THC Science

An induction or asynchronous motor is a type of AC motor where power is supplied to the rotor by means of electromagnetic induction, rather than a commutator or slip rings as in other types of motor. These motors are widely used in industrial drives, particularly polyphase induction motors, because they are rugged and have no brushes. Single-phase versions are used in small appliances. Their speed is determined by the frequency of the supply current, so they are most widely used in constant-speed applications, although variable speed versions, using variable frequency drives are becoming more common. The most common type is the squirrel cage motor.

History

The idea of a rotating magnetic field was developed by François Arago in 1824,^[1] and first implemented by Walter Baily.^[2] Based on this, practical alternating current induction motors seem to have been independently invented by Nikola Tesla and Galileo Ferraris. Ferraris demonstrated a working model of his motor in 1885 and Tesla built his working model in 1887 and demonstrated it at the American Institute of Electrical Engineers in 1888^[3]^[4] .^[5] (although Tesla claimed that he conceived the rotating magnetic field in 1882).^[6] In 1888, Ferraris published his research to the Royal Academy of Sciences in Turin, where he detailed the foundations of motor operation;^[7] Tesla, in the same year, was granted a United States patent for his own motor. ^[8] The modern system of a three-phase transformer and induction motor with a cage was invented by Mikhail Dolivo-Dobrovolsky in 1889/1890.^[9]

The influence of General Electric Company's Philip Alger on induction and synchronous motors is such that he is known to this date as Mr. Induction Motor. He is also known for his seminal 1928 paper, "The Calculation of Armature Reactance of Synchronous Machines". Alger is credited with presaging that industrial electrification depended to a large extent on the development of smaller and lighter motors and was instrumental in the creation of new NEMA standards for motors in the 1940s. Indeed, induction motor improvements were such that a 100 hp induction motor currently has the same mounting dimensions as a 7.5 hp in 1897.^[10]^[11]^[12]^[13]^[14]

The Park transformation has long been widely used in the analysis and study of synchronous and induction machines. The transformation is by far the single most important concept for an understanding of high-performance vector-controlled variable-frequency drives (VFD), which are becoming increasingly popular. The Park transformation was first conceptualized in a 1929 paper authored by Robert Park.^[15] Park's paper was ranked second most important in terms of impact from among all power engineering related papers published in the twentieth century. The novelty of Park's work involves his ability to transform any related machine's linear differential equation set from one with time varying coefficients to another set with time invariant coefficients.^[16] Through the Park transformation, a vector-controlled AC drive allows a three-phase induction motor to behavior like a separated excited DC motor with torque control decoupled from flux control.^[13]

Operation

A 3-phase power supply provides a rotating magnetic field in an induction motor.

In both induction and synchronous motors, the AC power supplied to the motor'sstator creates a magnetic field that rotates in time with the AC oscillations. Whereas a synchronous motor's rotor turns at the same rate as the stator field, a induction motor's rotor rotates at a slower speed than the stator field. The induction motor stator's magnetic field is therefore changing or rotating relative to the rotor. The induction motor's rotor has windings in the form of closed loops. The rotating magnetic flux induces currents in the windings of the rotor;^[17] in a manner similar to currents induced in transformer's secondary windings. These currents in turn create magnetic fields in the rotor that react against the stator field. Due to Lenz's Law, the direction of the magnetic field created will be such as to oppose the change in current through the windings. The cause of induced current in the rotor is the rotating stator magnetic field, so to oppose this the rotor will start to rotate in the direction of the rotating stator magnetic field. The rotor accelerates until the magnitude of induced rotor current and torque balances the applied load. Since rotation at synchronous speed would result in no induced rotor current, an induction motor always operates slower than synchronous speed. The difference between actual and synchronous speed or slip varies from about 0.5 to 5% for normal Design A and B torque curve induction motors.^[18]

For these currents to be induced, the speed of the physical rotor must be lower than that of the stator's rotating magnetic field ( $n_{s}$ ), or the magnetic field would not be moving relative to the rotor conductors and no currents would be induced. As the speed of the rotor drops below synchronous speed, the rotation rate of the magnetic field in the rotor increases, inducing more current in the windings and creating more torque. The ratio between the rotation rate of the magnetic field as seen by the rotor (slip speed) and the rotation rate of the stator's rotating field is called "slip". Under load, the speed drops and the slip increases enough to create sufficient torque to turn the load. For this reason, induction motors are sometimes referred to as asynchronous motors.^[19] An induction motor can be used as an induction generator, or it can be unrolled to form the linear induction motor which can directly generate linear motion.

Synchronous speed

The synchronous speed of an AC motor is the rotation rate of the rotating magnetic field created by the stator. It is always an integer fraction of the supply frequency. The synchronous speed n_s in revolutions per minute (RPM) is given by:

n_{s}={60\times {f} \over {p}}

where f is the frequency of the AC supply current in Hz and p is the number of magnetic pole pairs per phase. When using total number of poles, use 120 as constant instead of 60.;^[20] For example, a small 3-phase motor typically has six magnetic poles organized as three opposing pairs 120° apart, each powered by one phase of the supply current. So there is one pair of poles per phase, which means p = 1, and for a line frequency of 50 Hz the synchronous speed is 3000 RPM.

Slip

The slip s is defined as 'the difference between synchronous speed and operating speed, at the same frequency, expressed in rpm or in percent or ratio of synchronous speed'. Thus

s={\frac {n_{s}-n_{r}}{n_{s}}}\,

where $n_{s}$ is stator electrical speed, $n_{r}$ is rotor mechanical speed.^[21]^[5] Slip is zero at synchronous speed and 1 (100%) when the rotor is stationary. The slip determines the motor's torque. Since the short-circuited rotor windings have small resistance, a small slip induces a large current in the rotor and produces large torque.^[22] At full rated load, typical values of slip are 4-6% for small motors and 1.5-2% for large motors, so induction motors have good speed regulation and are considered constant-speed motors.

Torque curve

The torque exerted by the motor as a function of slip is given by a torque curve. Over a motor's normal load range, the torque line is close to a straight line, so the torque is proportional to slip.^[23] As the load increases above the rated load, increases in slip provide less additional torque, so the torque line begins to curve over. Finally at a slip of around 20%^[22] the motor reaches its maximum torque, called the "breakdown torque". If the load torque reaches this value, the motor will stall. At values of slip above this, the torque decreases. In 3-phase motors the torque drops but still remains high at a slip of 100% (stationary rotor), so these motors are self-starting. The starting torque of an induction motor is less than that of other types of motors, but still around 300% of rated torque.^[23] In 2-pole single-phase motors, the torque goes to zero at 100% slip (zero speed), so these require alterations to the stator such as shaded poles to provide starting torque.

Construction

Typical winding pattern for a 3 phase, 4 pole motor (phases here are labelled U, V, W). Note the interleaving of the pole windings and the resulting quadrupole field.

The stator of an induction motor consists of poles carrying supply current to induce a magnetic field that penetrates the rotor. To optimize the distribution of the magnetic field, the windings are distributed in slots around the stator, with the magnetic field having the same number of north and south poles. Induction motors are most commonly run on single-phase or three-phase power, but two-phase motors exist; in theory, induction motors can have any number of phases. Many single-phase motors having two windings can be viewed as two-phase motors, since a capacitor is used to generate a second power phase 90 degrees from the single-phase supply and feeds it to the second motor winding. Single-phase power is more widely available in residential buildings, but cannot produce a rotating field in the motor, so they must incorporate some kind of starting mechanism to produce a rotating field. There are three types of rotor: squirrel cage rotors made up of skewed (to reduce noise) bars of copper or aluminum that span the length of the rotor, slip ring rotors with windings connected to slip rings replacing the bars of the squirrel cage, and solid core rotors made from mild steel.^{[citation needed]} For information on die-cast copper rotors in energy-efficient induction motors, see: Copper die-cast rotors.

