Editing Electric motor (section)

==Operating principles==
===Force and torque===
An electric motor converts electrical energy to mechanical energy through the force between two opposed magnetic fields. At least one of the two magnetic fields must be created by an electromagnet through the magnetic field caused by an electrical current.

The force between a current <math>I</math> in a conductor of length <math>\ell</math> perpendicular to a magnetic field <math>\mathbf{B}</math> may be calculated using the [[Lorentz force law#Force on a current-carrying wire|Lorentz force law]]:

: <math>\mathbf{F} = I \ell \times \mathbf{B}</math>

Note: X denotes vector [[cross product]].

The most general approaches to calculating the forces in motors use tensor notation.<ref name="Kirtley (2005)2">{{cite conference|last=Kirtley| first=James L. Jr. |year=2005|title=Class Notes 1: Electromagnetic Forces|url=http://web.mit.edu/course/6/6.685/www/chapter1.pdf|publisher=MIT Dept of Electrical Engineering|archive-url=https://web.archive.org/web/20170104210907/http://web.mit.edu/course/6/6.685/www/chapter1.pdf|archive-date=4 January 2017|access-date=15 March 2013|book-title=6.6585 – Electric Machines|url-status=live}}</ref>

===Power===
Electric motor output power is given as <math display="block">P_\text{em} = T\omega = F v</math>where:

* <math> \omega </math>: shaft [[Angular frequency|angular speed]], [radians per second] 
* <math>T</math>: torque, [Newton-meters]
* <math>F</math>: force, [Newtons]
* <math>v</math>: velocity, [meters per second].

In [[Imperial units]] a motor's mechanical power output is given by,<ref>{{cite web|date=30 November 2011|title=DC Motor Calculations, part 1|url=http://zone.ni.com/devzone/cda/ph/p/id/46|archive-url=https://web.archive.org/web/20071012234329/http://zone.ni.com/devzone/cda/ph/p/id/46|archive-date=12 October 2007|access-date=7 December 2012|publisher=National Instruments}}</ref>

: <math>P_\text{em} = \frac {\omega_\text{rpm} T}{5252}</math> (horsepower)
where:

* <math>\omega_\text{rpm}</math>, shaft angular speed [<nowiki/>[[Revolutions per minute|rpm]]]
* <math>T</math>: torque, [foot-pounds].

In an asynchronous or induction motor, the relationship{{citation needed|date=July 2022}} between motor speed and air gap power{{clarify|reason=what is air gap power|date=July 2022}} is given by the following:

: <math>P_\text{airgap} = \frac{R_r}{s} I_r^{2}</math>, where
:: R<sub>r</sub> – rotor resistance
:: I<sub>r</sub><sup>2</sup> – square of current induced in the rotor
:: s – motor slip{{clarify|reason=what are the units of s|date=July 2022}}; i.e., difference between synchronous speed and slip speed, which provides the relative movement needed for current induction in the rotor.

===Back EMF===
{{Main|Electromotive force}}
The movement of armature windings of a direct-current or universal motor through a magnetic field, induce a voltage in them. This voltage tends to oppose the motor supply voltage and so is called "[[Electromotive force|back electromotive force (EMF)]]". The voltage is proportional to the running speed of the motor. The back EMF of the motor, plus the voltage drop across the winding internal resistance and brushes, must equal the voltage at the brushes. This provides the fundamental mechanism of speed regulation in a DC motor. If the mechanical load increases, the motor slows down; a lower back EMF results, and more current is drawn from the supply. This increased current provides the additional torque to balance the load.<ref name="Dwight (1949)2">{{cite book|last1=Dwight|first1=Herbert B.|title=§27 to §35A Electromagnetic Induction of EMF in Sec. 2 – Electric and Magnetic Circuits|last2=Fink|first2=D.G.|pages=36–41}} in {{harvnb|Knowlton|1949}}</ref>

In AC machines, it is sometimes useful to consider a back EMF source within the machine; this is of particular concern for close speed regulation of induction motors on VFDs.<ref name="Dwight (1949)2" />

===Losses===
[[Losses in electrical systems|Motor losses]] are mainly due to [[Joule heating|resistive losses]] in windings, core losses and mechanical losses in bearings, and aerodynamic losses, particularly where cooling fans are present, also occur.

Losses also occur in commutation, mechanical commutators spark; electronic commutators and also dissipate heat.

