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# power

[pou-er]
power, in mathematics: see exponent.
power, in physics, time rate of doing work or of producing or expending energy. The unit of power based on the English units of measurement is the horsepower, devised for describing mechanical power by James Watt, who estimated that a horse can do 550 ft-lb of work per sec; a foot-pound is the work done when a weight (force) of 1 lb is moved through a distance of 1 ft. The unit of power in the metric system is the watt, named in honor of James Watt and equal to 1 joule per sec; the watt is used for measuring electric power in most countries, even those still using English units for other quantities. In common usage, the terms power and energy have become synonymous; for example, electrical energy is usually referred to as electric power (see power, electric). See also energy, sources of.
power, electric, energy dissipated in an electrical or electronic circuit or device per unit of time. The electrical energy supplied by a current to an appliance enables it to do work or provide some other form of energy such as light or heat. Electric power is usually measured in Watts, kilowatts (1,000 watts), and megawatts (1,000,000 watts). The amount of electrical energy used by an appliance is found by multiplying its consumed power by the length of time of operation. The units of electrical energy are usually watt-seconds (joules), watt-hours, or kilowatt-hours. For commercial purposes the kilowatt-hour is the unit of choice.

## Sources of Electrical Energy

Electrical energy occurs naturally, but seldom in forms that can be used. For example, although the energy dissipated as lightning exceeds the world's demand for electricity by a large factor, lightning has not been put to practical use because of its unpredictability and other problems. Generally, practical electric-power-generating systems convert the mechanical energy of moving parts into electrical energy (see generator). While systems that operate without a mechanical step do exist, they are at present either excessively inefficient or expensive because of a dependence on elaborate technology. While some electric plants derive mechanical energy from moving water (hydroelectric power), the vast majority derive it from heat engines in which the working substance is steam. Roughly 89% of power in the United States is generated this way. The steam is generated with heat from combustion of fossil fuels or from nuclear fission (see nuclear energy; nuclear reactor).

Steam as an Energy Source

The conversion of mechanical energy to electrical energy can be accomplished with an efficiency of about 80%. In a hydroelectric plant, the losses occur in the turbines, bearings, penstocks, and generators. The basic limitations of thermodynamics fix the maximum efficiency obtainable in converting heat to electrical energy. The necessity of limiting the temperature to safe levels also helps to keep the efficiency down to about 41% for a fossil-fuel plant. Most nuclear plants use low-pressure, low-temperature steam operation, and have an even lower efficiency of about 30%. Nuclear plants have been able to achieve efficiency up to 40% with liquid-metal cooling. It is thought that by using magnetohydrodynamic "topping" generators in conjunction with normal steam turbines, the efficiency of conventional plants can be raised to close to 50%. These devices remove the restrictions imposed by the blade structure of turbines by using the steam or gasses produced by combustion as the working fluid.

Environmental Concerns

The heat generated by an electric-power plant that is not ultimately converted into electrical energy is called waste heat. The environmental impact of this waste is potentially catastrophic, especially when, as is often the case, the heat is absorbed by streams or other bodies of water. Cooling towers help to dispose waste heat into the atmosphere. Associated with nuclear plants, in addition to the problem of waste heat, are difficulties attending the disposal and confinement of reaction products that remain dangerously radioactive for many thousands of years and the adjustment of such plants to variable demands for power. Public concern about such issues—fueled in part by the accidents at the Three Mile Island nuclear plant in Harrisburg Pennsylvania in 1979, and the nuclear plant explosion in the Soviet Union at Chernobyl in 1986—forced the U.S. government to introduce extensive safety regulations for nuclear plants. Partly because of those regulations, nuclear plants are proving to be uneconomical. Several are being shut down and replaced by conventionally fueled plants.

Alternative Energy Sources

Fuel cells develop electricity by direct conversion of hydrogen, hydrocarbons, alcohol, or other fuels, with an efficiency of 50% to 60%. Although they have been used to produce electric power in space vehicles and some terrestrial locations, several problems have kept them from being widely used. Most important, the catalyst, which is an important component of a fuel cell, especially one that is operating at around room temperature, is very expensive. Controlled nuclear fusion could provide a virtually unlimited source of heat energy to produce steam in generating plants; however, many problems surround its development, and no appreciable contribution is expected from this source in the near future.

Solar energy has been recognized as a feasible alternative. It has been suggested that efficient collection of the solar energy incident on 14% of the western desert areas of the United States would provide enough electricity to satisfy current demands. Two main solar processes could be used. Photovoltaic cells (see solar cell) convert sunlight directly into electrical energy. Another method would use special coatings that absorb sunlight readily and emit infrared radiation slowly, making it possible to heat fluids to 1,000°F; (540°C;) by solar radiation. The heat in turn can be converted to electricity. Some of this heat would be stored to allow operation at night and during periods of heavy cloud cover. The projected efficiency of such a plant would be about 30%, but this fairly low efficiency must be balanced against the facts that energy from the sun costs nothing and that the waste heat from such a plant places virtually no additional burden on the environment. The principal problem with this and other exotic systems for generating electricity is that the time needed for their implementation may be considerable.