Speed control

Typical torque curves for different line frequencies. By varying the line frequency with an inverter, induction motors can be kept on the stable part of the torque curve above the peak over a wide range of rotation speeds. However, the inverters can be expensive, and fixed line frequencies and other start up schemes are often employed instead.

The theoretical unloaded speed (with slip approaching zero) of the induction motor is controlled by the number of pole pairs and the frequency of the supply voltage.

When driven from a fixed line frequency, loading the motor reduces the rotation speed. When used in this way, induction motors are usually run so that the shaft rotation speed is kept above the peak torque point; then the motor will tend to operate at reasonably constant speed. Below this point, the speed tends to be unstable and the motor may stall or run at reduced shaft speed, depending on the nature of the mechanical load.

Before the development of semiconductor power electronics, it was difficult to vary the frequency, and squirrel-cage induction motors were mainly used in fixed speed applications. Applications such as electric overhead cranes used wound rotor motors with slip rings to allow an external variable resistance to be inserted in the rotor circuit, allowing considerable range of speed control. Some very large slip-ring motor drives recovered energy from the rotor circuit, rectified it, and returned it to the system using an inverter. Many DC motor variable-speed applications can now be served by induction motors and accompanying inverters in industrial applications.The most common and efficient way to control the speed of asynchronous motors is using power inverters and that, in fact, is the only significant disadvantage of this kind of motors, because inverters are rather expensive and usually less reliable than motors themselves.

Rotation reversal

The method of changing the direction of rotation of an induction motor depends on whether it is a three-phase or single-phase machine. In the case of three phase, reversal is carried out by swapping the connection of any 2 out of the 3 power phases. In the case of a single-phase motor it is usually achieved by changing the connection of a starting capacitor from one section of a motor winding to the other. In this latter case both motor windings are usually similar (e.g. in washing machines).

Equivalent circuit

The equivalent circuit and associated key equations of the induction motor are as follows.^[24] ^[25]

Z_{m}=R_{s}+jX_{s}+{\frac {({\frac {R_{s}}{s}}+jX_{r})(jX_{m})}{{\frac {R_{r}}{s}}+j(X_{r}+X_{m})}}

I_{s}=V_{s}/(R_{s}+jX_{s}+{\frac {({\frac {R_{s}}{s}}+jX_{r})(jX_{m})}{{\frac {R_{r}}{s}}+j(X_{r}+X_{m})}})

I_{r}^{'}={\frac {jX_{m}}{{\frac {R_{r}}{s}}+j(X_{r}+X_{m})}}I_{s}

\omega _{r}={\frac {2{\pi }n_{s}}{60}}={\frac {4{\pi }f_{s}}{p}}

T_{em}={\frac {P_{em}}{\omega _{r}}}={\frac {\frac {P_{r}}{s}}{\omega _{r}}}={\frac {3I_{r}^{'2}R_{r}}{\omega _{r}s}}

where

$f_{s}$	stator synchronous frequency
$n_{s}$	synchronous speed in revolutions per second
$I_{s}$	stator current
$I_{r}^{'}$	rotor current referred to stator side
$j={\sqrt {-1}}$	imaginary number operator
$p$	number of motor poles
$P_{em}$	electromagnetic power
$P_{r}$	rotor copper losses
$R_{s},X_{s}$	stator resistance and leakage reactance
$R_{r}^{'},X_{r}^{'}$	rotor resistance and leakage reactance referred to the stator side
$s$	slip
$T_{em}$	electromagnetic torque
$V_{s}$	stator phase voltage
$X_{m}$	magnetizing reactance
$Z_{m}$	motor equivalent impedance
$\omega _{r}$	rotor speed in radians per second

Starting

A single phase induction motor is not self-starting; thus, it is necessary to provide a starting circuit and associated start windings to give the initial rotation in a single phase induction motor. The normal running windings within such a motor can cause the rotor to turn in either direction, so the starting circuit determines the operating direction.

A polyphase induction motor is self-starting and produces torque even at standstill. The four methods of starting an induction motor are direct on-line, reactor, auto-transformer and star-delta. Unlike a wound-rotor motor, the rotor circuit is inaccessible and it is not feasible to introduce extra resistance for starting or speed control.

For small single-phase shaded-pole motor of a few watts, starting is done by a shaded pole, with a turn of copper wire around part of the pole. The current induced in this turn lags behind the supply current, creating a delayed magnetic field around the shaded part of the pole face. This imparts sufficient rotational character to start the motor. These motors are typically used in applications such as desk fans and record players, as the starting torque is very low and low efficiency is not objectionable.

Larger single phase motors have a second stator winding fed with out-of-phase current; such currents may be created by feeding the winding through a capacitor or having it have different values of inductance and resistance from the main winding. In some designs, the second winding is disconnected once the motor is up to speed, usually either by a centrifugal switch acting on weights on the motor shaft or a thermistor which heats up and increases its resistance, reducing the current through the second winding to an insignificant level. Other designs keep the second winding on when running, improving torque.

Polyphase motors have rotor bars shaped to give different speed/torque characteristics. The current distribution within the rotor bars varies depending on the frequency of the induced current. At standstill, the rotor current is the same frequency as the stator current, and tends to travel at the outermost parts of the squirrel-cage rotor bars (the skin effect). The different bar shapes can give usefully different speed/torque characteristics as well as some control over the inrush current at startup. Polyphase motors can generate torque from standstill, so no extra mechanism is required to initiate rotation.

In a wound rotor motor, slip rings are provided and external resistance can be inserted in the rotor circuit, allowing the speed/torque characteristic to be changed for purposes of acceleration control and speed control. Generally, maximum torque is delivered when the reactance of the rotor circuit is equal to its resistance.

Linear induction motor

A linear induction motor (LIM) is an AC asynchronous linear motor that works by the same general principles as other induction motors but which has been designed to directly produce motion in a straight line.

Linear motors frequently run on a 3 phase power supply.

Their uses include magnetic levitation, linear propulsion, and linear actuators. They have also been used for pumping liquid metals.^[26]

Electrical energy efficiency

Various regulatory authorities in many countries have introduced and implemented legislation to encourage the manufacture and use of higher efficiency electric motors. There is existing and forthcoming legislation regarding the future mandatory use of premium-efficiency induction-type motors in defined equipment. For more information, see: Premium efficiency and Copper in energy efficient motors.

Sources

Henri Boy de la Tour (1906). The Induction Motor: Its Theory and Design, Set Forth By a Practical Method of Calculation. Translated Cyprien Odilon Mailloux. McGraw Pub. Co.
Benjamin Franklin Bailey (1911). The Induction Motor. McGraw-Hill.
Bernhard Arthur Behrend (1901). The Induction Motor: A Short Treatise on its Theory and Design, With Numerous Experimental Data and Diagrams. Electrical world and engineer.