===Efficiency===
To calculate a motor's efficiency, the mechanical output power is divided by the electrical input power:

: <math>\eta = \frac{P_\text{m}}{P_\text{e}}</math>,

where <math>\eta</math> is [[energy conversion efficiency]], <math>P_\text{e}</math> is electrical input power, and <math>P_\text{m}</math> is mechanical output power:

: <math>P_\text{e} = I V</math>

: <math>P_\text{m} = T \omega</math>

where <math>V</math> is input voltage, <math>I</math> is input current, <math>T</math> is output torque, and <math>\omega</math> is output angular velocity. It is possible to derive analytically the point of maximum efficiency. It is typically at less than 1/2 the [[stall torque]].{{citation needed|date=September 2013}}

Various national regulatory authorities have enacted legislation to encourage the manufacture and use of higher-efficiency motors. Electric motors have efficiencies ranging from around 15%-20% for shaded pole motors, up to 98% for permanent magnet motors,<ref>{{cite book | url=https://books.google.com/books?id=VCHumncaeAAC&dq=15%25+shaded+pole+motor&pg=PA141 | title=Handbook of Fractional-Horsepower Drives | isbn=978-3-540-73129-0 | last1=Stoelting | first1=Hans-Dieter | last2=Kallenbach | first2=Eberhard | last3=Amrhein | first3=Wolfgang | date=29 April 2008 | publisher=Springer }}</ref><ref>{{cite book | url=https://books.google.com/books?id=EBteVXBlF-sC&dq=shaded+pole+motor+efficiency&pg=PA72 | title=Small Electric Motors | isbn=978-0-85296-921-2 | last1=Moczala | first1=Helmut | year=1998 }}</ref><ref>{{Cite web|last=Ruffo|first=Gustavo Henrique|title=Magnax Yokeless Axial Flux Motor Promises 98 Percent Efficiency|url=https://insideevs.com/news/361185/magnax-axial-flux-electric-motor/|website=InsideEVs}}</ref> with efficiency also dependent on load. Peak efficiency is usually at 75% of the rated load. So (as an example) a 10 HP motor is most efficient when driving a load that requires 7.5 HP.<ref>{{cite web|title=Determining Electric Motor Load and Efficiency|url=https://www.energy.gov/sites/prod/files/2014/04/f15/10097517.pdf|url-status=live|access-date=22 July 2021|publisher=U.S. Department of Energy|archive-url=https://web.archive.org/web/20161130131125/http://www.energy.gov:80/sites/prod/files/2014/04/f15/10097517.pdf |archive-date=2016-11-30 }}</ref> Efficiency also depends on motor size; larger motors tend to be more efficient.<ref>{{cite web|date=September 2013|title=E3 Product Profile: Electric Motors|url=https://www.energyrating.gov.au/sites/default/files/documents/Product_profile_-_Electric_motors_September_2013.pdf|url-status=live|access-date=22 July 2021|series=E3 Equipment Energy Efficiency|publisher=Governments of Australia and New Zealand|archive-url=https://web.archive.org/web/20200328010904/https://www.energyrating.gov.au/sites/default/files/documents/Product_profile_-_Electric_motors_September_2013.pdf |archive-date=2020-03-28 }}</ref> Some motors can not operate continually for more than a specified period of time (e.g. for more than an hour per run)<ref>{{Cite web|title=Traction motors｜Transportation Systems Products｜Transportation Systems｜Products Information｜Toyo Denki Seizo K.K.|url=https://www.toyodenki.co.jp/en/products/transport/train/motor.php|website=www.toyodenki.co.jp}}</ref>

===Goodness factor===
{{main|Goodness factor}}
[[Eric Laithwaite]]<ref name="Laithwaite2">{{cite journal|last=Laithwaite|first=E.R.|date=February 1975|title=Linear electric machines – A personal view|journal=Proceedings of the IEEE|volume=63|issue=2|pages=250–90|bibcode=1975IEEEP..63..250L|doi=10.1109/PROC.1975.9734|s2cid=20400221}}</ref> proposed a metric to determine the 'goodness' of an electric motor:<ref name="Patterson (2003)2">{{cite conference|last=Patterson|first=D.J.|author2=Brice, C.W.|author3=Dougal, R.A.|author4=Kovuri, D.|date=1–4 June 2003|title=The "Goodness" of Small Contemporary Permanent Magnet Electric Machines|url=http://vtb.engr.sc.edu/vtbwebsite/downloads/publications/IEMDCpaper.pdf|publisher=IEEE|volume=2|pages=1195–200|doi=10.1109/IEMDC.2003.1210392|isbn=0-7803-7817-2 |archive-url=https://web.archive.org/web/20100613193529/http://vtb.engr.sc.edu/vtbwebsite/downloads/publications/IEMDCpaper.pdf|archive-date=13 June 2010|book-title=Electric Machines and Drives Conference, 2003. IEMDC'03|url-status=live}}</ref>
:<math>G = \frac {\omega} {\text{resistance} \times \text{reluctance}} = \frac {\omega \mu \sigma A_\text{m} A_\text{e}} {l_\text{m} l_\text{e}}</math>

Where:

: <math>G</math> is the goodness factor (factors above 1 are likely to be efficient)
: <math>A_\text{m}, A_\text{e}</math> are the cross sectional areas of the magnetic and electric circuit
: <math>l_\text{m}, l_\text{e}</math> are the lengths of the magnetic and electric circuits
: <math>\mu</math> is the permeability of the core
: <math>\omega</math> is the angular frequency the motor is driven at

From this, he showed that the most efficient motors are likely to have relatively large magnetic poles. However, the equation only directly relates to non PM motors.