time at ±V₀, so ⟨v²⟩ = V₀² and Vᵣₘₛ = V₀: crest factor 1. A triangular wave gives V₀/√3 = 0.577 V₀. A half-wave rectified sine keeps only half the cycle, halving the mean square to V₀²/4, so Vᵣₘₛ = V₀/2 — while a full-wave rectified sine, which keeps every squared value, is back at V₀/√2. This is why meters disagree. An average-responding meter measures ⟨|v|⟩ and multiplies by the sine form factor 0.707/0.637 = 1.11; fed a square wave it reads 11% high, and fed a triangle 3.8% low. A true-rms meter squares, averages, and roots, and is right for all of them.

Squaring destroys the sign, and with it the phase. Two 5.0 A rms currents of the same frequency can combine into anything from 0 to 10.0 A rms depending on their relative phase, so rms values never simply add. What does add is the mean square, and only when the parts are uncorrelated — a DC offset with an AC ripple, or harmonics of different frequency — giving Vᵣₘₛ² = V_DC² + V₁rms² + V₂rms² + … . The definition also assumes a window holding a whole number of cycles: average over three-quarters of a cycle and the answer depends on where the window opened. For a signal whose amplitude drifts, rms is only meaningful over a window long compared with T and short compared with the drift.