References

^ Babbage, C. (Jan. 1825). "Account of the Repetition of M. Arago's Experiments on the Magnetism Manifested by Various Substances during the Act of Rotation". Philosophical Transactions of the Royal Society of London. 115 (0): 467–496. doi:10.1098/rstl.1825.0023. Retrieved 2 December 2012. {{cite journal}}: Check date values in: |date= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ Thompson, Silvanus Phillips (1895). Polyphase Electric Currents and Alternate-Current Motors. London: E. & F.N. Spon. p. 261. Retrieved 2 December 2012.
^ Froehlich, Fritz E. Editor-in-Chief (1992). The Froehlich/Kent Encyclopedia of Telecommunications: Volume 17 - Television Technology to Wire Antennas (First ed.). New York: Marcel Dekker, Inc. p. 36. ISBN 0-8247-2902-1. Retrieved 2 December 2012. {{cite book}}: |first= has generic name (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ The Electrical Engineer (21 Sep. 1888). ". . . a new application of the alternating current in the production of rotary motion was made known almost simultaneously by two experimenters, Nikola Tesla and Galileo Ferraris, and the subject has attracted general attention from the fact that no commutator or connection of any kind with the armature was required. . . .". Vol. Volume II. London: Charles & Co. p. 239. {{cite book}}: |volume= has extra text (help); Check date values in: |year= (help)
^ ^a ^b Ferraris, Galileo (1885). "Electromagnetic Rotation with an Alternating Current". Electrican. 36: 360–375. Cite error: The named reference "Sravastava" was defined multiple times with different content (see the help page).
^ O'Neill, John J. (2007). Prodigal Genius : The Life of Nikola Tesla. San Diego, Calif.: Book Tree. p. 115. ISBN 978-1585093083.
^ The Case Files: Nikola Tesla. "Two-Phase Induction Motor". The Franklin Institute. Retrieved 2 December 2012.
^ Day, Lance (1996). Biographical Dictionary of the History of Technology. London: Routledge. p. 1204. ISBN 0-203-02829-5. Retrieved 2 December 2012. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ Hubbell, M.W. (2011). The Fundamentals of Nuclear Power Generation Questions & Answers. Authorhouse. p. 27. ISBN 978-1463424411.
^ Alger, Philip L. (1928). "The Calculation of Armature Reactance of Synchronous Machines". Trans. of American Institute of Electrical Engineers. 2. 47: 493–512. Retrieved 1 December 2012. {{cite journal}}: Unknown parameter |month= ignored (help)
^ Alger, Philip L. "Biography". IEEE Global History Network. Retrieved 1 December 2012.
^ Alger, P. L. (1979). "Correspondence: How I Earned the Sobriquet of Mr. Induction Motor". Electric Machines & Power Systems. 3 (3–4): 369–370. doi:10.1080/03616967908955353. Retrieved 1 December 2012.
^ ^a ^b Krishnan, R. (2011). "Prof. Joachim Holtz's Everlasting Contributions in Power Electronics and Electric Motor Drives". IEEE Industrial Electronics Magazine. 5 (2): 4–5. doi:10.1109/MIE.2011.941129. Retrieved 1 December 2012. {{cite journal}}: Unknown parameter |month= ignored (help)
^ Alger, P.L. (1976). "The History of Induction Motors in America". Proceedings of the IEEE. 64 (9): 1380–1383. doi:10.1109/PROC.1976.10329. {{cite journal}}: |access-date= requires |url= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ Park, Robert (1929). "Two Reaction Theory of Synchronous Machines". Trans. of the AIEE. 48: 716–730.
^ Heydt,, G. T. (2000). "High Impact Papers in Power Engineering, 1900-1999" (PDF). North American Power Symposium (NAPS) 2000: P-1 to P-7. Retrieved May 23, 2012. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)CS1 maint: extra punctuation (link)
^ "AC Motors". NSW HSC Online - Charles Sturt University. Retrieved 2 December 2012.
^ NEMA MG-1 Condensed (2007). Information Guide for General Purpose Industrial AC Small and Medium Squirrel-Cage Induction Motor Standards. Rosslyn, VA USA: NEMA. p. 29 (Table 11). Retrieved 2 December 2012.{{cite book}}: CS1 maint: numeric names: authors list (link)
^ "Induction (Asychronous) Motors" (PDF). Mississipi State University Dept of Electrical and Computer Engineering, Course ECE 3183, 'Electrical Engineering Systems for non-ECE majors'. Retrieved 2 December 2012.
^ Electric Motors Reference Center by Machine Design magazine. "Induction Motors". Penton Media, Inc.
^ NEMA Standards Publication (2007). Application Guide for AC Adjustable Speed Drive Systems. Rosslyn, VA USA: NEMA. p. 6. Retrieved 2 December 2012.
^ ^a ^b Herman, Stephen L. (2011). Alternating Current Fundamentals (8th ed.). USA: Cengage Learning. pp. 529–536. ISBN 1-111-03913-5. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)
^ ^a ^b Keljik, Jeffrey (2009). "Chapter 12 - The Three-Phase, Squirrel-Cage Induction Motor". Electricity 4 : AC/DC Motors, Controls, and Maintenance (9th ed.). Clifton Park, NY: Delmar, Cengage Learning. pp. 112–115. ISBN 1-4354-0031-3. {{cite conference}}: Cite has empty unknown parameter: |coauthors= (help); Unknown parameter |booktitle= ignored (|book-title= suggested) (help)
^ Liang, Xiaodong (2011). "Induction Motor Starting in Practical Industrial Applications". IEEE Transactions on Industry Applications. 47 (1): 271–280. doi:10.1109/TIA.2010.2090848. Retrieved 4 December 2012. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)
^ Jordan, Howard E. (1994). Energy-Efficient Electric Motors and their Applications (2nd ed.). New York: Plenum Press. ISBN 0-306-44698-7.
^ Bulletin of the Atomic Scientists. Educational Foundation for Atomic Science. 6 June 1973. Retrieved 8 August 2012.

External links

A drawing of an induction motor
Template:It Rotating magnetic fields: interactive
Construct your squirrelcage electromotor using povray
[1] More on Induction Motor.

Template:Link GA

[Babbage_(1825)-1] Babbage, C. (Jan. 1825). "Account of the Repetition of M. Arago's Experiments on the Magnetism Manifested by Various Substances during the Act of Rotation". Philosophical Transactions of the Royal Society of London. 115 (0): 467–496. doi:10.1098/rstl.1825.0023. Retrieved 2 December 2012. {{cite journal}}: Check date values in: |date= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

[Thompson_(1895)-2] Thompson, Silvanus Phillips (1895). Polyphase Electric Currents and Alternate-Current Motors. London: E. & F.N. Spon. p. 261. Retrieved 2 December 2012.

[Froehlich_(1992)-3] Froehlich, Fritz E. Editor-in-Chief (1992). The Froehlich/Kent Encyclopedia of Telecommunications: Volume 17 - Television Technology to Wire Antennas (First ed.). New York: Marcel Dekker, Inc. p. 36. ISBN 0-8247-2902-1. Retrieved 2 December 2012. {{cite book}}: |first= has generic name (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

[TEE_(1888)-4] The Electrical Engineer (21 Sep. 1888). ". . . a new application of the alternating current in the production of rotary motion was made known almost simultaneously by two experimenters, Nikola Tesla and Galileo Ferraris, and the subject has attracted general attention from the fact that no commutator or connection of any kind with the armature was required. . . .". Vol. Volume II. London: Charles & Co. p. 239. {{cite book}}: |volume= has extra text (help); Check date values in: |year= (help)

[Sravastava-5] Ferraris, Galileo (1885). "Electromagnetic Rotation with an Alternating Current". Electrican. 36: 360–375. Cite error: The named reference "Sravastava" was defined multiple times with different content (see the help page).

[O'Neill_(2007)-6] O'Neill, John J. (2007). Prodigal Genius : The Life of Nikola Tesla. San Diego, Calif.: Book Tree. p. 115. ISBN 978-1585093083.

[TFI_(now)-7] The Case Files: Nikola Tesla. "Two-Phase Induction Motor". The Franklin Institute. Retrieved 2 December 2012.

[Day_(1996)-8] Day, Lance (1996). Biographical Dictionary of the History of Technology. London: Routledge. p. 1204. ISBN 0-203-02829-5. Retrieved 2 December 2012. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

[9] Hubbell, M.W. (2011). The Fundamentals of Nuclear Power Generation Questions & Answers. Authorhouse. p. 27. ISBN 978-1463424411.

[Alger_(1928)-10] Alger, Philip L. (1928). "The Calculation of Armature Reactance of Synchronous Machines". Trans. of American Institute of Electrical Engineers. 2. 47: 493–512. Retrieved 1 December 2012. {{cite journal}}: Unknown parameter |month= ignored (help)

[Alger_(IEEE_GHN)-11] Alger, Philip L. "Biography". IEEE Global History Network. Retrieved 1 December 2012.

