Root Mean Square (RMS) Voltage

The DC voltage that dissipates the same average power in a resistor as the AC signal

© 2026 Theodore P. Pavlic
MIT License
AC Source V(t) varies
+ − R
V(t) = 0 V (V/Vpk)² = 0.000
≡
same
avg P
↑ Same Average Power in AC & DC ↓
DC Source (VRMS) constant
+ − R
VRMS = 0 V (Vrms/Vpk)² = 0.000
Normalized Instantaneous Power — 2 Cycles P(t) = [V(t)/Vpk]²
PAC(t) — instantaneous
PDC = (VRMS/Vpk)² — constant, equals average of AC
↔︎ drag either graph to move through the cycle
Vpk = 120√2 ≈ 170 V — matching the North American 120 VRMS standard for a sine wave. Changing the waveform shape shows how VRMS/Vpk depends only on shape, not on amplitude or resistance.
RMS Definition & Current Values
VRMS = √( 1⁄T ∫0T V(t)² dt )
Sine:  VRMS = Vpk/√2 = 120.0 V
Pavg = (VRMS/Vpk)² = 0.5000  = (1/2) Ppk (sine)
RMS in other contexts
🔈
Audio Engineering
Audio meters display RMS amplitude because perceived loudness tracks average power. A peak reading can be misleading; the RMS value reflects what you hear.
📐
Statistics (RMSE)
Root Mean Square Error is the RMS of residuals. For zero-mean errors it equals the standard deviation — the typical size of a single prediction error.
⚛️
Kinetic Theory
RMS molecular speed vrms = √(3kBT/m) — the speed whose kinetic energy equals the mean kinetic energy of the gas ensemble.