# Performance (physics)

Physical size
Surname power
Formula symbol ${\ displaystyle P}$ Derived from energy
Size and
unit system
unit dimension
SI W. M · L 2 · T −3
cgs erg · s −1 = 10 −7 W M · L 2 · T −3

The power as a physical quantity describes the energy converted in a period of time in relation to this period of time. Their symbol is mostly (from English power ), their SI unit is the watt with the unit symbol  W. ${\ displaystyle P}$ In the physical-technical context, the term performance is used in different meanings:

• as installed or maximum possible power (identifier of a device or system ; also called nominal power )
• as actual performance in an application
• the power supplied
• the performance submitted in terms of the task.

The power consumption and the useful power output for a specific application can differ considerably depending on the efficiency or waste heat .

## Definitions

The performance is the ratio of verrichteter work or for energy expended  and of the required time  : ${\ displaystyle P}$ ${\ displaystyle \ Delta W}$ ${\ displaystyle \ Delta E}$ ${\ displaystyle \ Delta t}$ ${\ displaystyle P = {\ frac {\ Delta E} {\ Delta t}} = {\ frac {\ Delta W} {\ Delta t}} \.}$ example
If an energy of 1 kilowatt hour is drawn in a period of 1 hour, then the output is 1 kilowatt.
If the same energy is drawn in a shorter time, then the output is greater; if 1 kilowatt hour is used in ½ hour, the output is 2 kilowatts.

In the case of power that changes over time, for example in the loudspeaker or in the electrical power supply network, there is an instantaneous power or instantaneous power that results from the limit value when the time segment  approaches zero: ${\ displaystyle P (t)}$ ${\ displaystyle \ Delta t}$ ${\ displaystyle P (t) = \ lim _ {\ Delta t \ rightarrow 0} {\ frac {\ Delta W} {\ Delta t}} \ {,}}$ so as a differential quotient

${\ displaystyle P (t) = {\ frac {\ mathrm {d} W (t)} {\ mathrm {d} t}} \.}$ An average power performed in a time interval of length is more measurable${\ displaystyle T = \ left [t_ {1}, t_ {2} \ right]}$ ${\ displaystyle {\ overline {P}}}$ ${\ displaystyle {\ overline {P}} = {\ frac {1} {T}} \ int _ {t_ {1}} ^ {t_ {2}} P (t) \ mathrm {d} t \,}$ This information is particularly important if changes periodically and the period duration is. ${\ displaystyle P (t)}$ ${\ displaystyle T}$ ## Mechanical performance

### Translation

The simplest case, with the force parallel to the direction of movement, is the drawbar power , it applies

${\ displaystyle P = Fv}$ with the power and the speed . ${\ displaystyle F}$ ${\ displaystyle v}$ Without this restriction, the corresponding vector equation applies

${\ displaystyle P = {\ vec {F}} \ cdot {\ vec {v}} \ ,.}$ This takes into account the angular dependence through the scalar product , as explained in the article work (physics) for “force times displacement”.

### rotation

The same applies to the rotation against a torque M

${\ displaystyle P = {\ vec {M}} \ cdot {\ vec {\ omega}} \,}$ where the angular velocity is about an axis parallel to the direction vector . ${\ displaystyle {\ vec {\ omega}} = {\ tfrac {\ mathrm {d} \ varphi} {\ mathrm {d} t}} \; {\ vec {e}}}$ ${\ displaystyle {\ vec {e}}}$ For a shaft with torque and speed , the shaft power results in ${\ displaystyle M}$ ${\ displaystyle n = {\ tfrac {\ omega} {2 \ pi}} \}$ ${\ displaystyle P = 2 \ pi \ Mn \ ,.}$ If for example one combustion engines common units kW, Nm min -1 sets based gets to the numerical value equation

${\ displaystyle \ {P \} = \ {M \} \, \ {n \} \, \ pi \ /30,000 \approx \ {M \} \, \ {n \} / 9550}$ ,

wherein the numerical value of the power in kW, the numerical value of the torque in Nm and the numerical value of the speed in rpm -1 is. ${\ displaystyle \ {P \}}$ ${\ displaystyle \ {M \}}$ ${\ displaystyle \ {n \}}$ ### Hydraulics

The hydraulic power through volume work is the product of the pressure difference and the volume flow . ${\ displaystyle \ Delta p}$ ${\ displaystyle Q = {\ frac {\ Delta V} {\ Delta t}}}$ ${\ displaystyle P = \ Delta p \, Q \ ,.}$ ## Electrical power