[Alger_(1979)-12] Alger, P. L. (1979). "Correspondence: How I Earned the Sobriquet of Mr. Induction Motor". Electric Machines & Power Systems. 3 (3–4): 369–370. doi:10.1080/03616967908955353. Retrieved 1 December 2012.

[Krishman_(2011)-13] Krishnan, R. (2011). "Prof. Joachim Holtz's Everlasting Contributions in Power Electronics and Electric Motor Drives". IEEE Industrial Electronics Magazine. 5 (2): 4–5. doi:10.1109/MIE.2011.941129. Retrieved 1 December 2012. {{cite journal}}: Unknown parameter |month= ignored (help)

[Alger_(1976)-14] Alger, P.L. (1976). "The History of Induction Motors in America". Proceedings of the IEEE. 64 (9): 1380–1383. doi:10.1109/PROC.1976.10329. {{cite journal}}: |access-date= requires |url= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

[Park_(1929)-15] Park, Robert (1929). "Two Reaction Theory of Synchronous Machines". Trans. of the AIEE. 48: 716–730.

[Heydt_(2000)-16] Heydt,, G. T. (2000). "High Impact Papers in Power Engineering, 1900-1999" (PDF). North American Power Symposium (NAPS) 2000: P-1 to P-7. Retrieved May 23, 2012. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)CS1 maint: extra punctuation (link)

[NSWHSCOnline-17] "AC Motors". NSW HSC Online - Charles Sturt University. Retrieved 2 December 2012.

[NEMAMG1C-18] NEMA MG-1 Condensed (2007). Information Guide for General Purpose Industrial AC Small and Medium Squirrel-Cage Induction Motor Standards. Rosslyn, VA USA: NEMA. p. 29 (Table 11). Retrieved 2 December 2012.{{cite book}}: CS1 maint: numeric names: authors list (link)

[19] "Induction (Asychronous) Motors" (PDF). Mississipi State University Dept of Electrical and Computer Engineering, Course ECE 3183, 'Electrical Engineering Systems for non-ECE majors'. Retrieved 2 December 2012.

[MDEMRC-20] Electric Motors Reference Center by Machine Design magazine. "Induction Motors". Penton Media, Inc.

[NEMAASD-21] NEMA Standards Publication (2007). Application Guide for AC Adjustable Speed Drive Systems. Rosslyn, VA USA: NEMA. p. 6. Retrieved 2 December 2012.

[Herman-22] Herman, Stephen L. (2011). Alternating Current Fundamentals (8th ed.). USA: Cengage Learning. pp. 529–536. ISBN 1-111-03913-5. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)

[Keljik-23] Keljik, Jeffrey (2009). "Chapter 12 - The Three-Phase, Squirrel-Cage Induction Motor". Electricity 4 : AC/DC Motors, Controls, and Maintenance (9th ed.). Clifton Park, NY: Delmar, Cengage Learning. pp. 112–115. ISBN 1-4354-0031-3. {{cite conference}}: Cite has empty unknown parameter: |coauthors= (help); Unknown parameter |booktitle= ignored (|book-title= suggested) (help)

[Liang_(2011)-24] Liang, Xiaodong (2011). "Induction Motor Starting in Practical Industrial Applications". IEEE Transactions on Industry Applications. 47 (1): 271–280. doi:10.1109/TIA.2010.2090848. Retrieved 4 December 2012. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)

[Jordan_(1994)-25] Jordan, Howard E. (1994). Energy-Efficient Electric Motors and their Applications (2nd ed.). New York: Plenum Press. ISBN 0-306-44698-7.

[BAS_(1973)-26] Bulletin of the Atomic Scientists. Educational Foundation for Atomic Science. 6 June 1973. Retrieved 8 August 2012.

