ECE 8830 Electric Drives

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ECE 8830 - Electric Drives
Topic 2 Fundamentals of Electric Motors
Spring 2004
2
Generic Electric Motor

• Below figure shows cartoon of induction
• motor. Most (but not all) machines have
• this structure.

Ref J.L. Kirtley, Jr. MIT Course 6.11s (June
2003)Course Notes
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Generic Electric Motor (contd)

• Rotor - mounted on shaft supported by
• bearings (usually rotor on inside
• - but not always)
• - shown with conductors but may
• have permanent magnets instead
• - sometimes just an oddly shaped
• piece of steel (variable reluctance
  
• machines)

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Generic Electric Motor (contd)

• Stator - armature (electrical input) on
• stator (opposite to DC brush
  motor)
• - on outside with windings
• In most electric motors, rotor and stator are
  made of highly magnetically permeable materials -
  steel or iron.
• In many common motors, rotor and stator are made
  of thin sheets of silicon steel (laminations).
  Punched into these sheets are slots which contain
  rotor and stator conductors.

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Generic Electric Motor (contd)

• Time-varying magnetic fields passing through
  ferromagnetic materials (iron/steel) - eddy
  currents to flow - energy loss (power
  dissipation). Laminations (thin sheets of steel)
  are used to minimize eddy current losses.
• Windings - many turns of Al/Cu conductor
  concentrically wound about a common axis.
• Field winding carries excitation flux.
• Armature winding carries electrical power.

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Basic Principles of Operation of Electric Motors

• Changes in flux linkage between rotor and stator
  creates torque and therefore relative motion
  between rotor and stator.
• Fq(vxB)
• F l(ixB)

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Basic Principles of Operation of Electric Motors
(contd)

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Electrical Radians and Synchronous Speed

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Electrical Radians and Synchronous Speed
(contd)
electrical rads.

• frequency of
• induced voltage
• where P of poles
• p of pole pairs and
• Nsynchronous speed of rotor (rpm)

electrical rads./sec.
Hz
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Flux per Pole

• Consider a sinusoidally distributed flux
  density, B(?e)Bpkcos ?e. The flux per pole is
  given by

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Induced Voltage

• Full-pitched coil w/N turns moving laterally
  w.r.t. sinusoidal flux density.

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Induced Voltage (contd)

• At t0 coils axis coincides w/flux density
  wave peak. Thus, at time t, flux linked by coil
  is given by
• ? induced voltage in full-pitch coil is
• given by

transformer voltage speed voltage
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RMS Value of Induced Voltage

• RMS value of sinusoidally varying speed
  voltage term is
• In high power ac machines may have distributed
  or short-pitch windings. Use distribution and
  pitch factors (kd and kp respectively) to account
  for these designs. The rms value of the induced
  voltage under these conditions becomes
• where kwkdkp is the winding factor.

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Distribution Factor

• Phase windings may have series/parallel
• coils under a different pole-pair. Within
• each pole-pair region, the coils of a
• distributed winding are spread out over
• several pairs of slots.

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Distribution Factor (contd)

• The voltages induced in component coils for a
  single phase winding occupying adjacent slots
  will be separated by the slot angle separating
  them ?se (electrical angle subtended by arc
  between two adjacent slots.)

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Distribution Factor (contd)

• The distribution factor can be defined as the
  ratio of the resultant voltage with coils
  distributed to resultant voltage if coils were in
  one location, i.e.
• kd Resultant voltage of coils under one
  pole-pair Epole
• Arithmetic sum of coil voltages ?i Eci
• If a phase winding has q coils/phase/pole,
  Epole 2REsin(q?se/2) and Eci2REsin(?se/2),
  and

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Pitch Factor

• Short-pitching is when coils with less than
  one pole-pitch are used.

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Pitch Factor (contd)

• Short-pitching is used in machines with
  fractional-slot windings (non-integral slots/pole
  or slots/pole/phase) in a double-layer winding
  arrangement. Allows for a finite set of stampings
  with a fixed number of slots to be used for
  different speed machines.
• Also, short-pitching can be used to suppress
  certain harmonics in the phase emfs. Although
  short-pitching also offers shorter end
  connections, the resultant fundamental phase emf
  is reduced.

