Power (physics)

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In physics, power is the rate of producing or consuming energy. It is also the rate of doing work. The watt (W), named after the Scottish engineer James Watt), is the SI unit for power and is defined as 1 Joule/second. Many products are rated by their ability to generate or use power. Household lightbulbs are usually rated in the 40-120 W range, as are many stereo systems. In the automotive industry, power is often expressed in units of horsepower, which is equivalent to 746 W or 550 ft•lbf/sec.

Electrical power

In direct current (DC) electonic circuits, in which a current I is flowing through a resistor R, and experiencing a voltage drop V, the power consumption can be expressed as I²R, VI or V²/R.

Calculating the power of AC circuits is more complex because both the voltage and the current are a function of time.

Given a sinusoidal time-dependent voltage, v(t) = V₀sin(wt), or current i(t)= V₀sin(wt)/R = I₀sin(wt), the instaneous power p(t) at any time is given by:

Instantaneous AC power: p(t) = I₀²Rsin²(wt).

The instantaneous power is of little use in most applications, so AC power is usually represented as a power averaged over time.

$P_{av} = \frac{1}{T}\int{p(t)}dt = \frac{1}{T}\int{v(t)i(t)dt}$

$= \frac{1}{T}\int{(V_0 sin(wt))(\frac{V_0 sin(wt)}{R})}dt$

$= \frac{1}{T}\frac{V_\mathrm{0}^2}{R}\int{sin(wt)sin(wt)}dt = \frac{V_\mathrm{0}^2}{2R}$

Since for a sinusoidal voltage, the rooot-mean-square power, V_rms is defined as

$V_{rms} = [\frac{1}{T}\int{v(t)v(t)dt}]^{1/2} = \frac{V_0}{\sqrt2}$ , the average power can be rewritten as

$P_{av} = \frac{V_\mathrm{rms}^2}{R} = I_\mathrm{rms}^2R$

The root-mean-squared voltage is the value typically thought of in regular usage. Thus, in the U.S., standard 110 voltage lines refer to the $V_{rms}$ value. The actual peak voltage is around 156 volts but it, and the $V_{rms}$ actually vary throughout the day depending on electrical demand and sometime they fall low enough to cause a brownout.