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 ==History==
 [[File:Squirrel cage.jpg|thumb|right|Squirrel cage rotor]]
-The idea of a [[rotating magnetic field]] was developed by [[François Arago]] in 1824,<ref>Babbage, C. and Herschel, J.W.F. (1825) "Account of the repetition of M. Arago's experiments on the magnetism manifested by various substances during the act of rotation," ''Philosophical Transactions of the Royal Society of London'', vol. 115, pages 467-496.</ref> and first implemented by [[Walter Baily]].<ref>[[Silvanus Phillips Thompson]], ''Polyphase electric currents and alternate-current motors'' (London, England :  E. & F.N. Span, 1895), [http://books.google.com/books?id=TvwHAAAAMAAJ&pg=PA84 Page 84].</ref> Based on this, practical alternating current induction motors seem to have been independently invented by [[Nikola Tesla]] and [[Galileo Ferraris]]. Ferraris demonstrated a working model of his motor in 1885 and Tesla built his working model in 1887 and demonstrated it at the ''American Institute of Electrical Engineers'' in 1888<ref>{{cite book|url=http://books.google.com/books?id=8j5bJ5OkGpgC&pg=PA36&lpg=PA36&dq=Westinghouse+tesla++Electrical+Engineers&source=bl&ots=5nHK8Hlhi1&sig=7NmfQ2iGm7IznasuUyl8Ce58bNQ&hl=en&sa=X&ei=1-xNUNb6Aonw0gGWv4H4Cg&ved=0CFMQ6AEwBg#v=onepage&q=Westinghouse%20tesla%20%20Electrical%20Engineers&f=false |title=Fritz E. Froehlich, Allen Kent, The Froehlich/Kent Encyclopedia of Telecommunications: Volume 17, page 36 |publisher=Books.google.com |date= |accessdate=2012-09-10}}</ref><ref>The Electrical Engineer. (1888). London: Biggs & Co. Pg., 239. [cf., "[...] new application of the alternating current in the production of rotary motion was made known almost simultaneously by two experimenters, Nikola Tesla and Galileo Ferraris, and the subject has attracted general attention from the fact that no commutator or connection of any kind with the armature was required."]</ref><ref>Galileo Ferraris, "Electromagnetic rotation with an alternating current," Electrican, Vol 36 [1885]. pg 360-75.</ref> (although Tesla claimed that he conceived the rotating magnetic field in 1882).<ref>[[Prodigal Genius: The Life of Nikola Tesla]]. Pg 115</ref>  In 1888, Ferraris published his research to the Royal Academy of Sciences in Turin, where he detailed the foundations of motor operation;<ref>[http://www.fi.edu/learn/case-files/tesla/motor.html "Two-Phase Induction Motor"] (2011), ''The Case Files: Nikola Tesla'', The Franklin Institute.</ref> Tesla, in the same year, was granted a United States patent for his own motor.<ref>[http://books.google.com/books?id=n--ivouMng8C&pg=PA1204&lpg=PA1204&dq=tesla+induction+motor+patent&source=bl&ots=CwZdCXFBMs&sig=yHtXcB6ukl3dO26c73h884URzsI&hl=en&sa=X&ei=1VpOUKCPAaLv0gGb14HwAw&ved=0CEMQ6AEwAw#v=onepage&q=tesla%20induction%20motor%20patent&f=false Lance Day, Biographical Dictionary of the History of Technology, page 1204]</ref> The modern system of a three-phase transformer and induction motor with a cage was invented by [[Mikhail Dolivo-Dobrovolsky]] in 1889/1890.<ref>M. W. Hubbell, The Fundamentals of Nuclear Power Generation: Questions & Answers - Page 27</ref>
+The idea of a [[rotating magnetic field]] was developed by [[François Arago]] in 1824,<ref name="Babbage (1825)">{{cite journal|last=Babbage|first=C.|coauthors=Herschel, J. F. W.|title=Account of the Repetition of M. Arago's Experiments on the Magnetism Manifested by Various Substances during the Act of Rotation|journal=Philosophical Transactions of the Royal Society of London|volume=115|issue=0|pages=467–496|doi=10.1098/rstl.1825.0023|url=http://archive.org/stream/philtrans03806447/03806447#page/n0/mode/2up|accessdate=2 December 2012|date=Jan. 1825}}</ref> and first implemented by [[Walter Baily]].<ref name="Thompson (1895)">{{cite book|last=[[Silvanus Phillips Thompson|Thompson]]|first=Silvanus Phillips|title=Polyphase Electric Currents and Alternate-Current Motors|year=1895|publisher=E. & F.N. Spon|pages=261|location=London|url=http://archive.org/stream/polyphaseelectri00thomuoft#page/n5/mode/2up|accessdate=2 December 2012}}</ref>  Based on this, practical alternating current induction motors seem to have been independently invented by [[Nikola Tesla]] and [[Galileo Ferraris]]. Ferraris demonstrated a working model of his motor in 1885 and Tesla built his working model in 1887 and demonstrated it at the ''American Institute of Electrical Engineers'' in 1888<ref name="Froehlich (1992)">{{cite book|last=Froehlich|first=Fritz E. Editor-in-Chief|coauthors=Allen Kent Co-Editor|title=The Froehlich/Kent Encyclopedia of Telecommunications: Volume 17 - Television Technology to Wire Antennas|year=1992|publisher=Marcel Dekker, Inc.|location=New York|isbn=0-8247-2902-1|url=http://www.amazon.com/Froehlich-Kent-Encyclopedia-Telecommunications-Television/dp/0824729153#reader_0824729153|edition=First|accessdate=2 December 2012|page=36}}</ref><ref name="TEE (1888)">{{cite book|last=The Electrical Engineer|year=21 Sep. 1888|title=". . . a new application of the alternating current in the production of rotary motion was made known almost simultaneously by two experimenters, Nikola Tesla and Galileo Ferraris, and the subject has attracted general attention from the fact that no commutator or connection of any kind with the armature was required. . . ."|publisher=Charles & Co.|volume=Volume II|location=London|page=239|url=http://books.google.ca/books?id=_KvmAAAAMAAJ&pg=PA239&lpg=PA239&dq=The+electrical+engineer+1888+by+two+experimenters,+Nikola+Tesla+and+Galileo+Ferraris&source=bl&ots=O9MmzKi-0t&sig=GQS21Uaduwa2VUfA55rO7bx7LgM&hl=en&sa=X&ei=fdG6UMrVNImBywHy44AI&ved=0CE0Q6AEwBg#v=onepage&q=The%20electrical%20engineer%201888%20by%20two%20experimenters%2C%20Nikola%20Tesla%20and%20Galileo%20Ferraris&f=false}}</ref> .<ref name=Sravastava>{{cite journal|title=Electromagnetic Rotation with an Alternating Current|first=Galileo|last=Ferraris|journal=Electrican|volume=36|date=1885|pages=360-375}}</ref> (although Tesla claimed that he conceived the rotating magnetic field in 1882).<ref name="O'Neill (2007)">{{cite book|last=O'Neill|first=John J.|title=Prodigal Genius : The Life of Nikola Tesla|year=2007|publisher=Book Tree|location=San Diego, Calif.|isbn=978-1585093083|url=http://www.amazon.ca/Prodigal-Genius-Life-Nikola-Tesla/dp/1585093084|page=115}}</ref>   In 1888, Ferraris published his research to the Royal Academy of Sciences in Turin, where he detailed the foundations of motor operation;<ref name="TFI (now)">{{cite web|last=The Case Files: Nikola Tesla|title=Two-Phase Induction Motor|url=http://www.fi.edu/learn/case-files/tesla/motor.html|publisher=The Franklin Institute|accessdate=2 December 2012}}</ref>
+Tesla, in the same year, was granted a United States patent for his own motor.
+<ref name="Day (1996)">{{cite book|last=Day|first=Lance |title=Biographical Dictionary of the History of Technology|year=1996|publisher=Routledge|location=London|isbn=0-203-02829-5|coauthors=McNeil, Ian; (Editors)|page=1204|accessdate=2 December 2012|url=http://books.google.ca/books?id=n--ivouMng8C&pg=PA1204&lpg=PA1204&dq=tesla+induction+motor+patent&source=bl&ots=CwZdCXFBMs&sig=yHtXcB6ukl3dO26c73h884URzsI&hl=en&sa=X&ei=1VpOUKCPAaLv0gGb14HwAw&redir_esc=y#v=onepage&q=tesla%20induction%20motor%20patent&f=false}}</ref>  The modern system of a three-phase transformer and induction motor with a cage was invented by [[Mikhail Dolivo-Dobrovolsky]] in 1889/1890.<ref>{{cite book|last=Hubbell|first=M.W.|date=2011|title=The Fundamentals of Nuclear Power Generation Questions & Answers.|publisher=Authorhouse|isbn=978-1463424411|url=http://www.amazon.com/Fundamentals-Nuclear-Power-Generation-Questions/dp/1463424418|page=27}}</ref>
+The influence of General Electric Company's Philip Alger on induction and synchronous motors is such that he is known to this date as Mr. Induction Motor. He is also known for his seminal 1928 paper, "The Calculation of Armature Reactance of Synchronous Machines". Alger is credited with presaging that industrial [[electrification]] depended to a large extent on the development of smaller and lighter motors and was instrumental in the creation of new [[NEMA]] standards for motors in the 1940s.  Indeed, induction motor improvements were such that a 100 hp induction motor currently has the same mounting dimensions as a 7.5 hp in 1897.<ref name="Alger (1928)">{{cite journal|last=Alger|first=Philip L.|title=The Calculation of Armature Reactance of Synchronous Machines|journal=Trans. of American Institute of Electrical Engineers|year=1928|month=April|volume=47|series=2|pages=493-512|url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5055008|accessdate=1 December 2012}}</ref><ref name="Alger (IEEE GHN)">{{cite web|last=Alger|first=Philip L.|title=Biography|url=http://www.ieeeghn.org/wiki/index.php/Philip_L._Alger|publisher=IEEE Global History Network|accessdate=1 December 2012}}</ref><ref name="Alger (1979)">{{cite journal|last=Alger|first=P. L.|title=Correspondence: How I Earned the Sobriquet of Mr. Induction Motor|journal=Electric Machines & Power Systems|year=1979|volume=3|issue=3-4|pages=369–370|doi=10.1080/03616967908955353|url=http://www.tandfonline.com/doi/abs/10.1080/03616967908955353|accessdate=1 December 2012}}</ref><ref name="Krishman (2011)">{{cite journal|last=Krishnan|first=R.|title=Prof. Joachim Holtz's Everlasting Contributions in Power Electronics and Electric Motor Drives|journal=IEEE Industrial Electronics Magazine|year=2011|month=June|volume=5|issue=2|pages=4–5|doi=10.1109/MIE.2011.941129|url=http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=05876645|accessdate=1 December 2012}}</ref><ref name="Alger (1976)">{{cite journal|last=Alger|first=P.L.|coauthors=Arnold, R.E.|title=The History of Induction Motors in America|journal=Proceedings of the IEEE|year=1976|volume=64|issue=9|pages=1380–1383|doi=10.1109/PROC.1976.10329|accessdate=1 December 2012}}</ref>