The electrical power that is converted in a component with the ohmic resistance  is the product of electrical voltage and current strength at constant values${\ displaystyle R}$ ${\ displaystyle U}$ ${\ displaystyle I}$ ${\ displaystyle P = UI = I ^ {2} R = {\ frac {U ^ {2}} {R}} \.}$ In the case of variables that change over time and the instantaneous value of the power is defined as ${\ displaystyle u (t)}$ ${\ displaystyle i (t)}$ ${\ displaystyle P (t)}$ ${\ displaystyle P (t) = u (t) \, i (t) \.}$ Instead of this fluctuating quantity, power specifications that are defined by averaging and that are constant over time for periodic alternating current quantities are used:

• Real power ,${\ displaystyle P}$ • Reactive power ,${\ displaystyle Q}$ • Apparent power .${\ displaystyle S}$ ## Performance data

### Accepted and delivered service

The manufacturers of electrical devices are obliged to indicate the maximum power consumption, i.e. the maximum power that is drawn from the power supply (mains, battery). This is always a larger numerical value than the power output, i.e. the power in the form that the user wants (e.g. mechanical power, light output). The power output can be much lower depending on the degree of efficiency , ie after subtracting the energy losses when converting the electrical energy into the desired type of energy. Heat losses, mechanical and other losses reduce the actual output e.g. B. a drill or a vacuum cleaner.

In the case of light sources, in addition to the power consumption in watts, the luminous flux in lumens must also be specified. Because of its definition via the physiology of the human eye , it cannot be compared directly to electrical output. Rather, the luminous efficacy can be given in units of lumen per watt. Efficiency can be roughly estimated by dividing the radiation power in the visible spectral range (approx. 400 to 700 nm ) by the consumption power. This would result in z. B. for incandescent lamps a value of about five percent. However, the boundaries between the visible and the infrared or ultraviolet range are fluid, so that such an efficiency would not be clearly defined. In addition, it does not take into account the different spectral sensitivity of the eye.

With lasers , however, the power actually contained in the laser beam is specified. The electrical consumption (connected load) of a laser beam source is always higher depending on the respective efficiency.

In household appliances, e.g. B. an electric lawn mower, the electrical power that is taken from the socket is indicated. The situation is different with electric motors with higher power. The mechanical power available on the motor shaft and the amount of apparent power consumed are also given on the nameplate . In the case of electric hand drills, the maximum power taken from the mains when the spindle is at a standstill is specified - so it has nothing to do with the mechanical power output. In the case of vacuum cleaners, the electrical power consumption is specified, which does not have to have much to do with the suction power. The (electrical) power consumption of a heater is always the same as the heat output.

#### Chillers

Fridges , freezers and heat pumps transport heat output from the cold to the warm side. The commonly used pump requires a drive, electric motors are common. The power consumption of the motor is usually less than the heat output. A heat pump heater can therefore provide 2.5 times the electrical power consumption as heat output, for example.

#### Heat exchanger

The heat output of heat exchangers is often proportional to the temperature difference. Also, heat sinks and heat-dissipating housing possess this characteristic. Their performance is therefore often specified in watts per Kelvin temperature difference (W / K).

### Continuous and short-term performance

The performance specification for a device can refer to a "KB xx min", i. H. Draw short operating time xx minutes. This is to avoid overheating due to limited heat capacities and heat conduction. Examples are electrical kitchen appliances, soldering guns or arc welding devices. They must cool down after the specified operating time at the latest. The same applies to the hourly output of electric locomotives, which can be delivered continuously over an hour.

In the case of baking ovens, the power rating can indicate the power when heating up, while the power later during baking etc. is much lower due to the temperature control.

Very high performances are possible for very short periods of time. For example, the PHELIX laser system delivers 0.5 petawatts (= 0.5 · 10 15 W) over a period of 2 picoseconds (= 2 · 10 −12 s).

## units

In the international system of units, power is given in watts. In addition to the CGS uniterg per second”, other units are also used. Some examples are given in the table:

watt Kilopond meters per second Horsepower Kilocalories per hour
1 W (= 1 kg · m² / s³) = 1 0.102 0.00136 0.860
1 kp m / s = 9.80665 1 0.01 3 8.4322
1 PS = 735.49875 75 1 632,415
1 kcal / h = 1.163 0.1186 0.00158 1