Power

Power
Power
Dimension	L 2 MT −3
Units
SI	Tue
GHS	erg s −1

Power is a scalar physical quantity that is generally equal to the rate of change, conversion, transmission, or energy consumption of a system. In a narrower sense, power is equal to the ratio of work performed over a certain period of time to this period of time ^[1] .

Content

Designations Used

Usually in the formulas of mechanics is indicated by the symbol N (the origin of the symbol is to be clarified).

In electrical engineering, it is usually indicated by the symbol P - from lat. p otestas (strength, power, effectiveness);

Sometimes the symbol W is used (from English watt).

Basic Formulas

Distinguish average power over a period of time $\Delta t$ ${\ displaystyle \ Delta t}$ :

N={\frac {\Delta A}{\Delta t}},

{\ displaystyle N = {\ frac {\ Delta A} {\ Delta t}},}

and instantaneous power at a given time:

N={\frac {dA}{dt}}.

{\ displaystyle N = {\ frac {dA} {dt}}.}

The time integral of the instantaneous power over a period of time is equal to the total energy transferred during this time:

\int _{t_{0}}^{t_{1}}Ndt=E.

{\ displaystyle \ int _ {t_ {0}} ^ {t_ {1}} Ndt = E.}

Units

In the International System of Units (SI), the unit of measure for power is a watt (W) equal to one joule per second (J / s). In theoretical physics , astrophysics , erg per second (erg / s) is often used as a unit for power.

Another common, but now obsolete unit of measure for power is horsepower . In its recommendations, the International Organization of Legal Metrology (OIML) defines horse power as a unit of measurement “which should be withdrawn from circulation as soon as possible where it is currently used, and which should not be introduced if it is not used” ^{[2 ]} .

Relations between power units
Units	Tue	kw	MW	kgf · m / s	erg / s	l with. (met.)	l with. (eng.)
1 watt	one	10 ⁻³	10 ⁻⁶	0.102	10 ⁷	1.36⋅10 ⁻³	1.34⋅10 ⁻³
1 kilowatt	10 ³	one	10 ⁻³	102	10 ¹⁰	1.36	1.34
1 megawatt	10 ⁶	10 ³	one	102⋅10 ³	10 ¹³	1.36⋅10 ³	1.34⋅10 ³
1 kilogram-force meter per second	9.81	9.81⋅10 ⁻³	9.81⋅10 ⁻⁶	one	9.81⋅10 ⁷	1.33⋅10 ⁻²	1.31⋅10 ⁻²
1 erg per second	10 ⁻⁷	10 ⁻¹⁰	10 ⁻¹³	1,02⋅10 ⁻⁸	one	1.36⋅10 ⁻¹⁰	1.34⋅10 ⁻¹⁰
1 horsepower (metric)	735.5	735.5⋅10 ⁻³	735.5⋅10 ⁻⁶	75	7,355⋅10 ⁹	one	0.9863
1 horsepower (English)	745.7	745.7⋅10 ⁻³	745.7⋅10 ⁻⁶	76.04	7,457⋅10 ⁹	1.014	one

Power in mechanics

If a force acts on a moving body, then this force does the work. Power in this case is equal to the scalar product of the force vector by the velocity vector with which the body moves:

N=\mathbf {F} \cdot \mathbf {v} =F\cdot v\cdot \cos \alpha ,

{\ displaystyle N = \ mathbf {F} \ cdot \ mathbf {v} = F \ cdot v \ cdot \ cos \ alpha,}

where F is the force vector; v is the velocity vector; $\alpha$ ${\ displaystyle \ alpha}$ - the angle between the velocity and force vector; F is the modulus of force; v is the velocity modulus.

A special case of power during rotational motion:

N=\mathbf {M} \cdot \mathbf {\omega } ={\frac {2\pi \cdot \mathbf {M} \cdot \mathbf {n} }{60}}

{\ displaystyle N = \ mathbf {M} \ cdot \ mathbf {\ omega} = {\ frac {2 \ pi \ cdot \ mathbf {M} \ cdot \ mathbf {n}} {60}}}

M is the moment of force , $\mathbf {\omega }$ ${\ displaystyle \ mathbf {\ omega}}$ - angular velocity $\pi$ ${\ displaystyle \ pi}$ Is the number of pi , n is the rotation frequency (number of revolutions per minute, rpm.).

Electric power

Electric power - a physical quantity that characterizes the speed of transmission or conversion of electrical energy .

Instant electrical power $P(t)$ ${\ displaystyle P (t)}$ electric circuit area:

P(t)=I(t)\cdot U(t)\,

{\ displaystyle P (t) = I (t) \ cdot U (t) \,}

Where

I(t)

{\ displaystyle I (t)}

- instantaneous current through the circuit;

U(t)

{\ displaystyle U (t)}

- instantaneous voltage in this area.