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Pitch Factor (contd)

• The pitch factor kp is defined by
• kp Resultant voltage in short-pitch coil
• Arithmetic sum of voltages induced in full coil
• With sinusoidal voltages, each coil voltage is
  the phasor sum of its two coil-side voltages.
  Thus, for coil a, Eca EaE-a

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Spatial MMF Distribution of a Winding

• A current i flowing through a single coil of
  nc turns creates a quasi-square wave mmf of
  amplitude F1 given by F1 nci/2.

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Spatial MMF Distribution of a Winding (contd)

• The fundamental component of this quasi-square
  wave mmf distribution is given by
• The fundamental component of the airgap flux
  density in a uniform airgap machine is
• where g is the airgap.

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Effect of Distributing Phase Coils

• Consider a three-phase distributed winding
  with 4 coils each per phase in a 2-layer
  arrangement in a 2-pole stator.

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Effect of Distributing Phase Coils (contd)

• If each coil has nc turns, the sum of the
  fundamental mmf components produced by coils a1
  and a2 is given by
• where ?e is the angle measured from the
  a-phase winding and ?s is the angle between the
  center lines of adjacent slots. Similarly for
  coils a3 and a4,

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Effect of Distributing Phase Coils (contd)

• Therefore distribution factor is

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Effect of Short-Pitching

• Consider a layout of windings that are
  short-pitched by one slot angle. Lets consider
  this to be made of four fictitious full-pitch
  coils (a1,-a3), (a4,-a2), (a2,-a1) and (a3,-a4).

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Effect of Short-Pitching (contd)

• The fundamental mmf component from these 4
  fictitious full-pitch coils is given by

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Effect of Short-Pitching (contd)

• Comparing this expression to
• and allowing for the distribution of the four
  full-pitch coils by a kd of cos(?s/2), the factor
  due to the short-pitching by one slot angle is
  given by

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Effect of Short-Pitching (contd)

• In general, the expression for the fundamental
  mmf component of a distributed winding with
  winding factor kw and a total of npole turns over
  a two-pole region is given by

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Effect of Short-Pitching (contd)

• Assuming total phase turns Nph in the P-pole
  machine are divided equally among P/2 pole-pair
  regions, number of turns per pole-pair Nph/P.
  In terms of Nph, fundamental mmf is
• The effective number of full-pitch concentric
  coils per pole-pair to achieve this same
  fundamental mmf is

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Winding Inductances

• Here we derive expressions for self- and
  mutual winding inductances for the elementary
  machine shown below.

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Winding Inductances (contd)

• Self-inductance of the stator winding,Lss,
  with Neffs turns per pole-pair linking ?pole
  (ignoring leakage inductances) is given by
• Similarly, the self-inductance of the rotor
  winding, Lrr, with Neffr turns per pole-pair is
  given by

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Winding Inductances (contd)

• An expression for the mutual inductance
  between the stator and rotor windings can be
  obtained by considering the flux linking the
  windings, ?rs which is given by
• ?

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Rotating Fields

• The fundamental component of space mmf for a
  single-phase winding carrying a sinusoidal
  current iIacos?t is given by
• where is the peak
  value
• of the fundamental mmf and ?a is the
  electrical angle measured in the
  counter-clockwise direction from the winding
  axis.

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Rotating Fields (contd)

• This equation may be rewritten as
• Two interpretations
• 1) pulsating standing wave
• 2) two counter-revolving mmf waves of half the
  amplitude of the resultant forward component
  rotates counter-clockwise, reverse component
  rotates clockwise.

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Rotating Fields (contd)

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Rotating Fields (contd)

• In a three-phase machine the axes of the
  windings are spaced 2?/3 apart. Assuming balanced
  operation (phase currents are of same magnitude)
  the currents are given by

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Rotating Fields (contd)

• The fundamental airgap mmfs of the three phases
  are given (in terms of ?a) by

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Rotating Fields (contd)

• The sum of these three winding mmfs is
• Therefore the resulting airgap mmfs is a
  constant amplitude sinusoidal wave rotating wave
  whose peak coincides with the magnetic axis of
  the a-phase winding at t0 and rotates with a
  speed ? in a direction corresponding to the
  sequence of peaking of the phase currents.

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Torque in a Uniform Airgap Machine

• From basic energy conversion principles, the
  torque developed in a machine is given by
• The co-energy is the complement of the field
  energy
• Wfld ?i - Wfld

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Torque in a Uniform Airgap Machine (contd)

• Example of three-phase machine
• (see text for other approaches/results)