Power (physics)
Power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. Power is a scalar quantity.
The output power of a motor is the product of the torque that the motor generates and the angular velocity of its output shaft. Likewise, the power dissipated in an electrical element of a circuit is the product of the current flowing through the element and of the voltage across the element.
Definition
Power is the rate with respect to time at which work is done or, more generally, the rate of change of total mechanical energy. It is given by:where is power, is the total mechanical energy, and is time.
For cases where only work is considered, power is also expressed as:
where is the work done on the system. However, in systems where potential energy changes without explicit work being done, the total energy definition is more general.
We will now show that the mechanical power generated by a force on a body moving at the velocity can be expressed as the product:
If a constant force is applied throughout a distance, the work done is defined as. In this case, power can be written as:
If instead the force is variable over a three-dimensional curve , then the work is expressed in terms of the line integral:
From the fundamental theorem of calculus, we know that Hence the formula is valid for any general situation.
In older works, power is sometimes called activity.
Units
The dimension of power is energy divided by time. In the International System of Units, the unit of power is the watt, which is equal to one joule per second. Other common and traditional measures are horsepower, comparing to the power of a horse; one mechanical horsepower equals about 745.7 watts. Other units of power include ergs per second, foot-pounds per minute, dBm, a logarithmic measure relative to a reference of 1 milliwatt, calories per hour, BTU per hour, and tons of refrigeration.Average power and instantaneous power
As a simple example, burning one kilogram of coal releases more energy than detonating a kilogram of TNT, but because the TNT reaction releases energy more quickly, it delivers more power than the coal.If is the amount of work performed during a period of time of duration, the average power over that period is given by the formula
It is the average amount of work done or energy converted per unit of time. Average power is often called "power" when the context makes it clear.
Instantaneous power is the limiting value of the average power as the time interval approaches zero.
When power is constant, the amount of work performed in time period can be calculated as
In the context of energy conversion, it is more customary to use the symbol rather than.
Mechanical power
Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or the product of a torque on a shaft and the shaft's angular velocity.Mechanical power is also described as the time derivative of work. In mechanics, the work done by a force on an object that travels along a curve is given by the line integral:
where defines the path and is the velocity along this path.
If the force is derivable from a potential, then applying the gradient theorem yields:
where and are the beginning and end of the path along which the work was done.
The power at any point along the curve is the time derivative:
In one dimension, this can be simplified to:
In rotational systems, power is the product of the torque and angular velocity,
where is angular frequency, measured in radians per second. The represents scalar product.
In fluid power systems such as hydraulic actuators, power is given by where is pressure in pascals or N/m2, and is volumetric flow rate in m3/s in SI units.
Mechanical advantage
If a mechanical system has no losses, then the input power must equal the output power. This provides a simple formula for the mechanical advantage of the system.Let the input power to a device be a force acting on a point that moves with velocity and the output power be a force acts on a point that moves with velocity. If there are no losses in the system, then
and the mechanical advantage of the system is given by
The similar relationship is obtained for rotating systems, where and are the torque and angular velocity of the input and and are the torque and angular velocity of the output. If there are no losses in the system, then
which yields the mechanical advantage
These relations are important because they define the maximum performance of a device in terms of velocity ratios determined by its physical dimensions. See for example gear ratios.
Electrical power
The instantaneous electrical power P delivered to a component is given bywhere
- is the instantaneous power, measured in watts,
- is the potential difference across the component, measured in volts, and
- is the current through it, measured in amperes.
where
is the electrical resistance, measured in ohms.
Peak power and duty cycle
In the case of a periodic signal of period, like a train of identical pulses, the instantaneous power is also a periodic function of period. The peak power is simply defined by:The peak power is not always readily measurable, however, and the measurement of the average power is more commonly performed by an instrument. If one defines the energy per pulse as
then the average power is
One may define the pulse length such that so that the ratios
are equal. These ratios are called the duty cycle of the pulse train.