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:P= rac{dE}{dt}= rac{dW}{dt}

where
P

E

W

t


The average power (often simply called "power" when the context makes it clear) is the average amount of work done or energy transferred per unit time. The '''instantaneous power''' is then the limiting value of the average power as the time interval ''Δt'' approaches zero.

:P=\lim_{\Delta t ightarrow 0} rac{\Delta W}{\Delta t} = \lim_{\Delta t ightarrow 0} P_\mathrm{avg}

When the rate of energy transfer or work is constant, all of this can be simplified to
:P= rac{W}{t} = rac{E}{t}\ ,
where ''W'' and ''E'' are, respectively, the work done or energy transferred in time ''t''.


UNITS

The units of power are units of energy divided by time. The SI unit of power is the Watt , which is equal to one Joule per Second . The power consumption of a Human is on average roughly 100 watts, ranging from 85 W during sleep to 800 W or more while playing a strenuous sport. Professional cyclists have been measured at 2000 W output for short periods of time.

Non-SI units of power include Horsepower (HP), Pferdestärke (PS), cheval vapeur (CV) and foot-pounds per minute. One unit of horsepower is equivalent to 33,000 foot-pounds per minute, or the power required to lift 550 Pound s one foot in one second, and is equivalent to about 746 watts. Other units include DBm , a logarithmic measure with 1 milliwatt as reference; and Kilocalorie s per hour (often referred to as ''Calories per hour'').


MECHANICAL POWER

In Mechanics , the Work done on an object is related to the forces acting on it by

:W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}
where
:F is Force
:s is the Displacement of the object.

This is often summarized by saying that work is equal to the force acting on an object times its displacement (how far the object moves while the force acts on it). Note that only motion that is in the same direction as the force "counts", however.

Differentiating by time gives that the instantaneous power is equal to the force times the object's Velocity v(''t''):
:P(t) = \mathbf{F}(t) \cdot \mathbf{v}(t).
The average power is then
:P_\mathrm{avg} = rac{1}{\Delta t}\int\mathbf{F} \cdot \mathbf{v}\;\mathrm{d}t.

This formula is important in characterizing Engine s—the power put out by an engine is equal to the force it exerts times its velocity.


ELECTRICAL POWER

''Main article: Electric Power ''

Instantaneous electrical power


The instantaneous electrical power ''P'' delivered to a component is given by
: P(t) = I(t) \cdot V(t) \,\!
where

P(t)


V(t)


I(t)


If the component is a Resistor , then:
: P=I^2 \cdot R \,\!

or
: P= rac{V^2}{R}

where

R



Average electrical power for sinusoidal voltages


The average power consumed by a two-terminal electrical device is a function of the Root Mean Square values of the Sinusoidal Voltage across the terminals and the sinusoidal Current passing through the device. That is,
:P=I \cdot V \cdot \cos\phi \,\!

where

P


I


V


φ


The amplitudes of sinusoidal voltages and currents, such as those used almost universally in mains electrical supplies, are normally specified in terms of root mean square values. This makes the above calculation a simple matter of multiplying the two stated numbers together.

This figure can also be called the Effective Power , as compared to the larger Apparent Power which is expressed in Volt-amperes Reactive (VAR) and does not include the cos ''φ'' term due to the current and voltage being out of phase. For simple domestic appliances or a purely resistive network, the cos ''φ'' term (called the Power Factor ) can often be assumed to be unity, and can therefore be omitted from the equation. In this case, the effective and apparent power are assumed to be equal.


Average electrical power for AC


:
P = {1 \over T} \int_{0}^{T} i(t) v(t)\, dt


Where v(t) and i(t) are, respectively, the instantaneous voltage and current as functions of time.


Electrical power transfer

The efficient transfer of electrical power is governed by the Maximum Power Theorem , which states that for the transfer of maximum power from a source with a fixed Internal Resistance to a load, the resistance of the load must be equal to that of the source.


Peak power and duty cycle




One may define the pulse length au such that P_0 au = \epsilon_\mathrm{pulse} so that the ratios

: rac{P_\mathrm{avg}}{P_0} = rac{ au}{T}

are equal. These ratios are called the ''duty cycle'' of the pulse train.


POWER IN OPTICS

See Also: Optical power


In Optics , the term ''power'' sometimes refers to the average rate of energy transport by electromagnetic radiation. The term "power" is also, however, used to express the ability of a Lens or other optical device to Focus light. It is measured in Dioptre s (inverse Metre s), and is equal to one over the Focal Length of the optical device.


SEE ALSO




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