When studying AC networks , in addition to the instantaneous power corresponding to the general physical definition, the following concepts are also introduced:

active power equal to the average over the period value of instantaneous power,
- instant active power:

$p(t)={1 \over 2}\cdot Um\cdot Im\cdot \cos(\varphi )-{1 \over 2}\cdot Um\cdot Im\cdot \cos(\varphi )cos(2\omega t)$ ${\ displaystyle p (t) = {1 \ over 2} \ cdot Um \ cdot Im \ cdot \ cos (\ varphi) - {1 \ over 2} \ cdot Um \ cdot Im \ cdot \ cos (\ varphi) cos (2 \ omega t)}$

reactive power , which corresponds to the energy circulating without dissipation from source to consumer and vice versa,
- instant reactive power:

at $\varphi >0$ ${\ displaystyle \ varphi> 0}$

$q(t)={\frac {1}{2}}\cdot Um\cdot Im\cdot \sin(\varphi )\cdot \cos {\Bigl (}2\omega t+{\frac {\pi }{2}}{\Bigr )}$ ${\ displaystyle q (t) = {\ frac {1} {2}} \ cdot Um \ cdot Im \ cdot \ sin (\ varphi) \ cdot \ cos {\ Bigl (} 2 \ omega t + {\ frac {\ pi} {2}} {\ Bigr)}}$

at $\varphi <0$ ${\ displaystyle \ varphi <0}$

$q(t)={\frac {1}{2}}\cdot Um\cdot Im\cdot \sin(\varphi )\cdot \cos {\Bigl (}2\omega t-{\frac {\pi }{2}}{\Bigr )}$ ${\ displaystyle q (t) = {\ frac {1} {2}} \ cdot Um \ cdot Im \ cdot \ sin (\ varphi) \ cdot \ cos {\ Bigl (} 2 \ omega t - {\ frac { \ pi} {2}} {\ Bigr)}}$

total power , calculated as the product of the effective values of current and voltage without taking into account the phase shift .
- instantaneous apparent power

$s(t)={1 \over 2}\cdot Um\cdot Im\cdot \cos(\varphi )-{1 \over 2}\cdot Um\cdot Im\cdot cos{\Bigl (}2\omega t-\varphi {\Bigr )}$ ${\ displaystyle s (t) = {1 \ over 2} \ cdot Um \ cdot Im \ cdot \ cos (\ varphi) - {1 \ over 2} \ cdot Um \ cdot Im \ cdot cos {\ Bigl (} 2 \ omega t- \ varphi {\ Bigr)}}$

Where

$I m$ ${\ displaystyle Im}$ - current amplitude;

$U m$ ${\ displaystyle Um}$ - voltage amplitude;

$\varphi$ ${\ displaystyle \ varphi}$ - the angle between the initial voltage angle $\psi _{u}$ ${\ displaystyle \ psi _ {u}}$ and initial current angle $\psi _{i}$ ${\ displaystyle \ psi _ {i}}$ - $(\varphi =\psi _{u}-\psi _{i})$ ${\ displaystyle (\ varphi = \ psi _ {u} - \ psi _ {i})}$

$\omega$ ${\ displaystyle \ omega}$ - angular velocity;

$t$ ${\ displaystyle t}$ - time.

Instruments for measuring electrical power

Wattmeters (including varometers )

Hydraulic Power

The power of the hydraulic machine or hydraulic cylinder is equal to the product of the differential pressure on the machine (pressure difference at the inlet and outlet) and the fluid flow:

N_{H}=Q_{H}\cdot P_{H},

{\ displaystyle N_ {H} = Q_ {H} \ cdot P_ {H},}

Where

Q_{H}

{\ displaystyle Q_ {H}}

- fluid flow rate, m ³ / s ;

P_{H}

{\ displaystyle P_ {H}}

- pressure drop, Pa .

For example, the NP-89D pump, standing on the Su-24 , Tu-134 and Tu-154 , has a capacity of 55 l / min (~ 0,000917 m ³ / s) at a pressure of 210 kgf / cm ² (21 MPa) ^{[ 3]} - therefore, its hydraulic power is approximately 19.25 kW.

Notes

↑ Power - an article from the Physical Encyclopedia
↑ OIML International Document D2. Legalized (officially approved for use) units. Appendix B (unopened) (inaccessible link) . Date of treatment April 5, 2013. Archived on October 14, 2013.
↑ NP-89D. Description. Specifications. Aggregates of production of OJSC MMZ Znamya

Links

[1] Power - an article from the Physical Encyclopedia

[2] OIML International Document D2. Legalized (officially approved for use) units. Appendix B (unopened) (inaccessible link) . Date of treatment April 5, 2013. Archived on October 14, 2013.

[3] NP-89D. Description. Specifications. Aggregates of production of OJSC MMZ Znamya

Power

Content

Designations Used

Basic Formulas

Units

Power in mechanics

Electric power

Instruments for measuring electrical power

Hydraulic Power

See also

Notes

Links

More articles:

Power
$N,P,W={\frac {dA}{dt}}$ ${\ displaystyle N, P, W = {\ frac {dA} {dt}}}$ $N, P, W = {\ frac {dA} {dt}}$
Dimension	L ² MT ⁻³
Units
SI	Tue
GHS	erg s ⁻¹