+The [[Park transform]]ation has long been widely used in the analysis and study of [[synchronous machine|synchronous]] and induction machines. The transformation is by far the single most important concept for an understanding of high-performance [[vector control (motor)|vector-controll]]ed [[variable-frequency drive]]s (VFD), which are becoming increasingly popular. The Park transformation was first conceptualized in a 1929 paper authored by Robert Park.<ref name="Park (1929)">{{cite journal|last=Park|first=Robert|title=Two Reaction Theory of Synchronous Machines|journal=Trans. of the AIEE|year=1929|volume=48|pages=716–730}}</ref> Park's paper was ranked second most important in terms of impact from among all power engineering related papers published in the twentieth century. The novelty of Park's work involves his ability to transform any related machine's linear [[differential equation]] set from one with time varying coefficients to another set with time ''invariant'' coefficients.<ref name="Heydt (2000)">{{cite journal|last=Heydt,|first=G. T.|coauthors=Venkata, S. S.; Balijepalli, N.|title=High Impact Papers in Power Engineering, 1900-1999|journal=North American Power Symposium (NAPS) 2000|year=2000|month=Oct. 23-24|pages=P-1 to P-7|url=http://www.ece.mtu.edu/faculty/ljbohman/peec/papers/Heydt.pdf|accessdate=May 23, 2012}}</ref>  Through the Park transformation, a vector-controlled AC drive allows a three-phase induction motor to behavior like a separated excited DC motor with torque control decoupled from flux control.<ref name="Krishman (2011)"/>
 ==Operation==
 [[Image:Rotatingfield.png|thumb|A 3-phase power supply provides a rotating magnetic field in an induction motor.]]
-In both induction and [[synchronous motor]]s, the [[stator]] is powered with alternating current ([[polyphase current]] in large machines) and designed to create a [[rotating magnetic field]] which rotates in time with the AC oscillations.  In a [[synchronous motor]], the rotor turns at the same rate as the stator field.  By contrast, in an induction motor the rotor rotates at a slower speed than the stator field.  Therefore the magnetic field through the rotor is changing (rotating).  The rotor has windings in the form of closed loops.  The rotating [[magnetic flux]] induces currents in the windings of the rotor;<ref>http://hsc.csu.edu.au/physics/core/motors/2698/Phy935net.htm {{verified|date=September 2012}}</ref> similar to a [[transformer]].  These currents in turn create magnetic fields in the rotor, that react against the stator field.   Due to [[Lenz's law]], the direction of the magnetic field created will be such as to oppose the change in current through the windings.  The cause of induced current in the rotor is the rotating stator magnetic field, so to oppose this the rotor will start to rotate in the direction of the rotating stator magnetic field. The rotor accelerates until the magnitude of induced rotor current and torque balances the applied load. Since rotation at synchronous speed would result in no induced rotor current, an induction motor always operates slower than synchronous speed. The difference between actual and synchronous speed is called "slip" and in practical motors varies from 1 to 5% at full load torque.
+In both induction and [[synchronous motor]]s, the AC power supplied to the motor's[[stator]] creates a [[rotating magnetic field|magnetic field]] that rotates in time with the AC oscillations.  Whereas a [[synchronous motor]]'s rotor turns at the same rate as the stator field, a induction motor's rotor rotates at a slower speed than the stator field.  The induction motor stator's magnetic field is therefore changing or rotating relative to the rotor.  The induction motor's rotor has windings in the form of closed loops.  The rotating [[magnetic flux]] induces currents in the windings of the rotor;<ref name=NSWHSCOnline>{{cite web|title=AC Motors|publisher=NSW HSC Online - Charles Sturt University|accessdate=2 December 2012|url=http://hsc.csu.edu.au/physics/core/motors/2698/Phy935net.htm }}</ref>  in a manner similar to currents induced in [[transformer]]'s secondary windings.  These currents in turn create magnetic fields in the rotor that react against the stator field.   Due to [[Lenz's Law]], the direction of the magnetic field created will be such as to oppose the change in current through the windings.  The cause of induced current in the rotor is the rotating stator magnetic field, so to oppose this the rotor will start to rotate in the direction of the rotating stator magnetic field. The rotor accelerates until the magnitude of induced rotor current and torque balances the applied load. Since rotation at synchronous speed would result in no induced rotor current, an induction motor always operates slower than synchronous speed. The difference between actual and synchronous speed or slip varies from about 0.5 to 5% for normal Design A and B torque curve induction motors.<ref name=NEMAMG1C>{{cite book |last=NEMA MG-1 Condensed |title= Information Guide for General Purpose Industrial AC Small and Medium Squirrel-Cage Induction Motor Standards | year=2007 | url=http://www.nema.org/Standards/Pages/Information-Guide-for-General-Purpose-Industrial-AC-Small-and-Medium-Squirrel-Cage-Induction-Motor-Standards.aspx | accessdate=2 December 2012| publisher=[[NEMA]]| location=Rosslyn, VA USA |page=29 (Table 11)}}</ref>
-For these currents to be induced, the speed of the physical rotor must be lower than that of the stator's rotating magnetic field (<math>n_s</math>), or the magnetic field would not be moving relative to the rotor conductors and no currents would be induced.  As the speed of the rotor drops below synchronous speed, the rotation rate of the magnetic field in the rotor increases, inducing more current in the windings and creating more torque.  The ratio between the rotation rate of the magnetic field as seen by the rotor (slip speed) and the rotation rate of the stator's rotating field is called "''slip''".  Under load, the speed drops and the slip increases enough to create sufficient torque to turn the load.  For this reason, induction motors are sometimes referred to as asynchronous motors.<ref>{{cite web|url=http://www.ece.msstate.edu/~donohoe/ece3183asynchronous_synchronous_machines.pdf|title=Induction (Asynchronous) Machines}}</ref> An induction motor can be used as an [[induction generator]], or it can be unrolled to form the [[linear induction motor]] which can directly generate linear motion.
+For these currents to be induced, the speed of the physical rotor must be lower than that of the stator's rotating magnetic field (<math>n_s</math>), or the magnetic field would not be moving relative to the rotor conductors and no currents would be induced.  As the speed of the rotor drops below synchronous speed, the rotation rate of the magnetic field in the rotor increases, inducing more current in the windings and creating more torque.  The ratio between the rotation rate of the magnetic field as seen by the rotor (slip speed) and the rotation rate of the stator's rotating field is called "''slip''".  Under load, the speed drops and the slip increases enough to create sufficient torque to turn the load.  For this reason, induction motors are sometimes referred to as asynchronous motors.<ref>{{cite web|title=Induction (Asychronous) Motors|url=http://www.ece.msstate.edu/~donohoe/ece3183asynchronous_synchronous_machines.pdf|publisher=Mississipi State University Dept of Electrical and Computer Engineering, Course ECE 3183, 'Electrical Engineering Systems for non-ECE majors'|accessdate=2 December 2012}}</ref>  An induction motor can be used as an [[induction generator]], or it can be unrolled to form the [[linear induction motor]] which can directly generate linear motion.
 ===Synchronous speed===
@@ Line 23: / Line 29: @@
 <!--
 -->
-where  ''f'' is the [[Utility frequency|frequency]] of the AC supply current in [[Hertz|Hz]] and ''p'' is the number of magnetic '''pole pairs''' per phase. When using total number of poles, use 120 as constant instead of 60.<ref>[http://www.electricmotors.machinedesign.com/guiEdits/Content/bdeee11/bdeee11_7.aspx "Induction Motors"] (2011), Electric Motors Reference Center, ''Machine Design'', Penton Media, Inc.</ref> For example, a small [[3-phase]] motor typically has six magnetic poles organized as three opposing pairs 120° apart, each powered by one phase of the supply current.  So there is one pair of poles per phase, which means ''p'' = 1, and for a line frequency of 50&nbsp;Hz the synchronous speed is 3000 RPM.
+where  ''f'' is the [[Utility frequency|frequency]] of the AC supply current in [[Hertz|Hz]] and ''p'' is the number of magnetic '''pole pairs''' per phase. When using total number of poles, use 120 as constant instead of 60.;<ref name=MDEMRC>{{cite web|url=http://www.electricmotors.machinedesign.com/guiEdits/Content/bdeee11/bdeee11_7.aspx|title=Induction Motors|last=Electric Motors Reference Center by ''Machine Design'' magazine|title=Induction Motors|publisher=Penton Media, Inc.}}</ref> For example, a small [[3-phase]] motor typically has six magnetic poles organized as three opposing pairs 120° apart, each powered by one phase of the supply current.  So there is one pair of poles per phase, which means ''p'' = 1, and for a line frequency of 50&nbsp;Hz the synchronous speed is 3000 RPM.
 ===Slip===
 [[File:Couple glissement MAs.svg|thumb|right|Typical torque curve as a function of slip (slip is represented by ''g'' here, which is proportional to ''s'' in the formula at left).]]
+The slip ''s'' is defined as 'the difference between synchronous speed and operating speed, at the same frequency, expressed in rpm or in percent or ratio of synchronous speed'. Thus
-Slip ''s'' is the rotation rate of the magnetic field, relative to the rotor, divided by the absolute rotation rate of the stator magnetic field
 :<math>s = \frac{n_s-n_r}{n_s}\,</math>
+where
-where <math>n_r</math> is the rotor rotation speed in rpm.<ref>TORQUE SLIP CHARACTERISTICS OF INDUCTION MOTOR By Avinash SrivastavaRavi Kumar (MTECH CAID MSRIT)</ref> Slip is zero at synchronous speed and 1 (100%) when the rotor is stationary.  The slip determines the motor's torque.  Since the short-circuited rotor windings have small resistance, a small slip induces a large current in the rotor and produces large torque.<ref name="Herman">{{cite book
+<math>n_s</math> is stator electrical speed, <math>n_r</math> is rotor mechanical speed.<ref name=NEMAASD>{{cite book |last=NEMA Standards Publication |title= Application Guide for AC Adjustable Speed Drive Systems | year=2007 | url=http://www.nema.org/stds/acadjustable.cfm | accessdate=2 December 2012| publisher=[[NEMA]]| location=Rosslyn, VA USA |page=6}}</ref><ref name=Sravastava>{{cite journal|title=Torque Slip Characteristics of Induction Motor|last=Srivastava|first=Avinash|coauthors=Kumar, Ravi|publisher=Malnad College Of Engineering|journal=Course notes}}</ref>  Slip is zero at synchronous speed and 1 (100%) when the rotor is stationary.  The slip determines the motor's torque.  Since the short-circuited rotor windings have small resistance, a small slip induces a large current in the rotor and produces large torque.<ref name="Herman">{{cite book
   | last = Herman
   | first = Stephen L.
   | authorlink =
   | coauthors =
-  | title = Alternating Current Fundamentals, 8th Ed.
+  | title = Alternating Current Fundamentals