Windmills, once widely used for pumping water, have become viable for electric-power generation because of advances in their design and the development of increasingly efficient generators. Windmill "farms," at which rows of windmills are joined together as the source of electrical energy, serve as a significant, though minor, source of electrical energy in coastal and plains areas. However, the vagaries of the wind make this a difficult solution to implement on a large scale.

## Transmission of Electrical Energy

Electrical energy is of little use unless it can be made available at the place where it is to be used. To minimize energy losses from heating of conductors and to economize on the material needed for conductors, electricity is usually transmitted at the highest voltages possible. As modern transformers are virtually loss free, the necessary steps upward or downward in voltage are easily accomplished. Transmission lines for alternating current using voltages as high as 765,000 volts are not uncommon. For voltages higher than this it is advantageous to transmit direct current rather than alternating current. Recent advances in rectifiers, which turn alternating current into direct current, and inverters, which convert direct into alternating, have made possible transmission lines that operate at 800,000 volts and above. Such lines are still very expensive, however.

Electric utilities are tied together by transmission lines into large systems called power grids. They are thus able to exchange power so that a utility with a low demand can assist another with a high demand to help prevent a blackout, which involves the partial or total shutdown of a utility. Under such a system a utility experiencing too great a load, as when peak demand coincides with equipment failure, must remove itself from the grid or endanger other utilities. During periods in which demand exceeds supply a utility can reduce the power drawn from it by lowering its voltage. These voltage reductions, which are normally of 3%, 5%, or 8%, result in power reductions, or brownouts, of about 6%, 10%, or 15%, causing inefficient operation of some electrical devices. The power distribution system, because of its generation of low-frequency electromagnetic fields, has been suggested as a possible source of health problems.

## Reactive Power

Reactive power is a concept used by engineers to describe the loss of power in a system arising from the production of electric and magnetic fields. Although reactive loads such as inductors and capacitors dissipate no power, they drop voltage and draw current, which creates the impression that they actually do. This "imaginary power" or "phantom power" is called reactive power. It is measured in a unit called Volt-Amps-Reactive (VAR). The actual amount of power being used, or dissipated, is called true power, and is measured in the unit of watts. The combination of reactive power and true power is called apparent power, and it is the product of a circuit's voltage and current. Apparent power is measured in the unit of Volt-Amps (VA). Devices which store energy by virtue of a magnetic field produced by a flow of current are said to absorb reactive power; those which store energy by virtue of electric fields are said to generate reactive power. Reactive power is significant because it must be provided and maintained to insure continuous, steady voltage on transmission networks. Reactive power thus is produced for maintenance of the system and not for end-use consumption. Power losses incurred in transmission from heat and electromagnetic emissions are included in the total reactive power requirement as are the needs of power hungry devices, such as electric motors, electromagnetic generators, and alternators. This power is supplied for many purposes by condensers, capacitors, and similar devices, which can react to changes in current flow by releasing energy to normalize the flow. If elements of the power grid cannot get the reactive power they need from nearby sources, they will pull it across transmission lines and destabilize the grid. In this way, poor management of reactive power can cause major blackouts.

## Bibliography

See K. W. Li and A. P. Priddy, Power Plant System Design (1985); L. F. Drbal et al., Power Plant Engineering (1996).

For optical fibers, a power-law index profile is an index of refraction profile characterized by

$n\left(r\right) =$
begin{cases} n_1 sqrt{1-2Deltaleft({r over alpha}right)^g} & r le alpha n_1 sqrt{1-2Delta} & r ge alpha end{cases} where $Delta = \left\{n_1^2 - n_2^2 over 2 n_1^2\right\},$

and $n\left(r\right)$ is the nominal refractive index as a function of distance from the fiber axis, $n_1$ is the nominal refractive index on axis, $n_2$ is the refractive index of the cladding, which is taken to be homogeneous ($n\left(r\right)=n_2 mathrm\left\{ for \right\} r ge alpha$), $alpha$ is the core radius, and $g$ is a parameter that defines the shape of the profile. $alpha$ is often used in place of $g$. Hence, this is sometimes called an alpha profile.

For this class of profiles, multimode distortion is smallest when $g$ takes a particular value depending on the material used. For most materials, this optimum value is approximately 2. In the limit of infinite $g$, the profile becomes a step-index profile.