Why 120 V AC Peaks at 170 V

Household voltage swings back and forth, so what does "120 volts" mean? It's the steady voltage that would deliver the same average power. Average power is an average function value: an integral divided by the time.

How it works

Average function value

In one sentence

The average value of an integrable $f$ on $[a, b]$ is $f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx$ ; if $f$ is continuous, $f(c) = f_{\text{avg}}$ for some $c$ (the Mean Value Theorem for integrals).

General formula

{\htmlData{sym=0}{\overline{P}}} = \frac{1}{\htmlData{sym=3}{{T}}}\int_0^{\htmlData{sym=3}{T}} \frac{{\htmlData{sym=1}{V_p}}^2\sin^2(\omega t)}{\htmlData{sym=2}{{R}}}\,dt = \frac{{\htmlData{sym=1}{V_p}}^2}{2{\htmlData{sym=2}{R}}} \quad\Rightarrow\quad {\htmlData{sym=4}{V_{\text{rms}}}} = \frac{{\htmlData{sym=1}{V_p}}}{\sqrt2}

Flatten the area under the curve into a rectangle on the same base: its height is the average value.

What the symbols mean

$\overline{P}$
Average power
The steady power that delivers the same energy per cycle.
$V_p$
Peak voltage
The highest the voltage swings.
$R$
Resistance
The load, like a heater.
$T$
Period
One full cycle.
$V_{\text{rms}}$
RMS voltage
The rated voltage: the steady voltage with the same average power.

The average value of $f$ on $[a, b]$ is $\frac{1}{b - a}\int_a^b f\,dx$ : the height of a rectangle with the same area.
Power in a resistor is $V^2/R$ , so with $V = V_p\sin\omega t$ the power is $\frac{V_p^2}{R}\sin^2\omega t$ .
Over a full cycle $\sin^2$ averages to $\frac12$ (it's $\frac{1 - \cos 2\omega t}{2}$ ).
So average power equals that of a steady $V_{\text{rms}} = V_p/\sqrt2$ . For 120 V rms, the peak is $120\sqrt2 \approx 170$ V.

Learn the math

Average Function ValueCalc I · Lesson 48, Applications of Integrals RMS Values and AC PowerE&M · Lesson 24, AC Circuits Energy, Intensity and the Poynting VectorE&M · Lesson 29, Maxwell's Equations and Light