+| edition= 8th
   | publisher = Cengage Learning
   | year = 2011
@@ Line 47: / Line 54: @@
 ===Torque curve===
-The torque exerted by the motor as a function of slip is given by a torque curve.  Over a motor's normal load range, the torque line is close to a straight line, so the torque is proportional to slip.<ref name="Keljik">{{cite book
+The torque exerted by the motor as a function of slip is given by a torque curve.  Over a motor's normal load range, the torque line is close to a straight line, so the torque is proportional to slip.<ref name="Keljik">{{cite conference
   | last = Keljik
-  | first = Jeffrey J. Keljik
+  | first = Jeffrey
+|chapter=12
+|title=Chapter 12 - The Three-Phase, Squirrel-Cage Induction Motor
   | authorlink =
   | coauthors =
-  | title = Electricity Four, 9th Ed.
+  | booktitle = Electricity 4 : AC/DC Motors, Controls, and Maintenance
-  | publisher = Cengage Learning
+  | publisher = Delmar, Cengage Learning
-  | year = 2008
+  | year = 2009
-  | location =
+  | edition = 9th
+  | location = Clifton Park, NY
   | pages = 112–115
   | url = http://books.google.com/books?id=y69O8PnwLbYC&pg=PA105&dq=squirrel+cage+motor+induction&hl=en&sa=X&ei=rP_6TqbTA6aciQLswMDQDg&ved=0CE4Q6AEwAA#v=onepage&q=squirrel%20cage%20motor%20induction&f=false
@@ Line 81: / Line 91: @@
 ==Equivalent circuit==
+The [[equivalent circuit]] and associated key equations of the induction motor are as follows.<ref name="Liang (2011)">{{cite journal|last=Liang|first=Xiaodong|coauthors=Ilochonwu, Obinna|title=Induction Motor Starting in Practical Industrial Applications|journal=IEEE Transactions on Industry Applications|year=2011|month=Jan|volume=47|issue=1|pages=271–280|doi=10.1109/TIA.2010.2090848|url=http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=5621895&url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F28%2F4957013%2F05621895.pdf%3Farnumber%3D5621895|accessdate=4 December 2012}}</ref> <ref name="Jordan (1994)">{{cite book|last=Jordan|first=Howard E.|title=Energy-Efficient Electric Motors and their Applications|url=http://books.google.co.uk/books?id=utWtW_9NMgcC&lpg=PA89&dq=induction%20motor%20power%20correction&pg=PA89#v=onepage&q=induction%20motor%20power%20correction&f=false|year=1994|publisher=Plenum Press|location=New York|isbn=0-306-44698-7|edition=2nd}}</ref>
-[[File:Induction equivalent.png|thumb|The equivalent circuit of an induction motor.]]
-The [[equivalent circuit]] of an induction motor has the equivalent resistance of the stator on the left, consisting of the copper and core resistance in series, as <math>R_s</math>. During operation, the stator induces reactance, represented by the inductor <math>X_s</math>. <math>X_r</math> represents the effect of the rotor passing through the stator's magnetic field. The effective resistance of the rotor, <math>R_r</math>, is composed of the equivalent value of the machine's power and the ohmic resistance of the stator windings and squirrel cage.
+[[File:IMEQCCT.jpg|thumb|center|400px|Induction motor equivalent circuit]]
+:<math>Z_m = R_s + jX_s + \frac{(\frac{R_s}{s} + jX_r)(jX_m)}{\frac{R_r}{s} + j(X_r + X_m)}</math>
+:<math>I_s = V_s/(R_s + jX_s + \frac{(\frac{R_s}{s} + jX_r)(jX_m)}{\frac{R_r}{s} + j(X_r + X_m)})</math>
+:<math>I_r^' = \frac{jX_m}{\frac{R_r}{s} + j(X_r + X_m)} I_s</math>
+:<math>\omega_r = \frac{2{\pi}n_s}{60} = \frac{4{\pi}f_s}{p}</math>
-The induction motor equivalent circuit when idle is approximately <math>R_s+X_s</math>, which is mostly reactive. Induction motors generally have a poor [[power factor]], which can be improved by a capacitive compensation network. However, resonant conditions must be avoided at all costs, and so capacitor networks are usually designed to lag. Further, because capacitors may be connected elsewhere across the supply, induction motors frequently need automatic disconnection during power failures.<ref>[http://books.google.co.uk/books?id=utWtW_9NMgcC&lpg=PA89&dq=induction%20motor%20power%20correction&pg=PA89#v=onepage&q=induction%20motor%20power%20correction&f=false Energy-Efficient Electric Motors and Their Applications By H.E. Jordan]</ref>
+:<math>T_{em} = \frac{P_{em}}{\omega_r} = \frac{\frac{P_r}{s}}{\omega_r} = \frac{3I_r^{'2} R_r}{\omega_r s}</math>
-The induction motor's idle current draw is often near the rated current, due to the copper and core losses existing without load. In these conditions, this is usually more than half the power loss at the rated load. If the torque against the motor spindle is increased, the active current in the rotor increases by <math>R_r</math>. Due to the construction of the induction motor, the two resistances induce magnetic flux, in contrast to synchronous machines where it is induced only by the reactive current in the stator windings.
-The current produces a voltage drop in the cage factor of <math>R_r</math> and a slightly higher one in the stator windings. Hence, the losses increase faster in the rotor than in the stator. <math>R_s</math> and the copper factor of <math>R_r</math> both cause <math>I^2R</math> losses, meaning the efficiency improves with increasing load and reduces with temperature.
+:<math>where</math>
-<math>X_s</math> gets smaller with smaller frequency and must be reduced by the delivered drive voltage. Thus, <math>R_s \over X_s+R_s</math> increases engine power losses. In continuous operation, this is an approximation because a nominal torque generated by the cooling of the rotor and stator is not included in the calculation. Above the rated speed or  frequency, induction motors are more effective at higher voltages. Today, <math>R_s</math> and <math>R_r</math> are measured automatically and thus can be used on a motor to automatically configure itself and thus protect it from overload. Holding torques and speeds close to zero can be achieved with vector controls. There can be problems with cooling here, since the fan is usually mounted on the rotor.
+{| class="wikitable"
+|-
+|<math>f_s</math> ||stator synchronous frequency
+|-
+|<math>n_s </math>||synchronous speed in revolutions per second
+|-
+|<math>I_s</math>||stator current
+|-
+|<math>I_r^'</math>||rotor current referred to stator side
+|-
+|<math>j = \sqrt{-1}</math>||imaginary number operator
+|-
+|<math>p</math> ||number of motor poles
+|-
+|<math>P_{em}</math>||electromagnetic power
+|-
+|<math>P_r</math>||rotor copper losses
+|-
+|<math>R_s, X_s</math>||stator resistance and leakage reactance
+|-
+|<math>R_r^', X_r^'</math>||rotor resistance and leakage reactance referred to the stator side
+|-
+|<math>s </math>||slip
+|-
+|<math>T_{em}</math>||electromagnetic torque
+|-
+|<math>V_s</math>||stator phase voltage
+|-
+|<math>X_m</math>||magnetizing reactance
+|-
+|<math>Z_m</math>||motor  equivalent impedance
+|-
+|<math>\omega_r </math>||rotor speed in radians per second
+|-
+|}
 ==Starting==
@@ Line 118: / Line 168: @@
 Linear motors frequently run on a 3 phase power supply.
-Their uses include [[magnetic levitation]], linear propulsion, and linear actuators. They have also been used for pumping liquid metals.<ref>{{cite book|url=http://books.google.co.uk/books?id=fgsAAAAAMBAJ&lpg=PA52&ots=NfAng_7A27&dq=einstein%20Linear%20induction%20motor&pg=PA52#v=onepage&q=einstein%20Linear%20induction%20motor&f=false |title=Bulletin of the Atomic Scientists - Google Books |publisher=Books.google.co.uk |date=1973-06-06 |accessdate=2012-08-08}}</ref>
+Their uses include [[magnetic levitation]], linear propulsion, and linear actuators. They have also been used for pumping liquid metals.<ref name="BAS (1973)">{{cite book|url=http://books.google.co.uk/books?id=fgsAAAAAMBAJ&lpg=PA52&ots=NfAng_7A27&dq=einstein%20Linear%20induction%20motor&pg=PA52#v=onepage&q=einstein%20Linear%20induction%20motor&f=false |title=Bulletin of the Atomic Scientists |year=6 June 1973|accessdate=8 August 2012|publisher=Educational Foundation for Atomic Science}}</ref>
 ==Electrical energy efficiency==
@@ Line 124: / Line 174: @@
 ==Sources==
-*{{cite book| url=http://books.google.com/books?id=hbM_AAAAYAAJ&printsec=frontcover&dq=induction+motor&source=bl&ots=_JgDsnjN2s&sig=LHXibhTQ9XXIOvzsWATRSHA-xkA&hl=en&ei=X1O3TOekFpCisAPomqGeCQ&sa=X&oi=book_result&ct=result&resnum=14&sqi=2&ved=0CFoQ6AEwDQ#v=onepage&q&f=false| title=The induction motor: its theory and design, set forth by a practical method of calculation| author=Henri Boy de la Tour| others=Translated Cyprien Odilon Mailloux| publisher=McGraw Pub. Co.| year= 1906 }}
+*{{cite book| url=http://books.google.com/books?id=hbM_AAAAYAAJ&printsec=frontcover&dq=induction+motor&source=bl&ots=_JgDsnjN2s&sig=LHXibhTQ9XXIOvzsWATRSHA-xkA&hl=en&ei=X1O3TOekFpCisAPomqGeCQ&sa=X&oi=book_result&ct=result&resnum=14&sqi=2&ved=0CFoQ6AEwDQ#v=onepage&q&f=false| title=The Induction Motor: Its Theory and Design, Set Forth By a Practical Method of Calculation| author=Henri Boy de la Tour| others=Translated Cyprien Odilon Mailloux| publisher=McGraw Pub. Co.| year= 1906 }}
-*{{cite book| url=http://books.google.com/books?id=r_dOAAAAMAAJ&printsec=frontcover&dq=induction+motor&source=bl&ots=g7Th09trR-&sig=onxjvgyC920oARs_LUDqnzV2kHg&hl=en&ei=1VS3TNTyNoKKlwfWwJ3MDA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDcQ6AEwAzgK#v=onepage&q&f=false| title=The induction motor| author=Benjamin Franklin Bailey| publisher=McGraw-Hill| year= 1911 }}
+*{{cite book| url=http://books.google.com/books?id=r_dOAAAAMAAJ&printsec=frontcover&dq=induction+motor&source=bl&ots=g7Th09trR-&sig=onxjvgyC920oARs_LUDqnzV2kHg&hl=en&ei=1VS3TNTyNoKKlwfWwJ3MDA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDcQ6AEwAzgK#v=onepage&q&f=false| title=The Induction Motor| author=Benjamin Franklin Bailey| publisher=McGraw-Hill| year= 1911 }}
-*{{cite book| url=http://books.google.com/books?id=ffpOAAAAMAAJ&printsec=frontcover&dq=induction+motor&source=bl&ots=AWJzYuRVCl&sig=Bm0VKBdRKgCfTPpeR5_YU3BCrso&hl=en&ei=1VS3TNTyNoKKlwfWwJ3MDA&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEUQ6AEwBjgK#v=onepage&q&f=false| title=The induction motor: A short treatise on its theory and design, with numerous experimental data and diagrams| author=Bernhard Arthur Behrend| publisher=Electrical world and engineer| year= 1901 }}
+*{{cite book| url=http://books.google.com/books?id=ffpOAAAAMAAJ&printsec=frontcover&dq=induction+motor&source=bl&ots=AWJzYuRVCl&sig=Bm0VKBdRKgCfTPpeR5_YU3BCrso&hl=en&ei=1VS3TNTyNoKKlwfWwJ3MDA&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEUQ6AEwBjgK#v=onepage&q&f=false| title=The Induction Motor: A Short Treatise on its Theory and Design, With Numerous Experimental Data and Diagrams| author=Bernhard Arthur Behrend| publisher=Electrical world and engineer| year= 1901 }}
 ==See also==

Electric machines
AC - Alternating current DC - Direct current PM - Permanent magnet SC - Self-commutated
Components and accessories	Armature Braking chopper Brush Coil winding technology Commutator Damper winding DC injection braking Field coil Rotor Slip ring Stator Winding
Generators	Alternator Electric generator
Motors	AC motor Induction motor Shaded-pole Dahlander pole changing motor Wound-rotor (WRIM) Linear induction Synchronous motor Repulsion DC motor Homopolar Brushed DC electric motor Brushless DC electric motor Unipolar Universal Switched reluctance (SRM) Reluctance Doubly fed Linear Permanent magnet Dual-rotor permanent magnet induction motor (DRPMIM) Servomotor Stepper Traction Electrostatic Piezoelectric Ultrasonic TEFC Axial flux
Motor controllers	AC-to-AC converter Cycloconverter Amplidyne Drives Variable-frequency drive Direct torque control Vector control Metadyne Motor soft starter Ward Leonard control
History, education, recreational use	Timeline of the electric motor Ball bearing motor Barlow's wheel Lynch motor Mendocino motor Mouse mill motor
Experimental, futuristic	Coilgun Railgun Superconducting machine
Related topics	Blocked-rotor test Circle diagram Electromagnetism Open-circuit test Open-loop controller Power-to-weight ratio Two-phase system Inchworm motor Starter Voltage controller
People	Arago Barlow Botto Davenport Davidson Dolivo-Dobrovolsky Faraday Ferraris Gramme Henry Jacobi Jedlik Lenz Maxwell Ørsted Park Pixii Saxton Siemens Sprague Steinmetz Sturgeon Tesla